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# 计算机代写|机器学习代写Machine Learning代考|Is the Model Statistically Significant? F-Test

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## 计算机代写|机器学习代写Machine Learning代考|Is the Model Statistically Significant? F-Test

It is customary to compute what is called a $p$-value when we perform a statistical test to indicate if the results obtained are statistically significant, i.e., they have any (statistical) merit. That is, whether the results can be trusted and are meaningful, and can really be used. The $p$-values are computed in different ways for different situations.

In addition, when we perform a statistical test, there is always an associated null hypothesis and an alternate hypothesis. For linear regression, the null hypothesis is that the coefficients associated with the independent variables (or, features) is 0 , i.e., the regression coefficients, except the constant or intercept, are all 0 . In other words, the null hypothesis says there is no relationship of any significance between the dependent and independent variables. The alternate hypothesis is that the coefficients are not equal to 0 , i.e., there is actually a linear relationship between the dependent variable and the independent variable(s). In particular, for linear regression with one independent variable, the null hypothesis would be that $\theta_1=0$. The alternate hypothesis is that $\theta_1 \neq 0$. We have to be able to reject the null hypothesis and accept the alternate hypothesis to accept the results obtained for linear regression by a system like R. We can reject the null hypothesis by performing what is called the F-test.

## 计算机代写|机器学习代写Machine Learning代考|The Role of Optimization in Regression

An astute reader may have noticed that we have to solve an optimization problem to obtain the least squared error line of fit, given a dataset with an independent variable and an associated dependent variable. This is not unsual at all in machine learning. A machine learning algorithm attempts to obtain a good model to that fits the data, keeping in mind that the purpose is to build a model that learns the essence of the particulars of the training dataset, but also focuses on generalizing to examples not in the training dataset. This process usually or always involves an underlying optimization algorithm. In fitting an LSRL, we used a simple optimization algorithm that we learn in calculus. It involves obtaining partial derivatives of an expression or function with respect to its variables, setting the derivatives to 0 , obtaining a system of linear equations and solving them. In the case where we have a single independent variable, this approach results in a quick solution to the optimization problem. However, we will see that the underlying optimization problem becomes more difficult as our machine learning algorithm becomes more complex. For example, it is possible that the number of independent variables becomes large, say in the hundreds or thousands. There is always one dependent variable. In such a situation, we will have to solve a system of linear equations in a large number of variables. Solving such a system and obtaining reliable and stable results is not straightfoward. This is where, we may start approaching the optimization problem for fitting a good line or plane or hyperplane or some other curve to a training dataset using different and more sophisticated algorithms. There are many ways to solving them. This is an active area of research in mathematics. A variety of approaches such as gradient descent, QR factorization, stochastic gradient descent, semi-definite optimization, and conic optimization can be used. We will discuss one or more of these algorithms later in the book as necessary. In particular, we should note that the $1 \mathrm{~m}$ implementation in $\mathrm{R}$ does not use the approach of partial derivatives and systems of linear equations discussed here. It uses a more sophisticated approach called QR matrix decomposition to solve an over-determined system of equations. We discussed the approach we choose here to introduce the idea that optimization plays a significant and all pervasive role in machine learning, and a simple algorithm that everyone understands from calculus is a good place to start. We will discuss a few other optimization approaches later in this book, as needed.

## 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。