## 统计代写|贝叶斯分析代写Bayesian Analysis代考|MAST90125

statistics-lab™ 为您的留学生涯保驾护航 在代写贝叶斯分析Bayesian Analysis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。我们的专家在代写贝叶斯分析Bayesian Analysis代写方面经验极为丰富，各种代写贝叶斯分析Bayesian Analysis相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• Advanced Probability Theory 高等概率论
• Advanced Mathematical Statistics 高等数理统计学
• (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|贝叶斯分析代写Bayesian Analysis代考|INDEPENDENT AND CONDITIONALLY INDEPENDENT

A pair of random variables $(X, Y)$ is said to be independent if for any $A$ and $B$,
$$p(X \in A \mid Y \in B)=p(X \in A),$$
or alternatively $p(Y \in B \mid X \in A)=p(Y \in B)$ (these two definitions are correct and equivalent under very mild conditions that prevent ill-formed conditioning on an event that has zero probability).

Using the chain rule, it can also be shown that the above two definitions are equivalent to the requirement that $p(X \in A, Y \in B)=p(X \in A) p(Y \in B)$ for all $A$ and $B$.

Independence between random variables implies that the random variables do not provide information about each other. This means that knowing the value of $X$ does not help us infer anything about the value of $Y$-in other words, it does not change the probability of $Y$. (Or vice-versa $-Y$ does not tell us anything about $X$.) While independence is an important concept in probability and statistics, in this book we will more frequently make use of a more refined notion of independence, called “conditional independence”-which is a generalization of the notion of independence described in the beginning of this section. A pair of random variables $(X, Y)$ is conditionally independent given a third random variable $Z$, if for any $A, B$ and $z$, it holds that $p(X \in A \mid Y \in B, Z=z)=p(X \in A \mid Z=z)$.

Conditional independence between two random variables (given a third one) implies that the two variables are not informative about each other, if the value of the third one is known. 3
Conditional independence (and independence) can be generalized to multiple random variables as well. We say that a set of random variables $X_{1}, \ldots, X_{n}$, are mutually conditionally independent given another set of random variables $Z_{1}, \ldots, Z_{m}$ if the following applies for any $A_{1}, \ldots, A_{n}$ and $z_{1}, \ldots, z_{m}:$
$$\begin{gathered} p\left(X_{1} \in A_{1}, \ldots, X_{n} \in A_{n} \mid Z_{1}=z_{1}, \ldots, Z_{m}=z_{m}\right)= \ \prod_{i=1}^{n} p\left(X_{i} \in A_{i} \mid Z_{1}=z_{1}, \ldots, Z_{m}=z_{m}\right) . \end{gathered}$$
This type of independence is weaker than pairwise independence for a set of random variables, in which only pairs of random variables are required to be independent. (Also see exercises.)

## 统计代写|贝叶斯分析代写Bayesian Analysis代考|EXCHANGEABLE RANDOM VARIABLES

Another type of relationship that can be present between random variables is that of exchangeability. A sequence of random variables $X_{1}, X_{2}, \ldots$ over $\Omega$ is said to be exchangeable, if for any finite subset, permuting the random variables in this finite subset, does not change their joint distribution. More formally, for any $S=\left{a_{1}, \ldots, a_{m}\right}$ where $a_{i} \geq 1$ is an integer, and for any permutation $\pi$ on ${1, \ldots, m}$, it holds that: ${ }^{4}$
$$p\left(x_{a_{1}}, \ldots, x_{a_{m}}\right)=p\left(x_{a_{\pi(1)}}, \ldots, x_{\left.a_{\pi(m)}\right)}\right) .$$
Due to a theorem by de Finetti (Finetti, 1980), exchangeability can be thought of as meaning “conditionally independent and identically distributed” in the following sense. De Finetti showed that if a sequence of random variables $X_{1}, X_{2}, \ldots$ is exchangeable, then under some regularity conditions, there exists a sample space $\Theta$ and a distribution over $\Theta, p(\theta)$, such that:
$$p\left(X_{a_{1}}, \ldots, X_{a_{m}}\right)=\int_{\theta} \prod_{i=1}^{m} p\left(X_{a_{i}} \mid \theta\right) p(\theta) d \theta,$$
for any set of $m$ integers, $\left{a_{1}, \ldots, a_{m}\right}$. The interpretation of this is that exchangeable random variables can be represented as a (potentially infinite) mixture distribution. This theorem is also called the “representation theorem.”

The frequentist approach assumes the existence of a fixed set of parameters from which the data were generated, while the Bayesian approach assumes that there is some prior distribution over the set of parameters that generated the data. (This will hecome clearer as the hook progresses.) De Finetti’s theorem provides another connection between the Bayesian approach and the frequentist one. The standard “independent and identically distributed” (i.i.d.) assumption in the frequentist setup can be asserted as a setup of exchangeability where $p(\theta)$ is a point-mass distribution over the unknown (but single) parameter from which the data are sampled. This leads to the observations being unconditionally independent and identically distributed. In the Bayesian setup, however, the observations are correlated, because $p(\theta)$ is not a point-mass distribution. The prior distribution plays the role of $p(\theta)$. For a detailed discussion of this similarity, see O’Neill (2009).

## 统计代写|贝叶斯分析代写Bayesian Analysis代考|EXPECTATIONS OF RANDOM VARIABLES

If we consider again the naive definition of random variables, as functions that map the sample space to real values, then it is also useful to consider various ways in which we can summarize these random variables. One way to get a summary of a random variable is by computing its expectation, which is its weighted mean value according to the underlying probability model.
It is easiest to first consider the expectation of a continuous random variable with a density function. Say $p(\theta)$ defines a distribution over the random variable $\theta$, then the expectation of $\theta$, denoted $E[\theta]$ would be defined as:
$$E[\theta]=\int_{\theta} p(\theta) \theta d \theta .$$
For the discrete random variables that we consider in this book, we usually consider expectations of functions over these random variables. As mentioned in Section 1.2, discrete random variable values often range over a set which is not numeric. In these cases, there is no “mean value” for the values that these random variables accept. Instead, we will compute the mean value of a real-function of these random variables.
With $f$ being such a function, the expectation $E[f(X)]$ is defined as:
$$E[f(X)]=\sum_{x} p(x) f(x)$$ For the linguistic structures that are used in this book, we will often use a function $f$ that indicates whether a certain property holds for the structure. For example, if the sample space of $X$ is a set of sentences, $f(x)$ can be an indicator function that states whether the word “spring” appears in the sentence $x$ or not; $f(x)=1$ if the word “spring” appears in $x$ and 0 , otherwise. In that case, $f(X)$ itself can be thought of as a Bernoulli random variable, i.e., a binary random variable that has a certain probability $\theta$ to be 1 , and probability $1-\theta$ to be 0 . The expectation $E[f(X)]$ gives the probability that this random variable is 1 . Alternatively, $f(x)$ can count how many times the word “spring” appears in the sentence $x$. In that case, it can be viewed as a sum of Bernoulli variables, each indicating whether a certain word in the sentence $x$ is “spring” or not.

