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# 计算机代写|机器学习代写Machine Learning代考|Ridge Regression

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## 计算机代写|机器学习代写Machine Learning代考|Ridge Regression

Ridge Regression attempts to dampen the coefficients of a linear regression fit. Ridge regression adds an extra component to the error or loss function. The objective of the optimization function becomes
\begin{aligned} L & =\sum_i^N\left(y^{(i)}-\theta_0-\sum_{j=1}^n \theta_i x_j^{(i)}\right)^2+\lambda \sum_{j=1}^n \theta_j^2 \ & =L S E+\lambda \sum_{j=1}^n \theta_j^2 \end{aligned}
where LSE is the Least Squared Error we have discussed before. The term $\lambda \sum_{j=1}^n \theta_j^2$ is called the shrinkage penalty. Since it uses the squares of the coefficients, it is also called an $L_2$-loss. Such an optimization objective has the effect of shrinking the sizes of the coefficients toward 0 , assuming $\lambda$ is positive. The values of the $\theta_i$ s can be positive or negative.

The constant $\lambda$ is a tuning parameter that controls or regulates the amount of shrinkage of the regression coefficients. For example, if $\lambda=0$, there is no shrinkage; it is simply the linear least squares regression. If $\lambda \rightarrow \infty$, the values of the $\theta_j$ s can be made arbitrarily small. Thus, the fitted equations that come out of Ridge Regression are not unique and depend on the value of $\lambda$, and it is important to use good values of $\lambda$. This is usually done using the approach called cross-validation.

The parameter $\theta_0$ is the intercept on the Y-axis of the model. If $\theta_0$ is shrunk or penalized, we may need or force the model to always have low intercept. Thus, it is recommended that $\theta_0$ be estimated separately. A recommended values for the intercept is
$$\theta_0=\bar{y}=\frac{1}{N} \sum_{i=1}^N y^{(i)} .$$

## 计算机代写|机器学习代写Machine Learning代考|Lasso Regression

In Ridge Regression, we see that as the value of the hyperparameter $\lambda$ becomes bigger, the parameters $\theta_0 \cdots \theta_n$ become smaller and smaller, but the way it has been designed the parameters never become 0 although they could become quite small. Thus, all the features are likely to matter in the final regression equation we come up with no matter what value of $\lambda$ we choose, although some may become marginal if the corresponding coefficient becomes really small in absolute value. The fact that all independent variables remain at the end, may not matter much in this example with three independent variables or features, but if we have a lot of features to begin with, say tens or hundreds or even more, Ridge Regression will keep them even though the coefficients may become tiny. Making some of the coefficients exactly 0 so that they do not matter at all in the final equation may improve interpretability of the regression equation since the number of variables will become smaller. For example, if there were 50 independent variables to begin with and we are left with only 10 at the end, it becomes much easier to visualize or understand the relationships between the ten independent variables and the dependent variable. Lasso Regression has been designed to achieve this type of reduction in the number of independent variables that matter as $\lambda$ becomes bigger. The effect of removing independent variables from consideration is called feature selection.

Lasso Regression is quite similar to Ridge Regression in formulation, but instead of an $L_2$-loss in Ridge, it uses absolute value regression.
$$L=\sum_i^N\left(y^{(i)}-\theta_0-\sum_{j=1}^n \theta_i x_j^{(i)}\right)^2+\lambda \sum_{j=1}^n\left|\theta_j\right|$$

## 计算机代写|机器学习代写Machine Learning代考|Ridge Regression

$$L=\sum_i^N\left(y^{(i)}-\theta_0-\sum_{j=1}^n \theta_i x_j^{(i)}\right)^2+\lambda \sum_{j=1}^n \theta_j^2 \quad=L S E+\lambda \sum_{j=1}^n \theta_j^2$$

$$\theta_0=\bar{y}=\frac{1}{N} \sum_{i=1}^N y^{(i)}$$

## 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。