# 线性代数网课代修|最小二乘法代写least squares method辅导|STAT7607

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• 矩阵论
• 优化理论
• 线性规划
• 逼近论

## 线性代数作业代写linear algebra代考|Obtaining the Least Squares Solution

The least squares solution is defined as the point in the “unknown parameter” space at which the objective function $S$ is minimized. Thus, if there are $p$ unknown parameters $\left(\boldsymbol{a}{k}, \boldsymbol{k}=\mathbf{1}\right.$ to $\left.\boldsymbol{p}\right)$, the solution yields the values of the $\boldsymbol{a}{k}$ ‘s that minimize $\boldsymbol{S}$. To find this minimum point we set the $\boldsymbol{p}$ partial derivatives of $S$ to zero yielding $\boldsymbol{p}$ equations for the $\boldsymbol{p}$ unknown values of $a_{k}$ :
$$\frac{\partial S}{\partial a_{k}}=0 \quad k=1 \text { to } p$$
In Section $2.2$ the following expression (Equation 2.2.3) for the objective function $\boldsymbol{S}$ was developed:
$$S=\sum_{i=1}^{i=n} w_{i}\left(Y_{i}-f\left(\mathbf{X}{i}\right)\right)^{2}$$ In this expression the independent variable $\mathbf{X}{i}$ can be either a scalar or a vector. The variable $Y_{i}$ can also be a vector but is usually a scalar. Using this expression and Equation 2.4.1, we get the following $\boldsymbol{p}$ equations:
$$-2 \sum_{i=1}^{i=n} w_{i}\left(Y_{i}-f\left(\mathbf{X}{i}\right)\right) \frac{\partial f\left(\mathbf{X}{i}\right)}{\partial a_{k}}=0 \quad k=1 \text { to } p$$
$$\sum_{i=1}^{i=n} w_{i} f\left(\mathbf{X}{i}\right) \frac{\partial f\left(\mathbf{X}{l}\right)}{\partial a_{k}}=\sum_{i=1}^{i=n} w_{i} Y_{i} \frac{\partial f\left(\mathbf{X}{i}\right)}{\partial a{k}} \quad k=1 \text { to } p$$

## 线性代数作业代写linear algebra代考|Uncertainty in the Model Parameters

In Section $2.4$ we developed the methodology for finding the set of $\boldsymbol{a}{\boldsymbol{k}}$ ‘s that minimize the objective function $S$. In this section we turn to the task of determining the uncertainties associated with the $a{k}$ ‘s. The usual measures of uncertainty are standard deviation (i.e., $\sigma$ ) or variance (i.e., $\sigma^{2}$ ) so we seek an expression that allows us to estimate the $\sigma_{a_{k}}$ ‘s. It can be shown [WO67, BA74, GA92] that the following expression gives us an unbiased estimate of $\boldsymbol{\sigma}{a{k}}$ :
\begin{aligned} \sigma_{a_{k}}^{2} &=S_{n-p} C_{k k}^{-1} \ \sigma_{a_{k}} &=\left(\frac{S}{n-p} C_{k k}^{-1}\right)^{1 / 2} \end{aligned}
We see from this equation that the unbiased estimate of $\sigma_{a_{k}}$ is related to the objective function $S$ and the $\boldsymbol{k}^{\text {th }}$ diagonal term of the inverse matrix $\boldsymbol{C}^{-1}$. The matrix $C^{-1}$ is required to find the least squares values of the $\boldsymbol{a}{k}$ ‘s and once these values have been determined, the final (i.e., minimum) value of $S$ can easily be computed. Thus the process of determining the $a{k}$ ‘s leads painlessly to a determination of the $\boldsymbol{\sigma}{a{k}}$ ‘s.

As an example, consider the data included in Table 2.3.4. In Section $2.4$ details were included for a straight-line fit to the data using unit weighting:
$$y=f(x)=a_{1}+a_{2} x=0.5786+5.5286 x$$

## 线性代数作业代写linear algebra代考|Obtaining the Least Squares Solution

$$\frac{\partial S}{\partial a_{k}}=0 \quad k=1 \text { to } p$$

$$S=\sum_{i=1}^{i=n} w_{i}\left(Y_{i}-f(\mathbf{X} i)\right)^{2}$$

$$-2 \sum_{i=1}^{i=n} w_{i}\left(Y_{i}-f(\mathbf{X} i)\right) \frac{\partial f(\mathbf{X} i)}{\partial a_{k}}=0 \quad k=1 \text { to } p$$
$$\sum_{i=1}^{i=n} w_{i} f(\mathbf{X} i) \frac{\partial f(\mathbf{X} l)}{\partial a_{k}}=\sum_{i=1}^{i=n} w_{i} Y_{i} \frac{\partial f(\mathbf{X} i)}{\partial a k} \quad k=1 \text { to } p$$

## 线性代数作业代写linear algebra代考|Uncertainty in the Model Parameters

$$\sigma_{a_{k}}^{2}=S_{n-p} C_{k k}^{-1} \sigma_{a_{k}}=\left(\frac{S}{n-p} C_{k k}^{-1}\right)^{1 / 2}$$

$$y=f(x)=a_{1}+a_{2} x=0.5786+5.5286 x$$

# 计量经济学代写

## 在这种情况下，如何学好线性代数？如何保证线性代数能获得高分呢？

1.1 mark on book

【重点的误解】划重点不是书上粗体，更不是每个定义，线代概念这么多，很多朋友强迫症似的把每个定义整整齐齐用荧光笔标出来，然后整本书都是重点，那期末怎么复习呀。我认为需要标出的重点为

A. 不懂，或是生涩，或是不熟悉的部分。这点很重要，有的定义浅显，但证明方法很奇怪。我会将晦涩的定义，证明方法标出。在看书时，所有例题将答案遮住，自己做，卡住了就说明不熟悉这个例题的方法，也标出。

B. 老师课上总结或强调的部分。这个没啥好讲的，跟着老师走就对了

C. 你自己做题过程中，发现模糊的知识点

1.2 take note

1.3 understand the relation between definitions