## 统计代写|贝叶斯分析代写Bayesian Analysis代考|INDEPENDENT AND CONDITIONALLY INDEPENDENT

$$p(X \in A \mid Y \in B)=p(X \in A),$$

$$p\left(X_{1} \in A_{1}, \ldots, X_{n} \in A_{n} \mid Z_{1}=z_{1}, \ldots, Z_{m}=z_{m}\right)=\prod_{i=1}^{n} p\left(X_{i} \in A_{i} \mid Z_{1}=z_{1}, \ldots, Z_{m}=z_{m}\right) \text {. }$$

## 统计代写|贝叶斯分析代写Bayesian Analysis代考|EXCHANGEABLE RANDOM VARIABLES

$\mathrm{S}=|$ left{a_{1}，Vdots, a_{m}|right} 在哪里 $a_{i} \geq 1$ 是一个整数，并且对于任何排列 $\pi$ 上 $1, \ldots, m$ ，它认为: ${ }^{4}$
$$p\left(x_{a_{1}}, \ldots, x_{a_{m}}\right)=p\left(x_{a_{\pi(1)}}, \ldots, x_{\left.a_{\pi(m)}\right)}\right) .$$

$$p\left(X_{a_{1}}, \ldots, X_{a_{m i}}\right)=\int_{\theta} \prod_{i=1}^{m} p\left(X_{a_{i}} \mid \theta\right) p(\theta) d \theta$$

## 统计代写|贝叶斯分析代写Bayesian Analysis代考|EXPECTATIONS OF RANDOM VARIABLES

$$E[\theta]=\int_{\theta} p(\theta) \theta d \theta$$

$$E[f(X)]=\sum_{x} p(x) f(x)$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 统计代写|贝叶斯分析代写Bayesian Analysis代考|MSH3

statistics-lab™ 为您的留学生涯保驾护航 在代写贝叶斯分析Bayesian Analysis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。我们的专家在代写贝叶斯分析Bayesian Analysis代写方面经验极为丰富，各种代写贝叶斯分析Bayesian Analysis相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• Advanced Probability Theory 高等概率论
• Advanced Mathematical Statistics 高等数理统计学
• (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|贝叶斯分析代写Bayesian Analysis代考|JOINT DISTRIBUTION OVER MULTIPLE RANDOM VARIABLES

It is possible to define several random variables on the same sample space. For example, for a discrete sample space, such as a set of words, we can define two random variables $X$ and $Y$ that take integer values-one could measure word length and the other could measure the count of vowels in a word. Given two such random variables, the joint distribution $P(X, Y)$ is a function that maps pairs of events $(A, B)$ as follows:
$$p(X \in A, Y \in B)=p\left(X^{-1}(A) \cap Y^{-1}(B)\right)$$
It is often the case that we take several sets $\left{\Omega_{1}, \ldots, \Omega_{m}\right}$ and combine them into a single sample space $\Omega=\Omega_{1} \times \ldots \times \Omega_{m}$. Each of the $\Omega_{i}$ is associated with a random variable. Based on this, a joint probability distribution can be defined for all of these random variables together. For example, consider $\Omega=V \times P$ where $V$ is a vocabulary of words and $P$ is a part-of-speech tag. This sample space enables us to define probabilities $p(x, y)$ where $X$ denotes a word associated with a part of speech $Y$. In this case, $x \in V$ and $y \in P$.

With any joint distribution, we can marginalize some of the random variables to get a distribution which is defined over a subset of the original random variables (so it could still be a joint distribution, only over a subset of the random variables). Marginalization is done using integration (for continuous variables) or summing (for discrete random variables). This operation of summation or integration eliminates the random variable from the joint distribution. The result is a joint distribution over the non-marginalized random variables.

For the simple part-of-speech example above, we could either get the marginal $p(x)=$ $\sum_{y \in P} p(x, y)$ or $p(y)=\sum_{x \in V} p(x, y)$. The marginals $p(X)$ and $p(Y)$ do not uniquely determine the joint distribution value $p(X, Y)$. Only the reverse is true. However, whenever $X$ and $Y$ are independent then the joint distribution can be determined using the marginals. More about this in Section 1.3.2.

## 统计代写|贝叶斯分析代写Bayesian Analysis代考|CONDITIONAL DISTRIBUTIONS

Joint probability distributions provide an answer to questions about the probability of several random variables to obtain specific values. Conditional distributions provide an answer to a different, but related question. They help to determine the values that a random variable can obtain, when other variables in the joint distribution are restricted to specific values (or when they are “clamped”).

Conditional distributions are derivable from joint distributions over the same set of random variables. Consider a pair of random variables $X$ and $Y$ (either continuous or discrete). If $A$ is an event from the sample space of $X$ and $y$ is a value in the sample space of $Y$, then:
$$p(X \in A \mid Y=y)=\frac{p(X \in A, Y=y)}{p(Y=y)}$$
is to be interpreted as a conditional distribution that determines the probability of $X \in A$ conditioned on $Y$ obtaining the value $y$. The bar denotes that we are clamping $Y$ to the value $y$ and identifying the distribution induced on $X$ in the restricted sample space. Informally, the conditional distribution takes the part of the sample space where $Y=y$ and re-normalizes the joint distribution such that the result is a probability distribution defined only over that part of the sample space.

When we consider the joint distribution in Equation $1.1$ to be a function that maps events to probabilities in the space of $X$, with $y$ being fixed, we note that the value of $p(Y=y)$ is actually a normalization constant that can be determined from the numerator $p(X \in A, Y=y)$. For example, if $X$ is discrete when using a PMF, then:
$$p(Y=y)=\sum_{x} p(X=x, Y=y) .$$

## 统计代写|贝叶斯分析代写Bayesian Analysis代考|BAYES’ RULE

Bayes’ rule is a basic result in probability that describes a relationship between two conditional distributions $p(X \mid Y)$ and $p(Y \mid X)$ for a pair of random variables (these random variables can also be continuous). More specifically, Bayes’ rule states that for any such pair of random variables, the following identity holds:
$$p(Y=y \mid X=x)=\frac{p(X-x \mid Y-y) p(Y-y)}{p(X=x)}$$

This result also generally holds true for any two events $A$ and $B$ with the conditional probability $p(X \in A \mid Y \in B)$.

The main advantage that Bayes’ rule offers is inversion of the conditional relationship between two random variables – therefore, if one variable is known, then the other can be calculated as well, assuming the marginal distributions $p(X=x)$ and $p(Y=y)$ are also known.
Bayes’ rule can be proven in several ways. One way to derive it is simply by using the chain rule twice. More specifically, we know that the joint distribution values can be rewritten as follows, using the chain rule, either first separating $X$ or first separating $Y$ :
\begin{aligned} p(X&=x, Y=y) \ &=p(X=x) p(Y=y \mid X=x) \ &=p(Y=y) p(X=x \mid Y=y) \end{aligned}
Taking the last equality above, $p(X=x) p(Y=y \mid X=x)=p(Y=y) p(X=x \mid Y=$ $y)$, and dividing both sides by $p(X=x)$ results in Bayes’ rule as described in Equation 1.2.
Bayes’ rule is the main pillar in Bayesian statistics for reasoning and learning from data. Bayes’ rule can invert the relationship between “observations” (the data) and the random variables we are interested in predicting. This makes it possible to infer target predictions from such observations. A more detailed description of these ideas is provided in Section 1.5, where statistical modeling is discussed.

