### 统计代写|统计模型作业代写Statistical Modelling代考|Examples Less Important for the Sequel

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• Foundations of Data Science 数据科学基础

## 统计代写|统计模型作业代写Statistical Modelling代考|Finite Markov chains

Suppose we observe a single realization of a finite Markov chain, from time 0 to a fixed time $n$. For simplicity of notation and calculations, let it have only two states, 0 and 1 . Let the unknown transition matrix be
$$\Gamma=\left(\begin{array}{ll} \gamma_{00} & \gamma_{01} \ \gamma_{10} & \gamma_{11} \end{array}\right) .$$
Thus, the one-step transition probabilities are $\operatorname{Pr}{0 \rightarrow 1}=\gamma_{01}=1-\gamma_{00}$ and $\operatorname{Pr}{1 \rightarrow 0}=\gamma_{10}=1-\gamma_{11}$, so the parameter dimension of $\Gamma$ is 2 .
The probability for observing a specific realized sequence, say $0 \rightarrow 0 \rightarrow$ $1 \rightarrow 0 \rightarrow 0 \cdots$, is the corresponding product of transition matrix probabilities, $\gamma_{00} \gamma_{01} \gamma_{10} \gamma_{00} \cdots$, or more generally,
$$\prod_{i=1}^{2} \prod_{j=1}^{2} \gamma_{i j}^{n_{i j}}=\exp \left{\sum_{i} \sum_{j}\left(\log \gamma_{i j}\right) n_{i j}\right}$$
where $n_{i j}$ is the observed number of consecutive pairs $(i, j)$ (the four possible transitions $i \rightarrow j$, including the case $i=j$ ).

The four $n_{i j}$-values satisfy the restriction $\sum_{i j} n_{i j}=n$, and it might appear

as if there are three linearly independent statistics but only two parameters. However, there is a near equality $n_{01} \approx n_{10}$, because after one transition the next transition must be in the opposite direction. More precisely, say that the chain starts in state 0 , then $n_{01}-n_{10}=0$ or $=1$. The outcome probability depends on the two parameters, but intuitively there is very little additional information in that outcome, at least for large $n$.

In analogy with the binomial we have one set of $n_{0}=n_{00}+n_{01}$ Bernoulli trials when the chain is in state 0 , with transition probability $\gamma_{01}\left(=1-\gamma_{00}\right)$, and another set of $n_{1}=n_{10}+n_{11}$ trials with another transition probability $\gamma_{10}\left(=1-\gamma_{11}\right)$. The main difference from the binomial is that the numbers $n_{0}$. and $n_{1}$. $=n-n_{0}$. of these trials are random, and the distributions of $n_{0}$. and $n_{1}$. do in fact depend on the $\gamma$-values. Another analogue is the time to first success situation (Example 2.6; or the negative binomial, see Exercise 2.2). Bernoulli trials are carried out until the first success, when a transition to the other state occurs. There the situation is repeated with a different probability for success, until the next transition, which means the chain restarts from the beginning. The inferential complication of the negative binomial when the numbers of successes $n_{01}$ and $n_{10}$ are regarded as fixed is that the number $n$ of repetitions then is random.

Note that the probability (2.26), regarded as a likelihood for $\Gamma$, looks the same as it would have done in another model, with fixed numbers $n_{0}$. and $n_{1}$. of Bernoulli trials. In particular, the ML estimator of $\Gamma$ is the same in the two models.

## 统计代写|统计模型作业代写Statistical Modelling代考|Von Mises and Fisher distributions for directional data

The von Mises and Fisher distributions, sometimes referred to together under the name von Mises-Fisher model, are distributions for directions in the plane and in space, respectively. Such directions can be represented by vectors of length one, that is, vectors on the unit circle and on the unit sphere, respectively. Therefore they fall outside the main stream here, by neither being counts or $k$-dimensional vectors in $\mathbb{R}^{k}$. Applications are found in biology, geology, meteorology and astronomy. Both distributions have a density of type
$$f(\boldsymbol{y} ; \boldsymbol{\theta})=\frac{1}{C(\boldsymbol{\theta})} \exp \left(\boldsymbol{\theta}^{T} \boldsymbol{y}\right),$$
where the exponent is the scalar product between the observed direction vector $y$ on the unit circle or sphere and an arbitrary parameter vector $\theta$ in $\mathbb{R}^{2}$ or $\mathbb{R}^{3}$, respectively. The special case $\boldsymbol{\theta}=\boldsymbol{0}$ yields a uniformly distributed direction.

The von Mises distribution is a symmetric, unimodal model for a sample

of directions in the plane. If we represent both $\boldsymbol{\theta}$ and $\boldsymbol{y}$ in polar coordinates, $\boldsymbol{\theta}=(\rho \cos \psi, \rho \sin \psi)$ and $\boldsymbol{y}=(\cos z, \sin z), 0 \leq z<2 \pi$, their scalar product $\boldsymbol{\theta}^{T} \boldsymbol{y}$ in the exponent may be written
$$\boldsymbol{\theta}^{T} \boldsymbol{y}=\rho{\cos z \cos \psi+\sin z \sin \psi}=\rho \cos (z-\psi)$$
The direction $\psi$ is the mean direction, whereas $\rho$ is a measure of the degree of variability around $\psi$, high values representing small variability. The norming constant per observation is the integral
$$\int_{0}^{2 \pi} e \rho \cos (z-\psi) \mathrm{d} z=\int_{0}^{2 \pi} e^{\rho \cos z} \mathrm{~d} z=2 \pi I_{0}(\rho)$$
where $I_{0}$ is called the modified Bessel function of the first kind and order 0 . Note that the norming constant is free from $\psi$. Except for this simplification, the norming constant and the structure function are analytically complicated. We will not further discuss this model, but the interested reader is referred to Martin-Löf (1970) and Mardia and Jupp (2000).

