2016年ACCA知识点：Linear regression

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linear regression知识点

linear regression知识点1.引言1.1 概述引言部分是文章的开头，用来介绍文章的背景和重要性。

在"概述"部分，我们可以对linear regression（线性回归）的基本概念和作用进行简单介绍。

概述：线性回归是机器学习领域中最简单且最常用的回归方法之一。

它是一种建立输入变量（自变量）和输出变量（因变量）之间线性关系的统计学模型。

线性回归可以帮助我们探索和理解数据，预测未知的因变量值，并在实际问题中做出决策。

线性回归的基本思想是基于已知的训练数据，通过拟合一条直线（或超平面）来近似描述输入和输出之间的关系。

这条直线可以用来做预测和回答各种问题。

线性回归的关键是通过最小化预测值与实际观测值之间的差距，找到最佳拟合直线。

线性回归不仅可以用于预测连续性数值型数据，还可以用于分类问题，例如将输出变量划分为两个或多个不同的类别。

尽管线性回归在实际问题中很常见，但它也有一些局限性，例如对于非线性关系的建模能力较弱。

为了克服这些局限性，研究人员还提出了各种改进方法。

本文将深入探讨线性回归的基本概念和原理，介绍线性回归模型的建立与求解过程，并探讨线性回归在实际应用中的场景和局限性，同时提出一些改进方法。

通过阅读本文，读者将能够全面了解线性回归的知识和应用，从而在实际问题中更好地应用和理解线性回归方法。

下面我们将详细介绍本文的结构和目的。

1.2 文章结构文章结构部分的内容可以描述整篇文章的组织和安排，可以按照以下内容进行阐述：在本篇文章中，我们将从引言、正文和结论三个部分来组织和阐述关于Linear Regression（线性回归）的知识点。

首先，在引言部分，我们将对线性回归进行概述，介绍其基本概念和原理。

同时，我们将阐明本篇文章的目的，即通过介绍线性回归的知识点，让读者对线性回归有一个全面的了解。

接着，在正文部分，我们将分为两个小节来详细讲解线性回归的知识点。

首先，我们将介绍线性回归的基本概念，包括线性回归的定义、特点以及模型表示等。

线性回归(Linear Regression)

1. #!/usr/bin/python2. # -*- coding: utf-8 -*-3.4. """5. author ： duanxxnj@6. time : 2016-06-19-20-487.8. 这个是线性回归的示例代码9. 使用一条直线对一个二维数据拟合10.11. """12. print(__doc__)13.14.15. import matplotlib.pyplot as plt16. import numpy as np17. from sklearn import datasets, linear_model18.19. # 加载用于回归模型的数据集20. # 这个数据集中一共有442个样本，特征向量维度为1021. # 特征向量每个变量为实数，变化范围（-.2 ，.2）22. # 目标输出为实数，变化范围（25 ，346）23. diabetes = datasets.load_diabetes()24.25. # 查看数据集的基本信息26. print diabetes.data.shape27. print diabetes.data.dtype28. print diabetes.target.shape29. print diabetes.target.dtype30.31. # 为了便于画图显示32. # 仅仅使用一维数据作为训练用的X33. # 这里使用np.newaxis的目的是让行向量变成列向量34. # 这样diabetes_X每一项都代表一个样本35. diabetes_X = diabetes.data[:, np.newaxis,2]36.37. # 此时diabetes_X的shape是(442L, 1L)38. # 如果上面一行代码是：diabetes_X = diabetes.data[:, 2]39. # 则diabetes_X的shape是(442L，)，是一个行向量40. print diabetes_X.shape41.42. # 人工将输入数据划分为训练集和测试集43. # 前400个样本作为训练用，后20个样本作为测试用44. diabetes_X_train = diabetes_X[:-20]45. diabetes_X_test = diabetes_X[-20:]46. diabetes_y_train = diabetes.target[:-20]47. diabetes_y_test = diabetes.target[-20:]48.49. # 初始化一个线性回归模型50. regr = linear_model.LinearRegression()51.52. # 基于训练数据，对线性回归模型进行训练53. regr.fit(diabetes_X_train, diabetes_y_train)54.55. # 模型的参数56. print'模型参数：', regr.coef_57. print'模型截距：', regr.intercept_58.59. # 模型在测试集上的均方差（mean square error）60. print("测试集上的均方差: %.2f"61. % np.mean((regr.predict(diabetes_X_test)- diabetes_y_test)**2))62. # 模型在测试集上的得分，得分结果在0到1之间，数值越大，说明模型越好63. print('模型得分: %.2f'% regr.score(diabetes_X_test, diabetes_y_test))64.65. # 绘制模型在测试集上的效果66. plt.scatter(diabetes_X_test, diabetes_y_test, color='black')67. plt.plot(diabetes_X_test, regr.predict(diabetes_X_test), color='blue',68. linewidth=3)69.70. plt.grid()71. plt.show()。

linearregression()相关系数

linearregression()相关系数一、介绍LinearRegression()函数是在Python的许多统计和机器学习库中常见的一种回归模型，它主要用于根据已知的数据点预测未知的数据点。

