第二次课 时序平滑相关预测方法 + 趋势与季节分析
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Naive rate of change model:
ˆ = Y ⋅ Yt Yt +1 t Yt −1
4 Naive models (for seasonal time series)
Naive seasonal model for quarterly data
ˆ Yt +1 = Yt −3
Naive trend and seasonal model for quarterly data
Lecture 2
时序平滑相关预测方法 趋势与季节分析
Part 1 Basic methods for time series forecasting
时间序列：一个变量按照固定时间间隔的一 序列观测值。 y0, y1, y2, ..., yt 基本预测方法： Naive Methods Moving Average Methods Smoothing Methods
7 Simple Exponential Smoothing
It combines the new info into the lagged forecast. ˆ ˆ Yt +1 = α ⋅ Yt + (1 − α ) ⋅ Yt It could be regarded as weighted moving average, giving more weight on new info while less wight on old ones. But it still could not deal with trend data well.
Practice Notes
Familiar with the procedure Familiar with the characteristics of different methods Check trends and seasonalitity Check and trace errors if forecasting
Trend ( T, level value), basic forces of population, price, tech., productivity changes and product life cycles. Cyclical (C, ratio %), economic cycles, climate cycles with length of several years Seasonal (S, ratio %) Irregular / Random (I, ratio %)
8* Practice: X12 procedure
Data: Malaysia Palm Oil monthly production and stock, 1980-2010, From: Malaysia Palm Oil Board (MPOB). File:po_stock_prod_malaysia_1980-2010.wf1
ˆ = M = (Yt + Yt −1 + ... + Yt − k +1 ) Yt +1 t k
6 Double Moving Average ( For trend data)
Double moving average:
'
( M t + M t −1 + ... + M t − k +1 ) Mt= k
Part 2 Decomposition of Time Series
1 Strategy underneath the approach, decomposition of time series One approach for analyzing time series data : To identify the component factors that influence each values in a series. Decomposition: 1) Each component is identified separately. 2) Projections of each components can be combined to produce forecasts. 3) Also used to simply display the underlying growth or decline of a series.
Understanding:
*Primary a tool for understanding changes in a series seasonal*Could produce seasonal-adjusted series, seasonal index.
3 Four components in a time series
9 Problems in forecasting
Model-based methods rely heavily on the data. But a subjective review still is important. The causal forces in the past will be still in effect? This must be continually reevaluated!
6 Seasonal and Irregular components
Divide the series Y by trend-cycle T, you get SI. Then by moving average, the rough estimate of seasonal index S be obtained. The irregular component I equals to SI / S.
9* Practice: Holt-Winter's smoothing (Multiplicative)
Data: Malaysia Palm Oil monthly production and stock, 1980-2010, From: Malaysia Palm Oil Board (MPOB). www.mpob.gov.my File:po_stock_prod_malaysia_1980-2010.wf1
7 Two type of decomposition models
The additive components model:
Yt = Tt + St + I t
The multiplicative components model: (used more frequently)
Yt = Tt × St × I t
St = γ ⋅
t
Yt Lt = α ⋅ + (1 − α )( Lt −1 + Tt −1 ) St − s
Lt
+ (1 − γ ) St − s
Forecast: Y = ( L + T ⋅ p) S ˆ t+ p t t t + p−s Parameters: 0 < α , β, γ < 1,
9 Holt-Winter's smoothing (Multiplicative, recursive） p357
Level estimate: Trend estimate: T = β ( L − L ) + (1 − β )T t t t −1 t −1 Seasonality estimate: Y
8 The census II decomposition method
Basic Idea: To isolate the trend-cycle, seasonal and irregular components one by one. For details, see U.S. Census Manual for X12. X12-ARIMA Reference Manual: Can be found in the DOCS subdirectory of your EViews directory in the PDF files (FINALPT1.PDF and FINALPT2.PDF).
8 Holt-Winter's Smoothing (Non-seasonal, recursive）p358
Level estimate: Lt = α ⋅ Yt + (1 − α )( Lt −1 + Tt −1 ) Trend estimate: Tt = β ( Lt − Lt −1 ) + (1 − β )Tt −1 ˆ Yt + p = Lt + Tt ⋅ p Forecast: Parameters: 0 < α , β < 1 Questions: 1)Initial values for L and T? 2)The optimal valuesα and β?
2 Naive Models
Most Important Benchmark for comparison For stable time series data:
3 Naive Models ( for time series with trend )
Naive Trend Model:
ˆ Yt +1 = Yt + (Yt − Yt −1 )
8* Practice: Holt-Winter's smoothing (Non-seasonal)
Data: Euro. Brent crude oil daily price, 2000-2010 From: EIA, www.eia.gov Energy Information Administration, File: eurobrentoi源自文库price(daily2000-2010).wf1
at = 2M t − M 't
bt = 2 ( M t − M 't ) k −1
Level estimate at the t-th period: Trend estimate at the t-th period: The p-periods ahead forecast:
ˆ Yt + p = at + bt ⋅ p
2 Decomposition method mainly for forecasting or understanding time series? Forecasting:
*Long history, but having lost its attractiveness in forecasting *Because of difficulty in getting accurate projections for the components
5 Trend-cycle (also denoted by T)
In practice, cycles are often difficult to identify or separated from the series. Frequently regarded as part of the trend. By moving average, a rough trend-cycle estimate could be obtained.
ˆ = Y + (Yt − Yt − 4 ) + (Yt −1 − Yt −5 ) + ... + (Yt −3 − Yt −7 ) Yt +1 t −3 4
5 Moving Average for k time periods
For stationary data, it behaves well. But it could not deal with trend data well. It gives lower forecasts facing trend series.
