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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

## 数学代写|信息论作业代写information theory代考|Mutual Information

Shannon’s entropy is a probabilistic quantity that measures information in terms of the number of bits needed to identify any typical realization of a random waveform, quantized at level $\epsilon$ on each dimension of the space, and subject to an expected norm per degree of freedom constraint. It is the stochastic analog of Kolmogorov’s entropy, which counts the number of bits needed to identify any waveform in the space, quantized at level $\epsilon$ and subject to an energy constraint on the whole signal.

Another information-theoretic quantity, analogous to Kolmogorov’s capacity, is used to quantify in a stochastic setting the largest amount of information that can be transported by any waveform in the space. In this case, the stochastic model assumes that observations are random variables governed by a probability distribution. It follows that the signals’ coefficients that are selected randomly at the transmitter and observed at the receiver have a joint probability density $g(x, y)$. Given any transmitted random coefficient $X$, the observed coefficient $Y$ has a conditional distribution $g(y \mid x)$, and we say that $\mathrm{X}$ is observed through the probabilistic channel depicted in Figure 1.18.

The conditional differential entropy represents the remaining uncertainty about the outcome of $Y$ given the outcome of $X$,
\begin{aligned} h_{Y \mid X} &=-\int_{\mathscr{X}} \int_{\mathscr{Y}} g(x, y) \log g(y \mid x) d x d y \ &=-\mathbb{E}{\mathrm{X}, \mathrm{Y}} \log g(\mathrm{Y} \mid \mathrm{X}) \end{aligned} A simple calculation yields the following chain rule: $$h{\mathrm{X}, \mathrm{Y}}=h_{\mathrm{X}}+h_{\mathrm{Y} \mid \mathrm{X}},$$
and when the two variables are independent, we have
$$h \mathrm{Y} \mid \mathrm{X}=h \mathrm{Y},$$
so that (1.53) reduces to (1.35).

## 数学代写|信息论作业代写information theory代考|Gaussian Noise

The Shannon capacity can be viewed as a limit due to an effective resolution level imposed by the noise inherent in the observation of the waveform. This makes the results above consistent with the deterministic view expressed by (1.26), showing that the capacity grows at most linearly with the number of degrees of freedom, but only logarithmically with the signal-to-noise ratio. We provide this interpretation in the context of Gaussian uncertainty, for which the Shannon capacity is expressed in terms of number of degrees of freedom, signal’s energy, and noise constraints.

The uncertainty in the signal’s observation is due to a variety of causes, but all of them can be traced back to the quantized nature of the world arising at the microscopic scale. At a very small scale, continuum fields are described by quantum particles whose configurations are uncertain, and repeated measurements appear to fluctuate randomly around their average values. A mathematical model for this situation adds a zero mean Gaussian random variable of standard deviation $\epsilon$ independently to each field’s coefficient, obtaining the channel depicted in Figure 1.23:
$$\mathrm{Y}{n}=\mathrm{X}{n}+\mathrm{Z}{n} .$$ The model is justified as follows. The entropy associated with the noise measures its statistical dispersion: low differential entropy implies that noise realizations are confined to a small effective volume, while high differential entropy indicates that outcomes are widely dispersed. The probability density function that maximizes the uncertainty of the observation, subject to the average constraint $$\mathbb{E}\left(Z{n}^{2}\right) \leq \epsilon^{2},$$
is the Gaussian one that achieves the maximum differential entropy
$$h \mathrm{Z}_{n}=\frac{1}{2} \log \left(2 \pi e \epsilon^{2}\right) .$$
This distribution provides the most surprising observation, for the given moment constraint. In addition, by the central limit theorem, the Gaussian assumption is valid in a large number of practical situations where the noise models the cumulative effect of a variety of random effects. A treatment of maximum entropy distributions and their relationship with noise modeling, statistical mechanics, and the second law is given in Chapter $11 .$

# 信息论代写

## 数学代写|信息论作业代写information theory代考|Mutual Information

$$h_{Y \mid X}=-\int_{\mathscr{X}} \int_{\mathscr{Y}} g(x, y) \log g(y \mid x) d x d y \quad=-\mathbb{E X}, \mathrm{Y} \log g(\mathrm{Y} \mid \mathrm{X})$$

$$h \mathrm{X}, \mathrm{Y}=h_{\mathrm{X}}+h_{\mathrm{Y} \mid \mathrm{X}},$$

$$h \mathrm{Y} \mid \mathrm{X}=h \mathrm{Y}$$

## 数学代写|信息论作业代写information theory代考|Gaussian Noise

$$\mathrm{Y} n=\mathrm{X} n+\mathrm{Z} n .$$

$$\mathbb{E}\left(Z n^{2}\right) \leq \epsilon^{2}$$

$$h \mathrm{Z}_{n}=\frac{1}{2} \log \left(2 \pi e \epsilon^{2}\right) .$$

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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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