models for time series and forecasting中小学PPT教学课件

合集下载

建筑施工计算手册(第二版)

建筑施工计算手册（第二版）建筑施工计算手册(第二版)l236岩土力学2007生【4】ASAOKAA.Observationalprocedureofsettlement predication[J].SoilsandFoundations,1978,l8(4):87一l01.【5】张仪萍,俞亚南,张土乔.时变参数灰色沉降预测模型及其应用【J】_浙江大学(工学版),2002,36(4):357—360.ZHANGYi—ping,YUYa—nan,ZHANGTu—qiao.Studyontime—dependentparametermodelofsettlementprediction withapplications[J].JournalofZhejiangUniversity (EngineeringScience),2002,36(4):357—360.【6】张仪萍,俞亚南,张土乔.时变参数预测模型及其在沉降预测中的应用【J】.土木工程,2003,36(6):83—86.ZHANGYi—ping,YUYa—nan,ZHANGTu—qiao.Anewpredictionmodelwithtime—dependentparametersused forsettlementprediction[J].ChinaCivilEngineering Journal,2003,36(6):83--86.【7】韩志刚.动态系统预报的一种新方法【J】.自动化, l983,9(3):16l一168.HANZhi—gang.Anewmethodofdynamicsystemprediction[J].ACTAAutomaticaSinica,1983,9(3): l6l—l68.【8】刘国林.顾及多因素影响的双线性变形模型及其动态参数估计【J】_测绘,1995,24(3):l83—191.LIUGuo—lin.Bilineardeformationmodelsinconsiderationofvariousfactorsanddynamicparametersestimation[J].ACTAGeodaeticaetCartographica Sinica,1995,24(3):183—191.【9】余天堂,王振波.基于多层递阶方法的边坡位移预测【J】岩上力学,2003,24(3):442--444.YUTian—tang,WANGZhen—bo.Predictionofslopedisplacementswithmulti—layerrecursivemethod[J].Rock andSoilMechanirs,2003,24(3):442,444.【l0】张仪萍,王士金,张土乔.沉降预测的多层递阶时问序列模型研究【J】.浙江大学(工学版),2005,39(7):983——986.ZHANGYi—ping,WANGShi-jin,ZHANGTu—qiao.Studyonmulti—layerrecursivetimeseriesmodelforsettlementprediction[J].JournalofZhejiangUniversity (EngineeringScience),2005,39(7):983--986. [1IJ姜继忱.关于一类递推算法的进一步探讨fJ1.黑龙江大学(自然科学版),1982,(1):20—29.【l2】吕培印,杨锦军,陈伟.时间序列分析法预报建筑物地基沉降『J】.辽宁工程学院,1996,16(4):27--30.LUPei—yin,YANGJin-jun,CHENWei.Timeseries methodforforecastingsettlementofbuildingfoundation[J].JournalofLiaoningInstituteof Technology,1996,l6(4):27—30.【l3】徐浩峰,应宏伟,朱向荣.时序分析预报基坑周围建筑物沉降【J】_建筑技术,2003,34(2):l09—110.XUHao—feng,YINGHong—wei,ZHUXiang—rong.Time—seriesmethodforforecastingsettlementofbuilding besidefoundationpits[J].ArchitectureTechnology,2003,34(2):109—110.建筑旅工计算手册(第二版)建筑科学类/江正荣N~:/4,16开/l000:~-W800页/精装/估价:88.~/2007年5月出版/ISBN:978—7—112—09l44—7本手册主要介绍工业与民用建筑施工经常遇到的各类有关施工计算问题.全书共有22章,主要包括:土方工程,基坑支护工程,地下水控制工程,地基处理工程,地基与基础工程,砌体与墙体工程,脚手架工程等,基本覆盖了建筑施工计算的主要应用领域.书中对每项计算公式均有详细的说明或推导,附有必要的图表资料,参考数据以及大量的典型计算实例.书后的两个附录是施工常用计算数据和施工常用结构计算用表及公式.读者对象:施工技术人员,管理人员,土建设计人员.(摘自新华书目报?科技新书目)。

holt-winter-multiplicative法 -回复

holt-winter-multiplicative法-回复Holt-Winters Multiplicative Model: A Comprehensive GuideIntroduction:Forecasting is an essential aspect of decision-making in various fields ranging from finance to marketing and supply chain management. There are multiple time series forecasting models available, and one prominent approach is the Holt-Winters Multiplicative Model. This model incorporates trends, seasonal patterns, and level shifts in the data to provide accurate and insightful forecasts. In this comprehensive guide, we will explore the steps involved in implementing the Holt-Winters Multiplicative Model and understand its strengths and limitations.Section 1: Understanding the Holt-Winters Multiplicative Model1.1 Definition:The Holt-Winters Multiplicative Model is a time series forecasting model that decomposes the data into three components - level, trend, and seasonality - to make predictions. Unlike simpler models, such as the moving average or exponential smoothing, this modelallows for dynamic adjustment of these three components, enabling accurate forecasting.1.2 Characteristics:The Holt-Winters Multiplicative Model is widely used due to its flexibility and ability to capture nonlinear trends and seasonal patterns in the data. It is particularly effective when the data exhibits a changing trend and fluctuating patterns. Moreover, this model provides robust forecasts by incorporating both short-term and long-term dependencies.Section 2: Implementing the Holt-Winters Multiplicative Model2.1 Data Preprocessing:Before implementing the Holt-Winters Multiplicative Model, it is essential to preprocess the data. This involves identifying and handling missing values, outliers, and other anomalies. Additionally, the time series data should be transformed if it exhibitsnon-constant variance or non-linear relationships.2.2 Model Initialization:The Holt-Winters Multiplicative Model requires initializing the threecomponents - level, trend, and seasonality. This can be done using various methods, such as simple averages, linear regression, or exponential smoothing. The choice of initialization method depends on the characteristics of the data and the analyst's expertise.2.3 Parameter Estimation:Once the model is initialized, the next step is to estimate the parameters. This involves finding the optimal values for the smoothing constants (alpha, beta, and gamma) and the length of the seasonal period. Parameter estimation can be performed using iterative optimization techniques, such as least squares or maximum likelihood estimation.2.4 Model Fitting and Forecasting:After estimating the parameters, the Holt-Winters Multiplicative Model is fitted to the data. This involves updating the level, trend, and seasonality components iteratively using the chosen optimization technique. Once the model is fitted, it can be used to forecast future values by extrapolating the trend and seasonal patterns.Section 3: Evaluating and Interpreting Holt-Winters Multiplicative Model3.1 Model Evaluation:To assess the performance of the Holt-Winters Multiplicative Model, various evaluation metrics can be employed. These include mean absolute error (MAE), mean squared error (MSE), root mean squared error (RMSE), and mean absolute percentage error (MAPE). Comparison with alternative models and techniques is also crucial to gauge the model's accuracy.3.2 Interpretation:Interpreting the results of the Holt-Winters Multiplicative Model is essential for decision-making. By analyzing the estimated components - level, trend, and seasonality - one can gain insights into the direction and magnitude of changes in the time series. This information can be leveraged to understand the underlying factors driving the data and make informed predictions.Section 4: Strengths and Limitations of Holt-Winters Multiplicative Model4.1 Strengths:The Holt-Winters Multiplicative Model has several strengths. It captures both trend and seasonal patterns, providing accurate forecasts for complex time series data. Additionally, it allows for flexible adjustment of the three components, enabling dynamic adaptation to changing patterns. Moreover, this model accommodates both short-term and long-term dependencies, enhancing its forecasting capabilities.4.2 Limitations:Despite its strengths, the Holt-Winters Multiplicative Model has some limitations. It assumes that the components - level, trend, and seasonality - are multiplicative, which may not always hold true in practice. Additionally, this model requires historical data that demonstrate stable patterns, making it less suitable for highly volatile or irregular time series. Moreover, this model has limited explanatory capabilities, focusing primarily on forecasting rather than detailed causal analysis.Conclusion:The Holt-Winters Multiplicative Model is a powerful forecastingapproach that incorporates trends, seasonality, and level shifts in the data. By understanding its characteristics, implementing the model step-by-step, and evaluating its performance, analysts can leverage this technique to make accurate predictions. However, it is essential to be aware of the model's limitations and interpret the results cautiously. With its ability to capture complex patterns, the Holt-Winters Multiplicative Model offers valuable insights for decision-making in various industries.。

时间序列与arima模型的关系

英文回复：Time—series data are observations or records over time that are important for analysing and predicting future trends，cyclicality and regularity。

As amon statistical method， time—series analysis is aimed at effectively predicting future developments through in—depth analysis of historical data。

For time series analysis， there are many models and methods，of which the ARIMA model is an effective one。

At this critical point， we should pursue an approach that closely integrates practical， integrated and scientific decision—making and promotes continuous innovation in time—series analysis theory and methodology to better serve our countries and peoples。

