线性代数网课代修|最小二乘法代写least squares method辅导|STAT615

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线性代数作业代写linear algebra代考|The Objective Function

The starting point for the method of least squares is the objective function. Minimization of this function yields the least squares solution. The simplest problems are those in which $\boldsymbol{y}$ (a scalar quantity) is related to an independent variable $\boldsymbol{x}$ (or variables $\boldsymbol{x}{\boldsymbol{j}}$ ‘s) and it can be assumed that there is no (or negligible) errors in the independent variable (or variables). The objective function for these cases is: $$S=\sum{i=1}^{i=n} w_{i} R_{i}^{2}=\sum_{i=1}^{i=n} w_{i}\left(Y_{i}-y_{i}\right)^{2}=\sum_{i=1}^{i=n} w_{i}\left(Y_{i}-f\left(\mathrm{X}{i}\right)\right)^{2}$$ In this equation $\boldsymbol{n}$ is the number of data points, $Y{i}$ is the $\boldsymbol{i}^{\text {th }}$ input value of the dependent variable and $y_{i}$ is the $i^{\text {th }}$ computed value of the dependent variable. The variable $\boldsymbol{R}{i}$ is called the $\boldsymbol{i}^{\boldsymbol{t h}}$ residual and is the difference between the input and computed values of $\boldsymbol{y}$ for the $i^{\text {th }}$ data point. The variable $\mathbf{X}{i}$ (unitalicized) represents the independent variables and is either a scalar if there is only one independent variable or a vector if there is more than one independent variable. The function $f$ is the equation used to express the relationship between $\mathbf{X}$ and $\boldsymbol{y}$. The variable $\boldsymbol{w}{i}$ is called the “weight” associated with the $i^{\text {th }}$ data point and is discussed in the next section. A schematic diagram of the variables for point $i$ is shown in Figure 2.2.1. In this diagram there is only a single independent variable so the notation $x$ is used instead of $\mathbf{X}$. The variable $\boldsymbol{E}{i}$ is the true but unknown error in the $\boldsymbol{i}^{\text {th }}$ value of $\boldsymbol{y}$. Note that neither the value of $\boldsymbol{Y}{i}$ nor $\boldsymbol{y}{i}$ is exactly equal to the unknown $\eta_{i}$ (the true value of $y$ ) at this value of $\boldsymbol{x}{\boldsymbol{i}}$. However, a fundamental assumption of the method of least squares is that if $\boldsymbol{Y}{\boldsymbol{i}}$ is determined many times, the average value would approach this true value.

线性代数作业代写linear algebra代考|Data Weighting

In Section 2.2, we noted that regardless of the choice of the objective function, a weight $w_{i}$ is specified for each point. The “weight” associated with a point is based upon the relative uncertainties associated with the different points. Clearly, we must place greater weight upon points that have smaller uncertainties, and less weight upon the points that have greater uncertainties. In other words the weight $\boldsymbol{w}{i}$ must be related in some way to the uncertainties $\boldsymbol{\sigma}{y_{i}}$ and $\boldsymbol{\sigma}{x{i}}$.

The alternative to using $\boldsymbol{w}{i}^{\prime}$ ‘s associated with the $\boldsymbol{\sigma}$ ‘s of the $i^{\text {th }}$ data point is to simply use unit weighting (i.e., $\boldsymbol{w}{\boldsymbol{i}}=1$ ) for all points. This is a reasonable have no idea regarding the values (actual or even relative) of $\boldsymbol{\sigma}$ for the different points. However, when the differences in the $\boldsymbol{\sigma}$ ‘s are significant, then use of unit weighting can lead to poor results. This point is illustrated in Figure 2.3.1. In this example, we fit a straight line to a set of data. Note that the line obtained when all points are equally weighted is very different than the line obtained when the points are “weighted” properly. Also note how far the unit weighting line is from the first few points.

线性代数作业代写linear algebra代考|The Objective Function

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

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