## 统计代写|贝叶斯分析代写Bayesian Analysis代考|JOINT DISTRIBUTION OVER MULTIPLE RANDOM VARIABLES

$$p(X \in A, Y \in B)=p\left(X^{-1}(A) \cap Y^{-1}(B)\right)$$
$\mathrm{~ 我 们 经 常 会 采 取 几 组 ~ V e f t { 1 O m e g a _ { 1 } , ~ \ d o t s , ~ I O m e g a _ { m }}$ $\Omega=\Omega_{1} \times \ldots \times \Omega_{m}$. 每个 $\Omega_{i}$ 与随机变量相关联。基于此，可以为所有这些随机变量一起定义联合概率分布。例 如，考虑 $\Omega=V \times P$ 在哪里 $V$ 是一个词汇表和 $P$ 是词性标签。这个样本空间使我们能够定义概率 $p(x, y)$ 在哪里 $X$ 表示与词性相关的词 $Y$. 在这种情况下， $x \in V$ 和 $y \in P$.

## 统计代写|贝叶斯分析代写Bayesian Analysis代考|CONDITIONAL DISTRIBUTIONS

$$p(X \in A \mid Y=y)=\frac{p(X \in A, Y=y)}{p(Y=y)}$$

$$p(Y=y)=\sum_{x} p(X=x, Y=y)$$

## 统计代写|贝叶斯分析代写Bayesian Analysis代考|BAYES’ RULE

$$p(Y=y \mid X=x)=\frac{p(X-x \mid Y-y) p(Y-y)}{p(X=x)}$$

$$p(X=x, Y=y) \quad=p(X=x) p(Y=y \mid X=x)=p(Y=y) p(X=x \mid Y=y)$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 统计代写|贝叶斯分析代写Bayesian Analysis代考|DATA5711

statistics-lab™ 为您的留学生涯保驾护航 在代写贝叶斯分析Bayesian Analysis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。我们的专家在代写贝叶斯分析Bayesian Analysis代写方面经验极为丰富，各种代写贝叶斯分析Bayesian Analysis相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• Advanced Probability Theory 高等概率论
• Advanced Mathematical Statistics 高等数理统计学
• (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|贝叶斯分析代写Bayesian Analysis代考|PROBABILITY MEASURES

At the core of probabilistic theory (and probabilistic modeling) lies the idea of a “sample space.” The sample space is a set $\Omega$ that consists of all possible elements over which we construct a probability distribution. In this book, the sample space most often consists of objects relating to language, such as words, phrase-structure trees, sentences, documents or sequences. As we see later, in the Bayesian setting, the sample space is defined to be a Cartesian product between a set of such objects and a set of model parameters (Section 1.5.1).

Once a sample space is determined, we can define a probability measure for that sample space. A probability measure $p$ is a function which attaches a real number to events-subsets of the sample space.
A probability measure has to satisfy three axiomatic properties:

• It has to be a non-negative function such that $p(A) \geq 0$ for any event $A$.
• For any countable disjoint sequence of events $A_{i} \subseteq \Omega, i \in{1, \ldots}$, if $A_{i} \cap A_{j}=\emptyset$ for $i \neq$ $j$, it should hold that $p\left(\bigcup_{i} A_{i}\right)=\sum_{i} p\left(A_{i}\right)$. This means that the sum of probabilities of disjoint events should equal the probability of the union of the events.
• The probability of $\Omega$ is $1: p(\Omega)=1$.

There are a few consequences from these three axiomatic properties. The first is that $p(\emptyset)=0$ (to see this, consider that $p(\Omega)+p(\emptyset)=p(\Omega \cup \emptyset)=p(\Omega)=1$ ). The second is that $p(A \cup B)=p(A)+p(B)-p(A \cap B)$ for any two events $A$ and $B$ (to see this, consider that $p(A \cup B)=p(A)+p(B \backslash(A \cap B))$ and that $p(B)=p(B \backslash(A \cap B))+p(A \cap B))$. And finally, the complement of an event $A, \Omega \backslash A$ is such that $p(\Omega \backslash A)=1-p(A)$ (to see this, consider that $1=p(\Omega)=p((\Omega \backslash A) \cup A)=p(\Omega \backslash A)+p(A)$ for any event $A)$.

In the general case, not every subset of the sample space should be considered an event.
From a measure-theoretic point of view for probability theory, an event must be a “measurable set.” The collection of measurable sets of a given sample space needs to satisfy some axiomatic properties. ${ }^{1}$ A discussion of measure theory is beyond the scope of this book, but see Ash and Doléans-Dade (2000) for a thorough investigation of this topic.

For our discrete sample spaces, consisting of linguistic structures or other language-related discrete objects, this distinction of measurable sets from arbitrary subsets of the sample space is not crucial. We will consider all subsets of the sample space to be measurable, which means they could be used as events. For continuous spaces, we will be using well-known probability measures that rely on Lebesgue’s measure. This means that the sample space will be a subset of a Euclidean space, and the set of events will be the subsets of this space that can be integrated over using Lebesgue’s integration.

## 统计代写|贝叶斯分析代写Bayesian Analysis代考|RANDOM VARIABLES

In their most basic form, random variables are functions that map each $w \in \Omega$ to a real value. They are often denoted by capital letters such as $X$ and $Z$. Once such a function is defined, under some regularity conditions, it induces a probability measure over the real numbers. More specifically, for any $A \subseteq \mathbb{R}$ such that the pre-image, $X^{-1}(A)$, defined as, ${\omega \in \Omega \mid X(\omega) \in A}$, is an event, its probability is:
$$p_{X}(A)=p(X \in A)=p\left(X^{-1}(A)\right),$$
where $p_{X}$ is the probability measure induced by the random variable $X$ and $p$ is a probability measure originally defined for $\Omega$. The sample space for $p_{X}$ is $\mathbb{R}$. The set of events for this sample space includes all $A \subseteq \mathbb{R}$ such that $X^{-1}(A)$ is an event in the original sample space $\Omega$ of $p$.
It is common to define a statistical model directly in terms of random variables, instead of explicitly defining a sample space and its corresponding real-value functions. In this case, random variables do not have to be interpreted as real-value functions and the sample space is understood to be a range of the random variable function. For example, if one wants to define a probability distribution over a language vocabulary, then one can define a random variable $X(\omega)=\omega$ with $\omega$ ranging over words in the vocabulary. Following this, the probability of a word in the vocabulary is denoted by $p(X \in{\omega})=p(X=\omega)$.