时间序列数据是指各种数据随着时间的推移所呈现出的观测结果或记录，其对于分析和预测未来的趋势、周期性和规律性具有重要意义。

时间序列分析作为一种常见的统计方法，通过对历史数据的深入剖析，旨在有效预测未来的发展走势。

针对时间序列分析，存在多种模型和方法，其中ARIMA模型为行之有效的一种。

值此关键节点，我们应坚持紧密结合实际、统筹兼顾、科学决策的方针，推动时间序列分析理论与方法的不断创新，以更好地服务于我们的国家和人民大众。

数据、模型与决策(运筹学)课后习题和案例答案013

CHAPTER 13FORECASTINGReview Questions13.1-1 Substantially underestimating demand is likely to lead to many lost sales, unhappycustomers, and perhaps allowing the competition to gain the upper hand in the marketplace. Significantly overestimating the demand is very costly due to excessive inventory costs, forced price reductions, unneeded production or storage capacity, and lost opportunity to market more profitable goods.13.1-2 A forecast of the demand for spare parts is needed to provide good maintenanceservice.13.1-3 In cases where the yield of a production process is less than 100%, it is useful toforecast the production yield in order to determine an appropriate value of reject allowance and, consequently, the appropriate size of the production run.13.1-4 Statistical models to forecast economic trends are commonly called econometricmodels.13.1-5 Providing too few agents leads to unhappy customers, lost calls, and perhaps lostbusiness. Too many agents cause excessive personnel costs.13.2-1 The company mails catalogs to its customers and prospective customers severaltimes per year, as well as publishing mini-catalogs in computer magazines. They then take orders for products over the phone at the company’s call center.13.2-2 Customers who receive a busy signal or are on hold too long may not call back andbusiness may be lost. If too many agents are on duty there may be idle time, which wastes money because of labor costs.13.2-3 The manager of the call center is Lydia Weigelt. Her current major frustration is thateach time she has used her procedure for setting staffing levels for the upcoming quarter, based on her forecast of the call volume, the forecast usually has turned out to be considerably off.13.2-4 Assume that each quarter’s cal l volume will be the same as for the preceding quarter,except for adding 25% for quarter 4.13.2-5 The average forecasting error is commonly called MAD, which stands for MeanAbsolute Deviation. Its formula is MAD = (Sum of forecasting errors) / (Number of forecasts)13.2-6 MSE is the mean square error. Its formula is (Sum of square of forecasting errors) /(Number of forecasts).13.2-7 A time series is a series of observations over time of some quantity of interest.13.3-1 In general, the seasonal factor for any period of a year measures how that periodcompares to the overall average for an entire year.13.3-2 Seasonally adjusted call volume = (Actual call volume) / (Seasonal factor).13.3-3 Actual forecast = (Seasonal factor)(Seasonally adjusted forecast)13.3-4 The last-value forecasting method sometimes is called the naive method becausestatisticians consider it naive to use just a sample size of one when additional relevant data are available.13.3-5 Conditions affecting the CCW call volume were changing significantly over the pastthree years.13.3-6 Rather than using old data that may no longer be relevant, this method averages thedata for only the most recent periods.13.3-7 This method modifies the moving-average method by placing the greatest weighton the last value in the time series and then progressively smaller weights on the older values.13.3-8 A small value is appropriate if conditions are remaining relatively stable. A largervalue is needed if significant changes in the conditions are occurring relatively frequently.13.3-9 Forecast = α(Last Value) + (1 –α)(Last forecast). Estimated trend is added to thisformula when using exponential smoothing with trend.13.3-10 T he one big factor that drives total sales up or down is whether there are any hotnew products being offered.13.4-1 CB Predictor uses the raw data to provide the best fit for all these inputs as well asthe forecasts.13.4-2 Each piece of data should have only a 5% chance of falling below the lower line and a5% chance of rising above the upper line.13.5-1 The next value that will occur in a time series is a random variable.13.5-2 The goal of time series forecasting methods is to estimate the mean of theunderlying probability distribution of the next value of the time series as closely as possible.13.5-3 No, the probability distribution is not the same for every quarter.13.5-4 Each of the forecasting methods, except for the last-value method, placed at leastsome weight on the observations from Year 1 to estimate the mean for each quarter in Year 2. These observations, however, provide a poor basis for estimating the mean of the Year 2 distribution.13.5-5 A time series is said to be stable if its underlying probability distribution usuallyremains the same from one time period to the next. A time series is unstable if both frequent and sizable shifts in the distribution tend to occur.13.5-6 Since sales drive call volume, the forecasting process should begin by forecastingsales.13.5-7 The major components are the relatively stable market base of numerous small-niche products and each of a few major new products.13.6-1 Causal forecasting obtains a forecast of the quantity of interest by relating it directlyto one or more other quantities that drive the quantity of interest.13.6-2 The dependent variable is call volume and the independent variable is sales.13.6-3 When doing causal forecasting with a single independent variable, linear regressioninvolves approximating the relationship between the dependent variable and the independent variable by a straight line.13.6-4 In general, the equation for the linear regression line has the form y = a + bx. Ifthere is more than one independent variable, then this regression equation has a term, a constant times the variable, added on the right-hand side for each of these variables.13.6-5 The procedure used to obtain a and b is called the method of least squares.13.6-6 The new procedure gives a MAD value of only 120 compared with the old MADvalue of 400 with the 25% rule.13.7-1 Statistical forecasting methods cannot be used if no data are available, or if the dataare not representative of current conditions.13.7-2 Even when good data are available, some managers prefer a judgmental methodinstead of a formal statistical method. In many other cases, a combination of the two may be used.13.7-3 The jury of executive opinion method involves a small group of high-level managerswho pool their best judgment to collectively make a forecast rather than just the opinion of a single manager.13.7-4 The sales force composite method begins with each salesperson providing anestimate of what sales will be in his or her region.13.7-5 A consumer market survey is helpful for designing new products and then indeveloping the initial forecasts of their sales. It is also helpful for planning a marketing campaign.13.7-6 The Delphi method normally is used only at the highest levels of a corporation orgovernment to develop long-range forecasts of broad trends.13.8-1 Generally speaking, judgmental forecasting methods are somewhat more widelyused than statistical methods.13.8-2 Among the judgmental methods, the most popular is a jury of executive opinion.Manager’s opinion is a close second.13.8-3 The survey indicates that the moving-average method and linear regression are themost widely used statistical forecasting methods.Problems13.1 a) Forecast = last value = 39b) Forecast = average of all data to date = (5 + 17 + 29 + 41 + 39) / 5 = 131 / 5 =26c) Forecast = average of last 3 values = (29 + 41 + 39) / 3 = 109 / 3 = 36d) It appears as if demand is rising so the average forecasting method seemsinappropriate because it uses older, out-of-date data.13.2 a) Forecast = last value = 13b) Forecast = average of all data to date = (15 + 18 + 12 + 17 + 13) / 5 = 75 / 5 =15c) Forecast = average of last 3 values = (12 + 17 + 13) / 3 = 42 / 3 = 14d) The averaging method seems best since all five months of data are relevant indetermining the forecast of sales for next month and the data appears relativelystable.13.3MAD = (Sum of forecasting errors) / (Number of forecasts) = (18 + 15 + 8 + 19) / 4 = 60 / 4 = 15 MSE = (Sum of squares of forecasting errors) / (Number of forecasts) = (182 + 152 + 82 + 192) / 4 = 974 / 4 = 243.513.4 a) Method 1 MAD = (258 + 499 + 560 + 809 + 609) / 5 = 2,735 / 5 = 547Method 2 MAD = (374 + 471 + 293 + 906 + 396) / 5 = 2,440 / 5 = 488Method 1 MSE = (2582 + 4992 + 5602 + 8092 + 6092) / 5 = 1,654,527 / 5 = 330,905Method 2 MSE = (3742 + 4712 + 2932 + 9062 + 3962) / 5 = 1,425,218 / 5 = 285,044Method 2 gives a lower MAD and MSE.b) She can use the older data to calculate more forecasting errors and compareMAD for a longer time span. She can also use the older data to forecast theprevious five months to see how the methods compare. This may make her feelmore comfortable with her decision.13.5 a)b)c)d)13.6 a)b)This progression indicatesthat the state’s economy is improving with the unemployment rate decreasing from 8% to 7% (seasonally adjusted) over the four quarters.13.7 a)b) Seasonally adjusted value for Y3(Q4)=28/1.04=27,Actual forecast for Y4(Q1) = (27)(0.84) = 23.c) Y4(Q1) = 23 as shown in partb Seasonally adjusted value for Y4(Q1) = 23 / 0.84 = 27 Actual forecast for Y4(Q2) = (27)(0.92) = 25Seasonally adjusted value for Y4(Q2) = 25 / 0.92 = 27 Actual forecast for Y4(Q3) = (27)(1.20) = 33Seasonally adjusted value for Y4(Q3) = 33/1.20 = 27Actual forecast for Y4(Q4) = (27)(1.04) = 28d)13.8 Forecast = 2,083 – (1,945 / 4) + (1,977 / 4) = 2,09113.9 Forecast = 782 – (805 / 3) + (793 / 3) = 77813.10 Forecast = 1,551 – (1,632 / 10) + (1,532 / 10) = 1,54113.11 Forecast(α) = α(last value) + (1 –α)(last forecast)Forecast(0.1) = (0.1)(792) + (1 –0.1)(782) = 783 Forecast(0.3) = (0.3)(792) + (1 –0.3)(782) = 785 Forecast(0.5) = (0.5)(792) + (1 – 0.5)(782) = 78713.12 Forecast(α) = α(last value) + (1 –α)(last forecast)Forecast(0.1) = (0.1)(1,973) + (1 –0.1)(2,083) = 2,072 Forecast(0.3) = (0.3)(1,973) + (1 –0.3)(2,083) = 2,050 Forecast(0.5) = (0.5)(1,973) + (1 – 0.5)(2,083) = 2,02813.13 a) Forecast(year 1) = initial estimate = 5000Forecast(year 2) = α(last value) + (1 –α)(last forecast)= (0.25)(4,600) + (1 –0.25)(5,000) = 4,900 Forecast(year 3) = (0.25)(5,300) + (1 – 0.25)(4,900) = 5,000b) MAD = (400 + 400 + 1,000) / 3 = 600MSE = (4002 + 4002 + 1,0002) / 3 = 440,000c) Forecast(next year) = (0.25)(6,000) + (1 – 0.25)(5,000) = 5,25013.14 Forecast = α(last value) + (1 –α)(last forecast) + Estimated trendEstimated trend = β(Latest trend) + (1 –β)(Latest estimate of trend) Latest trend = α(Last value – Next-to-last value) + (1 –α)(Last forecast – Next-to-last forecast)Forecast(year 1) = Initial average + Initial trend = 3,900 + 700 = 4,600Forecast (year 2) = (0.25)(4,600) + (1 –0.25)(4,600)+(0.25)[(0.25)(4,600 –3900) + (1 –0.25)(4,600 –3,900)] + (1 –0.25)(700) = 5,300Forecast (year 3) = (0.25)(5,300) + (1 – 0.25)(5,300) + (0.25)[(0.25)(5,300 – 4,600) + (1 – 0.25)(5,300 – 4,600)]+(1 – 0.25)(700) = 6,00013.15 Forecast = α(last value) + (1 –α)(last forecast) + Estimated trendEstimated trend = β(Latest trend) + (1 –β)(Latest estimate of trend) Latest trend = α(Last value – Next-to-last value) + (1 –α)(Last forecast – Next-to-last forecast)Forecast = (0.2)(550) + (1 – 0.2)(540) + (0.3)[(0.2)(550 – 535) + (1 – 0.2)(540 –530)] + (1 – 0.3)(10) = 55213.16 Forecast = α(last value) + (1 –α)(last forecast) + Estimated trendEstimated trend = β(Latest trend) + (1 –β)(Latest estimate of trend) Latest trend = α(Last value – Next-to-last value) + (1 –α)(Last forecast – Next-to-last forecast)Forecast = (0.1)(4,935) + (1 – 0.1)(4,975) + (0.2)[(0.1)(4,935 – 4,655) + (1 – 0.1) (4,975 – 4720)] + (1 – 0.2)(240) = 5,21513.17 a) Since sales are relatively stable, the averaging method would be appropriate forforecasting future sales. This method uses a larger sample size than the last-valuemethod, which should make it more accurate and since the older data is stillrelevant, it should not be excluded, as would be the case in the moving-averagemethod.b)c)d)e) Considering the MAD values (5.2, 3.0, and 3.9, respectively), the averagingmethod is the best one to use.f) Considering the MSE values (30.6, 11.1, and 17.4, respectively), the averagingmethod is the best one to use.g) Unless there is reason to believe that sales will not continue to be relatively stable,the averaging method should be the most accurate in the future as well.13.18 Using the template for exponential smoothing, with an initial estimate of 24, thefollowing forecast errors were obtained for various values of the smoothing constant α:use.13.19 a) Answers will vary. Averaging or Moving Average appear to do a better job thanLast Value.b) For Last Value, a change in April will only affect the May forecast.For Averaging, a change in April will affect all forecasts after April.For Moving Average, a change in April will affect the May, June, and July forecast.c) Answers will vary. Averaging or Moving Average appear to do a slightly better jobthan Last Value.d) Answers will vary. Averaging or Moving Average appear to do a slightly better jobthan Last Value.13.20 a) Since the sales level is shifting significantly from month to month, and there is noconsistent trend, the last-value method seems like it will perform well. Theaveraging method will not do as well because it places too much weight on olddata. The moving-average method will be better than the averaging method butwill lag any short-term trends. The exponential smoothing method will also lagtrends by placing too much weight on old data. Exponential smoothing withtrend will likely not do well because the trend is not consistent.b)Comparing MAD values (5.3, 10.0, and 8.1, respectively), the last-value method is the best to use of these three options.Comparing MSE values (36.2, 131.4, and 84.3, respectively), the last-value method is the best to use of these three options.c) Using the template for exponential smoothing, with an initial estimate of 120, thefollowing forecast errors were obtained for various values of the smoothingconstant α:constant is appropriate.d) Using the template for exponential smoothing with trend, using initial estimates of120 for the average value and 10 for the trend, the following forecast errors wereobtained for various values of the smoothing constants α and β:constants is appropriate.e) Management should use the last-value method to forecast sales. Using thismethod the forecast for January of the new year will be 166. Exponentialsmoothing with trend with high smoothing constants (e.g., α = 0.5 and β = 0.5)also works well. With this method, the forecast for January of the new year will be165.13.21 a) Shift in total sales may be due to the release of new products on top of a stableproduct base, as was seen in the CCW case study.b) Forecasting might be improved by breaking down total sales into stable and newproducts. Exponential smoothing with a relatively small smoothing constant canbe used for the stable product base. Exponential smoothing with trend, with arelatively large smoothing constant, can be used for forecasting sales of each newproduct.c) Managerial judgment is needed to provide the initial estimate of anticipated salesin the first month for new products. In addition, a manger should check theexponential smoothing forecasts and make any adjustments that may benecessary based on knowledge of the marketplace.13.22 a) Answers will vary. Last value seems to do the best, with exponential smoothingwith trend a close second.b) For last value, a change in April will only affect the May forecast.For averaging, a change in April will affect all forecasts after April.For moving average, a change in April will affect the May, June, and July forecast.For exponential smoothing, a change in April will affect all forecasts after April.For exponential smoothing with trend, a change in April will affect all forecastsafter April.c) Answers will vary. last value or exponential smoothing seem to do better than theaveraging or moving average.d) Answers will vary. last value or exponential smoothing seem to do better than theaveraging or moving average.13.23 a) Using the template for exponential smoothing, with an initial estimate of 50, thefollowing MAD values were obtained for various values of the smoothing constantα:Choose αb) Using the template for exponential smoothing, with an initial estimate of 50, thefollowing MAD values were obtained for various values of the smoothing constantα:Choose αc) Using the template for exponential smoothing, with an initial estimate of 50, thefollowing MAD values were obtained for various values of the smoothing constantα:13.24 a)b)Forecast = 51.c) Forecast = 54.13.25 a) Using the template for exponential smoothing with trend, with an initial estimatesof 50 for the average and 2 for the trend and α = 0.2, the following MAD values were obtained for various values of the smoothing constant β:Choose β = 0.1b) Using the template for exponential smoothing with trend, with an initial estimatesof 50 for the average and 2 for the trend and α = 0.2, the following MAD valueswere obtained for various values of the smoothing constant β:Choose β = 0.1c) Using the template for exponential smoothing with trend, with an initial estimatesof 50 for the average and 2 for the trend and α = 0.2, the following MAD valueswere obtained for various values of the smoothing constant β:13.26 a)b)0.582. Forecast = 74.c) = 0.999. Forecast = 79.13.27 a) The time series is not stable enough for the moving-average method. Thereappears to be an upward trend.b)c)d)e) Based on the MAD and MSE values, exponential smoothing with trend should beused in the future.β = 0.999.f)For exponential smoothing, the forecasts typically lie below the demands.For exponential smoothing with trend, the forecasts are at about the same level as demand (perhaps slightly above).This would indicate that exponential smoothing with trend is the best method to usehereafter.13.2913.30 a)factors:b)c) Winter = (49)(0.550) = 27Spring = (49)(1.027) = 50Summer = (49)(1.519) = 74Fall = (49)(0.904) = 44d)e)f)g) The exponential smoothing method results in the lowest MAD value (1.42) and thelowest MSE value (2.75).13.31 a)b)c)d)e)f)g) The last-value method with seasonality has the lowest MAD and MSE value. Usingthis method, the forecast for Q1 is 23 houses.h) Forecast(Q2) = (27)(0.92) = 25Forecast(Q3) = (27)(1.2) = 32Forecast(Q4) = (27)(1.04) = 2813.32 a)b) The moving-average method with seasonality has the lowest MAD value. Using13.33 a)b)c)d) Exponential smoothing with trend should be used.e) The best values for the smoothing constants are α = 0.3, β = 0.3, and γ = 0.001.C28:C38 below.13.34 a)b)c)d)e) Moving average results in the best MAD value (13.30) and the best MSE value(249.09).f)MAD = 14.17g) Moving average performed better than the average of all three so it should beused next year.h) The best method is exponential smoothing with seasonality and trend, using13.35 a)••••••••••0100200300400500600012345678910S a l e sMonthb)c)••••••••••0100200300400500600012345678910S a l e sMonthd) y = 410.33 + (17.63)(11) = 604 e) y = 410.33 + (17.63)(20) = 763f) The average growth in sales per month is 17.63.13.36 a)•••01000200030004000500060000123A p p l i c a t i o n sYearb)•••01000200030004000500060000123A p p l i c a t i o n sYearc)d) y (year 4) = 3,900+ (700)(4) = 6,700 y (year 5) = 3,900 + (700)(5) = 7,400 y (year 6) = 3,900 + (700)(6) = 8,100 y (year 7) = 3,900 + (700)(7) =8,800y (year 8) = 3,900 + (700)(8) = 9,500e) It does not make sense to use the forecast obtained earlier of 9,500. Therelationship between the variable has changed and, thus, the linear regression that was used is no longer appropriate.f)•••••••0100020003000400050006000700001234567A p p l i c a t i o n sYeary =5,229 +92.9x y =5,229+(92.9)(8)=5,971the forecast that it provides for year 8 is not likely to be accurate. It does not make sense to continue to use a linear regression line when changing conditions cause a large shift in the underlying trend in the data.g)Causal forecasting takes all the data into account, even the data from before changing conditions cause a shift. Exponential smoothing with trend adjusts to shifts in the underlying trend by placing more emphasis on the recent data.13.37 a)••••••••••50100150200250300350400450500012345678910A n n u a l D e m a n dYearb)c)••••••••••50100150200250300350400450500012345678910A n n u a l D e m a n dYeard) y = 380 + (8.15)(11) = 470 e) y = 380 = (8.15)(15) = 503f) The average growth per year is 8.15 tons.13.38 a) The amount of advertising is the independent variable and sales is the dependentvariable.b)•••••0510*******100200300400500S a l e s (t h o u s a n d s o f p a s s e n g e r s )Amount of Advertising ($1,000s)c)•••••0510*******100200300400500S a l e s (t h o u s a n d s o f p a s s e n g e r s )Amount of Advertising ($1,000s)d) y = 8.71 + (0.031)(300) = 18,000 passengers e) 22 = 8.71 + (0.031)(x ) x = $429,000f) An increase of 31 passengers can be attained.13.39 a) If the sales change from 16 to 19 when the amount of advertising is 225, then thelinear regression line shifts below this point (the line actually shifts up, but not as much as the data point has shifted up).b) If the sales change from 23 to 26 when the amount of advertising is 450, then the linear regression line shifts below this point (the line actually shifts up, but not as much as the data point has shifted up).c) If the sales change from 20 to 23 when the amount of advertising is 350, then the linear regression line shifts below this point (the line actually shifts up, but not as much as the data point has shifted up).13.40 a) The number of flying hours is the independent variable and the number of wingflaps needed is the dependent variable.b)••••••024*********100200W i n g F l a p s N e e d e dFlying Hours (thousands)c)d)••••••024*********100200W i n g F l a p s N e e d e dFlying Hours (thousands)e) y = -3.38 + (0.093)(150) = 11f) y = -3.38 + (0.093)(200) = 1513.41 Joe should use the linear regression line y = –9.95 + 0.10x to develop a forecast forCase13.1 a) We need to forecast the call volume for each day separately.1) To obtain the seasonally adjusted call volume for the past 13 weeks, we firsthave to determine the seasonal factors. Because call volumes follow seasonalpatterns within the week, we have to calculate a seasonal factor for Monday,Tuesday, Wednesday, Thursday, and Friday. We use the Template for SeasonalFactors. The 0 values for holidays should not factor into the average. Leaving themblank (rather than 0) accomplishes this. (A blank value does not factor into theAVERAGE function in Excel that is used to calculate the seasonal values.) Using thistemplate (shown on the following page, the seasonal factors for Monday, Tuesday,Wednesday, Thursday, and Friday are 1.238, 1.131, 0.999, 0.850, and 0.762,respectively.2) To forecast the call volume for the next week using the last-value forecasting method, we need to use the Last Value with Seasonality template. To forecast the next week, we need only start with the last Friday value since the Last Value method only looks at the previous day.The forecasted call volume for the next week is 5,045 calls: 1,254 calls are received on Monday, 1,148 calls are received on Tuesday, 1,012 calls are received on Wednesday, 860 calls are received on Thursday, and 771 calls are received on Friday.3) To forecast the call volume for the next week using the averaging forecasting method, we need to use the Averaging with Seasonality template.The forecasted call volume for the next week is 4,712 calls: 1,171 calls are received on Monday, 1,071 calls are received on Tuesday, 945 calls are received on Wednesday, 804 calls are received on Thursday, and 721 calls are received onFriday.4) To forecast the call volume for the next week using the moving-average forecasting method, we need to use the Moving Averaging with Seasonality template. Since only the past 5 days are used in the forecast, we start with Monday of the last week to forecast through Friday of the next week.The forecasted call volume for the next week is 4,124 calls: 985 calls are received on Monday, 914 calls are received on Tuesday, 835 calls are received on Wednesday, 732 calls are received on Thursday, and 658 calls are received on Friday.5) To forecast the call volume for the next week using the exponential smoothing forecasting method, we need to use the Exponential with Seasonality template. We start with the initial estimate of 1,125 calls (the average number of calls on non-holidays during the previous 13 weeks).The forecasted call volume for the next week is 4,322 calls: 1,074 calls are received on Monday, 982 calls are received on Tuesday, 867 calls are received onWednesday, 737 calls are received on Thursday, and 661 calls are received on Friday.b) To obtain the mean absolute deviation for each forecasting method, we simplyneed to subtract the true call volume from the forecasted call volume for each day in the sixth week. We then need to take the absolute value of the five differences.Finally, we need to take the average of these five absolute values to obtain the mean absolute deviation.1) The spreadsheet for the calculation of the mean absolute deviation for the last-value forecasting method follows.This method is the least effective of the four methods because this method depends heavily upon the average seasonality factors. If the average seasonality factors are not the true seasonality factors for week 6, a large error will appear because the average seasonality factors are used to transform the Friday call volume in week 5 to forecasts for all call volumes in week 6. We calculated in part(a) that the call volume for Friday is 0.762 times lower than the overall average callvolume. In week 6, however, the call volume for Friday is only 0.83 times lower than the average call volume over the week. Also, we calculated that the call volume for Monday is 1.34 times higher than the overall average call volume. In Week 6, however, the call volume for Monday is only 1.21 times higher than the average call volume over the week. These differences introduce a large error.。

T.W. ANDERSON (1971). The Statistical Analysis of Time Series. Series in Probability and Ma