相关系数是一种用于评估模型性能的重要指标，它可以帮助我们了解因变量和自变量之间的线性关系强度。

二、相关系数的计算相关系数是通过计算因变量和自变量之间的协方差，再除以因变量和自变量的标准差，得到的数值。

这个数值的范围在-1到1之间，其中1表示完全的正线性关系，-1表示完全的负线性关系，而0表示没有线性关系。

三、线性回归模型的建立在建立线性回归模型时，我们需要选择合适的自变量和因变量，并收集相关的数据。

在处理数据时，我们可能需要对其进行预处理，如缺失值的填补、异常值的处理以及多重共线性的检查和解决。

四、相关系数在模型评估中的应用相关系数可以用于评估线性回归模型的预测性能。

如果相关系数较大，说明因变量和自变量之间的线性关系较强，模型的预测效果较好。

反之，如果相关系数较小，说明因变量和自变量之间的线性关系较弱，模型的预测效果可能较差。

此外，相关系数还可以用于比较不同的模型或参数设置的效果。

五、常见问题及解决方案在使用相关系数评估模型时，可能会遇到一些问题，如数据缺失、异常值、多重共线性等。

对于数据缺失，我们可以使用插值或合并数据等方法进行填补。

对于异常值，我们可以进行剔除或使用适当的方法进行平滑。

对于多重共线性，我们可以使用主成分分析等方法进行降维。

六、结论相关系数是评估线性回归模型效果的重要指标之一。

通过了解因变量和自变量之间的线性关系强度，我们可以更好地理解模型的预测性能。

在实际应用中，我们需要选择合适的自变量和因变量，并进行适当的预处理，以提高模型的预测精度。

同时，我们也需要注意处理可能出现的问题，以保证结果的准确性和可靠性。

以上内容仅供参考，如需更具体信息请查询官方文档或相关资料。

线性回归LinearRegression

线性回归LinearRegression 成本函数（cost function）也叫损失函数（loss function），⽤来定义模型与观测值的误差。

模型预测的价格与训练集数据的差异称为残差（residuals）或训练误差（test errors）。

我们可以通过残差之和最⼩化实现最佳拟合，也就是说模型预测的值与训练集的数据最接近就是最佳拟合。

对模型的拟合度进⾏评估的函数称为残差平⽅和（residual sum of squares）成本函数。

就是让所有训练数据与模型的残差的平⽅之和最⼩。

我们⽤R⽅（r-squared）评估预测的效果。

R⽅也叫确定系数（coefficient of determination），表⽰模型对现实数据拟合的程度。

计算R⽅的⽅法有⼏种。

⼀元线性回归中R⽅等于⽪尔逊积矩相关系数（Pearson product moment correlation coefficient 或Pearson's r）的平⽅。

这种⽅法计算的R⽅⼀定介于0～1之间的正数。

其他计算⽅法，包括scikit-learn中的⽅法，不是⽤⽪尔逊积矩相关系数的平⽅计算的，因此当模型拟合效果很差的时候R⽅会是负值。

SStot是⽅差平⽅和 SSres是残差的平⽅和⼀元线性回归X_test = [[8], [9], [11], [16], [12]]y_test = [[11], [8.5], [15], [18], [11]]model = LinearRegression()model.fit(X, y)model.score(X_test, y_test）score⽅法计算R⽅多元线性回归最⼩⼆乘的代码from numpy.linalg import lstsqprint(lstsq(X, y)[0])多项式回归⼀种特殊的多元线性回归⽅法，增加了指数项（x 的次数⼤于1）。

现实世界中的曲线关系都是通过增加多项式实现的，其实现⽅式和多元线性回归类似。

经济计量学英文原版讲义2

SW Ch 4
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The OLS estimator solves:
min b0 ,b1 ∑ [Yi − (b0 + b1 X i )]2
i =1
n
• The OLS estimator minimizes the average squared difference between the actual values of Yi and the prediction (“predicted value”) based on the estimated line. • This minimization problem can be solved using calculus (App. 4.2). • The result is the OLS estimators of β 0 and β 1.
min b0 ,b1 ∑ [Yi − (b0 + b1 X i )]2
i =1 n
SW Ch 4
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Mechanics of OLS The population regression line: Test Score = β 0 + β 1STR
∆Test score β1 = = ?? ∆STR
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SW Ch 4
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Application to the California Test Score – Class Size data
ˆ = – 2.28 Estimated slope = β 1 ˆ = 698.9 Estimated intercept = β
0
SW Ch 4
ˆ Antelope = 657.8 – 654.8 = 3.0 u

linearregression用法

linearregression用法线性回归用法线性回归是一种常见的统计学习方法，用于预测两个或多个变量之间的关系。

在许多实际问题中，线性回归模型被广泛使用，因为它能够有效地描述变量之间的关系，并给出准确的预测结果。

一、线性回归模型线性回归模型是一种基于线性方程的模型，它通过拟合一组线性方程来描述两个或多个变量之间的关系。

线性回归模型的公式表示为：y=β0+β1x1+β2x2+...+βpxp+ε，其中y是因变量，x1,x2,...,xp是自变量，β0,β1,β2,...,βp是系数，ε是误差项。

