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

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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|Notation

We employ lowercase letters $x, s, v, \cdots$ in order to describe variables that can lie in specific domains such as real numbers $\mathbb{R}$, real positive $\mathbb{R}^{+}$, integers $\mathbb{Z}$, closed intervals $[\cdot,$, , open intervals (,$\left.\cdot\right)$, and so on. Often, the problems studied involve multiple variables that can be regrouped in arrays. A 1-D array or vector containing scalars is represented as
$$\mathbf{x}=\left[\begin{array}{c} x_{1} \ x_{2} \ \vdots \ x_{n} \end{array}\right]$$
By convention, a vector $\mathbf{x}$ implicitly refers to a $n \times 1$ column vector. For example, if each element $x_{i} \equiv[\mathbf{x}]{i}$ is a real number $[\mathbf{x}]{i} \in \mathbb{R}$ for all $i$ from 1 to $n$, then the vector belongs to the $n$-dimensional real domain $\mathbb{R}^{n}$. This last statement can be expressed mathematically as $[\mathbf{x}]{i} \in \mathbb{R}, \forall i \in{1: n} \rightarrow \mathbf{x} \in \mathbb{R}^{n}$. In machine learning, it is common to have 2-D arrays or matrices, $$\mathbf{X}=\left[\begin{array}{cccc} x{11} & x_{12} & \cdots & x_{1 n} \ x_{21} & x_{22} & \cdots & x_{2 n} \ \vdots & \vdots & \ddots & \vdots \ x_{m 1} & x_{m 2} & \cdots & x_{m n} \end{array}\right]$$
where, for example, if each $x_{i j} \equiv[\mathbf{X}]_{i j} \in \mathbb{R}, \forall i \in{1: m}, j \in{1:$ $n} \rightarrow \mathbf{X} \in \mathbb{R}^{m \times n}$. Arrays beyond two dimensions are referred to as tensors. Although tensors are widely employed in the field of neural networks, they will not be treated in this book.

There are several matrices with specific properties: A diagona. matrix is square and has only terms on its main diagonal.
$$\mathbf{Y}=\operatorname{diag}(\mathbf{x})=\left[\begin{array}{cccc} x_{1} & 0 & \cdots & 0 \ 0 & x_{2} & \cdots & 0 \ \vdots & \vdots & \ddots & \vdots \ 0 & 0 & \cdots & x_{n} \end{array}\right]{n \times n}$$ An identity matrix $\mathbf{I}$ is similar to a diagonal matrix except that elements on the main diagonal are 1 , and 0 everywhere else, $$\mathbf{I}=\left[\begin{array}{cccc} 1 & 0 & \cdots & 0 \ 0 & 1 & \cdots & 0 \ \vdots & \vdots & \ddots & \vdots \ 0 & 0 & \cdots & 1 \end{array}\right]{n \times n}$$
A block diagonal matrix concatenates several matrices on the ma diagonal of a single matrix,
$$\operatorname{blkdiag}(\mathbf{A}, \mathbf{B})=\left[\begin{array}{lr} \mathbf{A} & \mathbf{0} \ \mathbf{0} & \mathbf{B} \end{array}\right] \text {. }$$
We can manipulate the dimensions of matrices using the transposition operation so that indices are permuted $\left[\mathbf{X}^{\top}\right]{i j}=[\mathbf{X}]{j i}$. For example,
$$\mathbf{X}=\left[\begin{array}{lll} x_{11} & x_{12} & x_{13} \ x_{21} & x_{22} & x_{23} \end{array}\right] \rightarrow \mathbf{X}^{\boldsymbol{\top}}=\left[\begin{array}{ll} x_{11} & x_{21} \ x_{12} & x_{22} \ x_{13} & x_{23} \end{array}\right]$$
The trace of a square matrix $\mathbf{X}$ corresponds to the sum of the elements on its main diagonal,
$$\operatorname{tr}(\mathbf{X})=\sum_{i=1}^{n} x_{i i}$$