425 BibliographyH.A KAIKE(1974).Markovian representation of stochastic processes and its application to the analysis of autoregressive moving average processes.Annals Institute Statistical Mathematics,vol.26,pp.363-387. B.D.O.A NDERSON and J.B.M OORE(1979).Optimal rmation and System Sciences Series, Prentice Hall,Englewood Cliffs,NJ.T.W.A NDERSON(1971).The Statistical Analysis of Time Series.Series in Probability and Mathematical Statistics,Wiley,New York.R.A NDRE-O BRECHT(1988).A new statistical approach for the automatic segmentation of continuous speech signals.IEEE Trans.Acoustics,Speech,Signal Processing,vol.ASSP-36,no1,pp.29-40.R.A NDRE-O BRECHT(1990).Reconnaissance automatique de parole`a partir de segments acoustiques et de mod`e les de Markov cach´e s.Proc.Journ´e es Etude de la Parole,Montr´e al,May1990(in French).R.A NDRE-O BRECHT and H.Y.S U(1988).Three acoustic labellings for phoneme based continuous speech recognition.Proc.Speech’88,Edinburgh,UK,pp.943-950.U.A PPEL and A.VON B RANDT(1983).Adaptive sequential segmentation of piecewise stationary time rmation Sciences,vol.29,no1,pp.27-56.L.A.A ROIAN and H.L EVENE(1950).The effectiveness of quality control procedures.Jal American Statis-tical Association,vol.45,pp.520-529.K.J.A STR¨OM and B.W ITTENMARK(1984).Computer Controlled Systems:Theory and rma-tion and System Sciences Series,Prentice Hall,Englewood Cliffs,NJ.M.B AGSHAW and R.A.J OHNSON(1975a).The effect of serial correlation on the performance of CUSUM tests-Part II.Technometrics,vol.17,no1,pp.73-80.M.B AGSHAW and R.A.J OHNSON(1975b).The inﬂuence of reference values and estimated variance on the ARL of CUSUM tests.Jal Royal Statistical Society,vol.37(B),no3,pp.413-420.M.B AGSHAW and R.A.J OHNSON(1977).Sequential procedures for detecting parameter changes in a time-series model.Jal American Statistical Association,vol.72,no359,pp.593-597.R.K.B ANSAL and P.P APANTONI-K AZAKOS(1986).An algorithm for detecting a change in a stochastic process.IEEE rmation Theory,vol.IT-32,no2,pp.227-235.G.A.B ARNARD(1959).Control charts and stochastic processes.Jal Royal Statistical Society,vol.B.21, pp.239-271.A.E.B ASHARINOV andB.S.F LEISHMAN(1962).Methods of the statistical sequential analysis and their radiotechnical applications.Sovetskoe Radio,Moscow(in Russian).M.B ASSEVILLE(1978).D´e viations par rapport au maximum:formules d’arrˆe t et martingales associ´e es. Compte-rendus du S´e minaire de Probabilit´e s,Universit´e de Rennes I.M.B ASSEVILLE(1981).Edge detection using sequential methods for change in level-Part II:Sequential detection of change in mean.IEEE Trans.Acoustics,Speech,Signal Processing,vol.ASSP-29,no1,pp.32-50.426B IBLIOGRAPHY M.B ASSEVILLE(1982).A survey of statistical failure detection techniques.In Contribution`a la D´e tectionS´e quentielle de Ruptures de Mod`e les Statistiques,Th`e se d’Etat,Universit´e de Rennes I,France(in English). M.B ASSEVILLE(1986).The two-models approach for the on-line detection of changes in AR processes. In Detection of Abrupt Changes in Signals and Dynamical Systems(M.Basseville,A.Benveniste,eds.). Lecture Notes in Control and Information Sciences,LNCIS77,Springer,New York,pp.169-215.M.B ASSEVILLE(1988).Detecting changes in signals and systems-A survey.Automatica,vol.24,pp.309-326.M.B ASSEVILLE(1989).Distance measures for signal processing and pattern recognition.Signal Process-ing,vol.18,pp.349-369.M.B ASSEVILLE and A.B ENVENISTE(1983a).Design and comparative study of some sequential jump detection algorithms for digital signals.IEEE Trans.Acoustics,Speech,Signal Processing,vol.ASSP-31, no3,pp.521-535.M.B ASSEVILLE and A.B ENVENISTE(1983b).Sequential detection of abrupt changes in spectral charac-teristics of digital signals.IEEE rmation Theory,vol.IT-29,no5,pp.709-724.M.B ASSEVILLE and A.B ENVENISTE,eds.(1986).Detection of Abrupt Changes in Signals and Dynamical Systems.Lecture Notes in Control and Information Sciences,LNCIS77,Springer,New York.M.B ASSEVILLE and I.N IKIFOROV(1991).A uniﬁed framework for statistical change detection.Proc.30th IEEE Conference on Decision and Control,Brighton,UK.M.B ASSEVILLE,B.E SPIAU and J.G ASNIER(1981).Edge detection using sequential methods for change in level-Part I:A sequential edge detection algorithm.IEEE Trans.Acoustics,Speech,Signal Processing, vol.ASSP-29,no1,pp.24-31.M.B ASSEVILLE, A.B ENVENISTE and G.M OUSTAKIDES(1986).Detection and diagnosis of abrupt changes in modal characteristics of nonstationary digital signals.IEEE rmation Theory,vol.IT-32,no3,pp.412-417.M.B ASSEVILLE,A.B ENVENISTE,G.M OUSTAKIDES and A.R OUG´E E(1987a).Detection and diagnosis of changes in the eigenstructure of nonstationary multivariable systems.Automatica,vol.23,no3,pp.479-489. M.B ASSEVILLE,A.B ENVENISTE,G.M OUSTAKIDES and A.R OUG´E E(1987b).Optimal sensor location for detecting changes in dynamical behavior.IEEE Trans.Automatic Control,vol.AC-32,no12,pp.1067-1075.M.B ASSEVILLE,A.B ENVENISTE,B.G ACH-D EVAUCHELLE,M.G OURSAT,D.B ONNECASE,P.D OREY, M.P REVOSTO and M.O LAGNON(1993).Damage monitoring in vibration mechanics:issues in diagnos-tics and predictive maintenance.Mechanical Systems and Signal Processing,vol.7,no5,pp.401-423.R.V.B EARD(1971).Failure Accommodation in Linear Systems through Self-reorganization.Ph.D.Thesis, Dept.Aeronautics and Astronautics,MIT,Cambridge,MA.A.B ENVENISTE and J.J.F UCHS(1985).Single sample modal identiﬁcation of a nonstationary stochastic process.IEEE Trans.Automatic Control,vol.AC-30,no1,pp.66-74.A.B ENVENISTE,M.B ASSEVILLE and G.M OUSTAKIDES(1987).The asymptotic local approach to change detection and model validation.IEEE Trans.Automatic Control,vol.AC-32,no7,pp.583-592.A.B ENVENISTE,M.M ETIVIER and P.P RIOURET(1990).Adaptive Algorithms and Stochastic Approxima-tions.Series on Applications of Mathematics,(A.V.Balakrishnan,I.Karatzas,M.Yor,eds.).Springer,New York.A.B ENVENISTE,M.B ASSEVILLE,L.E L G HAOUI,R.N IKOUKHAH and A.S.W ILLSKY(1992).An optimum robust approach to statistical failure detection and identiﬁcation.IFAC World Conference,Sydney, July1993.B IBLIOGRAPHY427 R.H.B ERK(1973).Some asymptotic aspects of sequential analysis.Annals Statistics,vol.1,no6,pp.1126-1138.R.H.B ERK(1975).Locally most powerful sequential test.Annals Statistics,vol.3,no2,pp.373-381.P.B ILLINGSLEY(1968).Convergence of Probability Measures.Wiley,New York.A.F.B ISSELL(1969).Cusum techniques for quality control.Applied Statistics,vol.18,pp.1-30.M.E.B IVAIKOV(1991).Control of the sample size for recursive estimation of parameters subject to abrupt changes.Automation and Remote Control,no9,pp.96-103.R.E.B LAHUT(1987).Principles and Practice of Information Theory.Addison-Wesley,Reading,MA.I.F.B LAKE and W.C.L INDSEY(1973).Level-crossing problems for random processes.IEEE r-mation Theory,vol.IT-19,no3,pp.295-315.G.B ODENSTEIN and H.M.P RAETORIUS(1977).Feature extraction from the encephalogram by adaptive segmentation.Proc.IEEE,vol.65,pp.642-652.T.B OHLIN(1977).Analysis of EEG signals with changing spectra using a short word Kalman estimator. Mathematical Biosciences,vol.35,pp.221-259.W.B¨OHM and P.H ACKL(1990).Improved bounds for the average run length of control charts based on ﬁnite weighted sums.Annals Statistics,vol.18,no4,pp.1895-1899.T.B OJDECKI and J.H OSZA(1984).On a generalized disorder problem.Stochastic Processes and their Applications,vol.18,pp.349-359.L.I.B ORODKIN and V.V.M OTTL’(1976).Algorithm forﬁnding the jump times of random process equation parameters.Automation and Remote Control,vol.37,no6,Part1,pp.23-32.A.A.B OROVKOV(1984).Theory of Mathematical Statistics-Estimation and Hypotheses Testing,Naouka, Moscow(in Russian).Translated in French under the title Statistique Math´e matique-Estimation et Tests d’Hypoth`e ses,Mir,Paris,1987.G.E.P.B OX and G.M.J ENKINS(1970).Time Series Analysis,Forecasting and Control.Series in Time Series Analysis,Holden-Day,San Francisco.A.VON B RANDT(1983).Detecting and estimating parameters jumps using ladder algorithms and likelihood ratio test.Proc.ICASSP,Boston,MA,pp.1017-1020.A.VON B RANDT(1984).Modellierung von Signalen mit Sprunghaft Ver¨a nderlichem Leistungsspektrum durch Adaptive Segmentierung.Doctor-Engineer Dissertation,M¨u nchen,RFA(in German).S.B RAUN,ed.(1986).Mechanical Signature Analysis-Theory and Applications.Academic Press,London. L.B REIMAN(1968).Probability.Series in Statistics,Addison-Wesley,Reading,MA.G.S.B RITOV and L.A.M IRONOVSKI(1972).Diagnostics of linear systems of automatic regulation.Tekh. Kibernetics,vol.1,pp.76-83.B.E.B RODSKIY and B.S.D ARKHOVSKIY(1992).Nonparametric Methods in Change-point Problems. Kluwer Academic,Boston.L.D.B ROEMELING(1982).Jal Econometrics,vol.19,Special issue on structural change in Econometrics. L.D.B ROEMELING and H.T SURUMI(1987).Econometrics and Structural Change.Dekker,New York. D.B ROOK and D.A.E VANS(1972).An approach to the probability distribution of Cusum run length. Biometrika,vol.59,pp.539-550.J.B RUNET,D.J AUME,M.L ABARR`E RE,A.R AULT and M.V ERG´E(1990).D´e tection et Diagnostic de Pannes.Trait´e des Nouvelles Technologies,S´e rie Diagnostic et Maintenance,Herm`e s,Paris(in French).428B IBLIOGRAPHY S.P.B RUZZONE and M.K AVEH(1984).Information tradeoffs in using the sample autocorrelation function in ARMA parameter estimation.IEEE Trans.Acoustics,Speech,Signal Processing,vol.ASSP-32,no4, pp.701-715.A.K.C AGLAYAN(1980).Necessary and sufﬁcient conditions for detectability of jumps in linear systems. IEEE Trans.Automatic Control,vol.AC-25,no4,pp.833-834.A.K.C AGLAYAN and R.E.L ANCRAFT(1983).Reinitialization issues in fault tolerant systems.Proc.Amer-ican Control Conf.,pp.952-955.A.K.C AGLAYAN,S.M.A LLEN and K.W EHMULLER(1988).Evaluation of a second generation reconﬁgu-ration strategy for aircraftﬂight control systems subjected to actuator failure/surface damage.Proc.National Aerospace and Electronic Conference,Dayton,OH.P.E.C AINES(1988).Linear Stochastic Systems.Series in Probability and Mathematical Statistics,Wiley, New York.M.J.C HEN and J.P.N ORTON(1987).Estimation techniques for tracking rapid parameter changes.Intern. Jal Control,vol.45,no4,pp.1387-1398.W.K.C HIU(1974).The economic design of cusum charts for controlling normal mean.Applied Statistics, vol.23,no3,pp.420-433.E.Y.C HOW(1980).A Failure Detection System Design Methodology.Ph.D.Thesis,M.I.T.,L.I.D.S.,Cam-bridge,MA.E.Y.C HOW and A.S.W ILLSKY(1984).Analytical redundancy and the design of robust failure detection systems.IEEE Trans.Automatic Control,vol.AC-29,no3,pp.689-691.Y.S.C HOW,H.R OBBINS and D.S IEGMUND(1971).Great Expectations:The Theory of Optimal Stop-ping.Houghton-Mifﬂin,Boston.R.N.C LARK,D.C.F OSTH and V.M.W ALTON(1975).Detection of instrument malfunctions in control systems.IEEE Trans.Aerospace Electronic Systems,vol.AES-11,pp.465-473.A.C OHEN(1987).Biomedical Signal Processing-vol.1:Time and Frequency Domain Analysis;vol.2: Compression and Automatic Recognition.CRC Press,Boca Raton,FL.J.C ORGE and F.P UECH(1986).Analyse du rythme cardiaque foetal par des m´e thodes de d´e tection de ruptures.Proc.7th INRIA Int.Conf.Analysis and optimization of Systems.Antibes,FR(in French).D.R.C OX and D.V.H INKLEY(1986).Theoretical Statistics.Chapman and Hall,New York.D.R.C OX and H.D.M ILLER(1965).The Theory of Stochastic Processes.Wiley,New York.S.V.C ROWDER(1987).A simple method for studying run-length distributions of exponentially weighted moving average charts.Technometrics,vol.29,no4,pp.401-407.H.C S¨ORG¨O and L.H ORV´ATH(1988).Nonparametric methods for change point problems.In Handbook of Statistics(P.R.Krishnaiah,C.R.Rao,eds.),vol.7,Elsevier,New York,pp.403-425.R.B.D AVIES(1973).Asymptotic inference in stationary gaussian time series.Advances Applied Probability, vol.5,no3,pp.469-497.J.C.D ECKERT,M.N.D ESAI,J.J.D EYST and A.S.W ILLSKY(1977).F-8DFBW sensor failure identiﬁcation using analytical redundancy.IEEE Trans.Automatic Control,vol.AC-22,no5,pp.795-803.M.H.D E G ROOT(1970).Optimal Statistical Decisions.Series in Probability and Statistics,McGraw-Hill, New York.J.D ESHAYES and D.P ICARD(1979).Tests de ruptures dans un mod`e pte-Rendus de l’Acad´e mie des Sciences,vol.288,Ser.A,pp.563-566(in French).B IBLIOGRAPHY429 J.D ESHAYES and D.P ICARD(1983).Ruptures de Mod`e les en Statistique.Th`e ses d’Etat,Universit´e deParis-Sud,Orsay,France(in French).J.D ESHAYES and D.P ICARD(1986).Off-line statistical analysis of change-point models using non para-metric and likelihood methods.In Detection of Abrupt Changes in Signals and Dynamical Systems(M. Basseville,A.Benveniste,eds.).Lecture Notes in Control and Information Sciences,LNCIS77,Springer, New York,pp.103-168.B.D EVAUCHELLE-G ACH(1991).Diagnostic M´e canique des Fatigues sur les Structures Soumises`a des Vibrations en Ambiance de Travail.Th`e se de l’Universit´e Paris IX Dauphine(in French).B.D EVAUCHELLE-G ACH,M.B ASSEVILLE and A.B ENVENISTE(1991).Diagnosing mechanical changes in vibrating systems.Proc.SAFEPROCESS’91,Baden-Baden,FRG,pp.85-89.R.D I F RANCESCO(1990).Real-time speech segmentation using pitch and convexity jump models:applica-tion to variable rate speech coding.IEEE Trans.Acoustics,Speech,Signal Processing,vol.ASSP-38,no5, pp.741-748.X.D ING and P.M.F RANK(1990).Fault detection via factorization approach.Systems and Control Letters, vol.14,pp.431-436.J.L.D OOB(1953).Stochastic Processes.Wiley,New York.V.D RAGALIN(1988).Asymptotic solutions in detecting a change in distribution under an unknown param-eter.Statistical Problems of Control,Issue83,Vilnius,pp.45-52.B.D UBUISSON(1990).Diagnostic et Reconnaissance des Formes.Trait´e des Nouvelles Technologies,S´e rie Diagnostic et Maintenance,Herm`e s,Paris(in French).A.J.D UNCAN(1986).Quality Control and Industrial Statistics,5th edition.Richard D.Irwin,Inc.,Home-wood,IL.J.D URBIN(1971).Boundary-crossing probabilities for the Brownian motion and Poisson processes and techniques for computing the power of the Kolmogorov-Smirnov test.Jal Applied Probability,vol.8,pp.431-453.J.D URBIN(1985).Theﬁrst passage density of the crossing of a continuous Gaussian process to a general boundary.Jal Applied Probability,vol.22,no1,pp.99-122.A.E MAMI-N AEINI,M.M.A KHTER and S.M.R OCK(1988).Effect of model uncertainty on failure detec-tion:the threshold selector.IEEE Trans.Automatic Control,vol.AC-33,no12,pp.1106-1115.J.D.E SARY,F.P ROSCHAN and D.W.W ALKUP(1967).Association of random variables with applications. Annals Mathematical Statistics,vol.38,pp.1466-1474.W.D.E WAN and K.W.K EMP(1960).Sampling inspection of continuous processes with no autocorrelation between successive results.Biometrika,vol.47,pp.263-280.G.F AVIER and A.S MOLDERS(1984).Adaptive smoother-predictors for tracking maneuvering targets.Proc. 23rd Conf.Decision and Control,Las Vegas,NV,pp.831-836.W.F ELLER(1966).An Introduction to Probability Theory and Its Applications,vol.2.Series in Probability and Mathematical Statistics,Wiley,New York.R.A.F ISHER(1925).Theory of statistical estimation.Proc.Cambridge Philosophical Society,vol.22, pp.700-725.M.F ISHMAN(1988).Optimization of the algorithm for the detection of a disorder,based on the statistic of exponential smoothing.In Statistical Problems of Control,Issue83,Vilnius,pp.146-151.R.F LETCHER(1980).Practical Methods of Optimization,2volumes.Wiley,New York.P.M.F RANK(1990).Fault diagnosis in dynamic systems using analytical and knowledge based redundancy -A survey and new results.Automatica,vol.26,pp.459-474.430B IBLIOGRAPHY P.M.F RANK(1991).Enhancement of robustness in observer-based fault detection.Proc.SAFEPRO-CESS’91,Baden-Baden,FRG,pp.275-287.P.M.F RANK and J.W¨UNNENBERG(1989).Robust fault diagnosis using unknown input observer schemes. In Fault Diagnosis in Dynamic Systems-Theory and Application(R.Patton,P.Frank,R.Clark,eds.). International Series in Systems and Control Engineering,Prentice Hall International,London,UK,pp.47-98.K.F UKUNAGA(1990).Introduction to Statistical Pattern Recognition,2d ed.Academic Press,New York. S.I.G ASS(1958).Linear Programming:Methods and Applications.McGraw Hill,New York.W.G E and C.Z.F ANG(1989).Extended robust observation approach for failure isolation.Int.Jal Control, vol.49,no5,pp.1537-1553.W.G ERSCH(1986).Two applications of parametric time series modeling methods.In Mechanical Signature Analysis-Theory and Applications(S.Braun,ed.),chap.10.Academic Press,London.J.J.G ERTLER(1988).Survey of model-based failure detection and isolation in complex plants.IEEE Control Systems Magazine,vol.8,no6,pp.3-11.J.J.G ERTLER(1991).Analytical redundancy methods in fault detection and isolation.Proc.SAFEPRO-CESS’91,Baden-Baden,FRG,pp.9-22.B.K.G HOSH(1970).Sequential Tests of Statistical Hypotheses.Addison-Wesley,Cambridge,MA.I.N.G IBRA(1975).Recent developments in control charts techniques.Jal Quality Technology,vol.7, pp.183-192.J.P.G ILMORE and R.A.M C K ERN(1972).A redundant strapdown inertial reference unit(SIRU).Jal Space-craft,vol.9,pp.39-47.M.A.G IRSHICK and H.R UBIN(1952).A Bayes approach to a quality control model.Annals Mathematical Statistics,vol.23,pp.114-125.A.L.G OEL and S.M.W U(1971).Determination of the ARL and a contour nomogram for CUSUM charts to control normal mean.Technometrics,vol.13,no2,pp.221-230.P.L.G OLDSMITH and H.W HITFIELD(1961).Average run lengths in cumulative chart quality control schemes.Technometrics,vol.3,pp.11-20.G.C.G OODWIN and K.S.S IN(1984).Adaptive Filtering,Prediction and rmation and System Sciences Series,Prentice Hall,Englewood Cliffs,NJ.R.M.G RAY and L.D.D AVISSON(1986).Random Processes:a Mathematical Approach for Engineers. Information and System Sciences Series,Prentice Hall,Englewood Cliffs,NJ.C.G UEGUEN and L.L.S CHARF(1980).Exact maximum likelihood identiﬁcation for ARMA models:a signal processing perspective.Proc.1st EUSIPCO,Lausanne.D.E.G USTAFSON, A.S.W ILLSKY,J.Y.W ANG,M.C.L ANCASTER and J.H.T RIEBWASSER(1978). ECG/VCG rhythm diagnosis using statistical signal analysis.Part I:Identiﬁcation of persistent rhythms. Part II:Identiﬁcation of transient rhythms.IEEE Trans.Biomedical Engineering,vol.BME-25,pp.344-353 and353-361.F.G USTAFSSON(1991).Optimal segmentation of linear regression parameters.Proc.IFAC/IFORS Symp. Identiﬁcation and System Parameter Estimation,Budapest,pp.225-229.T.H¨AGGLUND(1983).New Estimation Techniques for Adaptive Control.Ph.D.Thesis,Lund Institute of Technology,Lund,Sweden.T.H¨AGGLUND(1984).Adaptive control of systems subject to large parameter changes.Proc.IFAC9th World Congress,Budapest.B IBLIOGRAPHY431 P.H ALL and C.C.H EYDE(1980).Martingale Limit Theory and its Application.Probability and Mathemat-ical Statistics,a Series of Monographs and Textbooks,Academic Press,New York.W.J.H ALL,R.A.W IJSMAN and J.K.G HOSH(1965).The relationship between sufﬁciency and invariance with applications in sequential analysis.Ann.Math.Statist.,vol.36,pp.576-614.E.J.H ANNAN and M.D EISTLER(1988).The Statistical Theory of Linear Systems.Series in Probability and Mathematical Statistics,Wiley,New York.J.D.H EALY(1987).A note on multivariate CuSum procedures.Technometrics,vol.29,pp.402-412.D.M.H IMMELBLAU(1970).Process Analysis by Statistical Methods.Wiley,New York.D.M.H IMMELBLAU(1978).Fault Detection and Diagnosis in Chemical and Petrochemical Processes. Chemical Engineering Monographs,vol.8,Elsevier,Amsterdam.W.G.S.H INES(1976a).A simple monitor of a system with sudden parameter changes.IEEE r-mation Theory,vol.IT-22,no2,pp.210-216.W.G.S.H INES(1976b).Improving a simple monitor of a system with sudden parameter changes.IEEE rmation Theory,vol.IT-22,no4,pp.496-499.D.V.H INKLEY(1969).Inference about the intersection in two-phase regression.Biometrika,vol.56,no3, pp.495-504.D.V.H INKLEY(1970).Inference about the change point in a sequence of random variables.Biometrika, vol.57,no1,pp.1-17.D.V.H INKLEY(1971).Inference about the change point from cumulative sum-tests.Biometrika,vol.58, no3,pp.509-523.D.V.H INKLEY(1971).Inference in two-phase regression.Jal American Statistical Association,vol.66, no336,pp.736-743.J.R.H UDDLE(1983).Inertial navigation system error-model considerations in Kalmanﬁltering applica-tions.In Control and Dynamic Systems(C.T.Leondes,ed.),Academic Press,New York,pp.293-339.J.S.H UNTER(1986).The exponentially weighted moving average.Jal Quality Technology,vol.18,pp.203-210.I.A.I BRAGIMOV and R.Z.K HASMINSKII(1981).Statistical Estimation-Asymptotic Theory.Applications of Mathematics Series,vol.16.Springer,New York.R.I SERMANN(1984).Process fault detection based on modeling and estimation methods-A survey.Auto-matica,vol.20,pp.387-404.N.I SHII,A.I WATA and N.S UZUMURA(1979).Segmentation of nonstationary time series.Int.Jal Systems Sciences,vol.10,pp.883-894.J.E.J ACKSON and R.A.B RADLEY(1961).Sequential and tests.Annals Mathematical Statistics, vol.32,pp.1063-1077.B.J AMES,K.L.J AMES and D.S IEGMUND(1988).Conditional boundary crossing probabilities with appli-cations to change-point problems.Annals Probability,vol.16,pp.825-839.M.K.J EERAGE(1990).Reliability analysis of fault-tolerant IMU architectures with redundant inertial sen-sors.IEEE Trans.Aerospace and Electronic Systems,vol.AES-5,no.7,pp.23-27.N.L.J OHNSON(1961).A simple theoretical approach to cumulative sum control charts.Jal American Sta-tistical Association,vol.56,pp.835-840.N.L.J OHNSON and F.C.L EONE(1962).Cumulative sum control charts:mathematical principles applied to their construction and use.Parts I,II,III.Industrial Quality Control,vol.18,pp.15-21;vol.19,pp.29-36; vol.20,pp.22-28.432B IBLIOGRAPHY R.A.J OHNSON and M.B AGSHAW(1974).The effect of serial correlation on the performance of CUSUM tests-Part I.Technometrics,vol.16,no.1,pp.103-112.H.L.J ONES(1973).Failure Detection in Linear Systems.Ph.D.Thesis,Dept.Aeronautics and Astronautics, MIT,Cambridge,MA.R.H.J ONES,D.H.C ROWELL and L.E.K APUNIAI(1970).Change detection model for serially correlated multivariate data.Biometrics,vol.26,no2,pp.269-280.M.J URGUTIS(1984).Comparison of the statistical properties of the estimates of the change times in an autoregressive process.In Statistical Problems of Control,Issue65,Vilnius,pp.234-243(in Russian).T.K AILATH(1980).Linear rmation and System Sciences Series,Prentice Hall,Englewood Cliffs,NJ.L.V.K ANTOROVICH and V.I.K RILOV(1958).Approximate Methods of Higher Analysis.Interscience,New York.S.K ARLIN and H.M.T AYLOR(1975).A First Course in Stochastic Processes,2d ed.Academic Press,New York.S.K ARLIN and H.M.T AYLOR(1981).A Second Course in Stochastic Processes.Academic Press,New York.D.K AZAKOS and P.P APANTONI-K AZAKOS(1980).Spectral distance measures between gaussian pro-cesses.IEEE Trans.Automatic Control,vol.AC-25,no5,pp.950-959.K.W.K EMP(1958).Formula for calculating the operating characteristic and average sample number of some sequential tests.Jal Royal Statistical Society,vol.B-20,no2,pp.379-386.K.W.K EMP(1961).The average run length of the cumulative sum chart when a V-mask is used.Jal Royal Statistical Society,vol.B-23,pp.149-153.K.W.K EMP(1967a).Formal expressions which can be used for the determination of operating character-istics and average sample number of a simple sequential test.Jal Royal Statistical Society,vol.B-29,no2, pp.248-262.K.W.K EMP(1967b).A simple procedure for determining upper and lower limits for the average sample run length of a cumulative sum scheme.Jal Royal Statistical Society,vol.B-29,no2,pp.263-265.D.P.K ENNEDY(1976).Some martingales related to cumulative sum tests and single server queues.Stochas-tic Processes and Appl.,vol.4,pp.261-269.T.H.K ERR(1980).Statistical analysis of two-ellipsoid overlap test for real time failure detection.IEEE Trans.Automatic Control,vol.AC-25,no4,pp.762-772.T.H.K ERR(1982).False alarm and correct detection probabilities over a time interval for restricted classes of failure detection algorithms.IEEE rmation Theory,vol.IT-24,pp.619-631.T.H.K ERR(1987).Decentralizedﬁltering and redundancy management for multisensor navigation.IEEE Trans.Aerospace and Electronic systems,vol.AES-23,pp.83-119.Minor corrections on p.412and p.599 (May and July issues,respectively).R.A.K HAN(1978).Wald’s approximations to the average run length in cusum procedures.Jal Statistical Planning and Inference,vol.2,no1,pp.63-77.R.A.K HAN(1979).Someﬁrst passage problems related to cusum procedures.Stochastic Processes and Applications,vol.9,no2,pp.207-215.R.A.K HAN(1981).A note on Page’s two-sided cumulative sum procedures.Biometrika,vol.68,no3, pp.717-719.B IBLIOGRAPHY433 V.K IREICHIKOV,V.M ANGUSHEV and I.N IKIFOROV(1990).Investigation and application of CUSUM algorithms to monitoring of sensors.In Statistical Problems of Control,Issue89,Vilnius,pp.124-130(in Russian).G.K ITAGAWA and W.G ERSCH(1985).A smoothness prior time-varying AR coefﬁcient modeling of non-stationary covariance time series.IEEE Trans.Automatic Control,vol.AC-30,no1,pp.48-56.N.K LIGIENE(1980).Probabilities of deviations of the change point estimate in statistical models.In Sta-tistical Problems of Control,Issue83,Vilnius,pp.80-86(in Russian).N.K LIGIENE and L.T ELKSNYS(1983).Methods of detecting instants of change of random process prop-erties.Automation and Remote Control,vol.44,no10,Part II,pp.1241-1283.J.K ORN,S.W.G ULLY and A.S.W ILLSKY(1982).Application of the generalized likelihood ratio algorithm to maneuver detection and estimation.Proc.American Control Conf.,Arlington,V A,pp.792-798.P.R.K RISHNAIAH and B.Q.M IAO(1988).Review about estimation of change points.In Handbook of Statistics(P.R.Krishnaiah,C.R.Rao,eds.),vol.7,Elsevier,New York,pp.375-402.P.K UDVA,N.V ISWANADHAM and A.R AMAKRISHNAN(1980).Observers for linear systems with unknown inputs.IEEE Trans.Automatic Control,vol.AC-25,no1,pp.113-115.S.K ULLBACK(1959).Information Theory and Statistics.Wiley,New York(also Dover,New York,1968). K.K UMAMARU,S.S AGARA and T.S¨ODERSTR¨OM(1989).Some statistical methods for fault diagnosis for dynamical systems.In Fault Diagnosis in Dynamic Systems-Theory and Application(R.Patton,P.Frank,R. Clark,eds.).International Series in Systems and Control Engineering,Prentice Hall International,London, UK,pp.439-476.A.K USHNIR,I.N IKIFOROV and I.S AVIN(1983).Statistical adaptive algorithms for automatic detection of seismic signals-Part I:One-dimensional case.In Earthquake Prediction and the Study of the Earth Structure,Naouka,Moscow(Computational Seismology,vol.15),pp.154-159(in Russian).L.L ADELLI(1990).Diffusion approximation for a pseudo-likelihood test process with application to de-tection of change in stochastic system.Stochastics and Stochastics Reports,vol.32,pp.1-25.T.L.L A¨I(1974).Control charts based on weighted sums.Annals Statistics,vol.2,no1,pp.134-147.T.L.L A¨I(1981).Asymptotic optimality of invariant sequential probability ratio tests.Annals Statistics, vol.9,no2,pp.318-333.D.G.L AINIOTIS(1971).Joint detection,estimation,and system identiﬁrmation and Control, vol.19,pp.75-92.M.R.L EADBETTER,G.L INDGREN and H.R OOTZEN(1983).Extremes and Related Properties of Random Sequences and Processes.Series in Statistics,Springer,New York.L.L E C AM(1960).Locally asymptotically normal families of distributions.Univ.California Publications in Statistics,vol.3,pp.37-98.L.L E C AM(1986).Asymptotic Methods in Statistical Decision Theory.Series in Statistics,Springer,New York.E.L.L EHMANN(1986).Testing Statistical Hypotheses,2d ed.Wiley,New York.J.P.L EHOCZKY(1977).Formulas for stopped diffusion processes with stopping times based on the maxi-mum.Annals Probability,vol.5,no4,pp.601-607.H.R.L ERCHE(1980).Boundary Crossing of Brownian Motion.Lecture Notes in Statistics,vol.40,Springer, New York.L.L JUNG(1987).System Identiﬁcation-Theory for the rmation and System Sciences Series, Prentice Hall,Englewood Cliffs,NJ.。