## 统计代写|贝叶斯分析代写Bayesian Analysis代考|CONTINUOUS AND DISCRETE RANDOM VARIABLES

This book uses the two most common kinds of random variables available in statistics: continuous and discrete. Continuous random variables take values in a continuous space, usually a subspace of $\mathbb{R}^{d}$ for $d \geq 1$. Discrete random variables, on the other hand, take values from a discrete, possibly countable set. In this book, discrete variables are usually denoted using capital letters such as $X, Y$ and $Z$, while continuous variables are denoted using greek letters, such as $\theta$ and $\mu$.

The continuous variables in this book are mostly used to define a prior over the parameters of a discrete distribution, as is usually done in the Bayesian setting. See Section $1.5 .2$ for a discussion of continuous variables. The discrete variables, on the other hand, are used to model structures that will be predicted (such as parse trees, part-of-speech tags, alignments, clusters) or structures which are observed (such as a sentence, a string over some language vocabulary or other such sequences).

The discrete variables discussed in this book are assumed to have an underlying probability mass function (PMF)-i.e., a function that attaches a weight to each element in the sample space, $p(x)$. This probability mass function induces the probability measure $p(X \in A)$, which satisfies:
$$p(X \in A)=\sum_{x \in A} p(x),$$
where $A$ is a subset of the possible values $X$ can take. Note that this equation is the result of the axiom of probability measures, where the probability of an event equals the sum of probabilities of disjoint events that precisely cover that event (singletons, in our case).

## 统计代写|贝叶斯分析代写Bayesian Analysis代考|PROBABILITY MEASURES

• 它必须是一个非负函数，使得 $p(A) \geq 0$ 对于任何事件 $A$.
• 对于任何可数的不相交的事件序列 $A_{i} \subseteq \Omega, i \in 1, \ldots$ ，如果 $A_{i} \cap A_{j}=\emptyset$ 为了 $i \neq j$ ，它应该认为 $p\left(\bigcup_{i} A_{i}\right)=\sum_{i} p\left(A_{i}\right)$. 这意味着不相交事件的概率之和应该等于事件联合的概率。
• 的概率 $\Omega$ 是 $1: p(\Omega)=1$.
这三个公理性质有一些结果。第一个是 $p(\emptyset)=0$ (要看到这一点，请考虑
$p(\Omega)+p(\emptyset)=p(\Omega \cup \emptyset)=p(\Omega)=1)$ 。第二个是 $p(A \cup B)=p(A)+p(B)-p(A \cap B)$ 对于任何两 个事件 $A$ 和 $B$ (要看到这一点，请考虑 $p(A \cup B)=p(A)+p(B \backslash(A \cap B))$ 然后
$p(B)=p(B \backslash(A \cap B))+p(A \cap B))$. 最后，事件的补充 $A, \Omega \backslash A$ 是这样的 $p(\Omega \backslash A)=1-p(A)$ (要看到 这一点，请考虑 $1=p(\Omega)=p((\Omega \backslash A) \cup A)=p(\Omega \backslash A)+p(A)$ 对于任何事件 $A)$.
在一般情况下，并非样本空间的每个子集都应被视为一个事件。
从概率论的测度论观点来看，一个事件必须是一个”可测集”。给定样本空间的可测量集的集合需要满足一些公理性 质。 1 测度论的讨论超出了本书的范围，但请参阅 Ash 和 Doléans-Dade (2000) 对该主题的深入研究。
对于我们的离散样本空间，由语言结构或其他与语言相关的离散对象组成，可测量集与样本空间任意子集的区别并 不重要。我们将认为样本空间的所有子集都是可测量的，这意味着它们可以用作事件。对于连续空间，我们将使用 依赖于 Lebesgue 测度的众所周知的概率测度。这意味着样本空间将是欧几里得空间的子集，而事件集将是该空间 的子集，可以使用 Lebesgue 积分进行积分。

## 统计代写|贝叶斯分析代写Bayesian Analysis代考|RANDOM VARIABLES

$$p_{X}(A)=p(X \in A)=p\left(X^{-1}(A)\right),$$

$$p(X \in \omega)=p(X=\omega) \text {. }$$

## 统计代写|贝叶斯分析代写Bayesian Analysis代考|CONTINUOUS AND DISCRETE RANDOM VARIABLES

$$p(X \in A)=\sum_{x \in A} p(x)$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 数学代写|拓扑学代写Topology代考|MTH 3002

statistics-lab™ 为您的留学生涯保驾护航 在代写拓扑学Topology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。我们的专家在代写拓扑学Topology代写方面经验极为丰富，各种代写拓扑学Topology相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• Advanced Probability Theory 高等概率论
• Advanced Mathematical Statistics 高等数理统计学
• (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|拓扑学代写Topology代考|Gradients and Critical Points

In what follows, for simplicity of presentation, we assume that we consider smooth ( $C^{\infty}$-continuous) functions and smooth manifolds embedded in $\mathbb{R}^{d}$, even though often we only require the functions (resp. manifolds) to be $C^{2}$ continuous (resp. $C^{2}$-smooth).

To provide intuition, let us start with a smooth scalar function defined on the real line, $f: \mathbb{R} \rightarrow \mathbb{R}$; the graph of such a function is shown in Figure $1.8(\mathrm{~b})$. Recall that the derivative of a function at a point $x \in \mathbb{R}$ is defined as
$$D f(x)=\frac{d}{d x} f(x)=\lim _{t \rightarrow 0} \frac{f(x+t)-f(x)}{t}$$

The value $D f(x)$ gives the rate of change of the value of $f$ at $x$. This can be visualized as the slope of the tangent line of the graph of $f$ at $(x, f(x))$. The critical points of $f$ are the set of points $x$ such that $D f(x)-0$. For a function defined on the real line, there are two types of critical points in the generic case: maxima and minima, as marked in Figure $1.8(b)$.

Now suppose we have a smooth function $f: \mathbb{R}^{d} \rightarrow \mathbb{R}$ defined on $\mathbb{R}^{d}$. Fix an arbitrary point $x \in \mathbb{R}^{d}$. As we move a little around $x$ within its local neighborhood, the rate of change of $f$ differs depending on which direction we move. This gives rise to the directional derivative $D_{v} f(x)$ at $x$ in direction (i.e., a unit vector) $v \in \mathbb{S}^{d-1}$, where $\mathbb{S}^{d-1}$ is the unit $(d-1)$-sphere, defined as
$$D_{v} f(x)=\lim _{t \rightarrow 0} \frac{f(x+t \cdot v)-f(x)}{t}$$
The gradient vector of $f$ at $x \in \mathbb{R}^{d}$ intuitively captures the direction of steepest increase of the function $f$. More precisely, we have the following.