二、线性回归的优缺点优点：1.线性回归模型能够有效地描述变量之间的关系，并给出准确的预测结果。

2.线性回归模型可以通过拟合多个自变量和系数的组合来提高预测精度。

3.线性回归模型易于理解和解释。

缺点：1.线性回归模型对异常值和噪声非常敏感，因此需要处理数据中的异常值和噪声。

2.线性回归模型的泛化能力取决于模型的复杂性和数据的多样性，因此需要选择合适的模型参数和正则化方法来提高模型的泛化能力。

三、线性回归的应用场景线性回归广泛应用于各个领域，如财务、市场营销、生物医学等。

在财务领域，线性回归可以用于预测公司的盈利、股票价格等；在市场营销中，线性回归可以用于预测客户的购买行为、广告效果等；在生物医学中，线性回归可以用于基因组学、疾病预测等领域。

四、线性回归的步骤1.数据收集和清洗：首先需要收集相关的数据，并对数据进行清洗和处理，包括去除异常值、缺失值、重复值等。

2.特征选择：根据问题的实际情况，选择合适的自变量进行建模。

3.模型训练：使用线性回归算法对数据进行拟合，得到系数的估计值。

4.模型评估：使用各种评估指标，如均方误差、R平方值等，来评估模型的性能。

5.预测：使用训练好的模型对新的数据进行预测，得到预测结果。

五、线性回归的常见算法和工具常见算法：最小二乘法、岭回归、Lasso回归、弹性网回归等。

常见工具：Python中的Scikit-learn库、R语言中的glmnet包等。

multiple linear名词解释

multiple linear名词解释Multiplelinearregression（多元线性回归）是一种经典的回归分析方法，也是最常用和最广泛应用的回归分析方法之一。

它用于探索多个自变量（也称为因子）与一个因变量之间的线性关系。

多元线性回归用于预测和预报，以及探索如何改变一个或多个因变量，从而对另一个因变量产生影响。

多元线性回归是一种基于向量计算的统计学方法，它可以探索多个因子之间的关系。

多元线性回归需要假设测量变量间的线性关系，并建立以该线性关系为基础的模型。

一个多元线性回归的模型可以写为 Y=0+1x1+β2x2+…+pxp，其中Y为标签变量（也称为因变量），x1，x2，…，xp为自变量（或因子），β0，1，β2，…，βp为参数，分别代表x1，x2，…，xp在因变量Y上的影响程度。

多元线性回归的研究是基于以下假设的：1.线性关系：假设自变量与因变量之间的关系是线性的；2.独立性：假设自变量之间是独立的，即一个自变量对另一个自变量没有影响；3.正态性：假设误差项服从正态分布。

Multiple linear regression可以用来衡量各自变量对因变量的影响，以及它们之间的相互作用，从而探索多个自变量在确定一个因变量上的重要性。

这是多元线性回归分析最重要的特点。

另外，多元线性回归可以有效处理复杂的实验或模拟数据，可以有效地去除多个自变量的干扰。

多元线性回归的结果由统计模型的拟合程度、R平方（R2）、变量的系数等决定。

模型的拟合程度可以用SSE（残差平方和）和SSR （拟合平方和）来衡量，R2是模型的拟合程度的一种衡量方式，测量的是模型的拟合程度，它的取值范围从0到1，越大表示模型越好；而变量的系数代表了因变量受自变量影响的程度，变量系数是正值则表示自变量对因变量有正向影响，反之亦然。

多元线性回归分析在市场研究、财务分析、投资决策、经济分析、社会科学研究、药物研究、生物科学研究、人机交互研究、人际关系研究、营销策略分析等方面都有重要的应用，在各个领域的研究中都有广泛的应用，为研究人员提供了一种有效的研究工具。