acf和adf代码

ACF and ADF CodesIntroductionIn time series analysis, two commonly used models for forecasting are the autoregressive conditional heteroskedasticity (ARCH) model and the autoregressive integrated moving average (ARIMA) model. These models are widely used in finance, economics, and other fields to predict future values of a time series based on its past behavior. In this article, we will explore the ACF (Autocorrelation Function) and ADF (Augmented Dickey-Fuller) codes, which are essential tools for analyzing and diagnosing time series data.Autocorrelation Function (ACF)The ACF measures the correlation between a time series and its lagged values. It helps us understand the relationship between the current value of a series and its past values at different lags. The ACF is a crucial tool for identifying patterns, seasonality, and dependencies in time series data.The ACF code calculates the correlation coefficient for each lag and plots it against the lag. The code snippet below demonstrates how to calculate and plot the ACF for a given time series using Python:import pandas as pdimport matplotlib.pyplot as pltfrom statsmodels.graphics.tsaplots import plot_acf# Load the time series datadata = pd.read_csv('time_series_data.csv')# Calculate and plot the ACFplot_acf(data, lags=20)plt.show()In the code above, we first load the time series data from a CSV file using the pandas library. Then, we use the plot_acf function from the statsmodels.graphics.tsaplots module to calculate and plot the ACF. The lags parameter specifies the number of lags to include in the plot. Finally, we use plt.show() to display the plot.The ACF plot helps us identify significant lags in the time series data. If a lag has a correlation coefficient close to 1 or -1, it indicates a strong positive or negative correlation, respectively. On the other hand, if the correlation coefficient is close to 0, it suggests no correlation.Augmented Dickey-Fuller (ADF) TestThe ADF test is a statistical test used to determine if a time series is stationary or not. Stationarity is a key assumption in many time series models, including ARIMA. A stationary time series has constant mean and variance over time, and its properties do not depend on the specifictime period.The ADF code allows us to perform the ADF test on a given time series to check for stationarity. The following code snippet demonstrates how to perform the ADF test using Python:import pandas as pdfrom statsmodels.tsa.stattools import adfuller# Load the time series datadata = pd.read_csv('time_series_data.csv')# Perform the ADF testresult = adfuller(data)# Extract and print the test statisticsprint('ADF Statistic:', result[0])print('p-value:', result[1])print('Critical Values:', result[4])In the code above, we first load the time series data from a CSV file using the pandas library. Then, we use the adfuller function from the statsmodels.tsa.stattools module to perform the ADF test. The function returns a tuple containing the test statistics, p-value, and critical values.The ADF Statistic is the test statistic of the ADF test. The p-value represents the probability that the null hypothesis of non-stationarity is true. If the p-value is less than a chosen significance level (e.g., 0.05), we can reject the null hypothesis and conclude that the time series is stationary. The Critical Values represent the threshold values for different significance levels.ConclusionThe ACF and ADF codes are essential tools for analyzing and diagnosing time series data. The ACF helps us understand the correlation between a time series and its lagged values, while the ADF test allows us to check for stationarity. By using these codes, we can gain insights into the patterns, dependencies, and stationarity of time series data, which are crucial for accurate forecasting and decision-making in various fields.。

计量经济学Test-bank-questions-Chapter-5

Multiple Choice Test Bank Questions No Feedback – Chapter 5 Correct answers denoted by an asterisk.1. Consider the following model estimated for a time seriesy t = 0.3 + 0.5 y t-1 - 0.4εt-1 +εtwhere εt is a zero mean error process.What is the (unconditional) mean of the series, y t ?(a) * 0.6(b) 0.3(c) 0.0(d) 0.42. Consider the following single exponential smoothing model:S t = α X t + (1-α) S t-1You are given the following data:αˆ=0.1, X t=0.5,S t-1=0.2If we believe that the true DGP can be approximated by the exponential smoothing model, what would be an appropriate 2-step ahead forecast for X? (i.e. a forecast of X t+2 made at time t)(a) 0.2(b) * 0.23(c) 0.5(d) There is insufficient information given in the question to form more than a one step ahead forecast.3. Consider the following MA(3) process.y t = 0.1 + 0.4u t-1 + 0.2u t-2– 0.1u t-3 + u tWhat is the optimal forecast for y t, 3 steps into the future (i.e. for time t+2 if all information until time t-1 is available), if you have the following data?u t-1 = 0.3; u t-2 = -0.6; u t-3 = -0.3(a)0.4(b)0.0(c)* 0.07(d)–0.14. Which of the following sets of characteristics would usually best describe an autoregressive process of order 3 (i.e. an AR(3))?(a) * A slowly decaying acf, and a pacf with 3 significant spikes(b) A slowly decaying pacf and an acf with 3 significant spikes(c) A slowly decaying acf and pacf(d) An acf and a pacf with 3 significant spikes5. A process, x t, which has a constant mean and variance, and zero autocovariance for all non-zero lags is best described as(a) * A white noise process(b) A covariance stationary process(c) An autocorrelated process(d) A moving average process6. Which of the following conditions must hold for the autoregressive part of an ARMA model to be stationary?(a) * All roots of the characteristic equation must lie outside the unit circle(b) All roots of the characteristic equation must lie inside the unit circle(c) All roots must be smaller than unity(d) At least one of the roots must be bigger than one in absolute value.7. Which of the following statements are true concerning time-series forecasting?(i) All time-series forecasting methods are essentially extrapolative.(ii) Forecasting models are prone to perform poorly following a structural break in a series.(iii) Forecasting accuracy often declines with prediction horizon.(iv) The mean squared errors of forecasts are usually very highly correlated with the profitability of employing those forecasts in a trading strategy.(a) (i), (ii), (iii), and (iv)(b) * (i), (ii) and (iii) only(c) (ii), (iii) only(d) (ii) and (iv) only8. If a series, y t, follows a random walk (with no drift), what is the optimal 1-step ahead forecast for y?(a)* The current value of y.(b)Zero.(c)The historical unweighted average of y.(d)An exponentially weighted average of previous values of y.9. Consider a series that follows an MA(1) with zero mean and a moving average coefficient of 0.4. What is the value of the autocorrelation function at lag 1?(a)0.4(b)1(c)*0.34(d)It is not possible to determine the value of the autocovariances without knowing thedisturbance variance.10. Which of the following statements are true?(i)An MA(q) can be expressed as an AR(infinity) if it is invertible(ii)An AR(p) can be written as an MA(infinity) if it is stationary(iii)The (unconditional) mean of an ARMA process will depend only on the intercept and on the AR coefficients and not on the MA coefficients(iv) A random walk series will have zero pacf except at lag 1(a)(ii) and (iv) only(b)(i) and (iii) only(c)(i), (ii), and (iii) only(d)* (i), (ii), (iii), and (iv).11. Consider the following picture and suggest the model from the following list that best characterises the process:(a)An AR(1)(b)An AR(2)(c)* An ARMA(1,1)(d)An MA(3)The acf is clearly declining very slowly in this case, which is consistent with their being an autoregressive part to the appropriate model. The pacf is clearly significant for lags one and two, but the question is does it them become insignificant for lags 2 and 4,indicating an AR(2) process, or does it remain significant, which would be more consistent with a mixed ARMA process? Well, given the huge size of the sample that gave rise to this acf and pacf, even a pacf value of 0.001 would still be statistically significant. Thus an ARMA process is the most likely candidate, although note that it would not be possible to tell from the acf and pacf which model from the ARMA family was more appropriate. The DGP for the data that generated this plot was y_t = 0.9 y_(t-1) – 0.3 u_(t-1) + u_t.12. Which of the following models can be estimated using ordinary least squares?(i)An AR(1)(ii) An ARMA(2,0)(iii) An MA(1)(iv) An ARMA(1,1)(a)(i) only(b)* (i) and (ii) only(c)(i), (ii), and (iii) only(d)(i), (ii), (iii), and (iv).13. If a series, y, is described as “mea n-reverting”, which model from the following list is likely to produce the best long-term forecasts for that series y?(a)A random walk(b)* The long term mean of the series(c)A model from the ARMA family(d)A random walk with drift14. Consider the following AR(2) model. What is the optimal 2-step ahead forecast for y if all information available is up to and including time t, if the values of y at time t, t-1 and t-2 are –0.3, 0.4 and –0.1 respectively, and the value of u at time t-1 is 0.3?y t = -0.1 + 0.75y t-1 - 0.125y t-2 + u t(a)-0.1(b)0.27(c)* -0.34(d)0.3015. What is the optimal three-step ahead forecast from the AR(2) model given in question 14?(a)-0.1(b)0.27(c)-0.34(d)* -0.3116. Suppose you had to guess at the most likely value of a one hundred step-ahead forecast for the AR(2) model given in question 14 – what would your forecast be?(a)-0.1(b)0.7(c)* –0.27(d)0.75。