Definition 1.25. (Gradient for functions on $\mathbb{R}^{d}$ ) Given a smooth function $f$ : $\mathbb{R}^{d} \rightarrow \mathbb{R}$, the gradient vector field $\nabla f: \mathbb{R}^{d} \rightarrow \mathbb{R}^{d}$ is defined as follows: for any $x \in \mathbb{R}^{d}$
$$\nabla f(x)=\left[\frac{\partial f}{\partial x_{1}}(x), \frac{\partial f}{\partial x_{2}}(x), \ldots, \frac{\partial f}{\partial x_{d}}(x)\right]^{\mathrm{T}}$$
where $\left(x_{1}, x_{2}, \ldots, x_{d}\right)$ represents an orthonormal coordinate system for $\mathbb{R}^{d}$. The vector $\nabla f(x) \in \mathbb{R}^{d}$ is called the gradient vector of $f$ at $x$. A point $x \in \mathbb{R}^{d}$ is a critical point if $\nabla f(x)=\left[\begin{array}{llll}0 & 0 & \ldots\end{array}\right]^{\mathrm{T}}$; otherwise, $x$ is regular.

## 数学代写|拓扑学代写Topology代考|Connection to Topology

We now characterize how critical points influence the topology of $M$ induced by the scalar function $f: M \rightarrow \mathbb{R}$.

Detinition 1.29. (Interval, sublevel, and superlevel sets) Given $f: M \rightarrow \mathbb{K}$ and $I \subseteq \mathbb{R}$, the interval levelset of $f$ with respect to $I$ is defined as
$$M_{I}=f^{-1}(I)={x \in M \mid f(x) \in I}$$
The case for $I=(-\infty, a]$ is also referred to as the sublevel set $M_{\leq a}:=$ $f^{-1}((-\infty, a])$ of $f$, while $M_{\geq a}:=f^{-1}([a, \infty))$ is called the superlevel set; and $f^{-1}(a)$ is called the levelset of $f$ at $a \in \mathbb{R} .$

Given $f: M \rightarrow \mathbb{R}$, imagine sweeping $M$ with increasing function values of $f$. It turns out that the topology of the sublevel sets can only change when we sweep through critical values of $f$. More precisely, we have the following classical result, where a diffeomorphism is a homeomorphism that is smooth in both directions.

Theorem 1.3. (Homotopy type of sublevel sets) Let $f: M \rightarrow \mathbb{R}$ be a smooth function defined on a manifold $M$. Given $a<b$, suppose the interval levelset $M_{[a, b]}=f^{-1}([a, b])$ is compact and contains no critical points of $f$. Then $M_{\leq a}$ is diffeomorphic to $M_{\leq b}$.

Furthermore, $M_{\leq a}$ is a deformation retract of $M_{\leq b}$, and the inclusion map $i: M_{\leq a} \hookrightarrow M_{\leq b}$ is a homotopy equivalence .

As an illustration, consider the example of height function $f: M \rightarrow \mathbb{R}$ defined on a vertical torus as shown in Figure $1.10(a)$. There are four critical points for the height function $f, u$ (minimum), $v, w$ (saddles), and $z$ (maximum). We have that $M_{\leq a}$ is: (i) empty for $af(z)$.
Theorem $1.3$ states that the homotopy type of the sublevel set remains the same until it passes a critical point. For Morse functions, we can also characterize the homotopy type of sublevel sets around critical points, captured by attaching $k$-cells.

## 数学代写|拓扑学代写Topology代考|Complexes and Homology Groups

This chapter introduces two very basic tools on which topological data analysis (TDA) is built. One is simplicial complexes and the other is homology groups. Data supplied as a discrete set of points do not have an interesting topology. Usually, we construct a scaffold on top of the data which is commonly taken as a simplicial complex. It consists of vertices at the data points, edges connecting them, and triangles, tetrahedra, and their higher-dimensional analogues that establish higher-order connectivity. Section $2.1$ formalizes this construction. There are different kinds of simplicial complexes. Some are easier to compute, but take more space. Others are more sparse, but take more time to compute. Section $2.2$ presents an important construction called the nerve and a complex called the Cech complex which is defined on this construction. This section also presents a commonly used complex in topological data analysis called the Vietoris-Rips complex that interleaves with the Cech complexes in terms of containment. In Section 2.3, we introduce some of the complexes which are sparser in size than the Vietoris-Rips or Čech complexes.

The second topic of this chapter, the homology groups of a simplicial complex, are the essential algebraic structures with which TDA analyzes data. Homology groups of a topological space capture the space of cycles up to those called boundaries that bound “higher-dimensional” subsets. For simplicity, we introduce the concept in the context of simplicial complexes instead of topological spaces. This is called simplicial homology. The essential entities for defining the homology groups are chains, cycles, and boundaries which we cover in Section 2.4. For simplicity and also for relevance in TDA, we define these structures under $\mathbb{Z}_{2}$-additions.

Section $2.5$ defines the simplicial homology group of a simplicial complex as the quotient space of the cycles with respect to the boundaries. Some of the concepts related to homology groups, such as induced homology under a map, singular homology groups for general topological spaces, relative homology groups of a complex with respect to a subcomplex, and the dual concept of homology groups, called cohomology groups are also introduced in this section.

## 数学代写|拓扑学代写Topology代考|Gradients and Critical Points

$$D f(x)=\frac{d}{d x} f(x)=\lim {t \rightarrow 0} \frac{f(x+t)-f(x)}{t}$$ 价值 $D f(x)$ 给出值的变化率 $f$ 在 $x$. 这可以可视化为图形的切线的斜率 $f$ 在 $(x, f(x))$. 的关键点 $f$ 是点的集合 $x$ 这样 $D f(x)-0$. 对于定义在实线上的函数，一般情况下有两种临界点：极大值和极小值，如图所示 $1.8(b)$. 现在假设我们有一个平滑函数 $f: \mathbb{R}^{d} \rightarrow \mathbb{R}$ 定义于 $\mathbb{R}^{d}$. 修复任意点 $x \in \mathbb{R}^{d}$. 当我们稍微移动一下 $x$ 在其当地社区 内，变化率 $f$ 根据我们移动的方向而有所不同。这产生了方向导数 $D{v} f(x)$ 在 $x$ 在方向 (即，单位向量) $v \in \mathbb{S}^{d-1}$ ，在哪里S ${ }^{d-1}$ 是单位 $(d-1)$-sphere，定义为
$$D_{v} f(x)=\lim {t \rightarrow 0} \frac{f(x+t \cdot v)-f(x)}{t}$$ 的梯度向量 $f$ 在 $x \in \mathbb{R}^{d}$ 直观地捕捉到函数增长最陡的方向 $f$. 更准确地说，我们有以下内容。 定义 1.25。(函数的梯度 $\mathbb{R}^{d}$ ) 给定一个平滑函数 $f: \mathbb{R}^{d} \rightarrow \mathbb{R}$, 梯度向量场 $\nabla f: \mathbb{R}^{d} \rightarrow \mathbb{R}^{d}$ 定义如下: 对于任何 $x \in \mathbb{R}^{d}$ $$\nabla f(x)=\left[\frac{\partial f}{\partial x{1}}(x), \frac{\partial f}{\partial x_{2}}(x), \ldots, \frac{\partial f}{\partial x_{d}}(x)\right]^{\mathrm{T}}$$

## 数学代写|拓扑学代写Topology代考|Connection to Topology

$$M_{I}=f^{-1}(I)=x \in M \mid f(x) \in I$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 数学代写|拓扑学代写Topology代考|MAST90023

statistics-lab™ 为您的留学生涯保驾护航 在代写拓扑学Topology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。我们的专家在代写拓扑学Topology代写方面经验极为丰富，各种代写拓扑学Topology相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• Advanced Probability Theory 高等概率论
• Advanced Mathematical Statistics 高等数理统计学
• (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|拓扑学代写Topology代考|Manifolds

A manifold is a topological space that is locally connected in a particular way. A 1-manifold has this local connectivity looking like a segment. A 2manifold (with boundary) has the local connectivity looking like a complete or partial disk. In layman’s terms. a 2-manifold has the structure of a piece of paper or rubber sheet, possibly with the boundaries glued together to form a closed surface – a category that includes disks, spheres, tori, and Möhius bands.