ACCA笔记 SBL笔记11 Financial Decision Making

ACCA笔记| SBL笔记11 | Financial Decision Making今日笔记是关于 Financial Decision Making 的内容~1. Financial objectives of stakeholders- Financial strategy often focuses on shareholders wealth. But there are financial expectations from other stakeholders which must also be considered.- When setting strategy for the organisation, the financial expectations of other stakeholders need also be considered.2. Funding strategy- Funding for Non-for-profit organisations Most non-for-profit organisations need their core costs to be covered.- Funding SBUs and strategic choices using BCG matrix- Alternative sources of financeequity, debt, others (eg. government grants)3. Financial analysis and decision-making techniques- Break even analysis 1) To ascertain how much each $ sold actually contributes towards the fixed costs.C/S ratio = Contribution per unit / selling price per units= total contribution / total sales revenue2) To make zero profit, sales volume should be atBreakeven = FC/(sales price – unit variable cost)= FC/ CPU3) The margin of safety indicates by how much sales can decrease before a loss occurs Margin of safety= budgeted sales –BEP A large margin of safety indicates a low risk of making loss, whereas a small margin of safety might indicate a fairly high risk of loss.- Marginal analysisMarginal analysis refers to situations where we use contribution to make decisions.The key is that only cost which vary with the decision should be included in the analysis of decision.Relevant costs are future cash flows arising as a direct consequence of the decision under consideration.Marginal analysis can be used in key areas of decision making such as:1) Make or buy decisions2) Shut-down decisions3) Further processing decisions- Long-term decision making1) Accounting rate of return (ARR) Average profits / initial investment2) Payback period Determine how quickly the original cash injection is recovered3) Net present value (NPV) Use a cost of capital and discount factors to discount future cash flows to give the present value4) Internal rate of return (IRR) Determine the cost of capital that provides a zero NPV4. Cost and management accounting- Standard costing assumes that business operate in a stable environment where, for example, a standard amount of materials will be used in the production of each product and that a standard price can be attached to the price of those materials.- Forecast 1) Linear regression Linear regression measures the relationship between two variablesThe strength of the relationship is measured by correlation coefficient - 'r', which can range from +1 (perfect positive linear correlation) through 0 (no correlation) to -1 (perfect negative linear correlation).2) Time series analysis Time series analysis aims to separate seasonal and cyclical fluctuations from long-term underlying trends.- Budget 1) Benefits of budgets Promotes forward thinking Helps to co-ordinate the various aspects of the organisation Motivates performance Provides a basis for a system of control Provides a system of authorisation2) Limitations of budgets Employees may be demotivated if they believe the budget to be unattainable. Slack may be built in by managers to make the budget more achievable.Focuses on the short-term results rather than the underlying causes.Unrealistic budgets may cause managers to make decisions that are detrimental to the company.3) Alternative budgeting models Top-down (Imposed budgeting) VS Bottom-up (Participativebudgeting)Fixed budget VS Flexible budget Periodic budgets VS Rolling (Continuous) Budgets Incremental VS Zero based budgets。

讲义_6 & lecture_6

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. A planar response surface may not always be appropriate, but even when not it is often a good approximate descriptor of the regression function in “local” regions of the input space . The meaning of the parameters can be determined by taking partials 2
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ACCA_P3_关键模型汇总-精品资料

内部资料请勿外传ACCA P3 关键模型汇总汇编：詹也浙江财经大学2014年快速浏览法Part A（战略位势）1.1The strategy lenses（战略维度）2.Strategy as design.The design lens views strategy as the deliberate positioning of an organisation asthe result of some ‘rational, analytical, structured and directive process’.responsibility of top management to plan the destiny of the organisation. Lowerlevels of management carry out the operational actions required by the strategy.The design lens is associated with objective setting and a plan for moving theorganisation towards these objectives.1Strategy as experience.The experience lens views strategy development as the combination of individualand collective experience together with the taken-for-granted assumptions ofcultural influences. Strategy as experience seems innately conservative. It couldwork well when a small incremental change is required within a stableenvironment. However, this view may become a major barrier to developinginnovative strategies as experience may become rigid.2Strategy as ideas.It has a central role for innovation and new ideas. It sees s trategy as emergingfrom the variety and diversity in an organisation. It is as likely to come from the bottom of the organisation as from the top. Consequently, the organization should foster conditions that allow ideas to emerge and to be considered for inclusion ina ‘mainstream strategy’.1.2PESTEL应用范围：当题目要求做“environmental analysis”、“analysis of the macro-environmental”或“analysis of the position of company，都可以用这个模型。

linearregression 截距和系数 -回复

linearregression 截距和系数-回复什么是线性回归？线性回归是一种用于建立变量之间线性关系的统计模型。

它通过在自变量和因变量之间拟合一条直线来预测变量之间的关系。

线性回归模型通常使用最小化残差平方和的方法来估计截距和系数。

截距和系数的定义在线性回归模型中，截距代表了当自变量为零时，因变量的预测值。

系数代表了自变量的单位变化对因变量的影响。

直观来说，截距和系数描述了线性回归模型中直线的位置和斜率。

最小二乘法线性回归模型的估计通常使用最小二乘法。

最小二乘法的目标是选择使得预测值与真实值之间的残差平方和最小化的截距和系数。

残差是预测值与真实值之间的差异。

回归方程在线性回归模型中，回归方程用于描述自变量和因变量之间的关系。

回归方程可以表示为：Y = β0 + β1X1 + β2X2 + ... + βnXn，其中Y是因变量，X1到Xn是自变量，β0是截距，β1到βn是系数。

截距的计算截距表示当所有自变量为零时，因变量的预测值。

在最小二乘法中，截距可以通过以下公式进行计算：β0 = y平均- (β1x1平均+ β2x2平均+ ... + βnxn平均)，其中y平均是因变量的平均值，x1平均到xn平均是自变量的平均值。