Models for Time Series and ForecastingPPT

CHAPTER 18 Models for Time Series and Forecasting
to accompany
Introduction to Business Statistics
fourth edition, by Ronald M. Weiers
Presentation by Priscilla Chaffe-Stengel
© 2002 The Wadsworth Group
? = b0 + b1x y • Linear: ? = b + b x + b x2 • Quadratic: y 0 1 2
Trend Equations
? = the trend line estimate of y y x = time period b0, b1, and b2 are coefficients that are selected to minimize the deviations between the ? and the actual data values trend estimates y y for the past time periods. Regression methods are used to determine the best values for the coefficients.
• Trend equation • Moving average • Exponential smoothing
ቤተ መጻሕፍቲ ባይዱ
• Seasonal index • Ratio to moving average method • Deseasonalizing • MAD criterion • MSE criterion • Constructing an index using the CPI • Shifting the base of an index

Principles of Marketing Engineering - Ch5 - Forecasting

A study by Van den Bulte and Stremersch (2004) suggests an average value of 0.03 for p and an average value of 0.42 for q, The average was taken across a couple of hundred categories.
© CUMT2011 Principles Chapter 6: Forecasting - 14
6 巴斯模型的参数估计方法
（1）以历史销售数据（Past /Historical
Data）为依据校准模型
线性回归非线性最小二乘法
（2）以主观判断为依据校准模型
相似产品进行类比（Analogy）市场调查
N (t ) n(t ) [ N N (t )] p q N N(t) : Cumulative number of adopters until time t.
© CUMT2011 Principles Chapter 6: Forecasting - 9
3 巴斯（Bass）模型参数
This chapter focuses mainly on new
product forecasting
Bass diffusion model（巴斯扩散模型）
© CUMT2011 Principles Chapter 6: Forecasting - 4
第二节新产品预测的Bass 模型
© CUMT2011 Principles Chapter 6: Forecasting - 16
6 Generalized Bass Model

【免费下载】Wiley Series in Probability and Statistics

Wiley Series in Probability and Statistics1. A Course in Time Series Analysis2. A Primer on Experiments with Mixtures3. A Probabilistic Analysis of the Sacco and Vanzetti Evidence4. A User's Guide to Principal Components5. A Weak Convergence Approach to the Theory of Large Deviations6.Accelerated Testing: Statistical Models, Test Plans, and Data Analysis7.Advanced Calculus with Applications in Statistics, Second Edition8.Alternative Methods of Regression9.An Elementary Introduction to Statistical Learning Theory10.An Introduction to Categorical Data Analysis, Second Edition11.An Introduction to Probability and Statistics, Second Edition12.An Introduction to Regression Graphics13.Analysis of Financial Time Series14.Analysis of Financial Time Series, Second Edition15.Analysis of Financial Time Series, Third Edition, Third Edition16.Analysis of Ordinal Categorical Data, Second Edition17.Analyzing Microarray Gene Expression Data18.Applications of Statistics to Industrial Experimentation19.Applied Bayesian Modeling and Causal Inference from Incomplete-DataPerspectives: An Essential Journey with Donald Rubin's Statistical Family20.Applied Bayesian Modelling21.Applied Life Data Analysis22.Applied Linear Regression, Third Edition23.Applied MANOVA and Discriminant Analysis, Second Edition24.Applied Multiway Data Analysis25.Applied Regression Including Computing and Graphics26.Applied Spatial Statistics for Public Health Data27.Applied Survival Analysis: Regression Modeling of Time-to-Event Data, SecondEdition28.Approximate Dynamic Programming: Solving the Curses of Dimensionality29.Approximate Dynamic Programming: Solving the Curses of Dimensionality, SecondEdition30.Approximation Theorems of Mathematical Statistics31.Aspects of Multivariate Statistical Theory32.Aspects of Statistical Inference33.Batch Effects and Noise in Microarray Experiments: Sources and Solutions34.Bayes Linear Statistics: Theory and Methods35.Bayesian Analysis for the Social Sciences36.Bayesian Analysis of Stochastic Process Models37.Bayesian Methods and Ethics in a Clinical Trial Design38.Bayesian Models for Categorical Data39.Bayesian Networks: An Introduction40.Bayesian Statistical Modelling, Second Edition41.Bayesian Statistics and Marketing42.Bayesian Theory43.Bias and Causation44.Biostatistical Methods in Epidemiology45.Biostatistical Methods: The Assessment of Relative Risks46.Biostatistical Methods: The Assessment of Relative Risks, Second Edition47.Biostatistics: A Methodology for the Health Sciences, Second Edition48.Bootstrap Methods: A Guide for Practitioners and Researchers, Second Edition49.Business Survey Methods50.Case Studies in Reliability and Maintenance51.Categorical Data Analysis, Second Edition52.Causality: Statistical Perspectives and Applications53.Clinical Trial Design: Bayesian and Frequentist Adaptive Methods54.Clinical Trials: A Methodologic Perspective, Second Edition55.Cluster Analysis, 5th Editionbinatorial Methods in Discrete Distributionsparative Statistical Inference, Third Edition58.Constrained Statistical Inference: Inequality, Order, and Shape Restrictions59.Contemporary Bayesian Econometrics and Statistics60.Continuous Multivariate Distributions: Models and Applications, Volume 1,Second Edition61.Convergence of Probability Measures, Second Edition62.Counting Processes and Survival Analysis63.Decision Theory64.Design and Analysis of Clinical Trials: Concepts and Methodologies, SecondEdition65.Design and Analysis of Experiments: Advanced Experimental Design, Volume 266.Design and Analysis of Experiments: Introduction to Experimental Design, Volume1, Second Edition67.Design and Analysis of Experiments: Special Designs and Applications, Volume 368.Directional Statistics69.Dirichlet and Related Distributions: Theory, Methods and Applications70.Discrete Distributions: Applications in the Health Sciences71.Discriminant Analysis and Statistical Pattern Recognition72.Empirical Model Building73.Empirical Model Building: Data, Models, and Reality, Second Edition74.Environmental Statistics: Methods and Applications75.Exploration and Analysis of DNA Microarray and Protein Array Data76.Exploratory Data Mining and Data Cleaning77.Exploring Data Tables, Trends, and Shapes78.Financial Derivatives in Theory and Practice79.Finding Groups in Data: An Introduction to Cluster Analysis80.Finite Mixture Models81.Flowgraph Models for Multistate Time-to-Event Data82.Forecasting with Dynamic Regression Models83.Forecasting with Univariate Box-Jenkins Models: Concepts and Cases84.Fourier Analysis of Time Series: An Introduction, Second Edition85.Fractal-Based Point Processes86.Fractional Factorial Plans87.Fundamentals of Exploratory Analysis of Variance88.Generalized Least Squares89.Generalized Linear Models: With Applications in Engineering and the Sciences,Second Edition90.Generalized, Linear, and Mixed Models91.Geometrical Foundations of Asymptotic Inference92.Geostatistics: Modeling Spatial Uncertainty93.Geostatistics: Modeling Spatial Uncertainty, Second Edition94.High-Dimensional Covariance Estimation95.Hilbert Space Methods in Probability and Statistical Inference96.History of Probability and Statistics and Their Applications before 175097.Image Processing and Jump Regression Analysis98.Inference and Prediction in Large Dimensions99.Introduction to Nonparametric Regression100.Introduction to Statistical Time Series, Second Edition101.Introductory Stochastic Analysis for Finance and Insurancetent Class and Latent Transition Analysis: With Applications in the Social, Behavioral, and Health Sciencestent Curve Models: A Structural Equation Perspectivetent Variable Models and Factor Analysis: A Unified Approach, 3rd Edition105.Leading Personalities in Statistical Sciences: From the Seventeenth Century to the Present106.Lévy Processes in Finance: Pricing Financial Derivatives107.Linear Models: The Theory and Application of Analysis of Variance108.Linear Statistical Inference and its Applications: Second Editon109.Linear Statistical Models110.LISP-STAT: An Object-Oriented Environment for Statistical Computing and Dynamic Graphics111.Longitudinal Data Analysis112.Long-Memory Time Series: Theory and Methods113.Loss Distributions114.Loss Models: From Data to Decisions, Third Edition115.Management of Data in Clinical Trials, Second Edition116.Markov Decision Processes: Discrete Stochastic Dynamic Programming117.Markov Processes and Applications118.Markov Processes: Characterization and Convergence119.Mathematics of Chance120.Measurement Error Models121.Measurement Errors in Surveys122.Meta Analysis: A Guide to Calibrating and Combining Statistical Evidence 123.Methods and Applications of Linear Models: Regression and the Analysis of Variance124.Methods for Statistical Data Analysis of Multivariate Observations, Second Edition125.Methods of Multivariate Analysis, Second Edition126.Methods of Multivariate Analysis, Third Edition127.Mixed Models: Theory and Applications128.Mixtures: Estimation and Applications129.Modelling Under Risk and Uncertainty: An Introduction to Statistical, Phenomenological and Computational Methods130.Models for Investors in Real World Markets131.Models for Probability and Statistical Inference: Theory and Applications 132.Modern Applied U-Statistics133.Modern Experimental Design134.Modes of Parametric Statistical Inference135.Multilevel Statistical Models, 4th Edition136.Multiple Comparison Procedures137.Multiple Imputation for Nonresponse in Surveys138.Multiple Time Series139.Multistate Systems Reliability Theory with Applications140.Multivariable Model-Building: A pragmatic approach to regression analysis based on fractional polynomials for modelling continuous variables141.Multivariate Density Estimation: Theory, Practice, and Visualization142.Multivariate Observations143.Multivariate Statistics: High-Dimensional and Large-Sample Approximations 144.Nonlinear Regression145.Nonlinear Regression Analysis and Its Applications146.Nonlinear Statistical Models147.Nonparametric Analysis of Univariate Heavy-Tailed Data: Research and Practice148.Nonparametric Regression Methods for Longitudinal Data Analysis: Mixed-Effects Modeling Approaches149.Nonparametric Statistics with Applications to Science and Engineering 150.Numerical Issues in Statistical Computing for the Social Scientist151.Operational Risk: Modeling Analytics152.Optimal Learning153.Order Statistics, Third Edition154.Periodically Correlated Random Sequences: Spectral Theory and Practice 155.Permutation Tests for Complex Data: Theory, Applications and Software 156.Planning and Analysis of Observational Studies157.Planning, Construction, and Statistical Analysis of Comparative Experiments158.Precedence-Type Tests and Applications159.Preparing for the Worst: Incorporating Downside Risk in Stock Market Investments160.Probability and Finance: It's Only a Game!161.Probability: A Survey of the Mathematical Theory, Second Edition162.Quantitative Methods in Population Health: Extensions of Ordinary Regression163.Random Data: Analysis and Measurement Procedures, Fourth Edition164.Random Graphs for Statistical Pattern Recognition165.Randomization in Clinical Trials: Theory and Practice166.Recent Advances in Quantitative Methods in Cancer and Human Health Risk Assessment167.Records168.Regression Analysis by Example, Fourth Edition169.Regression Diagnostics: Identifying Influential Data and Sources of Collinearity170.Regression Graphics: Ideas for Studying Regressions Through Graphics171.Regression Models for Time Series Analysis172.Regression with Social Data: Modeling Continuous and Limited Response Variables173.Reliability and Risk: A Bayesian Perspective174.Reliability: Modeling, Prediction, and Optimization175.Response Surfaces, Mixtures, and Ridge Analyses, Second Edition176.Robust Estimation & Testing177.Robust Methods in Biostatistics178.Robust Regression and Outlier Detection179.Robust Statistics180.Robust Statistics, Second Edition181.Robust Statistics: Theory and Methods182.Runs and Scans with Applications183.Sampling, Third Edition184.Sensitivity Analysis in Linear Regression185.Sequential Estimation186.Shape & Shape Theory187.Simulation and the Monte Carlo Method188.Simulation and the Monte Carlo Method, Second Edition189.Simulation and the Monte Carlo Method: Solutions Manual to Accompany, Second Edition190.Simulation: A Modeler's Approach191.Smoothing and Regression: Approaches, Computation, and Application192.Smoothing of Multivariate Data: Density Estimation and Visualization193.Spatial Statistics194.Spatial Statistics and Spatio-Temporal Data: Covariance Functions and Directional Properties195.Spatial Tessellations: Concepts and Applications of Voronoi Diagrams, Second Edition196.Stage-Wise Adaptive Designs197.Statistical Advances in the Biomedical Sciences: Clinical Trials, Epidemiology, Survival Analysis, and Bioinformatics198.Statistical Analysis of Profile Monitoring199.Statistical Design and Analysis of Experiments: With Applications to Engineering and Science, Second Edition200.Statistical Factor Analysis and Related Methods: Theory and Applications 201.Statistical Inference for Fractional Diffusion Processes202.Statistical Meta-Analysis with Applications203.Statistical Methods for Comparative Studies: Techniques for Bias Reduction 204.Statistical Methods for Forecasting205.Statistical Methods for Quality Improvement, Third Edition206.Statistical Methods for Rates and Proportions, Third Edition207.Statistical Methods for Survival Data Analysis, Third Edition208.Statistical Methods for the Analysis of Biomedical Data, Second Edition 209.Statistical Methods in Diagnostic Medicine210.Statistical Methods in Diagnostic Medicine, Second Edition211.Statistical Methods in Engineering and Quality Assurance212.Statistical Methods in Spatial Epidemiology, Second Edition213.Statistical Modeling by Wavelets214.Statistical Models and Methods for Lifetime Data, Second Edition215.Statistical Rules of Thumb, Second Edition216.Statistical Size Distributions in Economics and Actuarial Sciences217.Statistical Texts for Mixed Linear Models218.Statistical Tolerance Regions: Theory, Applications, and Computation219.Statistics for Imaging, Optics, and Photonics220.Statistics for Research, Third Edition221.Statistics of Extremes: Theory and Applications222.Statistics: A Biomedical Introduction223.Stochastic Dynamic Programming and the Control of Queueing Systems224.Stochastic Processes for Insurance & Finance225.Stochastic Simulation226.Structural Equation Modeling: A Bayesian Approach227.Subjective and Objective Bayesian Statistics: Principles, Models, and Applications, Second Edition228.Survey Errors and Survey Costs229.System Reliability Theory: Models, Statistical Methods, and Applications, Second Edition230.The Analysis of Covariance and Alternatives: Statistical Methods for Experiments, Quasi-Experiments, and Single-Case Studies, Second Edition231.The Construction of Optimal Stated Choice Experiments: Theory and Methods 232.The EM Algorithm and Extensions, Second Edition233.The Statistical Analysis of Failure Time Data, Second Edition234.The Subjectivity of Scientists and the Bayesian Approach235.The Theory of Measures and Integration236.The Theory of Response-Adaptive Randomization in Clinical Trials237.Theory of Preliminary Test and Stein-Type Estimation With Applications 238.Time Series Analysis and Forecasting by Example239.Time Series: Applications to Finance with R and S-Plus, Second Edition 240.Uncertainty Analysis with High Dimensional Dependence Modelling241.Univariate Discrete Distributions, Third Editioning the Weibull Distribution: Reliability, Modeling, and Inference243.Variance Components244.Variations on Split Plot and Split Block Experiment Designs245.Visual Statistics: Seeing Data with Dynamic Interactive Graphics246.Weibull Models。

利用Excel进行统计分析-Chapter16-Time Series Forecasting and Index Numbers

Second average:
Y2 + Y3 + Y4 + Y5 + Y6 MA(5) = 5
Chap 16-15
Example: Annual Data
Year 1 2 3 4 5 6 7 8 9 10 11 etc… Sales 23 40 25 27 32 48 33 37 37 50 40 etc…
1 Cycle Sales
Year
Chap 16-9
Irregular Component
Unpredictable, random, “residual” fluctuations Due to random variations of
Nature Accidents or unusual events
Time-Series Components
Time Series
Trend Component Seasonal Component Cyclical Component Irregular Component
Chap 16-5
Trend Component
Long-run increase or decrease over time (overall upward or downward movement) Data taken over a long period of time
The weight is:
Close to 0 for smoothing out unwanted cyclical and irregular components Close to 1 for forecasting
Chap 16-20
Exponential Smoothing Model