Fisher’s distribution for directions in space (Fisher, 1953), is the corresponding density when $\boldsymbol{y}$ is the direction vector on the sphere, and $\boldsymbol{\theta}$ is a parameter vector in $\mathbb{R}^{3}$. In this case, the expression in polar coordinates becomes somewhat longer and more complicated, so we abstain from details and refer to Mardia and Jupp $(2000)$, see also Diaconis $(1988$, Sec. 9B). As in two dimensions, the density is symmetrical around the mean direction. This is the direction of $\boldsymbol{\theta}$, and the length of $\boldsymbol{\theta}$ is a concentration parameter.
$\Delta$

## 统计代写|统计模型作业代写Statistical Modelling代考|Maxwell–Boltzmann model in statistical physics

Already in 1859 James Clerk Maxwell gave the distribution of kinetic energy among particles in an ideal gas under thermal equilibrium, later established and extended by Ludwig Boltzmann. This distribution now goes under the name Maxwell-Boltzmann distribution. On the so-called microcanonical scale the particles are assumed to interact and change velocities by collisions but move with constant velocity vector between collisions. The Maxwell-Boltzmann distribution can then be derived as describing on the so-called canonical scale the distribution of velocity vectors $v$ among the particles in the gas. Let $v=\left(v_{1}, v_{2}, v_{3}\right) \in \mathbb{R}^{3}$, where the components are the velocities (with sign) in three orthogonal directions. Then the MaxwellBoltzmann distribution for the vector $v$ is given by the density
$$f_{v}\left(v_{1}, v_{2}, v_{3} ; T\right)=\left(\frac{m}{2 \pi k T}\right)^{3 / 2} e^{-\frac{m|v|^{2}}{2 k T}}$$
where $|v|^{2}=v^{T} v=v_{1}^{2}+v_{2}^{2}+v_{3}^{2}$ is the speed squared, neglecting its direction,

$m$ is the particle mass (assumed known), and $k$ and $T$ are other constants (see the next paragraph for their interpretations).

It is clear that the density $(2.27$ ) is a three-dimensional Gaussian distribution for mutually independent components $v_{1}, v_{2}$ and $v_{3}$, each being $\mathrm{N}\left(0, \sigma^{2}\right)$ with $\sigma^{2}=k T / \mathrm{m}$. It is also clear that we have an exponential family with $|v|^{2}$ as canonical statistic. Equivalently we could use the kinetic energy $E=m|v|^{2} / 2$ as canonical statistic, with the corresponding canonical parameter $\theta=-1 /(k T)$. Here $T$ is the thermodynamic temperature of the gas, which is regarded as a parameter, whereas $k$ is the Boltzmann constant, whose role is to transform the unit of temperature to the unit of energy. In statistical thermodynamics the notation $\beta$ for $-\theta=1 /(k T)$ is standard, called the thermodynamic beta.

From the normal distribution for $v$ we can for example easily find the mean energy per particle, in usual physics notation $\langle E\rangle$, as
$$\langle E\rangle=\frac{m}{2}\left\langle|v|^{2}\right\rangle=\frac{m}{2} 3 \sigma^{2}=\frac{3}{2} k T$$
Note that in thermodynamics the only available measurements will typically be of macroscopic characteristics such as temperature, thus representing the mean value $\langle E\rangle$. The sample size will be enormous, a laboratory quantity of gas of reasonable density containing say $n=10^{20}$ particles, so the corresponding sampling error will be negligible.

## 统计代写|统计模型作业代写Statistical Modelling代考|Finite Markov chains

Γ=(C00C01 C10C11).

\prod_{i=1}^{2} \prod_{j=1}^{2} \gamma_{i j}^{n_{i j}}=\exp \left{\sum_{i} \sum_{j} \left(\log \gamma_{i j}\right) n_{i j}\right}\prod_{i=1}^{2} \prod_{j=1}^{2} \gamma_{i j}^{n_{i j}}=\exp \left{\sum_{i} \sum_{j} \left(\log \gamma_{i j}\right) n_{i j}\right}

## 统计代写|统计模型作业代写Statistical Modelling代考|Von Mises and Fisher distributions for directional data

von Mises 和 Fisher 分布，有时一起称为 von Mises-Fisher 模型，分别是平面和空间方向的分布。这样的方向可以用长度为 1 的向量表示，即分别在单位圆和单位球面上的向量。因此，它们不属于这里的主流，既不重要也不重要ķ维向量Rķ. 应用在生物学、地质学、气象学和天文学中。两种分布都具有类型密度
F(是;θ)=1C(θ)经验⁡(θ吨是),

von Mises 分布是样本的对称单峰模型

θ吨是=ρ因⁡和因⁡ψ+罪⁡和罪⁡ψ=ρ因⁡(和−ψ)

∫02圆周率和ρ因⁡(和−ψ)d和=∫02圆周率和ρ因⁡和 d和=2圆周率一世0(ρ)

Δ

## 统计代写|统计模型作业代写Statistical Modelling代考|Maxwell–Boltzmann model in statistical physics

F在(在1,在2,在3;吨)=(米2圆周率ķ吨)3/2和−米|在|22ķ吨

⟨和⟩=米2⟨|在|2⟩=米23σ2=32ķ吨

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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。