截距反映了回归线与y轴的交点。

系数的计算系数代表了自变量单位变化对因变量的影响。

系数的计算也可以通过最小二乘法来实现。

系数估计的数学公式为：β= (X'X)^-1X'y，其中β是系数，X是自变量数据矩阵，X'是X的转置，y是因变量向量。

通过计算系数，我们可以了解每个自变量对因变量的影响程度。

解释系数和截距在线性回归模型中，系数和截距对于解释因变量与自变量之间的关系非常重要。

系数可以告诉我们单位自变量变化对因变量的预测变化。

截距可以告诉我们自变量为零时因变量的预测值。

通过对系数和截距的解释，我们可以了解哪些自变量对因变量有显著影响，以及这种影响的方向和大小。

线性回归分析(Linear Regression)

线性回归分析（Linear Regression ）是描述一个因变量（Dependent variable ）Y 与一个或多个自变量（Independent variable ）X 间的线性依存关系。

可以根据一批样本值来估计这种线性关系，建立回归方程。

用回归方程可以进行预测、控制以及由易测变量X 求得难测变量Y 等等。

多元线性回归还可起到对影响因素的识别作用。

回归分析要求应变量Y 服从正态分布，X 可以是随机变动的，也可以是人为取值的变量。

Linear 过程用于建立回归方程；回归方程的配合适度检验包括回归方程和回归系数（或偏回归系数）的假设检验、残差分析；直线回归的区间估计和直线相关及偏相关分析。

直线回归方程：y = a + b x步骤 1描述 2散点图3回归方程 b=sum((X-Xmean)(Y-Ymean))/sum(X-Xmean) 2 a=Ymean-bXmean4检验方程是否成立：方差分析数据准备及过程结果：RegressionDescriptive Statistics2.9025.41441249.33335.280012肺活量升体重公斤Mean Std. DeviationN统计表Correlations1.000.749.7491.000..003.003.12121212肺活量升体重公斤肺活量升体重公斤肺活量升体重公斤Pearson Correlation Sig. (1-tailed)N 肺活量升体重公斤PEARSON 相关系数r=0.749,体重公斤2.503.003.50肺活量升✌✌✌✌✌✌✌✌✌✌✌✌相关系数假设检验H0: ρ=0 两变量无直线相关关系H1: ρ≠0 两变量有直线相关关系a=0.05t=r/sqrt((1-r2)/n-2)t=3.58 v=10 0.005>p>0.002,按a=0.05水平拒绝H0,接受H1，体重与肺活量间成正直线关系引入或剔险变量表模型摘要表SS总（TOTAL SQUARES）=SS回（REGRESSION）+SS剩（RESIDUAL）假设 H0 β总体回归系数=0 无直线关系H1 β≠0 有直线关系a=0.05方程： Y肺活量=0.000413（constant）+0.058833X(体重)****PEMS 结果出现重大偏倚****│直线回归│数据文件名：E:\医学统计\学习笔记\直线回归.xls自变量X的变量名: F1因变量Y的变量名: F2样本例数: n=11均数和标准差───────────────────────变量均数标准差───────────────────────X 50.0000 4.9800Y 2.9345 0.4188───────────────────────直线回归方程:Y=-0.134+0.0614X直线回归的假设检验:方差分析表─────────────────────────────────────变异来源离均差平方和自由度均方 F P ─────────────────────────────────────总 1.7537 10回归 0.9341 1 0.9341 10.2569 0.0108 剩余 0.8196 9 0.0911─────────────────────────────────────【本分析结果完毕】。

2016年ACCA F9《财务管理》重要知识点讲解(2)

