19th Ave New York, NY 95822, USA

# 计算机代写|机器学习代写Machine Learning代考|Adjusted $R^2$

my-assignmentexpert™提供最专业的一站式服务：Essay代写，Dissertation代写，Assignment代写，Paper代写，Proposal代写，Proposal代写，Literature Review代写，Online Course，Exam代考等等。my-assignmentexpert™专注为留学生提供Essay代写服务，拥有各个专业的博硕教师团队帮您代写，免费修改及辅导，保证成果完成的效率和质量。同时有多家检测平台帐号，包括Turnitin高级账户，检测论文不会留痕，写好后检测修改，放心可靠，经得起任何考验！

## 计算机代写|机器学习代写Machine Learning代考|Adjusted $R^2$

The $R^2$ metric is simple and has many associated problems as a metric for goodness of fit of a regression model. One complaint about $R^2$ is that it has been observed as the number of independent variables (also called explanatory variables in statistics, we call them features in machine learning) increases, the value of $R^2$ increases, possibly weakly. To take this into account, a metric called Adjusted $R^2$ or $R_{a d j}^2$ is also often computed. To motivate $R_{a d j}^2$, we revisit the formula for $R^2$ and write it out a bit differently as follows:
\begin{aligned} R^2 & =1-\frac{S S E}{S S T} \ & =1-\frac{\frac{1}{N} S S E}{\frac{1}{N} S S T} \ & =1-\frac{V A R_{S S E}}{V A R_{S S T}} . \end{aligned}
Here, $\frac{1}{N} S S E=\frac{1}{N} \sum_{i=1}^n\left(y^{(i)}-\hat{y}^{(i)}\right)^2$ is the standard definition of variance of the residuals or errors, which we can call $V A R_{S S E}$, also written as $s_{S S E}^2$. We consider all $N$ residuals in computing $V A R_{S S E}$. Also, $\frac{1}{N} S S T=$ $\frac{1}{N} \sum_{i=1}^n\left(y^{(i)}-\bar{y}\right)^2=V A R_{S S T}$ where $V A R_{S S T}$ (also written as $s_{S S T}^2$ ) is the variance of the total error terms or errors from the baseline or horizontal mean $y$ model. There are two variances in the formula for $R^2$ above. To write an improved version of $R^2$, which we call $R_{a d j}^2$, we bring in the degrees of freedom of these variances, instead of simply writing $N$ in the denominator in the variance definitions.
$$R_{a d j}^2=1-\frac{\frac{V A R_{S S E}}{d f_e}}{\frac{V A R_{S S T}}{d f_t}}$$

where $d f_e=N-1$ is the number of degrees of freedom when computing variance of $N$ things, and $d f_t=N-q-1$ is the number of degrees of freedom when computing the error variance with respect to a regression model (here, linear), where $q$ is the number of explanatory (or independent) variables, which is actually the number of features in our parlance. The degrees of freedom for the error terms with respect to a regression model must take into account the number of parameters used by the model. For the linear regression model in this case, there is one independent variable $x$. Thus, $q=1$. As a result, the formula for $R_{a d j}^2$ for LSRL is:
$$R_{a d j}^2=1-\frac{\frac{V A R_{S S E}}{N-1}}{\frac{V A R_{S S T}}{N-2}} .$$
Obviously, if $N$ is large, having $N$ in the denominator or $N-1$ or $N-2$ does not make much difference. But, if $N$ is small, there may be a visible difference between the values of $R^2$ and $R_{a d j}^2$.

In our example, $\mathrm{R}$ computes $R_{a d j}^2$ as 0.8534 , which is smaller than $R^2$ whose value is 0.8778 .

## 计算机代写|机器学习代写Machine Learning代考|F-Statistic

The F-Statistic is another value that is commonly computed to gauge the quality of a regression fit. We will first discuss how F-statistic is computed, and then how it is used. Earlier, we defined the Sum of Squared Errors, SSE as
$$S S E=\sum_{i=1}^N\left(y^{(i)}-\hat{y}^{(i)}\right)^2 .$$
We can compute the Mean Squared Error, MSE as
$$M S E=\frac{S S E}{N-p}$$
where $p$ is the number of parameters, which are $\theta_1$ and $\theta_0$. Thus, $p=q+1$, where $p$ is the number of independent variables or features. $p=2$ for linear regression in our example. Then, for linear regression,
$$M S E=\frac{S S E}{N-2},$$
and we say that it has a degree of freedom of $N-2$. This definition reflects the fact that to begin with we have $N$ variables or degrees of freedom in the description of the data. Since there are two coefficients, two degrees of freedom are covered by the linear regression line. We can define a quantity called Sum of Squares Regression, SSR, which is the sum of squares of errors for each predicted $y$ value from the baseline “worst” model, as follows:
$$S S R=\sum_{i=1}^n\left(\hat{y}^{(i)}-\bar{y}\right)^2 .$$

## 机器学习代写

alsocalledexplanatoryvariablesinstatistics, wecallthemfeaturesinmachinelearning增加，价值 $R^2$ 增加，可能溦弱。考虑到这一点，一 个名为 Adjusted 的指标 $R^2$ 或者 $R_{a d j}^2$ 也经常被计算。激励 $R_{a d j}^2$ ，我们重新审视公式 $R^2$ 并以不同的方式写出来，如下所示:
$$R^2=1-\frac{S S E}{S S T} \quad=1-\frac{\frac{1}{N} S S E}{\frac{1}{N} S S T}=1-\frac{V A R_{S S E}}{V A R_{S S T}} .$$

$$S S R=\sum_{i=1}^n\left(\hat{y}^{(i)}-\bar{y}\right)^2$$

## 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。