ˆ = Y ⋅ Yt Yt +1 t Yt −1
4 Naive models (for seasonal time series)
Naive seasonal model for quarterly data
ˆ Yt +1 = Yt −3
Naive trend and seasonal model for quarterly data
Lecture 2
时序平滑相关预测方法 趋势与季节分析
Part 1 Basic methods for time series forecasting
时间序列：一个变量按照固定时间间隔的一 序列观测值。 y0, y1, y2, ..., yt 基本预测方法： Naive Methods Moving Average Methods Smoothing Methods
7 Simple Exponential Smoothing
It combines the new info into the lagged forecast. ˆ ˆ Yt +1 = α ⋅ Yt + (1 − α ) ⋅ Yt It could be regarded as weighted moving average, giving more weight on new info while less wight on old ones. But it still could not deal with trend data well.
Practice Notes
Familiar with the procedure Familiar with the characteristics of different methods Check trends and seasonalitity Check and trace errors if forecasting
Trend ( T, level value), basic forces of population, price, tech., productivity changes and product life cycles. Cyclical (C, ratio %), economic cycles, climate cycles with length of several years Seasonal (S, ratio %) Irregular / Random (I, ratio %)
8* Practice: X12 procedure
Data: Malaysia Palm Oil monthly production and stock, 1980-2010, From: Malaysia Palm Oil Board (MPOB). File:po_stock_prod_malaysia_1980-2010.wf1
ˆ = M = (Yt + Yt −1 + ... + Yt − k +1 ) Yt +1 t k
6 Double Moving Average ( For trend data)
Double moving average:
'
( M t + M t −1 + ... + M t − k +1 ) Mt= k
Part 2 Decomposition of Time Series
1 Strategy underneath the approach, decomposition of time series One approach for analyzing time series data : To identify the component factors that influence each values in a series. Decomposition: 1) Each component is identified separately. 2) Projections of each components can be combined to produce forecasts. 3) Also used to simply display the underlying growth or decline of a series.
Understanding:
*Primary a tool for understanding changes in a series seasonal*Could produce seasonal-adjusted series, seasonal index.
3 Four components in a time series
9 Problems in forecasting
Model-based methods rely heavily on the data. But a subjective review still is important. The causal forces in the past will be still in effect? This must be continually reevaluated!
6 Seasonal and Irregular components
Divide the series Y by trend-cycle T, you get SI. Then by moving average, the rough estimate of seasonal index S be obtained. The irregular component I equals to SI / S.
9* Practice: Holt-Winter's smoothing (Multiplicative)
Data: Malaysia Palm Oil monthly production and stock, 1980-2010, From: Malaysia Palm Oil Board (MPOB). www.mpob.gov.my File:po_stock_prod_malaysia_1980-2010.wf1
7 Two type of decomposition models
The additive components model:
Yt = Tt + St + I t
The multiplicative components model: (used more frequently)
Yt = Tt × St × I t
St = γ ⋅
t
Yt Lt = α ⋅ + (1 − α )( Lt −1 + Tt −1 ) St − s
Lt
+ (1 − γ ) St − s
Forecast: Y = ( L + T ⋅ p) S ˆ t+ p t t t + p−s Parameters: 0 < α , β, γ < 1,
9 Holt-Winter's smoothing (Multiplicative, recursive） p357
Level estimate: Trend estimate: T = β ( L − L ) + (1 − β )T t t t −1 t −1 Seasonality estimate: Y
8 The census II decomposition method
Basic Idea: To isolate the trend-cycle, seasonal and irregular components one by one. For details, see U.S. Census Manual for X12. X12-ARIMA Reference Manual: Can be found in the DOCS subdirectory of your EViews directory in the PDF files (FINALPT1.PDF and FINALPT2.PDF).
8 Holt-Winter's Smoothing (Non-seasonal, recursive）p358
Level estimate: Lt = α ⋅ Yt + (1 − α )( Lt −1 + Tt −1 ) Trend estimate: Tt = β ( Lt − Lt −1 ) + (1 − β )Tt −1 ˆ Yt + p = Lt + Tt ⋅ p Forecast: Parameters: 0 < α , β < 1 Questions: 1)Initial values for L and T? 2)The optimal valuesα and β?
2 Naive Models
Most Important Benchmark for comparison For stable time series data:
3 Naive Models ( for time series with trend )
Naive Trend Model:
ˆ Yt +1 = Yt + (Yt − Yt −1 )
8* Practice: Holt-Winter's smoothing (Non-seasonal)
Data: Euro. Brent crude oil daily price, 2000-2010 From: EIA, www.eia.gov Energy Information Administration, File: eurobrentoi源自文库price(daily2000-2010).wf1
at = 2M t − M 't
bt = 2 ( M t − M 't ) k −1
Level estimate at the t-th period: Trend estimate at the t-th period: The p-periods ahead forecast:
ˆ Yt + p = at + bt ⋅ p
2 Decomposition method mainly for forecasting or understanding time series? Forecasting:
*Long history, but having lost its attractiveness in forecasting *Because of difficulty in getting accurate projections for the components
5 Trend-cycle (also denoted by T)
In practice, cycles are often difficult to identify or separated from the series. Frequently regarded as part of the trend. By moving average, a rough trend-cycle estimate could be obtained.
ˆ = Y + (Yt − Yt − 4 ) + (Yt −1 − Yt −5 ) + ... + (Yt −3 − Yt −7 ) Yt +1 t −3 4
5 Moving Average for k time periods
For stationary data, it behaves well. But it could not deal with trend data well. It gives lower forecasts facing trend series.