# 数学代写|数值分析代写numerical analysis代考|Regression and Curve Fitting

#### Doug I. Jones

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## 数学代写|数值分析代写numerical analysis代考|Regression and Curve Fitting

Regression and curve fitting are also important computational tasks in statistics. A common approach is to use least squares to fit a set of data $\left(\boldsymbol{x}i, y_i\right)$, $i=1,2, \ldots, N$, with a function $x \mapsto \sum{j=1}^m c_j \varphi_j(x)$. We assume a statistical model $y_i=\sum_{j=1}^m c_j \varphi_j\left(x_i\right)+\epsilon_i$ where $\epsilon_i$ follows a $\operatorname{Normal}\left(0, \sigma^2\right)$ distribution and the $\epsilon_i$ ‘s are mutually independent. The likelihood function is \begin{aligned} L(\boldsymbol{c}) & =\prod_{i=1}^N(2 \pi)^{-1 / 2} \exp \left(-\epsilon_i^2 / \sigma^2\right) \ & =(2 \pi)^{-N / 2} \exp \left(-\epsilon^T \epsilon / \sigma^2\right), \quad \text { so } \ \ln L(\boldsymbol{c}) & =-\frac{N}{2} \ln (2 \pi)-\frac{1}{\sigma^2} \boldsymbol{\epsilon}^T \epsilon=-\frac{N}{2} \ln (2 \pi)-\frac{1}{\sigma^2}(\Phi \boldsymbol{c}-\boldsymbol{y})^T(\Phi \boldsymbol{c}-\boldsymbol{y}) \end{aligned}
where $\Phi_{i j}=\varphi_i\left(\boldsymbol{x}_j\right)$. So maximizing $L(\boldsymbol{c})$ for $\boldsymbol{c}$ is equivalent to minimizing $(\Phi \boldsymbol{c}-\boldsymbol{y})^T(\Phi \boldsymbol{c}-\boldsymbol{y})$. That is, we are minimizing the sum of the squares of the errors. This can be done using either normal equations or the $\mathrm{QR}$ factorization. The optimal $\boldsymbol{c}$ is given by the normal equations $\Phi^T \Phi \widehat{\boldsymbol{c}}=\Phi^T \boldsymbol{y}$. If $\boldsymbol{c}$ is the true set coefficients, $\boldsymbol{y}=\Phi \boldsymbol{c}+\boldsymbol{\epsilon}$ and so $\Phi^T \Phi \widehat{\boldsymbol{c}}=\Phi^T \Phi \boldsymbol{c}+\Phi^T \boldsymbol{\epsilon}$ and therefore $\widehat{\boldsymbol{c}}=\boldsymbol{c}+\left(\Phi^T \Phi\right)^{-1} \Phi^T \boldsymbol{\epsilon}$. The error in the computed coefficients $\widehat{\boldsymbol{c}}-\boldsymbol{c}=\left(\Phi^T \Phi\right)^{-1} \Phi^T \boldsymbol{\epsilon}$ is distributed according to the $\operatorname{Normal}\left(\mathbf{0}, \sigma^2\left(\Phi^T \Phi\right)^{-1}\right)$ distribution. This estimate is unbiased: $\mathbb{E}[\widehat{\boldsymbol{c}}]=\boldsymbol{c}$. We might want to estimate the variance $\sigma^2$ of the $\epsilon_i$ ‘s by using $\widehat{\boldsymbol{\epsilon}}=\boldsymbol{y}-\Phi \widehat{\boldsymbol{c}}$. However, this will lead to a biased estimate of $\sigma^2$ :
\begin{aligned} \widehat{\boldsymbol{\epsilon}} & =\boldsymbol{y}-\Phi \widehat{\boldsymbol{c}}=\boldsymbol{y}-\Phi\left(\Phi^T \Phi\right)^{-1} \Phi^T \boldsymbol{y} \ & =\left[I-\Phi\left(\Phi^T \Phi\right)^{-1} \Phi^T\right] \boldsymbol{y} . \end{aligned}
The matrix $P:=I-\Phi\left(\Phi^T \Phi\right)^{-1} \Phi^T$ is the orthogonal projection onto the orthogonal complement of range $\Phi$. Since $\boldsymbol{y}=\Phi \boldsymbol{c}+\boldsymbol{\epsilon}$,
$$\widehat{\epsilon}=P(\Phi c+\epsilon)=P \epsilon$$
so $\widehat{\boldsymbol{\epsilon}}$ is in (range $\Phi)^{\perp}$. We can consider $\widehat{\boldsymbol{\epsilon}}$ to be distributed according to the $\operatorname{Normal}\left(\mathbf{0}, \sigma^2 P\right)$ distribution as $P^T=P=P^2=P^T P$, understood as the limit of the distribution of $\operatorname{Normal}\left(0, \sigma^2 P+\alpha I\right)$ as $\alpha \downarrow 0$. Also
$$\mathbb{E}\left[\widehat{\boldsymbol{\epsilon}}^T \widehat{\boldsymbol{\epsilon}}\right]=\mathbb{E}\left[\boldsymbol{\epsilon}^T P^T P \boldsymbol{\epsilon}\right]=\mathbb{E}\left[\boldsymbol{\epsilon}^T P \boldsymbol{\epsilon}\right]=\operatorname{trace}(P) \sigma^2$$

