2. 简单线性回归

The simple linear regression model consists of the mean function and the variance function.
简单线性回归是由均值函数和方差函数组成的。

公式2.1

The parameters in the mean function are the intercept β0, which is the value of E(Y |X = x) when x equals zero, and the slope β1, which is the rate of change in E(Y |X = x) for a unit change in X; see Figure 2.1. By varying the parameters, we can get all possible straight lines. In most applications, parameters are unknown and must be estimated using data. The variance function in (2.1) is assumed to be constant, with a positive value σ^2 that is usually unknown.

在均值函数中的参数是截距β0和斜率β1。根据公式2.1，我们可以通过改变参数，得到所有可能的直线。在大多数应用中，参数都是未知的，我们必须通过数据来进行估计（参数估计）。方差函数被假设是常数（未知且且是正值）。

Because the variance σ^2 > 0, the observed value of the ith response yi will typically not equal its expected value E(Y |X = xi). To account for this difference between the observed data and the expected value, statisticians have invented a quantity called a statistical error, or ei, for case i defined implicitly by the equation yi = E(Y |X = xi) + ei or explicitly by ei = yi − E(Y |X = xi).The errors ei depend on unknown parameters in the mean function and so are not observable quantities. They are random variables and correspond to the vertical distance between the point yi and the mean function E(Y |X = xi). In the heights data, page 2, the errors are the differences between the heights of particular daughters and the average height of all daughters with mothers of a given fixed height。
注：这里的observed data指的是实际值，所以后面直接用实际值来代替。

因为方差 σ^2 > 0，所以第i个响应的观测值yi通常不等于期望。考虑到实际值和期望值之间的差异，统计学家提出了统计误差的概念。ei = yi − E(Y |X = xi)。误差ei取决于均值函数中的未知参数，因此是不可观察的变量。他们是随机变量，并且对应的是点yi和均值函数之间的垂直距离。在第2页的高度数据中，误差是女儿的高度(yi)与具有给定高度(xi)的母亲的所有女儿的平均身高之间即 E(Y |X = xi)的差异。

If the assumed mean function is incorrect, then the difference between the
observed data and the incorrect mean function will have a non random component,
as illustrated in Figure 2.2

We make two important assumptions concerning the errors. First, we assume
that E(ei|xi) = 0, so if we could draw a scatterplot of the ei versus the xi, we
would have a null scatterplot, with no patterns. The second assumption is that the
errors are all independent, meaning that the value of the error for one case gives no information about the value of the error for another case. This is likely to be
true in the examples in Chapter 1, although this assumption will not hold in all
problems.