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chapter 3: Linear Methods for Regression
第3章:回归的线性方法
3.1 Introduction
A linear regression model assumes that the regression function
E(Y∣X) is linear in the inputsX1,…,Xp . Linear models were largely developed in the precomputer age of statistics, but even in today’s computer era there are still good reasons to study and use them. They are simple and often provide an adequate and interpretable description of how the inputs affect the output. For prediction purposes they can sometimes outperform fancier nonlinear models, especially in situations with small numbers of training cases, low signal-to-noise ratio or sparse data. Finally, linear methods can be applied to transformations of the inputs and this considerably expands their scope. These generalizations are sometimes called basis-function methods, and are discussed in Chapter 5.
线性回归模型假设输入为
In this chapter we describe linear methods for regression, while in the next chapter we discuss linear methods for classification. On some topics we go into considerable detail, as it is our firm belief that an understanding of linear methods is essential for understanding nonlinear ones. In fact, many nonlinear techniques are direct generalizations of the linear methods discussed here.
在这一章中我们描述回归的线性方法,下一章我们将讨论分类的线性方法。在某些主题上我们将从细节上讨论,因为理解线性方法对理解非线性至关重要是我们坚定的信仰。事实上,许多非线性技巧是这里讨论的线性方法的直接概括。
3.2 Linear Regression Models and Least Squares
3.2 线性回归模型和最小二乘法
As introduced in Chapter 2, we have an input vector
XT=(X1,X2,…,Xp) , and want to predict a real-valued outputY . The linear regression model has the form
f(X)=β_0+∑_j=1pX_jβ_j(3.1)
The linear model either assumes that the regression functionE(Y∣X) is linear, or that the linear model is a reasonable approximation. Here theβj ’s are unknown parameters or coefficients, and the variablesXj can come from different sources:
- quantitative inputs;
- transformations of quantitative inputs, such as log, square-root or square;
- basis expansions, such asX_2=X_12,X_3=X_13 , leading to a polynomial representation;
- numeric or “dummy” coding of the levels of qualitative inputs. For example, ifG is a five-level factor input, we might createX_j,j=1,…,5 , such thatX_j=I(G=j) . Together this group ofX_j represents the effect ofG by a set of level-dependent constants, since in∑_j=15X_jβ_j , one of theX_j s is one, and the others are zero.
- interactions between variables, for example,X_3=X_1⋅X_2
正如第二章介绍的那样,我们有输入向量
线性模型要么假设回归函数
- 定量的输入
- 定量输入的变换,比如对数,平方根或者平方
- 基函数展开,比如
- 定性输入变量水平的数值或“虚拟”编码。举个例子,如果
- 变量之间的相交,举个例子,
无论
一般地,我们有一系列用来估计参数
从统计学的观点来看,如果训练观测值
我们怎样最小化(3.2)记
这是含
假设
得到唯一解
在输入向量
其中,
对向量求导的问题
图3.2展示了在
可能会出现
截至目前我们已经对数据的真实分布做了很少的假设。为了约束
经常通过下式来估计方差
分母是
为了对参数和模型进行推断,需要一些额外的假设。我们现在假设式
其中误差
由式(3.9),可以很简单地证明
这是一个有上述均值向量和方差-协方差矩阵的多变量正态分布。同时有
是一个自由度为
为了检验系数
其中
我们经常需要同时检验系数集体的显著性。举个例子,检验有
其中
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