2016年ACCA F9《财务管理》重要知识点讲解（2）ACCA考试F9内含报酬率法Internal Rate of Return (IRR) methodOften an entity would want to establish its internal rate of a project for various reasons e.g., for decision-making purposes. By IRR we mean a rate that will be used to discount future cash inflows to make the total of the present values equal the cost of the project. The attempt being made under IRR is to find a rate that will equate the NPV of a project to be zero. The IRR therefore is the maximum rate of discount that will be used to finance a project without making a loss from it. Again in a mutually exclusive situation, the project that has the highest IRR is the one to recommend. The reason being that the IRR is showing the highest rate that can be used to finance a project without incurring a loss from it. Students often suggest to the author that the IRR should be called the break-even rate. His reply to them is often, if this makes you understand IRR you should use that term for it.In order to find the IRR manually, this is done by the trial and error approach. By this we mean using a discount rate which gives a positive NPV and another rate which gives a negative NPV. Then apply the formula:IRR = A + a x (B ? A)a + bWhere:A = The lower discount rate which gives the positive NPVB = The higher discount rate which gives the negative NPVa = The value of the positive NPVb = The value of the negative NPVPlease note that a and b should be added together as the negative sign in b is ignored. Now let us demonstrate these various methods using a case from a fictitious company we shall call AICO Plc.CaseSenior management of AICO Plc have identified that there is a strategic need for a replacement machine to be acquired in one of their production departments. Theyhave to make a choice between two models ofthe machine ? model 1 is called Super and model 2 is called Deluxe. They are unsure as to which of the two models they should buy. They have given you the following profiles of the two models. They want you to use the four investment appraisal techniques discussed above. You are required to recommend which of the two models is better under each appraisal technique and to explain briefly why you have recommended one in place of the others under each technique. You are told that funds are only available for only one model.The cost of capital is 12% and AICO depreciates all its fixed assets on thestraight-line basis. By cost of capital is meant what it costs to raise the required finance for the projectBackgroundThere is no unequivocal definition of what is meant by an SME. McLaney (2000) identifies three characteristics:1. firms are likely to be unquoted;2. ownership of the business is restricted to few individuals, typically a family group; and3. they are not micro businesses that are normally regarded as those very small businesses that act as a medium for self-employment of the owners. However, this too is an important sub-group.The characteristics of SME’s can change as the business develops. T hus, for growing businesses a floatation on a market like AIM is a possibility in order to secure appropriate financing. In fact, venture capital support is usually preconditioned on such an assumption.The SME sector is important in terms of contribution to the economy and this is likely to be a characteristic of SME’s across the world. According to the Bank of England (1998), SME’s accounted for 45% of UK employment and 40% of sales turnover of all UK firms. This situation is similar across the EU.Future developments mean that the importance of the SME sector will continue, if not develop. The growth in small, new technology businesses servicing particular market segments and the shift from manufacturing to service industries, at least in Western economies, means that economies of scale are no longer as important as they once were and, hence, the necessity for scale in operations is no longer an imperative. We know, also, that innovation flourishes in the smaller organisation and that this will be an important characteristic of the business in the future.Payback methodSuper Model Cost ￡500,000YearCash inflows Accumulated Cash inflows 1250,000250,0002100,000350,0003100,000450,000450,000500,000Payback Period= 4 yearsDeluxe ModelCostYearCash inflowsAccumulated Cash inflows1150,000150,0002200,000350,0003250,000600,0004100,000700,0005100,000800,000Payback= 5 yearsBy comparing four years for the super model with five years for the deluxe model, one would recommend AICO Plc to buy the former, as its payback period is shorter than the latter.The Accounting Rateof ReturnThe information provided by AICO plcisbasedonhy pothetical cashflows.This informatio nisinsufficient forestablishing the ARR of thetwomodels;therefore it will be necessary tocalculate theannual depreciation figures for thetwomachines. Toarriveat theannual accounting profits will necessitateadeductionfromeach year,snetcashinflows,theannualdepreciationfigure.SupermodelDepreciationperannum =CostlessscrapvalueLifeofmachine=￡500,000less20,0006=￡80,000perannumDeluxemodelDepreciationperannum =￡800,000less80,0006=￡120,000perannumNowthatwehavecalculatedtheannualdepreciationforeachmachine,letusestablishtheannualaccountingprofitorlossforeachmachine.SupermodelYear Cashinflow less Depreciation =AccountingProfit/(Loss)1 250,000 ? 80,000 =170,0002 100,000 ? 80,000 =20,0003 100,000 ? 80,000 =20,0004 50,000 ? 80,000 =(30,000)5 150,000 ? 80,000 =70,0006 100,000 ? 80,000 =20,000Totalaccountingprofit 270,000 Thereforetheaverageaccountingprofit =￡270,000_______6=￡45,000AverageCapitalEmployed =￡500,000+20,0002=￡260,000ARR =45,000x100260,000=17?30%DeluxeModelYear Cashinflow less Depreciation =AccountingProfit/(Loss)1 150,000 120,000 =30,0002 200,000 120,000 =80,0003 250,000 120,000 =130,0004 100,000 120,000 =(20,000)5 100,000 120,000 =(20,000)6 250,000 120,000 =130,000Totalaccountingprofits 330,000 Averageaccountingprofit =￡330,0006=￡55,000Averagecapitalemployed =￡800,000+80,0002=￡440,000ThereforeARR =￡55,000x100￡440,000=12?50%UsingARR,clearlythesupermodelwitha17?30% will berecommende dinstead of the deluxemodel, which hasalowerrate of returnof12?50%。

linear regression

linear regressionLinear regression is a statistical method used to analyze relationships between two or more variables. It is used to understand the linear relationship between two or more variables, such as the relationship between the income of a person and the value of their house. Linear regression is a tool for understanding the relationship between one dependent variable (the outcome) and one or more independent variables (the predictors). The independent variables help explain why the dependent variable changes.Linear regression is a type of statistical modeling technique that looks for linear relationships among variables. This means that the value of one variable is associated with the value of another variable, and the relationship between them can be predicted and estimated using linear equations.In linear regression, the predictor variables and theresponse variable (also known as the target variable) are related in a linear way. The goal of linear regression is to find the best-fitting line through a set of data points.Linear regression is used in many fields, including economics, finance, and psychology. It can be used to studythe relationships between two or more financial assets, the relationship between the size of a company and its profits,or the relationship between students' grades and their test scores. It is a powerful and flexible tool that can be usedto gain insight into large and complex datasets.The most commonly used type of linear regression is simple linear regression, which involves finding the best-fitting line through a set of data points. It involves using a linear equation to find the best-fitting line through a set of data. If a linear equation cannot be found, then non-linear regression can be used. In the linear regression model, the fitted line should be as close as possible to all of the data points. To find the best-fitting line, we use an optimization procedure called least squares.Another type of linear regression is multiple linear regression. This type of linear regression involves the relationships between multiple independent variables and one dependent variable. This type of regression is useful for analyzing the effects of multiple independent variables on a single dependent variable. It is important to note that, unlike simple linear regression, a multiple linear regression model has more than one line of best fit.Linear regression is a powerful and useful tool for understanding the relationships between variables. It can be used for prediction, for exploration of data, or for gaining insight into complex datasets. It is important to choose the appropriate technique for the data, and to understand the limitations of linear regression models.。