Modeling and Forecasting realized volatility

MODELING AND FORECASTING REALIZED VOLATILITY *by Torben G. Andersen a , Tim Bollerslev b , Francis X. Diebold c and Paul Labys dFirst Draft: January 1999Revised: January 2001, January 2002We provide a general framework for integration of high-frequency intraday data into the measurement,modeling, and forecasting of daily and lower frequency return volatilities and return distributions. Most procedures for modeling and forecasting financial asset return volatilities, correlations, and distributions rely on potentially restrictive and complicated parametric multivariate ARCH or stochastic volatilitymodels. Use of realized volatility constructed from high-frequency intraday returns, in contrast, permits the use of traditional time-series methods for modeling and forecasting. Building on the theory ofcontinuous-time arbitrage-free price processes and the theory of quadratic variation, we develop formal links between realized volatility and the conditional covariance matrix. Next, using continuouslyrecorded observations for the Deutschemark / Dollar and Yen / Dollar spot exchange rates covering more than a decade, we find that forecasts from a simple long-memory Gaussian vector autoregression for the logarithmic daily realized volatilities perform admirably compared to a variety of popular daily ARCH and more complicated high-frequency models. Moreover, the vector autoregressive volatility forecast,coupled with a parametric lognormal-normal mixture distribution implied by the theoretically andempirically grounded assumption of normally distributed standardized returns, produces well-calibrated density forecasts of future returns, and correspondingly accurate quantile predictions. Our results hold promise for practical modeling and forecasting of the large covariance matrices relevant in asset pricing,asset allocation and financial risk management applications.K EYWORDS : Continuous-time methods, quadratic variation, realized volatility, realized correlation, high-frequency data, exchange rates, vector autoregression, long memory, volatility forecasting, correlation forecasting, density forecasting, risk management, value at risk._________________* This research was supported by the National Science Foundation. We are grateful to Olsen and Associates, who generously made available their intraday exchange rate data. For insightful suggestions and comments we thank three anonymous referees and the Co-Editor, as well as Kobi Bodoukh, Sean Campbell, Rob Engle, Eric Ghysels, Atsushi Inoue, Eric Renault, Jeff Russell, Neil Shephard, Til Schuermann, Clara Vega, Ken West, and seminar participants at BIS (Basel), Chicago, CIRANO/Montreal, Emory,Iowa, Michigan, Minnesota, NYU, Penn, Rice, UCLA, UCSB, the June 2000 Meeting of the Western Finance Association, the July 2001 NSF/NBER Conference on Forecasting and Empirical Methods in Macroeconomics and Finance, the November 2001 NBER Meeting on Financial Risk Management, and the January 2002 North American Meeting of the Econometric Society.a Department of Finance, Kellogg School of Management, Northwestern University, Evanston, IL 60208, and NBER,phone: 847-467-1285, e-mail: t-andersen@bDepartment of Economics, Duke University, Durham, NC 27708, and NBER,phone: 919-660-1846, e-mail: boller@ c Department of Economics, University of Pennsylvania, Philadelphia, PA 19104, and NBER,phone: 215-898-1507, e-mail: fdiebold@dGraduate Group in Economics, University of Pennsylvania, 3718 Locust Walk, Philadelphia, PA 19104,phone: 801-536-1511, e-mail: labys@ Copyright © 2000-2002 T.G. Andersen, T. Bollerslev, F.X. Diebold and P. LabysAndersen, T., Bollerslev, T., Diebold, F.X. and Labys, P. (2003),"Modeling and Forecasting Realized Volatility,"Econometrica, 71, 529-626.1. INTRODUCTIONThe joint distributional characteristics of asset returns are pivotal for many issues in financial economics. They are the key ingredients for the pricing of financial instruments, and they speak directly to the risk-return tradeoff central to portfolio allocation, performance evaluation, and managerial decision-making. Moreover, they are intimately related to the fractiles of conditional portfolio return distributions, which govern the likelihood of extreme shifts in portfolio value and are therefore central to financial risk management, figuring prominently in both regulatory and private-sector initiatives.The most critical feature of the conditional return distribution is arguably its second moment structure, which is empirically the dominant time-varying characteristic of the distribution. This fact has spurred an enormous literature on the modeling and forecasting of return volatility.1 Over time, the availability of data for increasingly shorter return horizons has allowed the focus to shift from modeling at quarterly and monthly frequencies to the weekly and daily horizons. Forecasting performance has improved with the incorporation of more data, not only because high-frequency volatility turns out to be highly predictable, but also because the information in high-frequency data proves useful for forecasting at longer horizons, such as monthly or quarterly.In some respects, however, progress in volatility modeling has slowed in the last decade. First, the availability of truly high-frequency intraday data has made scant impact on the modeling of, say, daily return volatility. It has become apparent that standard volatility models used for forecasting at the daily level cannot readily accommodate the information in intraday data, and models specified directly for the intraday data generally fail to capture the longer interdaily volatility movements sufficiently well. As a result, standard practice is still to produce forecasts of daily volatility from daily return observations, even when higher-frequency data are available. Second, the focus of volatility modeling continues to be decidedly very low-dimensional, if not universally univariate. Many multivariate ARCH and stochastic volatility models for time-varying return volatilities and conditional distributions have, of course, been proposed (see, for example, the surveys by Bollerslev, Engle and Nelson (1994) and Ghysels, Harvey and Renault (1996)), but those models generally suffer from a curse-of-dimensionality problem that severely constrains their practical application. Consequently, it is rare to see substantive applications of those multivariate models dealing with more than a few assets simultaneously.In view of such difficulties, finance practitioners have largely eschewed formal volatility modeling and forecasting in the higher-dimensional situations of practical relevance, relying instead on1 Here and throughout, we use the generic term “volatilities” in reference both to variances (or standard deviations)ad hoc methods, such as simple exponential smoothing coupled with an assumption of conditionally normally distributed returns.2 Although such methods rely on counterfactual assumptions and are almost surely suboptimal, practitioners have been swayed by considerations of feasibility, simplicity and speed of implementation in high-dimensional environments.Set against this rather discouraging background, we seek to improve matters. We propose a new and rigorous framework for volatility forecasting and conditional return fractile, or value-at-risk (VaR), calculation, with two key properties. First, it efficiently exploits the information in intraday return data, without having to explicitly model the intraday data, producing significant improvements in predictive performance relative to standard procedures that rely on daily data alone. Second, it achieves a simplicity and ease of implementation, which, for example, holds promise for high-dimensional return volatility modeling.We progress by focusing on an empirical measure of daily return variability called realized volatility, which is easily computed from high-frequency intra-period returns. The theory of quadratic variation suggests that, under suitable conditions, realized volatility is an unbiased and highly efficient estimator of return volatility, as discussed in Andersen, Bollerslev, Diebold and Labys (2001) (henceforth ABDL) as well as in concurrent work by Barndorff-Nielsen and Shephard (2002, 2001a).3 Building on the notion of continuous-time arbitrage-free price processes, we advance in several directions, including rigorous theoretical foundations, multivariate emphasis, explicit focus on forecasting, and links to modern risk management via modeling of the entire conditional density.Empirically, by treating volatility as observed rather than latent, our approach facilitates modeling and forecasting using simple methods based directly on observable variables.4 We illustrate the ideas using the highly liquid U.S. dollar ($), Deutschemark (DM), and Japanese yen (¥) spot exchange rate markets. Our full sample consists of nearly thirteen years of continuously recorded spot quotations from 1986 through 1999. During that period, the dollar, Deutschemark and yen constituted2This approach is exemplified by the highly influential “RiskMetrics” of J.P. Morgan (1997).3 Earlier work by Comte and Renault (1998), within the context of estimation of a long-memory stochastic volatility model, helped to elevate the discussion of realized and integrated volatility to a more rigorous theoretical level.4 The direct modeling of observable volatility proxies was pioneered by Taylor (1986), who fit ARMA models to absolute and squared returns. Subsequent empirical work exploiting related univariate approaches based on improved realized volatility measures from a heuristic perspective includes French, Schwert and Stambaugh (1987) and Schwert (1989), who rely on daily returns to estimate models for monthly realized U.S. equity volatility, and Hsieh (1991), who fits an AR(5) model to a time series of daily realized logarithmic volatilities constructed from 15-minute S&P500 returns.the main axes of the international financial system, and thus spanned the majority of the systematic currency risk faced by large institutional investors and international corporations.We break the sample into a ten-year "in-sample" estimation period, and a subsequent two-and-a-half-year "out-of-sample" forecasting period. The basic distributional and dynamic characteristics of the foreign exchange returns and realized volatilities during the in-sample period have been analyzed in detail by ABDL (2000a, 2001).5 Three pieces of their results form the foundation on which the empirical analysis of this paper is built. First, although raw returns are clearly leptokurtic, returns standardized by realized volatilities are approximately Gaussian. Second, although the distributions of realized volatilities are clearly right-skewed, the distributions of the logarithms of realized volatilities are approximately Gaussian. Third, the long-run dynamics of realized logarithmic volatilities are well approximated by a fractionally-integrated long-memory process.Motivated by the three ABDL empirical regularities, we proceed to estimate and evaluate a multivariate model for the logarithmic realized volatilities: a fractionally-integrated Gaussian vector autoregression (VAR) . Importantly, our approach explicitly permits measurement errors in the realized volatilities. Comparing the resulting volatility forecasts to those obtained from currently popular daily volatility models and more complicated high-frequency models, we find that our simple Gaussian VAR forecasts generally produce superior forecasts. Furthermore, we show that, given the theoretically motivated and empirically plausible assumption of normally distributed returns conditional on the realized volatilities, the resulting lognormal-normal mixture forecast distribution provides conditionally well-calibrated density forecasts of returns, from which we obtain accurate estimates of conditional return quantiles.In the remainder of this paper, we proceed as follows. We begin in section 2 by formally developing the relevant quadratic variation theory within a standard frictionless arbitrage-free multivariate pricing environment. In section 3 we discuss the practical construction of realized volatilities from high-frequency foreign exchange returns. Next, in section 4 we summarize the salient distributional features of returns and volatilities, which motivate the long-memory trivariate Gaussian VAR that we estimate in section 5. In section 6 we compare the resulting volatility point forecasts to those obtained from more traditional volatility models. We also evaluate the success of the density forecasts and corresponding VaR estimates generated from the long-memory Gaussian VAR in5 Strikingly similar and hence confirmatory qualitative findings have been obtained from a separate sample consisting of individual U.S. stock returns in Andersen, Bollerslev, Diebold and Ebens (2001).conjunction with a lognormal-normal mixture distribution. In section 7 we conclude with suggestions for future research and discussion of issues related to the practical implementation of our approach for other financial instruments and markets.2. QUADRATIC RETURN VARIATION AND REALIZED VOLATILITYWe consider an n -dimensional price process defined on a complete probability space, (,Û, P), evolvingin continuous time over the interval [0,T], where T denotes a positive integer. We further consider an information filtration, i.e., an increasing family of -fields, (Ût )t 0[0,T] f Û , which satisfies the usual conditions of P -completeness and right continuity. Finally, we assume that the asset prices through time t , including the relevant state variables, are included in the information set Ût .Under the standard assumptions that the return process does not allow for arbitrage and has afinite instantaneous mean the asset price process, as well as smooth transformations thereof, belong to the class of special semi-martingales, as detailed by Back (1991). A fundamental result of stochastic integration theory states that such processes permit a unique canonical decomposition. In particular, we have the following characterization of the logarithmic asset price vector process, p = (p(t))t 0[0,T].PROPOSITION 1: For any n-dimensional arbitrage-free vector price process with finite mean, the logarithmic vector price process, p, may be written uniquely as the sum of a finite variation and predictable mean component, A = (A 1 , ... , A n ), and a local martingale, M = (M 1 , ... , M n ). These may each be decomposed into a continuous sample-path and jump part,p(t) = p(0) + A(t) + M(t) = p(0) + A c (t) + )A(t) + M c (t) + )M(t),(1)where the finite-variation predictable components, A c and )A, are respectively continuous and pure jump processes, while the local martingales, M c and )M, are respectively continuous sample-path and compensated jump processes, and by definition M(0) / A(0) / 0. Moreover, the predictable jumps are associated with genuine jump risk, in the sense that if )A(t) ú 0, thenP [ sgn( )A(t) ) = - sgn( )A(t)+)M(t) ) ] > 0 ,(2)where sgn(x) / 1 for x $0 and sgn(x) / -1 for x < 0.Equation (1) is standard, see, for example, Protter (1992), chapter 3. Equation (2) is an implication of6 This does not appear particularly restrictive. For example, if an announcement is pending, a natural way to model the arrival time is according to a continuous hazard function. Then the probability of a jump within each (infinitesimal)instant of time is zero - there is no discrete probability mass - and by arbitrage there cannot be a predictable jump.the no-arbitrage condition. Whenever )A(t) ú 0, there is a predictable jump in the price - the timing and size of the jump is perfectly known (just) prior to the jump event - and hence there is a trivial arbitrage (with probability one) unless there is a simultaneous jump in the martingale component, )M(t) ú 0. Moreover, the concurrent martingale jump must be large enough (with strictly positive probability) to overturn the gain associated with a position dictated by sgn()A(t)).Proposition 1 provides a general characterization of the asset return process. We denote the(continuously compounded) return over [t-h,t] by r(t,h) = p(t) - p(t-h). The cumulative return process from t=0 onward, r = (r(t))t 0[0,T] , is then r(t) / r(t,t) = p(t) - p(0) = A(t) + M(t). Clearly, r(t) inherits all the main properties of p(t) and may likewise be decomposed uniquely into the predictable andintegrable mean component, A , and the local martingale, M . The predictability of A still allows for quite general properties in the (instantaneous) mean process, for example it may evolve stochastically and display jumps. Nonetheless, the continuous component of the mean return must have smooth sample paths compared to those of a non-constant continuous martingale - such as a Brownian motion - and any jump in the mean must be accompanied by a corresponding predictable jump (of unknown magnitude) in the compensated jump martingale, )M . Consequently, there are two types of jumps in the return process, namely, predictable jumps where )A(t)ú0 and equation (2) applies, and purely unanticipated jumps where )A(t)=0 but )M(t)ú0. The latter jump event will typically occur when unanticipated news hit the market. In contrast, the former type of predictable jump may be associated with the release of information according to a predetermined schedule, such as macroeconomic news releases or company earnings reports. Nonetheless, it is worth noting that any slight uncertainty about the precise timing of the news (even to within a fraction of a second) invalidates the assumption of predictability and removes the jump in the mean process. If there are no such perfectly anticipated news releases, the predictable,finite variation mean return, A , may still evolve stochastically, but it will have continuous sample paths. This constraint is implicitly invoked in the vast majority of the continuous-time models employed in the literature.6Because the return process is a semi-martingale it has an associated quadratic variation process. Quadratic variation plays a critical role in our theoretical developments. The following proposition7 All of the properties in Proposition 2 follow, for example, from Protter (1992), chapter 2.8 In the general case with predictable jumps the last term in equation (4) is simply replaced by0#s #tr i (s)r j (s),where r i (s) /A i (s) + M i (s) explicitly incorporates both types of jumps. However, as discussed above, this case is arguable of little interest from a practical empirical perspective.enumerates some essential properties of the quadratic return variation process.7PROPOSITION 2: For any n-dimensional arbitrage-free price process with finite mean, the quadratic variation nxn matrix process of the associated return process, [r,r] = { [r,r]t }t 0[0,T] , is well-defined. The i’th diagonal element is called the quadratic variation process of the i’th asset return while the ij’th off-diagonal element, [r i ,r j ], is called the quadratic covariation process between asset returns i and j. The quadratic variation and covariation processes have the following properties:(i)For an increasing sequence of random partitions of [0,T], 0 = J m,0 # J m,1 # ..., such thatsup j $1(J m,j+1 - J m,j )60 and sup j $1 J m,j 6T for m 64 with probability one, we have thatlim m 64 { E j $1 [r(t v J m,j ) - r(t v J m,j-1)] [r(t v J m,j ) - r(t v J m,j-1)]’ } 6 [r,r]t ,(3)where t v J / min(t,J ), t 0 [0,T], and the convergence is uniform on [0,T] in probability.(ii)If the finite variation component, A, in the canonical return decomposition in Proposition 1 iscontinuous, then[r i ,r j ]t = [M i ,M j ]t = [M i c ,M j c ]t + E 0#s #t )M i (s) )M j (s) .(4)The terminology of quadratic variation is justified by property (i) of Proposition 2. Property (ii) reflects the fact that the quadratic variation of continuous finite-variation processes is zero, so the meancomponent becomes irrelevant for the quadratic variation.8 Moreover, jump components only contribute to the quadratic covariation if there are simultaneous jumps in the price path for the i ’th and j ’th asset,whereas the squared jump size contributes one-for-one to the quadratic variation. The quadratic variation process measures the realized sample-path variation of the squared return processes. Under the weak auxiliary condition ensuring property (ii), this variation is exclusively induced by the innovations to the return process. As such, the quadratic covariation constitutes, in theory, a unique and invariant ex-post realized volatility measure that is essentially model free. Notice that property (i) also suggests that we9 This has previously been discussed by Comte and Renault (1998) in the context of estimating the spot volatility for a stochastic volatility model corresponding to the derivative of the quadratic variation (integrated volatility) process. 10 This same intuition underlies the consistent filtering results for continuous sample path diffusions in Merton (1980)and Nelson and Foster (1995).may approximate the quadratic variation by cumulating cross-products of high-frequency returns.9 We refer to such measures, obtained from actual high-frequency data, as realized volatilities .The above results suggest that the quadratic variation is the dominant determinant of the return covariance matrix, especially for shorter horizons. Specifically, the variation induced by the genuine return innovations, represented by the martingale component, locally is an order of magnitude larger than the return variation caused by changes in the conditional mean.10 We have the following theorem which generalizes previous results in ABDL (2001).THEOREM 1: Consider an n-dimensional square-integrable arbitrage-free logarithmic price process with a continuous mean return, as in property (ii) of Proposition 2. The conditional return covariance matrix at time t over [t, t+h], where 0 # t # t+h # T, is then given byCov(r(t+h,h)*Ût ) = E([r,r ]t+h - [r,r ]t *Ût ) + 'A (t+h,h) + 'AM (t+h,h) + 'AM ’(t+h,h),(5)where 'A (t+h,h) = Cov(A(t+h) - A(t) * Ût ) and 'AM (t+h,h) = E(A(t+h) [M(t+h) - M(t)]’ *Ût ).PROOF: From equation (1), r(t+h,h) = [ A(t+h) - A(t) ] + [ M(t+h) - M(t) ]. The martingale property implies E( M(t+h) - M(t) *Ût ) = E( [M(t+h) - M(t)] A(t) *Ût ) = 0, so, for i,j 0 {1, ..., n}, Cov( [A i (t+h)- A i (t)], [M j (t+h) - M j (t)] * Ût ) = E( A i (t+h) [M j (t+h) - M j (t)] * Ût ). It therefore follows that Cov(r(t+h,h) * Ût ) = Cov( M(t+h) - M(t) * Ût ) + 'A (t+h,h) + 'AM (t+h,h) + 'AM ’(t+h,h). Hence, it only remains to show that the conditional covariance of the martingale term equals the expected value of the quadratic variation. We proceed by verifying the equality for an arbitrary element of the covariancematrix. If this is the i ’th diagonal element, we are studying a univariate square-integrable martingale and by Protter (1992), chapter II.6, corollary 3, we have E[M i 2(t+h)] = E( [M i ,M i ]t+h ), so Var(M i (t+h) -M i (t) * Ût ) = E( [M i ,M i ]t+h - [M i ,M i ]t * Ût ) = E( [r i ,r i ]t+h - [r i ,r i ]t * Ût ), where the second equality follows from equation (3) of Proposition 2. This confirms the result for the diagonal elements of the covariance matrix. An identical argument works for the off-diagonal terms by noting that the sum of two square-integrable martingales remains a square-integrable martingale and then applying the reasoning toeach component of the polarization identity, [M i ,M j ]t = ½ ( [M i +M j , M i +M j ]t - [M i ,M i ]t - [M j ,M j ]t ). In particular, it follows as above that E( [M i ,M j ]t+h - [M i ,M j ]t * Ût ) = ½ [ Var( [M i (t+h)+M j (t+h)] -[(M i (t)+M j (t)]* Ût ) - Var( M i (t+h) - M i (t)*Ût ) - Var( M j (t+h) - M j (t)*Ût ) ]= Cov( [M i (t+h) - M i (t)],[M j (t+h) - M j (t)]*Ût ). Equation (3) of Proposition 2 again ensures that this equals E( [r i ,r j ]t+h - [r i ,r j ]t * Ût ). 9Two scenarios highlight the role of the quadratic variation in driving the return volatility process. These important special cases are collected in a corollary which follows immediately from Theorem 1.COROLLARY 1: Consider an n-dimensional square-integrable arbitrage-free logarithmic price process, as described in Theorem 1. If the mean process, {A(s) - A(t)}s 0[t,t+h] , conditional on information at time t is independent of the return innovation process, {M(u)}u 0[t,t+h], then the conditional return covariance matrix reduces to the conditional expectation of the quadratic return variation plus the conditional variance of the mean component, i.e., for 0 # t # t+h # T,Cov( r(t+h,h) * Ût ) = E( [r,r ]t+h - [r,r ]t * Ût ) + 'A (t+h,h).If the mean process, {A(s) - A(t)}s 0[t,t+h], conditional on information at time t is a predetermined function over [t, t+h], then the conditional return covariance matrix equals the conditional expectation of the quadratic return variation process, i.e., for 0 # t # t+h # T,Cov( r(t+h,h) * Ût ) = E( [r,r ]t+h - [r,r ]t * Ût ).(6)Under the conditions leading to equation (6), the quadratic variation is the critical ingredient in volatility measurement and forecasting. This follows as the quadratic variation represents the actual variability of the return innovations, and the conditional covariance matrix is the conditional expectation of this quantity. Moreover, it implies that the time t+h ex-post realized quadratic variation is an unbiased estimator for the return covariance matrix conditional on information at time t .Although the corollary’s strong implications rely upon specific assumptions, these sufficientconditions are not as restrictive as an initial assessment may suggest, and they are satisfied for a wide set of popular models. For example, a constant mean is frequently invoked in daily or weekly return models. Equation (6) further allows for deterministic intra-period variation in the conditional mean,11 Merton (1982) provides a similar intuitive account of the continuous record h-asymptotics . These limiting results are also closely related to the theory rationalizing the quadratic variation formulas in Proposition 2 and Theorem 1.induced by time-of-day or other calendar effects. Of course, equation (6) also accommodates a stochastic mean process as long as it remains a function, over the interval [t, t+h], of variables in the time tinformation set. Specification (6) does, however, preclude feedback effects from the random intra-period evolution of the system to the instantaneous mean. Although such feedback effects may be present in high-frequency returns, they are likely trivial in magnitude over daily or weekly frequencies, as we argue subsequently. It is also worth stressing that (6) is compatible with the existence of an asymmetric return-volatility relation (sometimes called a leverage effect), which arises from a correlation between the return innovations, measured as deviations from the conditional mean, and the innovations to the volatility process. In other words, the leverage effect is separate from a contemporaneous correlation between the return innovations and the instantaneous mean return. Furthermore, as emphasized above,equation (6) does allow for the return innovations over [t-h, t] to impact the conditional mean over [t,t+h] and onwards, so that the intra-period evolution of the system may still impact the future expected returns. In fact, this is how potential interaction between risk and return is captured in discrete-time stochastic volatility or ARCH models with leverage effects.In contrast to equation (6), the first expression in Corollary 1 involving 'A explicitlyaccommodates continually evolving random variation in the conditional mean process, although the random mean variation must be independent of the return innovations. Even with this feature present,the quadratic variation is likely an order of magnitude larger than the mean variation, and hence the former remains the critical determinant of the return volatility over shorter horizons. This observation follows from the fact that over horizons of length h , with h small, the variance of the mean return is of order h 2, while the quadratic variation is of order h . It is an empirical question whether these results are a good guide for volatility measurement at relevant frequencies.11 To illustrate the implications at a daily horizon, consider an asset return with standard deviation of 1% daily, or 15.8% annually, and a (large)mean return of 0.1%, or about 25% annually. The squared mean return is still only one-hundredth of the variance. The expected daily variation of the mean return is obviously smaller yet, unless the required daily return is assumed to behave truly erratically within the day. In fact, we would generally expect the within-day variance of the expected daily return to be much smaller than the expected daily return itself. Hence, the daily return fluctuations induced by within-day variations in the mean return are almostcertainly trivial. For a weekly horizon, similar calculations suggest that the identical conclusion applies.。

