## 数学代写|基础数据分析代写Elementary data Analysis代考|ST309

2022年10月11日

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## 数学代写|基础数据分析代写Elementary data Analysis代考|Estimating the Optimal Linear Predictor

To actually estimate $\beta$ from data, we need to make some probabilistic assumptions about where the data comes from. A fairly weak but often sufficient assumption is that observations $\left(\vec{X}_i, Y_i\right)$ are independent for different values of $i$, with unchanging covariances. Then if we look at the sample covariances, they will, by the law of large numbers, converge on the true covariances:
\begin{aligned} &\frac{1}{n} \mathbf{X}^T \mathbf{Y} \rightarrow \operatorname{Cov}[\vec{X}, Y] \ &\frac{1}{n} \mathbf{X}^T \mathbf{X} \rightarrow \mathbf{v} \end{aligned}
where as before $\mathbf{X}$ is the data-frame matrix with one row for each data point and one column for each variable, and similarly for $\mathbf{Y}$.
So, by continuity,
$$\hat{\beta}=\left(\mathbf{X}^T \mathbf{X}\right)^{-1} \mathbf{X}^T \mathbf{Y} \rightarrow \beta$$
and we have a consistent estimator.

On the other hand, we could start with the residual sum of squares
$$\operatorname{RSS}(\beta) \equiv \sum_{i=1}^n\left(y_i-\vec{x}_i \cdot \beta\right)^2$$
and try to minimize it. The minimizer is the same $\hat{\beta}$ we got by plugging in the sample covariances. No probabilistic assumption is needed to minimize the RSS, but it doesn’t let us say anything about the convergence of $\hat{\beta}$. For that, we do need some assumptions about $\vec{X}$ and $Y$ coming from distributions with unchanging covariances.
(One can also show that the least-squares estimate is the linear predictor with the minimax prediction risk. That is, its worst-case performance, when everything goes wrong and the data are horrible, will be better than any other linear method. This is some comfort, especially if you have a gloomy and pessimistic view of data, but other methods of estimation may work better in less-than-worst-case scenarios.)

## 数学代写|基础数据分析代写Elementary data Analysis代考|Omitted Variables and Shifting Distributions

That the optimal regression coefficients can change with the distribution of the predictor features is annoying, but one could after all notice that the distribution has shifted, and so be cautious about relying on the old regression. More subtle is that the regression coefficients can depend on variables which you do not measure, and those can shift without your noticing anything.
Mathematically, the issue is that
$$\mathbb{E}[Y \mid \vec{X}]=\mathbb{E}[\mathbb{E}[Y \mid Z, \vec{X}] \mid \vec{X}]$$
Now, if $Y$ is independent of $Z$ given $\vec{X}$, then the extra conditioning in the inner expectation does nothing and changing $Z$ doesn’t alter our predictions. But in general there will be plenty of variables $Z$ which we don’t measure (so they’re not included in $\vec{X}$ ) but which have some non-redundant information about the response (so that $Y$ depends on $Z$ even conditional on $\vec{X}$ ). If the distribution of $\vec{X}$ given $Z$ changes, then the optimal regression of $Y$ on $\vec{X}$ should change too.

Here’s an example. $X$ and $Z$ are both $\mathscr{N}(0,1)$, but with a positive correlation of $0.1$. In reality, $Y \sim \mathscr{N}(X+Z, 0.01)$. Figure $2.2$ shows a scatterplot of all three variables together $(n=100)$.

Now I change the correlation between $X$ and $Z$ to $-0.1$. This leaves both marginal distributions alone, and is barely detectable by eye (Figure 2.3).

Figure $2.4$ shows just the $X$ and $Y$ values from the two data sets, in black for the points with a positive correlation between $X$ and $Z$, and in blue when the correlation is negative. Looking by eye at the points and at the axis tick-marks, one sees that, as promised, there is very little change in the marginal distribution of either variable. Furthermore, the correlation between $X$ and $Y$ doesn’t change much, going only from $0.74$ to $0.63$. On the other hand, the regression lines are noticeably different. When $\operatorname{Cov}[X, Z]=0.1$, the slope of the regression line is $0.96$ – high values for $X$ tend to indicate high values for $Z$, which also increases $Y$. When $\operatorname{Cov}[X, Z]=-0.1$, the slope of the regression line is $0.84$, because now extreme values of $X$ are signs that $Z$ is at the opposite extreme, bringing $Y$ closer back to its mean. But, to repeat, the difference here is due to a change in the correlation between $X$ and $Z$, not how those variables themselves relate to $Y$. If I regress $Y$ on $X$ and $Z$, I get $\hat{\beta}=1,1$ in the first case and $\widehat{\beta}=1,1$ in the second.

# 基础数据分析代考

## 数学代写|基础数据分析代写基本数据分析代考|估计最优线性预测器

\begin{aligned} &\frac{1}{n} \mathbf{X}^T \mathbf{Y} \rightarrow \operatorname{Cov}[\vec{X}, Y] \ &\frac{1}{n} \mathbf{X}^T \mathbf{X} \rightarrow \mathbf{v} \end{aligned}
where as before $\mathbf{X}$ 数据框架矩阵是否对每个数据点有一行，对每个变量有一列 $\mathbf{Y}$

$$\hat{\beta}=\left(\mathbf{X}^T \mathbf{X}\right)^{-1} \mathbf{X}^T \mathbf{Y} \rightarrow \beta$$

$$\operatorname{RSS}(\beta) \equiv \sum_{i=1}^n\left(y_i-\vec{x}_i \cdot \beta\right)^2$$

## 数学代写|基础数据分析代写基本数据分析代考|省略变量和移动分布

.

$$\mathbb{E}[Y \mid \vec{X}]=\mathbb{E}[\mathbb{E}[Y \mid Z, \vec{X}] \mid \vec{X}]$$

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## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。