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

In the context of machine learning, linear algebra is employed because of its capacity to model linear systems of equations in a format that is compact and well suited for computer calculations. In a 1-D case, such as the one represented in figure 2.1, the $x$ space is mapped into the $y$ space, $\mathbb{R} \rightarrow \mathbb{R}$, through a linear (i.e., affine) function. Figure $2.2$ presents an example of a 2-D linear function where the $\mathbf{x}$ space is mapped into the $y$ space, $\mathbb{R}^{2} \rightarrow \mathbb{R}$. This can be generalized to linear systems $\mathbf{y}=\mathbf{A x}+\mathbf{b}$, defining a mapping so that $\mathbb{R}^{n} \rightarrow \mathbb{R}^{m}$, where $\mathbf{x}$ and $\mathbf{y}$ are respectively $n \times 1$ and $m \times 1$

vectors. The product of the matrix $\mathbf{A}$ with the vector $\mathbf{x}$ is defined as $[\mathbf{A x}]{i}=\sum{j}[\mathbf{A}]{i j} \cdot[\mathbf{x}]{j}$.

In more general cases, linear algebra is employed to multiply a matrix A of size $n \times k$ with another matrix $\mathbf{B}$ of size $k \times m$, so the result is a $n \times m$ matrix,
\begin{aligned} \mathbf{C} &=\mathbf{A B} \ &=\mathbf{A} \times \mathbf{B} \end{aligned}
The matrix multiplication operation follows $[\mathbf{C}]{i j}=\sum{k}[\mathbf{A}]{i k} \cdot[\mathbf{B}]{k j}$, as illustrated in figure $2.3$. Following the requirement on the size of the matrices multiplied, this operation is not generally commutative so that $\mathbf{A B} \neq \mathbf{B A}$. Matrix multiplication follows several properties such as the following:
Distributivity
\begin{aligned} \mathbf{A}(\mathbf{B}+\mathbf{C}) &=\mathbf{A B}+\mathbf{A} \mathbf{C} \ \mathbf{A}(\mathbf{B} \mathbf{C}) &=(\mathbf{A B}) \mathbf{C} \(\mathbf{A B})^{\top} &=\mathbf{B}^{\boldsymbol{\top}} \mathbf{A}^{\top} . \end{aligned}
When the matrix multiplication operator is applied to $n \times 1$ vectors, it reduces to the inner product,
\begin{aligned} \mathbf{x}^{\boldsymbol{\top}} \mathbf{y} & \equiv \mathbf{x} \cdot \mathbf{y} \ &=\left[\begin{array}{lll} x_{1} & \cdots & x_{n} \end{array}\right] \times\left[\begin{array}{c} y_{1} \ \vdots \ y_{n} \end{array}\right] \ &=\sum_{\mathbf{i}=1}^{n} x_{i} y_{k} \end{aligned}
Another common operation is the Hadamar product or elementwise product, which is represented by the symbol $\odot$. It consists in multiplying each term from matrices $\mathbf{A}{m \times n}$ and $\mathbf{B}{m \times n}$ in order to obtain $\mathbf{C}{m \times n}$, \begin{aligned} \mathbf{C} &=\mathbf{A} \odot \mathbf{B} \ {[\mathbf{C}]{i j} } &=[\mathbf{A}]{i j} \cdot[\mathbf{B}]{i j} \end{aligned}

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

Norms measure how large a vector is. In a generic way, the $L^{\rho}$-norm is defined as
$$|\mathbf{x}|_{p}=\left(\sum_{i}\left|[\mathbf{x}]{i}\right|^{p}\right)^{1 / p}$$ Special cases of interest are $$\begin{array}{ll} |\mathbf{x}|{2}=\sqrt{\sum_{i}[\mathbf{x}]{i}^{2}} \equiv \sqrt{\mathbf{x}^{\top} \mathbf{x}} & \text { (Euclidian norm) } \ |\mathbf{x}|{1}=\sum_{i}\left|[\mathbf{x}]{i}\right| & \text { (Manhattan norm) } \ |\mathbf{x}|{\infty}=\max {i}\left|[\mathbf{x}]{i}\right| . & (\text { Max norm) } \end{array}$$

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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。