Definition 1.22. (Manifold) A topological space $M$ is an m-manifold, or simply a manifold, if every point $x \in M$ has a neighborhood homeomorphic to $\mathbb{B}_{o}^{m}$ or $\mathrm{H}^{m}$. The dimension of $M$ is $m$.

Every manifold can be partitioned into boundary and interior points. Observe that these words mean very different things for a manifold than they do for a metric space or topological space.

Definition 1.23. (Boundary; Interior) The interior Int $M$ of an $m$-manifold $M$ is the set of points in $M$ that have a neighborhood homeomorphic to $\mathbb{B}_{o}^{m}$. The boundary $\mathrm{Bd} M$ of $M$ is the set of points $M \backslash$ Int $M$. The boundary $\mathrm{Bd} M$, if not empty, consists of the points that have a neighborhood homeomorphic to $\mathrm{H}^{m}$. If $\mathrm{Bd} M$ is the empty set, we say that $M$ is without boundary.

## 数学代写|拓扑学代写Topology代考|Smooth Manifolds

A purely topological manifold has no geometry. But if we embed it in a Euclidean space, it could appear smooth or wrinkled. We now introduce a “geometric” manifold by imposing a differential structure on it. For the rest of this chapter, we focus on only manifolds without boundary.

Consider a map $\phi: U \rightarrow W$ where $U$ and $W$ are open sets in $\mathbb{R}^{k}$ and $\mathbb{R}^{d}$, respectively. The map $\phi$ has $d$ components, namely $\phi(x)=$ $\left(\phi_{1}(x), \phi_{2}(x), \ldots, \phi_{d}(x)\right)$, where $x=\left(x_{1}, x_{2}, \ldots, x_{k}\right)$ denotes a point in $\mathbb{R}^{k}$. The Jacobian of $\phi$ at $x$ is the $d \times k$ matrix of the first-order partial derivatives
$$\left[\begin{array}{ccc} \frac{\partial \phi_{1}(x)}{\partial x_{1}} & \cdots & \frac{\partial \phi_{1}(x)}{\partial x_{k}} \ \vdots & \ddots & \vdots \ \frac{\partial \phi_{d}(x)}{\partial x_{1}} & \cdots & \frac{\partial \phi_{d}(x)}{\partial x_{k}} \end{array}\right]$$
The map $\phi$ is regular if its Jacobian has rank $k$ at every point in $U$. The map $\phi$ is $C^{i}$-continuous if the $i$-th-order partial derivatives of $\phi$ are continuous.

The reader may be familiar with parametric surfaces, for which $U$ is a twodimensional parameter space and its image $\phi(U)$ in $d$-dimensional space is a parametric surface. Unfortunately, a single parametric surface cannot easily represent a manifold with a complicated topology. However, for a manifold to be smooth, it suffices that each point on the manifold has a neighborhood that looks like a smooth parametric surface.

## 数学代写|拓扑学代写Topology代考|Functions on Smooth Manifolds

In previous sections, we introduced topological spaces, including the special case of (smooth) manifolds. Very often, a space can be equipped with continuous functions defined on it. In this section, we focus on real-valued functions of the form $f: X \rightarrow \mathbb{R}$ defined on a topological space $X$, also called scalar functions; see Figure 1.8(a) for the graph of a function $f: \mathbb{R}^{2} \rightarrow \mathbb{R}$. Scalar functions appear commonly in practice that describe space/data of interest (e.g., the elevation function defined on the surface of the Earth). We are interested in the topological structures behind scalar functions. In this section, we limit our discussion to nicely behaved scalar functions (called Morse functions) defined on smooth manifolds. Their topological structures are characterized by the so-called critical points which we will introduce below. Later in the book we will also discuss scalar functions on simplicial complex domains, as well as more complex maps defined on a space $X$, for example, a multivariate function $f: X \rightarrow \mathbb{R}^{d}$

## 数学代写|拓扑学代写Topology代考|Smooth Manifolds

[∂φ1(X)∂X1⋯∂φ1(X)∂Xķ ⋮⋱⋮ ∂φd(X)∂X1⋯∂φd(X)∂Xķ]

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 数学代写|拓扑学代写Topology代考|MATH3061

statistics-lab™ 为您的留学生涯保驾护航 在代写拓扑学Topology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。我们的专家在代写拓扑学Topology代写方面经验极为丰富，各种代写拓扑学Topology相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• Advanced Probability Theory 高等概率论
• Advanced Mathematical Statistics 高等数理统计学
• (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|拓扑学代写Topology代考|Topological Space

The basic object in a topological space is a ground set whose elements are called points. A topology on these points specifies how they are connected by listing what points constitute a neighborhood – the so-called open set.

The expression “rubber-sheet topology” commonly associated with the term “topology” exemplifies this idea of connectivity of neighborhoods. If we bend and stretch a sheet of rubber, it changes shape but always preserves the neighborhoods in terms of the points and how they are connected.

We first introduce basic notions from point set topology. These notions are prerequisites for more sophisticated topological ideas – manifolds, homeomorphism, isotopy, and other maps – used later to study algorithms for topological data analysis. Homeomorphisms, for example, offer a rigorous way to state that an operation preserves the topology of a domain, and isotopy offers a rigorous way to state that the domain can be deformed into a shape without ever colliding with itself.

Perhaps it is more intuitive to understand the concept of topology in the presence of a metric because then we can use the metric balls such as Euclidean balls in a Euclidean space to define neighborhoods – the open sets. Topological spaces provide a way to abstract out this idea without a metric or point coordinates, so they are more general than metric spaces. In place of a metric, we encode the connectivity of a point set by supplying a list of all of the open sets. This list is called a system of subsets of the point set. The point set and its system together describe a topological space.

## 数学代写|拓扑学代写Topology代考|Metric Space Topology

Metric spaces are a special type of topological space commonly encountered in practice. Such a space admits a metric that specifies the scalar distance between every pair of points satisfying certain axioms.

