### 计算机代写|机器学习代写machine learning代考|COMP30027

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 计算机代写|机器学习代写machine learning代考|Evaluating Regression Models

When developing the earlier linear models, we were somewhat imprecise about what is meant by a ‘line of best fit’ (or generally a model of best fit). Indeed, the pseudoinverse is not a ‘solution’ to the system of equations given in Equation (2.8), but is merely an approximation (naturally, the line of best fit does not pass through all points exactly).

Here, we would like to be more precise about what it means for a model to be ‘good.’ This is a key issue when fitting and evaluating any machine learning model: one needs a way of quantifying how closely a model fits the given data. Given a desired measure of success, we can compare alternative models against this measure and design optimization schemes that optimize the desired measure directly.

A commonly used evaluation criterion when evaluating regression algorithms is called the mean squared error, or MSE. The MSE between a model $f_\theta(X)$ and a set of labels $y$ is defined as
$$\operatorname{MSE}\left(y, f_\theta(X)\right)=\frac{1}{|y|} \sum_{i=1}^{|y|}\left(f_\theta\left(x_i\right)-y_i\right)^2,$$
in other words, the average squared difference between the model’s predictions and the labels. Often reported is also the root mean squared error (RMSE), that is, $\sqrt{\operatorname{MSE}\left(y, f_\theta(X)\right)}$; the RMSE is sometimes preferable as it is consistent in scale with the original labels.

With some effort, it can be shown that the linear model $f_\theta(X)$ that minimizes the MSE compared to the labels $y$ is given by using the pseudoinverse as in Equation (2.10). We leave this as an exercise (Exercise 2.6).

## 计算机代写|机器学习代写machine learning代考|Why the Mean Squared Error

Although the MSE has a convenient relationship with the pseudoinverse, it may otherwise seem a somewhat arbitrary choice of error measure. For instance, it may seem more obvious at first to compute an error measure such as the mean absolute error (or MAF):
$$\operatorname{MAE}\left(y, f_\theta(X)\right)=\frac{1}{|y|} \sum_{i=1}^{|y|}\left|f_\theta\left(x_i\right)-y_i\right| \text {. }$$
Or, why not count the number of times the model is wrong by more than one star? For that matter, why not measure the mean cubed error?

To defend the MSE as a reasonable choice, we need to characterize what types of errors are more ‘likely’ than others. Essentially, the MSE assigns very small penalties to small errors and very large penalties to large errors. This is in contrast to, say, the MAE, which assigns penalties precisely in proportion to how large the error is. What the MSE therefore seems to be assuming is that small errors are common and large errors are particularly uncommon.

What we are talking about informally here is a notion of how errors are distributed under some model. Formally, we say that the labels are equal to our model’s predictions, plus some error:
$$y=\underbrace{f_\theta(X)}{\text {prediction }}+\underbrace{\epsilon}{\text {error }},$$
and that our error follows some probability distribution. Our argument here said that small errors are common and large errors are very rare. This suggests that errors may be distributed following a bell curve, which we could capture with a Gaussian (or ‘Normal’) distribution:
$$\epsilon \sim \mathcal{N}\left(0, \sigma^2\right) .$$
The density function for a (zero mean) Gaussian distribution is given by
$$f^{\prime}\left(x^{\prime}\right)=\frac{1}{\sigma \sqrt{2 \pi}} e^{-\frac{1}{2}\left(\frac{2}{\sigma}\right)^2}$$

# 机器学习代考

## 计算机代写|机器学习代写machine learning代考|Evaluating Regression Models

$$\operatorname{MSE}\left(y, f_\theta(X)\right)=\frac{1}{|y|} \sum_{i=1}^{|y|}\left(f_\theta\left(x_i\right)-y_i\right)^2,$$

## 计算机代写|机器学习代写machine learning代考|Why the Mean Squared Error

$$\operatorname{MAE}\left(y, f_\theta(X)\right)=\frac{1}{|y|} \sum_{i=1}^{|y|}\left|f_\theta\left(x_i\right)-y_i\right| .$$

$$y=\underbrace{f_\theta(X)} \text { prediction }+\underbrace{\epsilon} \text { error, }$$

$$\epsilon \sim \mathcal{N}\left(0, \sigma^2\right) .$$
(零均值) 高斯分布的密度函数由下式给出
$$f^{\prime}\left(x^{\prime}\right)=\frac{1}{\sigma \sqrt{2 \pi}} e^{-\frac{1}{2}\left(\frac{2}{\sigma}\right)^2}$$

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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。