univariate linear regression定义

univariate linear regression定义Univariate Linear Regression: An Introduction and StepbyStep GuideIntroduction:Univariate Linear Regression is a statistical technique used to model the relationship between a single independent variable and a dependent variable. It is a simple yet powerful tool that helps us understand the relationship between two variables and make predictions based on the observed data. In this article, we will dive deep into the concept of Univariate Linear Regression and explore its various aspects through a stepbystep guide.1. Understanding Variables:Before diving into the details of Univariate Linear Regression, it is essential to understand the variables involved. In this case, we have a single independent variable (X) and a dependent variable (Y). The independent variable represents the input or predictor variable, whereas the dependent variable represents the output or response variable.2. Assumptions of Linear Regression:Linear Regression relies on several assumptions that need to be satisfied for accurate predictions. These assumptions include linearity, independence, homoscedasticity, normality, and lack of multicollinearity. Understanding these assumptions is crucial for performing a reliable linear regression analysis.3. Building the Regression Model:The first step in Univariate Linear Regression is to build a model that captures the relationship between the independent and dependent variables. This is done by fitting a line to the data that minimizes the sum of squared errors. The line is represented by the equation: Y = β0 + β1X, where β0 is the intercept and β1 is the slope coefficient.4. Estimating the Coefficients:To estimate the coefficients β0 and β1, we utilize a technique called Ordinary Least Squares (OLS). OLS minimizes the sum of squared errors between the predicted and actual values of the dependent variable. By solving the OLS equation, we obtain the coefficients that best fit thedata and describe the linear relationship.5. Assessing Model Fit:Once the coefficients are estimated, it is crucial to assess the fit of the model. This involves evaluating key metrics such as the coefficient of determination (Rsquared), standard error, tstatistic, pvalue, and confidence intervals. These metrics provide insights into the strength and significance of the relationship between the independent and dependent variables.6. Interpreting Coefficients:Interpreting the coefficients is an essential step in Univariate Linear Regression. The slope coefficient (β1) denotes the change in the dependent variable for a oneunit increase in the independent variable, while the intercept (β0) represents the value of the dependent variable when the independent variable is zero. Understanding the coefficients allows us to make meaningful interpretations and draw insights from the model.7. Making Predictions:One of the primary purposes of Univariate Linear Regression is to make predictions based on the model. By plugging in values of the independent variable into the regression equation, we can estimate the corresponding values of the dependent variable. It is important to note that the accuracy of the predictions depends on the quality and representativeness of the data used to build the model.8. Model Evaluation:To evaluate the performance of the model, we need to assess its predictive capability. This is done by testing the model on a separate dataset, preferably one that the model has not been trained on. Various metrics such as Mean Squared Error, Root Mean Squared Error, and Mean Absolute Error are used to evaluate the model's accuracy and compare it to alternative models or benchmarks.Conclusion:Univariate Linear Regression is a powerful tool that allows us tounderstand the relationship between a single independent variable and a dependent variable. By following the stepbystep guide outlined in this article, we can build a regression model, estimate coefficients, assess model fit, interpret results, make predictions, and evaluate the model's performance. Putting this technique into practice helps uncover important insights, make accurate predictions, and inform decisionmaking processes in a wide range of fields and industries.。

linearregression对象的主要方法中

linearregression对象的主要方法中LinearRegression对象是用于线性回归模型的主要工具之一、它帮助我们建立线性回归模型并进行预测。

常见的LinearRegression对象的主要方法包括以下几个：1. fit(X, y)：该方法用于拟合训练数据，其中X代表特征变量的输入数据，y代表对应的目标变量。

