### 统计代写|统计推断代写Statistical inference代考|STAT 3023

statistics-lab™ 为您的留学生涯保驾护航 在代写统计推断Statistical inference方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写统计推断Statistical inference代写方面经验极为丰富，各种代写统计推断Statistical inference相关的作业也就用不着说。

• 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

## 统计代写|统计推断代写Statistical inference代考|Signal Recovery Problem

One of the basic problems in Signal Processing is the problem of recovering a signal $x \in \mathbf{R}^{n}$ from noisy observations
$$y=A x+\eta$$
of a linear image of the signal under a given sensing mapping $x \mapsto A x: \mathbf{R}^{n} \rightarrow \mathbf{R}^{m}$; in (1.1), $\eta$ is the observation error. Matrix $A$ in (1.1) is called sensing matrix.
Recovery problems of the outlined types arise in many applications, including, but by far not reducing to,

• communications, where $x$ is the signal sent by the transmitter, $y$ is the signal recorded by the receiver, and $A$ represents the communication channel (reflecting, e.g., dependencies of decays in the signals’ amplitude on the transmitter-receiver distances); 7 here typically is modeled as the standard (zero mean, unit covariance matrix) $m$-dimensional Gaussian noise; ${ }^{1}$
• image reconstruction, where the signal $x$ is an image a $2 \mathrm{D}$ array in the usual photography, or a 3D array in tomography – and $y$ is data acquired by the imaging device. Here $\eta$ in many cases (although not always) can again be modeled as the standard Gaussian noise;
• linear regression, arising in a wide range of applications. In linear regression, one is given $m$ pairs “input $a^{i} \in \mathbf{R}^{n n}$ to a “black box,” with output $y_{i} \in \mathbf{R}$. Sometimes we have reason to believe that the output is a corrupted by noise version of the “existing in nature,” but unobservable, “ideal output” $y_{i}^{*}=x^{T} a^{i}$ which is just a linear function of the input (this is called “linear regression model,” with inputs $a^{i}$ called “regressors”). Our goal is to convert actual observations $\left(a^{i}, y_{i}\right), 1 \leq i \leq m$, into estimates of the unknown “true” vector of parameters $x$. Denoting by $A$ the matrix with the rows $\left[a^{i}\right]^{T}$ and assembling individual observations $y_{i}$ into a single observation $y=\left[y_{1}, \ldots, y_{m}\right] \in \mathbf{R}^{m}$, we arrive at the problem of recovering vector $x$ from noisy observations of $A x$. Here again the most popular model for $\eta$ is the standard Gaussian noise.

## 统计代写|统计推断代写Statistical inference代考|Signal Recovery: Parametric and nonparametric cases

Recovering signal $x$ from observation $y$ would be easy if there were no observation noise $(\eta=0)$ and the rank of matrix $A$ were equal to the dimension $n$ of the signals. In this case, which arises only when $m \geq n$ (“more observations than unknown parameters”), and is typical in this range of $m$ and $n$, the desired $x$ would be the unique solution to the system of linear equations, and to find $x$ would be a simple problem of Linear Algebra. Aside from this trivial “enough observations, no noise” case, people over the years have looked at the following two versions of the recovery problem:

Parametric case: $m \gg n, \eta$ is nontrivial noise with zero mean, say, standard Gaussian. This is the classical statistical setup with the emphasis on how to use numerous available observations in order to suppress in the recovery, to the extent possible, the influence of observation noise.

Nonparametric case: $m \ll n .^{2}$ If addressed literally, this case seems to be senseless: when the number of observations is less that the number of unknown parameters, even in the noiseless case we arrive at the necessity to solve an undetermined (fewer equations than unknowns) system of linear equations. Linear Algebra says that if solvable, the system has infinitely many solutions. Moreover, the solution set (an affine subspace of positive dimension) is unbounded, meaning that the solutions are in no sense close to each other. A typical way to make the case of $m \ll n$ meaningful is to add to the observations (1.1) some a priori information about the signal. In traditional Nonparametric Statistics, this additional information is summarized in a bounded convex set $X \subset \mathbf{R}^{n}$, given to us in advance, known to contain the true signal $x$. This set usually is such that every signal $x \in X$ can be approximated by a linear combination of $s=1,2, \ldots, n$ vectors from a properly selected basis known to us in advance (“dictionary” in the slang of signal processing) within accuracy $\delta(s)$, where $\delta(s)$ is a function, known in advance. approaching 0 as $s \rightarrow \infty$. In this situation, with appropriate $A$ (e.g., just the unit matrix, as in the denoising problem), we can select some $s \ll m$ and try to recover $x$ as if it were a vector from the linear span $E_{s}$ of the first $s$ vectors of the outlined basis $[54,86,124,112,208]$.

## 统计代写|统计推断代写Statistical inference代考|Signal Recovery Problem

$$y=A x+\eta$$

• 通讯，在哪里 $x$ 是发射机发送的信号， $y$ 是接收器记录的信号，并且 $A$ 表示通信信道 (反映，例如，信号幅 度衰减对发射机-接收机距离的依赖性) ；7 这里通常被建模为标准 (零均值，单位协方差矩阵) $m$-维高斯 噪声; 1
• 图像重建，其中信号 $x$ 是一个图像 $2 \mathrm{D}$ 通常摄影中的阵列，或断层扫描中的 3D 阵列 – 和 $y$ 是成像设备获取的 数据。这里 $\eta$ 在许多情况下 (尽管并非总是如此) 可以再次建模为标准高斯噪声；
• 线性回归，在广泛的应用中出现。在线性回归中，给出一个 $m$ 对“输入 $a^{i} \in \mathbf{R}^{n n}$ 到一个“黑匣子”，输出 $y_{i} \in \mathbf{R}$. 有时我们有理由相信输出是“存在于自然界”但不可观察的“理想输出”的噪声版本 $y_{i}^{*}=x^{T} a^{i}$ 这只是 输入的线性函数（这称为“线性回归模型”，输入 $a^{i}$ 称为“回归器”)。我们的目标是转换实际观察结果 $\left(a^{i}, y_{i}\right), 1 \leq i \leq m$, 估计末知的“真实”参数向量 $x$. 表示 $A$ 具有行的矩阵 $\left[a^{i}\right]^{T}$ 并收集个人观察结果 $y_{i}-$ 次观察 $y=\left[y_{1}, \ldots, y_{m}\right] \in \mathbf{R}^{m}$ ，我们得到了恢复向量的问题 $x$ 从嘈杂的观察 $A x$. 这里又是最受欢迎的模 型 $\eta$ 是标准高斯噪声。

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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。