Definition 1.8. (Metric space) A metric space is a pair ( $\mathbb{d}, \mathrm{d}$ ) where $\mathbb{T}$ is a set and $d$ is a distance function $d: \mathbb{I} \times \mathbb{T} \rightarrow \mathbb{R}$ satisfying the following properties:

• $\mathrm{d}(p, q)=0$ if and only if $p=q$ for all $p \in \mathbb{T}$;
• $\mathrm{d}(p, q)=\mathrm{d}(q, p)$ for all $p, q \in \mathbb{T}$;
• $\mathrm{d}(p, q) \leq \mathrm{d}(p, r)+\mathrm{d}(r, q)$ for all $p, q, r \in \mathbb{T}$.
It can be shown that the three axioms above imply that $\mathrm{d}(p, q) \geq 0$ for every pair $p, q \in \mathbb{T}$. In a metric space $\mathbb{T}$, an open metric ball with center $c$ and radius $r$ is defined to be the point set $B_{0}(c, r)={p \in \mathbb{T}: \mathrm{d}(p, c)<r}$. Metric balls definé a topology on a metric spacé.

Definition 1.9. (Metric space topology) Given a metric space $\mathbb{T}$, all metric balls $\left{B_{o}(c, r) \mid c \in \mathbb{T}\right.$ and $\left.0<r \leq \infty\right}$ and their union constituting the open sets define a topology on $\mathbb{T}$.

All definitions for general topological spaces apply to metric spaces with the above defined topology. However, we give alternative definitions using the concept of limit points which may be more intuitive.

As we have mentioned already, the heart of topology is the question of what it means for a set of points to be connected. After all, two distinct points cannot be adjacent to each other; they can only be connected to one another by passing through uncountably many intermediate points. The idea of limit points helps express this concept more concretely, specifically in the case of metric spaces.
We use the notation $\mathrm{d}(\cdot, \cdot)$ to express minimum distances between point sets $P, Q \subseteq \mathbb{T}$
\begin{aligned} \mathrm{d}(p, Q) &=\inf {\mathrm{d}(p, q): q \in Q} \ \mathrm{d}(P, Q) &=\inf {\mathrm{d}(p, q): p \in P, q \in Q} \end{aligned}

## 数学代写|拓扑学代写Topology代考|Maps, Homeomorphisms, and Homotopies

The equivalence of two topological spaces is determined by how the points that comprise them are connected. For example, the surface of a cube can be deformed into a sphere without cutting or gluing it because they are connected the same way. They have the same topology. This notion of topological equivalence can be formalized via functions that send the points of one space to points of the other while preserving the connectivity.

This preservation of connectivity is achieved by preserving the open sets. A function from one space to another that preserves the open sets is called a continuous function or a map. Continuity is a vehicle to define topological equivalence, because a continuous function can send many points to a single point in the target space, or send no points to a given point in the target space. If the former does not happen, that is, when the function is injective, we call it an embedding of the domain into the target space. True equivalence is given by a homeomorphism, a bijective function from one space to another which has continuity as well as a continuous inverse. This ensures that open sets are preserved in both directions.

A topological space can be embedded into a Euclidean space by assigning coordinates to its points so that the assignment is continuous and injective. For example, drawing a triangle on paper is an embedding of $\mathbb{S}^{1}$ into $\mathbb{R}^{2}$. There are topological spaces that cannot be embedded into a Euclidean space, or even into a metric space – these spaces cannot be represented by any metric.

Next we define a homeomorphism that connects two spaces that have essentially the same topology.

## 数学代写|拓扑学代写Topology代考|Metric Space Topology

• $\mathrm{d}(p, q)=0$ 当且仅当 $p=q$ 对所有人 $p \in \mathbb{T}$;
• $\mathrm{d}(p, q)=\mathrm{d}(q, p)$ 对所有人 $p, q \in \mathbb{T}$;
• $\mathrm{d}(p, q) \leq \mathrm{d}(p, r)+\mathrm{d}(r, q)$ 对所有人 $p, q, r \in \mathbb{T}$.
可以证明，上面的三个公理意味着 $\mathrm{d}(p, q) \geq 0$ 对于每一对 $p, q \in \mathbb{T}$. 在度量空间中 $\mathbb{T}$,一个带中心的开放公制 球 $c$ 和半径 $r$ 被定义为点集 $B_{0}(c, r)=p \in \mathbb{T}: \mathrm{d}(p, c)<r$. 度量球定义度量空间上的拓扑。
定义 1.9。 (度量空间拓扑) 给定一个度量空间 $\mathbb{T}$ ，所有公制球
$\mathrm{~ U l e f t { B _ { o } ( c , r )}$
一般拓扑空间的所有定义都适用于具有上述定义的拓扑的度量空间。但是，我们使用可能更直观的极限点概念给出 了替代定义。
正如我们已经提到的，拓扑的核心是连接一组点意味着什么的问题。毕竟，两个不同的点不能彼此相邻；它们只有 通过无数的中间点才能相互连接。极限点的概念有助于更具体地表达这个概念，特别是在度量空间的情况下。 我们使用符号 $\mathrm{d}(\cdot, \cdot)$ 表示点集之间的最小距离 $P, Q \subseteq \mathbb{T}$
$$\mathrm{d}(p, Q)=\inf \mathrm{d}(p, q): q \in Q \mathrm{~d}(P, Q) \quad=\inf \mathrm{d}(p, q): p \in P, q \in Q$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 金融代写|量化风险管理代写Quantitative Risk Management代考|BUSA90315

statistics-lab™ 为您的留学生涯保驾护航 在代写量化风险管理Quantitative Risk Management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。我们的专家在代写量化风险管理Quantitative Risk Management代写方面经验极为丰富，各种代写量化风险管理Quantitative Risk Management相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• Advanced Probability Theory 高等概率论
• Advanced Mathematical Statistics 高等数理统计学
• (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

## 金融代写|量化风险管理代写Quantitative Risk Management代考|Market Risk

Market risk is risk associated with changing asset values. Market risk is most often associated with assets that trade in liquid financial markets, such as stocks and bonds. During trading hours, the prices of stocks and bonds constantly fluctuate. An asset’s price will change as new information becomes available and investors reassess the value of that asset. An asset’s value can also change due to changes in supply and demand.

All financial assets have market risk. Even if an asset is not traded on an exchange, its value can change over time. Firms that use mark-to-market accounting recognize this change explicitly. For these firms, the change in value of exchange-traded assets will be based on market prices. Other assets will either be marked to model -that is, their prices will be determined based on financial models with inputs that may include market prices-or their prices will be based on broker quotes – that is, their prices will be based on the price at which another party expresses their willingness to buy or sell the assets. Firms that use historical cost accounting, or book value accounting, will normally only realize a profit or a loss when an asset is sold. Even if the value of the asset is not being updated on a regular basis, the asset still has market risk. For this reason, most firms that employ historical cost accounting will reassess the value of their portfolios when they have reason to believe that there has been a significant change in the value of their assets.