拟合过程中，模型会计算特定的系数，以使得特征变量和目标变量之间的线性关系最佳地表示出来。

2. predict(X)：该方法用于使用训练好的模型进行预测，其中X代表特征变量的输入数据。

模型会基于之前计算得到的系数，预测目标变量的值。

3. score(X, y)：该方法用于评估模型的准确性。

其中X代表特征变量的输入数据，y代表对应的目标变量。

评分结果一般为拟合得到的回归直线与真实值之间相关性的程度，一般使用R平方（R-squared）来表示。

4. coef_属性：该属性用于获取模型的回归系数。

回归系数表示特征变量与目标变量之间的关系。

如果有多个特征变量，coef_属性将返回一个数组，每个元素表示每个特征变量的回归系数。

5. intercept_属性：该属性用于获取模型的截距。

截距表示回归直线与y轴的交点。

6. get_params(：该方法用于获取模型的参数。

返回一个字典，包含模型的各个参数及其对应的值。

以上就是LinearRegression对象的主要方法。

使用这些方法，我们可以拟合数据、预测结果、评估模型准确性，并获取模型参数。

这些方法为我们进行线性回归建模和分析提供了很大的便利。

linearregression 截距和系数 -回复

linearregression 截距和系数-回复中括号内的内容为主题，写一篇1500-2000字文章，一步一步回答：[linear regression 截距和系数]一、简介线性回归（linear regression）是统计学中常用的一种方法，用于分析自变量（x）与因变量（y）之间的关系。

它通过学习数据集中的样本点，确定一条最佳拟合直线来描述x和y之间的线性关系。

这条直线可以由截距和系数来表示，本文将逐步介绍截距和系数的概念、计算方法以及它们在线性回归中的意义和应用。

二、截距（intercept）截距是线性回归模型中的一个重要参数，通常用字母b来表示。

它表示了当自变量为0时，因变量的取值。

在一维线性回归中，截距就是直线与y 轴的交点。

如果截距为正，表示y轴上的截距点高于原点；如果截距为负，表示截距点低于原点。

截距的计算通过最小二乘法来进行。

最小二乘法是一种常见的优化算法，用于拟合数据点与拟合直线之间的误差。

通过使误差的平方和最小化，可以求得最佳的截距。

具体地，最小二乘法的计算公式如下：b = (Σxy - n̄x̄ȳ) / (Σx^2 - n̄x̄^2)其中，Σxy表示x和y的乘积之和，n为样本数量，x̄和ȳ分别为x和y的平均值。

三、系数（coefficient）系数是线性回归模型中的另一个重要参数，通常用字母a来表示。

它表示了x每单位变化时，y的变化量。

在一维线性回归中，系数就是直线的斜率。

如果系数为正，表示y随着x的增加而增加；如果系数为负，表示y 随着x的增加而减小。

系数的计算也通过最小二乘法来进行。

根据最小二乘法的原理，系数可以通过下式计算：a = (Σxy - n̄x̄ȳ) / (Σx^2 - n̄x̄^2)其中，Σxy、n、x̄和ȳ的含义与截距的计算公式中相同。

四、截距和系数的意义截距和系数在线性回归中具有重要的意义。

截距代表了在自变量为0时，因变量的取值。

例如，在分析一个销售模型时，截距可以表示没有进行任何销售活动时的预期销售额。

[医药卫生]线性相关与回归

线性相关系数是说明具有直线关系的两个变量间相关密切程度和相关方向的统计量。皮尔森(Pearson)相关系数的计算公式为：
rr rXY ( X X )(Y Y ) ( X X ) (Y Y )
2 i i

2
LXY LXX .LYY
相关系数 r 没有测量单位，其数值为-1≤ r ≤1
LYY Y
2
5002 22810 82.727 11
LXY XY
( X )(Y ) n
86185
1891 500 230.455 11
按公式（10-1）计算相关系数 r
r 230.455 0.8012 1000 .909 82.727
三、相关系数的显著性检验
为直观地判断两个变量之间的关系，可在直角坐标系中
把每对（Xi,Yi）值所代表的点绘出来，形成散点图。例如12
名男青年身高与前臂长资料绘制的散点图如图所示：
52 50
48
前臂长
46
44
42
40 150
160
170
180
190
身高
若一个变量X由小到大（或由大到小），另
一变量Y亦相应地由小到大或由大到小，则两个
tr 0.8012 0 1 0.8012 11 2
2
4.017
ν =11-2=9 3. 查界值表，得统计结论查 t 界值表，得 t0.005(9) 3.690， t r t 0.005(9) ，P ＜ 0.005，结果与查 r 界值表一致。
四、进行线性相关分析的注意事项
变量的散点图呈直线趋势，称“共变”，也就是这两个变量之间有“相关关系”。男青年身高与前臂长散点呈直线趋势，即男

ACCAchap4

4
1. Correlation
1.2 Positive and Negative correlation Positive correlation means that low values of one variable are associated with low values of the other, and high values of one variable are associated with high values of the other. Negative correlation means that low values of one variable are associated with high values of the other, and high values of one variable with low values of the other.
7
8
Correlation in a time series Sales of product A between 20X7 and 20Y1 were as follows.
Required Determine whether there is a trend in sales. In other words, decide whether there is any correlation between the year and the number of units sold.
10
2. The correlation coefficient and the coefficient of determination
2.1 The correlation coefficient o Unless the correlation coefficient r is exactly +1, 0 or –1, its meaning or significance is a little unclear. o For example, if the correlation coefficient for two variables is 0.9, this would tell us that the variables are positively correlated, but the correlation is not perfect. It would not really tell us much else. o We calculate instead the square of the correlation coefficient ( r ), which is called the coefficient of determination (r2).