基于ARIMA和LSTM混合模型的时间序列预测

第３８卷第２期计算机应用与软件Ｖｏｌ３８Ｎｏ．２２０２１年２月ＣｏｍｐｕｔｅｒＡｐｐｌｉｃａｔｉｏｎｓａｎｄＳｏｆｔｗａｒｅＦｅｂ．２０２１基于ＡＲＩＭＡ和ＬＳＴＭ混合模型的时间序列预测王英伟　马树才（辽宁大学经济学院　辽宁沈阳１１００３６）收稿日期：２０１９－０７－２４。

王英伟，博士，主研领域：大数据，人工智能。

马树才，教授。

摘　要由于现实中的时间序列通常同时具有线性和非线性特征，传统ＡＲＩＭＡ模型在时间序列建模中常表现出一定局限性。

对此，提出基于ＡＲＩＭＡ和ＬＳＴＭ混合模型进行时间序列预测。

应用线性ＡＲＩＭＡ模型进行时间序列预测，用支持向量回归（ＳＶＲ）模型对误差序列进行预测，采用深度ＬＳＴＭ模型对ＡＲＩＭＡ模型和ＳＶＲ模型的预测结果组合，并将贝叶斯优化算法用于选择深度ＬＳＴＭ模型的超参数。

实验结果表明，与其他混合模型相比，该模型在五种不同时间序列预测中能够有效提高预测精度。

关键词ＡＲＩＭＡ模型　ＳＶＲ模型　深度ＬＳＴＭ模型　贝叶斯优化算法　时间序列预测中图分类号　ＴＰ３０２．７文献标志码　ＡＤＯＩ：１０．３９６９／ｊ．ｉｓｓｎ．１０００３８６ｘ．２０２１．０２．０４７ＴＩＭＥＳＥＲＩＥＳＦＯＲＥＣＡＳＴＩＮＧＢＡＳＥＤＯＮＡＲＩＭＡ＿ＤＬＳＴＭＨＹＢＲＩＤＭＯＤＥＬＷａｎｇＹｉｎｇｗｅｉ　ＭａＳｈｕｃａｉ（ＩｎｓｔｉｔｕｔｅｏｆＥｃｏｎｏｍｉｃｓ，ＬｉａｏｎｉｎｇＵｎｉｖｅｒｓｉｔｙ，Ｓｈｅｎｙａｎｇ１１００３６，Ｌｉａｏｎｉｎｇ，Ｃｈｉｎａ）ＡｂｓｔｒａｃｔＢｅｃａｕｓｅｒｅａｌｗｏｒｌｄｔｉｍｅｓｅｒｉｅｓｕｓｕａｌｌｙｃｏｎｔａｉｎｂｏｔｈｌｉｎｅａｒａｎｄｎｏｎｌｉｎｅａｒｐａｔｔｅｒｎｓ，ｔｒａｄｉｔｉｏｎａｌＡＲＩＭＡｍｏｄｅｌｈａｓａｌｉｍｉｔｅｄｐｅｒｆｏｒｍａｎｃｅｉｎｔｈｅｔｉｍｅｓｅｒｉｅｓｍｏｄｅｌｉｎｇ．Ｉｎｖｉｅｗｏｆｔｈｉｓ，ｗｅｐｒｏｐｏｓｅＡＲＩＭＡ＿ＤＬＳＴＭｈｙｂｒｉｄｍｏｄｅｌｆｏｒｔｉｍｅｓｅｒｉｅｓｆｏｒｅｃａｓｔｉｎｇ．ＬｉｎｅａｒＡＲＩＭＡｍｏｄｅｌｗａｓｕｓｅｄｆｏｒｔｉｍｅｓｅｒｉｅｓｐｒｅｄｉｃｔｉｏｎｆｉｒｓｔｌｙ，ａｎｄｔｈｅｎｓｕｐｐｏｒｔｖｅｃｔｏｒｒｅｇｒｅｓｓｉｏｎ（ＳＶＲ）ｗａｓｕｓｅｄｆｏｒｅｒｒｏｒｓｅｒｉｅｓｐｒｅｄｉｃｔｉｏｎ．ＴｈｅｄｅｅｐＬＳＴＭｍｏｄｅｌｗａｓｉｎｔｒｏｄｕｃｅｄｔｏｃｏｍｂｉｎｅｔｈｅｆｏｒｅｃａｓｔｓｏｆＡＲＩＭＡｍｏｄｅｌａｎｄＳＶＲｍｏｄｅｌ，ａｎｄＢａｙｅｓｉａｎｏｐｔｉｍｉｚａｔｉｏｎａｌｇｏｒｉｔｈｍｗａｓａｄｏｐｔｅｄｔｏｏｂｔａｉｎｔｈｅｏｐｔｉｍａｌｈｙｐｅｒｐａｒａｍｅｔｅｒｏｆｄｅｅｐＬＳＴＭｍｏｄｅｌ．ＴｈｅｅｘｐｅｒｉｍｅｎｔａｌｒｅｓｕｌｔｓｏｆｆｉｖｅｔｉｍｅｓｅｒｉｅｓｆｏｒｅｃａｓｔｉｎｇｓｈｏｗｔｈａｔＡＲＩＭＡ＿ＤＬＳＴＭｍｏｄｅｌｃａｎｅｆｆｅｃｔｉｖｅｌｙｉｍｐｒｏｖｅｔｈｅｐｒｅｄｉｃｔｉｏｎａｃｃｕｒａｃｙｃｏｍｐａｒｅｄｗｉｔｈｏｔｈｅｒｈｙｂｒｉｄｍｏｄｅｌｓ．ＫｅｙｗｏｒｄｓＡＲＩＭＡｍｏｄｅｌ　ＳＶＲｍｏｄｅｌ　ＤｅｅｐＬＳＴＭｍｏｄｅｌ　Ｂａｙｅｓｉａｎｏｐｔｉｍｉｚａｔｉｏｎａｌｇｏｒｉｔｈｍ　Ｔｉｍｅｓｅｒｉｅｓｆｏｒｅｃａｓｔｉｎｇ０　引　言时间序列预测在众多领域有广泛应用，如金融、经济、工程和航空等，并成为机器学习领域的重要研究课题［１］。

时间序列预测模型的书籍案例

时间序列预测模型的书籍案例时间序列预测模型是一种用于分析和预测时间序列数据的统计模型。

它基于时间序列的历史数据，通过建立数学模型来预测未来的趋势和变化。

时间序列预测模型在许多领域都有广泛的应用，如经济学、金融学、气象学等。

下面是一些关于时间序列预测模型的书籍案例，它们涵盖了不同的领域和方法：1. 《时间序列分析》（Time Series Analysis）- George E.P. Box, Gwilym M. Jenkins和Gregory C. Reinsel这本经典著作是时间序列分析领域的权威之作，介绍了时间序列模型的理论基础和实践应用。

它对传统的ARIMA模型和季节性时间序列模型进行了详细的讲解。

2. 《时间序列分析与预测》（Time Series Analysis and Forecasting）- Example Smith, Navdeep Gill和Walter Liggett 这本教材介绍了时间序列分析和预测的基本原理和方法。

它包括了ARIMA、ARCH/GARCH等常用模型，并提供了实际案例和R语言代码。

3. 《金融时间序列分析与预测》（Financial Time Series Analysis and Forecasting）- Ruey S. Tsay这本书重点介绍了在金融领域中应用时间序列分析和预测的方法。

它包括了ARCH/GARCH模型、VAR模型、协整模型等，并通过实际金融数据进行案例分析。

4. 《商业预测：原理与实践》（Business Forecasting: Principles and Practice）- Rob J. Hyndman和George Athanasopoulos这本书是一本实用的商业预测教材，介绍了时间序列预测的基本原理和常用方法。

它使用R语言进行案例分析，并提供了实际业务中的预测应用示例。

5. 《Python时间序列分析》（Python for Time Series Analysis）- Alan Elliott和Wayne A. Woodward这本书介绍了使用Python进行时间序列分析的方法和工具。

利用Excel进行统计分析-Chapter16-Time-Series-Forecasting-and-Index-Numbers

Time Series Plot
A time-series plot is a twodimensional plot of time-series data
the vertical axis 16.00 measures the variable 14.00
of interest
12.00 10.00
Data taken over a long period of time
Sales
Time
Chap 16-6
Chap 16-7
Seasonal Component
Sales
Short-term regular wave-like patterns Observed within 1 year Often monthly or quarterly
Statistics for Managers Using Microsoft® Excel
5th Edition
Chapter 16 Time Series Forecasting and Index
Numbers
Chap 16-1
Learning Objectives
In this chapter, you learn: About seven different time-series forecasting models:
8.00
the horizontal axis
6.00
corresponds to the
4.00 2.00
time periods
0.00
U.S. Inflation Rate
Year
Chap 16-4
Time-Series Components

统计学论文(精选6篇

统计学论文（精选6篇1. "A Bayesian Approach to Modeling Mixed-Methods Survey Data"This paper discusses the use of Bayesian methods for analyzing mixed-methods survey data, where both quantitative and qualitative data are collected from a sample. The authors present a hierarchical Bayesian model that allows for the incorporation of both types of data, and demonstrate its usefulness through simulations and a real-world application.2. "Network Analysis of Financial Risk"This paper uses network analysis to evaluate the interconnectedness of financial institutions and the potential for systemic risk. The authors construct a network of financial institutions based on their credit exposures, and analyze the network for patterns of vulnerability. The results suggest that the network is highly interconnected, and that some institutions are more central and influential than others.3. "A Comparison of Machine Learning Algorithms for Prediction of Patient Outcomes"This paper compares the performance of several machine learning algorithms for predicting patient outcomes, using data from electronic health records. The authors find that Random Forests and Support Vector Machines perform the best, and suggest that these models could be used in clinical decision-making.4. "Spatial Analysis of Crime Rates"This paper uses spatial analysis techniques to explore patterns of crime in a particular city. The authors use a spatial autocorrelationtest to identify areas of high and low crime rates, and then conduct a regression analysis to identify factors that may be contributing to the patterns. The results suggest that socio-economic factors are strongly correlated with crime rates.5. "Bayesian Inference for Time Series Forecasting"This paper presents a Bayesian approach to forecasting time series data, using a state-space model and Markov Chain Monte Carlo techniques for parameter estimation. The authors demonstrate the method using data on monthly inflation rates, and show that it outperforms traditional methods such as ARIMA.6. "Identifying Subpopulations with Latent Trait Models"This paper presents a method for identifying subpopulations within a larger sample, based on latent trait models. The authors use a mixture model to identify subgroups with different characteristics, and then conduct a regression analysis to explore the factors that are associated with membership in each subgroup. The method is applied to data on adolescent substance use, and the results suggest that there are different patterns of substance use among subgroups of adolescents.。

中英文双语外文文献翻译：一种基于...