## 数学代写|数值分析代写numerical analysis代考|Bayesian Inference

Given a data set $D$, how likely is a hypothesis $H$ ? This is the conditional probability $\operatorname{Pr}[H \mid D]$. But this is usually very difficult to compute directly. Instead, it is much easier to compute $\operatorname{Pr}[D \mid H]$ as the hypothesis $H$ is a statement about the nature of the data $D$. From Bayes’ theorem,
$$\operatorname{Pr}[H \mid D]=\frac{\operatorname{Pr}[D \& H]}{\operatorname{Pr}[D]}=\frac{\operatorname{Pr}[D \mid H] \operatorname{Pr}[H]}{\sum_{H^{\prime}} \operatorname{Pr}\left[D \mid H^{\prime}\right] \operatorname{Pr}\left[H^{\prime}\right]}$$
where $H^{\prime}$ ranges over all plausible hypotheses. The value $\operatorname{Pr}[H]$ is the probability that hypothesis $H$ is true before we have any data about it. This probability $\operatorname{Pr}[H]$ is the a priori probability, while $\operatorname{Pr}[H \mid D]$ is the a posteriori probability of hypothesis $H$. Estimating $\operatorname{Pr}[H]$ is often a subjective matter. Consider the question, “What is the probability that the sun will rise every 24 hours?” We might have personally observed these occurring tens of thousands of times and have historical records going back hundreds of thousands of times before that. But what should we assign to this hypothesis before we have evidence, or before we know anything about the sun? Without evidence we have very little basis for any computation of $\operatorname{Pr}[H]$. We can make an arbitrary assignment $\operatorname{Pr}[H]=\frac{1}{2}$. The observed data of centuries of observing the sun rise every day would then give $\operatorname{Pr}[H \mid D]$ very close to one.

# 数值分析代考

## 数学代写|数值分析代写numerical analysis代考|Regression and Curve Fitting

$x \mapsto \sum j=1^m c_j \varphi_j(x)$. 我们假设一个统计模型 $y_i=\sum_{j=1}^m c_j \varphi_j\left(x_i\right)+\epsilon_i$ 在哪里 $\epsilon_i$ 跟随一个
$\operatorname{Normal}\left(0, \sigma^2\right)$ 分布和 $\epsilon_i$ 的是相互独立的。似然函数是
$$L(\boldsymbol{c})=\prod_{i=1}^N(2 \pi)^{-1 / 2} \exp \left(-\epsilon_i^2 / \sigma^2\right) \quad=(2 \pi)^{-N / 2}$$

$\hat{\boldsymbol{c}}-\boldsymbol{c}=\left(\Phi^T \Phi\right)^{-1} \Phi^T \boldsymbol{\epsilon}$ 是根据分布
$\operatorname{Normal}\left(\mathbf{0}, \sigma^2\left(\Phi^T \Phi\right)^{-1}\right)$ 分配。这个估计是无偏

$$\hat{\boldsymbol{\epsilon}}=\boldsymbol{y}-\Phi \hat{\boldsymbol{c}}=\boldsymbol{y}-\Phi\left(\Phi^T \Phi\right)^{-1} \Phi^T \boldsymbol{y} \quad=[I-\Phi$$

$$\hat{\epsilon}=P(\Phi c+\epsilon)=P \epsilon$$

$$\mathbb{E}\left[\hat{\boldsymbol{\epsilon}}^T \hat{\boldsymbol{\epsilon}}\right]=\mathbb{E}\left[\boldsymbol{\epsilon}^T P^T P \boldsymbol{\epsilon}\right]=\mathbb{E}\left[\boldsymbol{\epsilon}^T P \boldsymbol{\epsilon}\right]=\operatorname{trace}(P) \sigma^2$$

## 数学代写|数值分析代写numerical analysis代考|Bayesian Inference

$$\operatorname{Pr}[H \mid D]=\frac{\operatorname{Pr}[D \& H]}{\operatorname{Pr}[D]}=\frac{\operatorname{Pr}[D \mid H] \operatorname{Pr}[H]}{\sum_{H^{\prime}} \operatorname{Pr}\left[D \mid H^{\prime}\right] \operatorname{Pr}\left[H^{\prime}\right]}$$

## 有限元方法代写

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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。

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