中英⽂双语外⽂⽂献翻译：⼀种基于...此⽂档是毕业设计外⽂翻译成品（含英⽂原⽂+中⽂翻译），⽆需调整复杂的格式！下载之后直接可⽤，⽅便快捷！本⽂价格不贵,也就⼏⼗块钱！⼀辈⼦也就⼀次的事！英⽂3890单词，20217字符(字符就是印刷符)，中⽂6398汉字。

A Novel Divide-and-Conquer Model for CPI Prediction UsingARIMA, Gray Model and BPNNAbstract:This paper proposes a novel divide-and-conquer model for CPI prediction with the existing compilation method of the Consumer Price Index (CPI) in China. Historical national CPI time series is preliminary divided into eight sub-indexes including food, articles for smoking and drinking, clothing, household facilities, articles and maintenance services, health care and personal articles, transportation and communication, recreation, education and culture articles and services, and residence. Three models including back propagation neural network (BPNN) model, grey forecasting model (GM (1, 1)) and autoregressive integrated moving average (ARIMA) model are established to predict each sub-index, respectively. Then the best predicting result among the three models’for each sub-index is identified. To further improve the performance, special modification in predicting method is done to sub-CPIs whose forecasting results are not satisfying enough. After improvement and error adjustment, we get the advanced predicting results of the sub-CPIs. Eventually, the best predicting results of each sub-index are integrated to form the forecasting results of the national CPI. Empirical analysis demonstrates that the accuracy and stability of the introduced method in this paper is better than many commonly adopted forecasting methods, which indicates the proposed method is an effective and alternative one for national CPI prediction in China.1.IntroductionThe Consumer Price Index (CPI) is a widely used measurement of cost of living. It not only affects the government monetary, fiscal, consumption, prices, wages, social security, but also closely relates to the residents’daily life. As an indicator of inflation in China economy, the change of CPI undergoes intense scrutiny. For instance, The People's Bank of China raised the deposit reserve ratio in January, 2008 before the CPI of 2007 was announced, for it is estimated that the CPI in 2008 will increase significantly if no action is taken. Therefore, precisely forecasting the change of CPI is significant to many aspects of economics, some examples include fiscal policy, financial markets and productivity. Also, building a stable and accurate model to forecast the CPI will have great significance for the public, policymakers and research scholars.Previous studies have already proposed many methods and models to predict economic time series or indexes such as CPI. Some previous studies make use of factors that influence the value of the index and forecast it by investigating the relationship between the data of those factors and the index. These forecasts are realized by models such as Vector autoregressive (VAR)model1 and genetic algorithms-support vector machine (GA-SVM) 2.However, these factor-based methods, although effective to some extent, simply rely on the correlation between the value of the index and limited number of exogenous variables (factors) and basically ignore the inherent rules of the variation of the time series. As a time series itself contains significant amount of information3, often more than a limited number of factors can do, time series-based models are often more effective in the field of prediction than factor-based models.Various time series models have been proposed to find the inherent rules of the variation in the series. Many researchers have applied different time series models to forecasting the CPI and other time series data. For example, the ARIMA model once served as a practical method in predicting the CPI4. It was also applied to predict submicron particle concentrations frommeteorological factors at a busy roadside in Hangzhou, China5. What’s more, the ARIMA model was adopted to analyse the trend of pre-monsoon rainfall data forwestern India6. Besides the ARIMA model, other models such as the neural network, gray model are also widely used in the field of prediction. Hwang used the neural-network to forecast time series corresponding to ARMA (p, q) structures and found that the BPNNs generally perform well and consistently when a particular noise level is considered during the network training7. Aiken also used a neural network to predict the level of CPI and reached a high degree of accuracy8. Apart from the neural network models, a seasonal discrete grey forecasting model for fashion retailing was proposed and was found practical for fashion retail sales forecasting with short historical data and better than other state-of-art forecastingtechniques9. Similarly, a discrete Grey Correlation Model was also used in CPI prediction10. Also, Ma et al. used gray model optimized by particle swarm optimization algorithm to forecast iron ore import and consumption of China11. Furthermore, to deal with the nonlinear condition, a modified Radial Basis Function (RBF) was proposed by researchers.In this paper, we propose a new method called “divide-and-conquer model”for the prediction of the CPI.We divide the total CPI into eight categories according to the CPI construction and then forecast the eight sub- CPIs using the GM (1, 1) model, the ARIMA model and the BPNN. To further improve the performance, we again make prediction of the sub-CPIs whoseforecasting results are not satisfying enough by adopting new forecasting methods. After improvement and error adjustment, we get the advanced predicting results of the sub-CPIs. Finally we get the total CPI prediction by integrating the best forecasting results of each sub-CPI.The rest of this paper is organized as follows. In section 2, we give a brief introduction of the three models mentioned above. And then the proposed model will be demonstrated in the section 3. In section 4 we provide the forecasting results of our model and in section 5 we make special improvement by adjusting the forecasting methods of sub-CPIs whose predicting results are not satisfying enough. And in section 6 we give elaborate discussion and evaluation of the proposed model. Finally, the conclusion is summarized in section 7.2.Introduction to GM(1,1), ARIMA & BPNNIntroduction to GM(1,1)The grey system theory is first presented by Deng in 1980s. In the grey forecasting model, the time series can be predicted accurately even with a small sample by directly estimating the interrelation of data. The GM(1,1) model is one type of the grey forecasting which is widely adopted. It is a differential equation model of which the order is 1 and the number of variable is 1, too. The differential equation is:Introduction to ARIMAAutoregressive Integrated Moving Average (ARIMA) model was first put forward by Box and Jenkins in 1970. The model has been very successful by taking full advantage of time series data in the past and present. ARIMA model is usually described as ARIMA (p, d, q), p refers to the order of the autoregressive variable, while d and q refer to integrated, and moving average parts of the model respectively. When one of the three parameters is zero, the model is changed to model “AR”, “MR”or “ARMR”. When none of the three parameters is zero, the model is given by:where L is the lag number,?t is the error term.Introduction to BPNNArtificial Neural Network (ANN) is a mathematical and computational model which imitates the operation of neural networks of human brain. ANN consists of several layers of neurons. Neurons of contiguous layers are connected with each other. The values of connections between neurons are called “weight”. Back Propagation Neural Network (BPNN) is one of the most widely employed neural network among various types of ANN. BPNN was put forward by Rumelhart and McClelland in 1985. It is a common supervised learning network well suited for prediction. BPNN consists of three parts including one input layer, several hidden layers and one output layer, as is demonstrated in Fig 1. The learning process of BPNN is modifying the weights of connections between neurons based on the deviation between the actual output and the target output until the overall error is in the acceptable range.Fig. 1. Back-propagation Neural Network3.The Proposed MethodThe framework of the dividing-integration modelThe process of forecasting national CPI using the dividing-integration model is demonstrated in Fig 2.Fig. 2.The framework of the dividing-integration modelAs can be seen from Fig. 2, the process of the proposed method can be divided into the following steps: Step1: Data collection. The monthly CPI data including total CPI and eight sub-CPIs are collected from the official website of China’s State Statistics Bureau (/doc/d62de4b46d175f0e7cd184254b35eefdc9d31514.html /).Step2: Dividing the total CPI into eight sub-CPIs. In this step, the respective weight coefficient of eight sub- CPIs in forming the total CPI is decided by consulting authoritative source .(/doc/d62de4b46d175f0e7cd184254b35eefdc9d31514.html /). The eight sub-CPIs are as follows: 1. Food CPI; 2. Articles for Smoking and Drinking CPI; 3. Clothing CPI; 4. Household Facilities, Articles and Maintenance Services CPI; 5. Health Care and Personal Articles CPI; 6. Transportation and Communication CPI;7. Recreation, Education and Culture Articles and Services CPI; 8. Residence CPI. The weight coefficient of each sub-CPI is shown in Table 8.Table 1. 8 sub-CPIs weight coefficient in the total indexNote: The index number stands for the corresponding type of sub-CPI mentioned before. Other indexes appearing in this paper in such form have the same meaning as this one.So the decomposition formula is presented as follows:where TI is the total index; Ii (i 1,2, ,8) are eight sub-CPIs. To verify the formula, we substitute historical numeric CPI and sub-CPI values obtained in Step1 into the formula and find the formula is accurate.Step3: The construction of the GM (1, 1) model, the ARIMA (p, d, q) model and the BPNN model. The three models are established to predict the eight sub-CPIs respectively.Step4: Forecasting the eight sub-CPIs using the three models mentioned in Step3 and choosing the best forecasting result for each sub-CPI based on the errors of the data obtained from the three models.Step5: Making special improvement by adjusting the forecasting methods of sub-CPIs whose predicting results are not satisfying enough and get advanced predicting results of total CPI. Step6: Integrating the best forecasting results of 8 sub-CPIs to form the prediction of total CPI with the decomposition formula in Step2.In this way, the whole process of the prediction by the dividing-integration model is accomplished.3.2. The construction of the GM(1,1) modelThe process of GM (1, 1) model is represented in the following steps:Step1: The original sequence:Step2: Estimate the parameters a and u using the ordinary least square (OLS). Step3: Solve equation as follows.Step4: Test the model using the variance ratio and small error possibility.The construction of the ARIMA modelFirstly, ADF unit root test is used to test the stationarity of the time series. If the initial time series is not stationary, a differencing transformation of the data is necessary to make it stationary. Then the values of p and q are determined by observing the autocorrelation graph, partial correlation graph and the R-squared value.After the model is built, additional judge should be done to guarantee that the residual error is white noise through hypothesis testing. Finally the model is used to forecast the future trend ofthe variable.The construction of the BPNN modelThe first thing is to decide the basic structure of BP neural network. After experiments, we consider 3 input nodes and 1 output nodes to be the best for the BPNN model. This means we use the CPI data of time , ,toforecast the CPI of time .The hidden layer level and the number of hidden neurons should also be defined. Since the single-hidden- layer BPNN are very good at non-liner mapping, the model is adopted in this paper. Based on the Kolmogorov theorem and testing results, we define 5 to be the best number of hidden neurons. Thus the 3-5-1 BPNN structure is determined.As for transferring function and training algorithm, we select ‘tansig’as the transferring function for middle layer, ‘logsig’for input layer and ‘traingd’as training algorithm. The selection is based on the actual performance of these functions, as there are no existing standards to decide which ones are definitely better than others.Eventually, we decide the training times to be 35000 and the goal or the acceptable error to be 0.01.4.Empirical AnalysisCPI data from Jan. 2012 to Mar. 2013 are used to build the three models and the data from Apr. 2013 to Sept. 2013 are used to test the accuracy and stability of these models. What’s more, the MAPE is adopted to evaluate the performance of models. The MAPE is calculated by the equation:Data sourceAn appropriate empirical analysis based on the above discussion can be performed using suitably disaggregated data. We collect the monthly data of sub-CPIs from the website of National Bureau of Statistics of China(/doc/d62de4b46d175f0e7cd184254b35eefdc9d31514.html /).Particularly, sub-CPI data from Jan. 2012 to Mar. 2013 are used to build the three models and the data from Apr. 2013 to Sept. 2013 are used to test the accuracy and stability of these models.Experimental resultsWe use MATLAB to build the GM (1,1) model and the BPNN model, and Eviews 6.0 to build the ARIMA model. The relative predicting errors of sub-CPIs are shown in Table 2.Table 2.Error of Sub-CPIs of the 3 ModelsFrom the table above, we find that the performance of different models varies a lot, because the characteristic of the sub-CPIs are different. Some sub-CPIs like the Food CPI changes drastically with time while some do not have much fluctuation, like the Clothing CPI. We use different models to predict the sub- CPIs and combine them by equation 7.Where Y refers to the predicted rate of the total CPI, is the weight of the sub-CPI which has already been shown in Table1and is the predicted value of the sub-CPI which has the minimum error among the three models mentioned above. The model chosen will be demonstrated in Table 3:Table 3.The model used to forecastAfter calculating, the error of the total CPI forecasting by the dividing-integration model is 0.0034.5.Model Improvement & Error AdjustmentAs we can see from Table 3, the prediction errors of sub-CPIs are mostly below 0.004 except for two sub- CPIs: Food CPI whose error reaches 0.0059 and Transportation & Communication CPI 0.0047.In order to further improve our forecasting results, we modify the prediction errors of the two aforementioned sub-CPIs by adopting other forecasting methods or models to predict them. The specific methods are as follows.Error adjustment of food CPIIn previous prediction, we predict the Food CPI using the BPNN model directly. However, the BPNN model is not sensitive enough to investigate the variation in the values of the data. For instance, although the Food CPI varies a lot from month to month, the forecasting values of it are nearly all around 103.5, which fails to make meaningful prediction.We ascribe this problem to the feature of the training data. As we can see from the original sub-CPI data on the website of National Bureau of Statistics of China, nearly all values of sub-CPIs are around 100. As for Food CPI, although it does have more absolute variations than others, its changes are still very small relative to the large magnitude of the data (100). Thus it will be more difficult for the BPNN model to detect the rules of variations in training data and the forecastingresults are marred.Therefore, we use the first-order difference series of Food CPI instead of the original series to magnify the relative variation of the series forecasted by the BPNN. The training data and testing data are the same as that in previous prediction. The parameters and functions of BPNN are automatically decided by the software, SPSS.We make 100 tests and find the average forecasting error of Food CPI by this method is 0.0028. The part of the forecasting errors in our tests is shown as follows in Table 4:Table 4.The forecasting errors in BPNN testError adjustment of transportation &communication CPIWe use the Moving Average (MA) model to make new prediction of the Transportation and Communication CPI because the curve of the series is quite smooth with only a few fluctuations. We have the following equation(s):where X1, X2…Xn is the time series of the Transportation and Communication CPI, is the value of moving average at time t, is a free parameter which should be decided through experiment.To get the optimal model, we range the value of from 0 to 1. Finally we find that when the value of a is 0.95, the forecasting error is the smallest, which is 0.0039.The predicting outcomes are shown as follows in Table5:Table 5.The Predicting Outcomes of MA modelAdvanced results after adjustment to the modelsAfter making some adjustment to our previous model, we obtain the advanced results as follows in Table 6: Table 6.The model used to forecast and the Relative ErrorAfter calculating, the error of the total CPI forecasting by the dividing-integration model is 0.2359.6.Further DiscussionTo validate the dividing-integration model proposed in this paper, we compare the results of our model with the forecasting results of models that do not adopt the dividing-integration method. For instance, we use the ARIMA model, the GM (1, 1) model, the SARIMA model, the BRF neural network (BRFNN) model, the Verhulst model and the Vector Autoregression (VAR) model respectively to forecast the total CPI directly without the process of decomposition and integration. The forecasting results are shown as follows in Table7.From Table 7, we come to the conclusion that the introduction of dividing-integration method enhances the accuracy of prediction to a great extent. The results of model comparison indicate that the proposed method is not only novel but also valid and effective.The strengths of the proposed forecasting model are obvious. Every sub-CPI time series have different fluctuation characteristics. Some are relatively volatile and have sharp fluctuations such as the Food CPI while others are relatively gentle and quiet such as the Clothing CPI. As a result, by dividing the total CPI into several sub-CPIs, we are able to make use of the characteristics of each sub-CPI series and choose the best forecasting model among several models for every sub-CPI’s prediction. Moreover, the overall prediction error is provided in the following formula:where TE refers to the overall prediction error of the total CPI, is the weight of the sub-CPI shown in table 1 and is the forecasting error of corresponding sub-CPI.In conclusion, the dividing-integration model aims at minimizing the overall prediction errors by minimizing the forecasting errors of sub-CPIs.7.Conclusions and future workThis paper creatively transforms the forecasting of national CPI into the forecasting of 8 sub-CPIs. In the prediction of 8 sub-CPIs, we adopt three widely used models: the GM (1, 1) model, the ARIMA model and the BPNN model. Thus we can obtain the best forecasting results for each sub-CPI. Furthermore, we make special improvement by adjusting the forecasting methods of sub-CPIs whose predicting results are not satisfying enough and get the advanced predicting results of them. Finally, the advanced predicting results of the 8 sub- CPIs are integrated to formthe forecasting results of the total CPI.Furthermore, the proposed method also has several weaknesses and needs improving. Firstly, The proposed model only uses the information of the CPI time series itself. If the model can make use of other information such as the information provided by factors which make great impact on the fluctuation of sub-CPIs, we have every reason to believe that the accuracy and stability of the model can be enhanced. For instance, the price of pork is a major factor in shaping the Food CPI. If this factor is taken into consideration in the prediction of Food CPI, the forecasting results will probably be improved to a great extent. Second, since these models forecast the future by looking at the past, they are not able to sense the sudden or recent change of the environment. So if the model can take web news or quick public reactions with account, it will react much faster to sudden incidence and affairs. Finally, the performance of sub-CPIs prediction can be higher. In this paper we use GM (1, 1), ARIMA and BPNN to forecast sub-CPIs. Some new method for prediction can be used. For instance, besides BPNN, there are other neural networks like genetic algorithm neural network (GANN) and wavelet neural network (WNN), which might have better performance in prediction of sub-CPIs. Other methods such as the VAR model and the SARIMA model should also be taken into consideration so as to enhance the accuracy of prediction.References1.Wang W, Wang T, and Shi Y. Factor analysis on consumer price index rising in China from 2005 to 2008. Management and service science 2009; p. 1-4.2.Qin F, Ma T, and Wang J. The CPI forecast based on GA-SVM. Information networking and automation 2010; p. 142-147.3.George EPB, Gwilym MJ, and Gregory CR. Time series analysis: forecasting and control. 4th ed. Canada: Wiley; 20084.Weng D. The consumer price index forecast based on ARIMA model. WASE International conferenceon information engineering 2010;p. 307-310.5.Jian L, Zhao Y, Zhu YP, Zhang MB, Bertolatti D. An application of ARIMA model to predict submicron particle concentrations from meteorological factors at a busy roadside in Hangzhou, China. Science of total enviroment2012;426:336-345.6.Priya N, Ashoke B, Sumana S, Kamna S. Trend analysis and ARIMA modelling of pre-monsoon rainfall data forwestern India. Comptesrendus geoscience 2013;345:22-27.7.Hwang HB. Insights into neural-network forecasting of time seriescorresponding to ARMA(p; q) structures. Omega2001;29:273-289./doc/d62de4b46d175f0e7cd184254b35eefdc9d31514.html am A. Using a neural network to forecast inflation. Industrial management & data systems 1999;7:296-301.9.Min X, Wong WK. A seasonal discrete grey forecasting model for fashion retailing. Knowledge based systems 2014;57:119-126.11. Weimin M, Xiaoxi Z, Miaomiao W. Forecasting iron ore import and consumption of China using grey model optimized by particleswarm optimization algorithm. Resources policy 2013;38:613-620.12. Zhen D, and Feng S. A novel DGM (1, 1) model for consumer price index forecasting. Greysystems and intelligent services (GSIS)2009; p. 303-307.13. Yu W, and Xu D. Prediction and analysis of Chinese CPI based on RBF neural network. Information technology and applications2009;3:530-533.14. Zhang GP. Time series forecasting using a hybrid ARIMA and neural network model. Neurocomputing 2003;50:159-175.15. Pai PF, Lin CS. A hybrid ARIMA and support vector machines model in stock price forecasting. Omega 2005;33(6):497-505.16. Tseng FM, Yu HC, Tzeng GH. Combining neural network model with seasonal time series ARIMA model. Technological forecastingand social change 2002;69(1):71-87.17.Cho MY, Hwang JC, Chen CS. Customer short term load forecasting by using ARIMA transfer function model. Energy management and power delivery, proceedings of EMPD'95. 1995 international conference on IEEE, 1995;1:317-322.译⽂：⼀种基于ARIMA、灰⾊模型和BPNN对CPI（消费物价指数）进⾏预测的新型分治模型摘要:在本⽂中，利⽤我国现有的消费者价格指数（CPI）的计算⽅法，提出了⼀种新的CPI预测分治模型。

物流系统2-forecast

• Disadvantages of Moving Average Method
– Requires saving lots of past data points: at least the N periods used in the moving average computation – Lags behind a trend – Ignores complex relationships in data
Ft = (∑i wi ·Ai) = (wt-1·At-1 + wt-2·At-2 + ….. wt-n·At-n) ∑i wi =1
19
Summary of Moving Averages
• Advantages of Moving Average Method
– Easily understood – Easily computed – Provides stable forecasts
4
Forecast Horizons in Operation Planning – Fhat to Forecast: – Demand/Price/Wage/Supply/Labor/Sales/... – Economic Growth/Technology Development/.... Impact of Inaccurate Forecasting: (If Plans are based on your Forecasting) – If Forecasting is Consistently Higher than Actual: – If Forecasting is Consistently Lower than Actual:
15