For most financial instruments, we expect price changes to be relatively smooth and continuous most of the time, and large and discontinuous rarely. Because of this, market risk models often involve continuous distribution. Market risk models can also have a relatively high frequency (i.e., daily or even intraday). For many financial instruments, we will have a large amount of historical market data that we can use to evaluate market risk.

## 金融代写|量化风险管理代写Quantitative Risk Management代考|Credit Risk

Credit risk is the risk that one party in a financial transaction will fail to pay the other party. Credit risk can arise in a number of different settings. Firms may extend credit to suppliers and customers. Credit card debt and home mortgages create credit risk. One of the most common forms of credit risk is the risk that a corporation or government will fail to make interest payments or to fully repay the principal on bonds they have issued. This type of risk is known as default risk, and in the case of national governments it is also referred to as sovereign risk. Defaults occur infrequently, and the simplest models of default risk are based on discrete distributions. Although bond markets are large and credit rating agencies have been in existence for a long time, default events are rare. Because of this, we have much less historical data to work with when developing credit models, compared to market risk models.

For financial firms, counterparty credit risk is another important source of credit risk. While credit risk always involves two counterparties, when risk managers talk about counterparty credit risk they are usually talking about the risk arising from a significant long-term relationship between two counterparties. Prime brokers will often provide loans to investment firms, provide them with access to emergency credit lines, and allow them to purchase securities on margin. Assessing the credit risk of a financial firm can be difficult, time consuming, and costly. Because of this, when credit risk is involved, financial firms often enter into long-term relationships based on complex legal contracts. Counterparty risk specialists help design these contracts and play a lead role in assessing and monitoring the risk of counterparties.

Derivatives contracts can also lead to credit risk. A derivative is essentially a contract between two parties, that specifies that certain payments be made based on the value of an underlying security or securities. Derivatives include futures, forwards, swaps, and options. As the value of the underlying asset changes, so too will the value of the derivative. As the value of the derivative changes, so too will the amount of money that the counterparties owe each other. This leads to credit risk.

## 金融代写|量化风险管理代写Quantitative Risk Management代考|Enterprise Risk

The enterprise risk management group of a firm, as the name suggests, is responsible for the risk of the entire firm. At large financial firms, this often means overseeing market, credit, liquidity, and operations risk groups, and combining information from those groups into summary reports. In addition to this aggregation role, enterprise risk management tends to look at overall business risk. Large financial companies will often have a number of business units (e.g., capital markets, corporate finance, commercial banking, retail banking, asset management, etc.). Some of these business units will work very closely with risk management (e.g. capital markets, asset management), while others may have very little day-to-day interaction with risk (e.g. corporate finance). Regardless, enterprise risk management would assess how each business unit contributes to the overall profitability of the firm in order to assess the overall risk to the firm’s revenue, income, and capital.

Operational risk is risk arising from all aspects of a firm’s business activities. Put simply, it is the risk that people will make mistakes and that systems will fail. Operational risk is a risk that all financial firms must deal with.

Just as the number of activities that businesses carry out is extremely large, so too are the potential sources of operational risk. That said, there are broad categories on which risk managers tend to focus. These include legal risk (most often risk arising from contracts, which may be poorly specified or misinterpreted), systems risk (risk arising from computer systems) and model risk (risk arising from pricing and risk models, which may contain errors, or may be used inappropriately).

As with credit risk, operational risk tends to be concerned with rare but significant events. Operational risk presents additional challenges in that the sources of operational risk are often difficult to identify, define, and quantify.

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 金融代写|量化风险管理代写Quantitative Risk Management代考|MKTG 7023

statistics-lab™ 为您的留学生涯保驾护航 在代写量化风险管理Quantitative Risk Management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。我们的专家在代写量化风险管理Quantitative Risk Management代写方面经验极为丰富，各种代写量化风险管理Quantitative Risk Management相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• Advanced Probability Theory 高等概率论
• Advanced Mathematical Statistics 高等数理统计学
• (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

## 金融代写|量化风险管理代写Quantitative Risk Management代考|Intrinsic and Extrinsic Risk

Some financial professionals talk about risk versus uncertainty. A better approach might be to contrast intrinsic risk and extrinsic risk.

When evaluating financial instruments, there are some risks that we consider to be intrinsic. No matter how much we know about the financial instrument we are evaluating, there is nothing we can do to reduce this intrinsic risk (other than reducing the size of our investment).

In other circumstances risk is due only to our own ignorance. In theory, this extrinsic risk can be eliminated by gathering additional information through research and analysis.

As an example, an investor in a hedge fund may be subject to both extrinsic and intrinsic risk. A hedge fund investor will typically not know the exact holdings of a hedge fund in which they are invested. Not knowing what securities are in a fund is extrinsic risk.

For various reasons, the hedge fund manager may not want to reveal the fund’s holdings, but, at least in theory, this extrinsic risk could be eliminated by revealing the fund’s holdings to the investor. At the same time, even if the investor did know what securities were in the fund, the returns of the fund would still not be fully predictable because the returns of the securities in the fund’s portfolio are inherently uncertain. This inherent uncertainty of the security returns represents intrinsic risk and it cannot be eliminated, no matter how much information is provided to the investor.

Interestingly, a risk manager could reduce a hedge fund investor’s extrinsic risk by explaining the hedge fund’s risk guidelines. The risk guidelines could help the investor gain a better understanding of what might be in the fund’s portfolio, without revealing the portfolio’s precise composition.

Differentiating between these two fundamental types of risk is important in financial risk management. In practice, financial risk management is as much about reducing extrinsic risk as it is about managing intrinsic risk.

## 金融代写|量化风险管理代写Quantitative Risk Management代考|Risk and Standard Deviation

At the start of this chapter, we said that risk could be defined in terms of possible deviations from expectations. This definition is very close to the definition of standard deviation in statistics. The variance of a random variable is the expected value of squared deviations from the mean, and standard deviation is just the square root of variance. This is indeed very close to our definition of risk, and in finance risk is often equated with standard deviation.

While the two definitions are similar, they are not exactly the same. Standard deviation only describes what we expect the deviations will look like on average. Two random variables can have the same standard deviation, but very different return profiles. As we will see, risk managers need to consider other aspects of the distribution of expected deviations, not just its standard deviation.

## 金融代写|量化风险管理代写Quantitative Risk Management代考|WHAT IS FINANCIAL RISK MANAGEMENT

In finance and in this book, we often talk about risk management, when it is understood that we are talking about financial risk management. Risk managers are found in a number of fields outside of finance, including engineering, manufacturing, and medicine.

When civil engineers are designing a levee to hold back flood waters, their risk analysis will likely include a forecast of the distribution of peak water levels. An engineer will often describe the probability that water levels will exceed the height of the levee in terms similar to those used by financial risk managers to describe the probability that losses in a portfolio will exceed a certain threshold. In manufacturing, engineers will use risk management to assess the frequency of manufacturing defects. Motorola popularized the term Six Sigma to describe its goal to establish a manufacturing process where manufacturing defects were kept below $3.4$ defects per million. (Confusingly the goal corresponds to $4.5$ standard deviations for a normal distribution, not 6 standard deviations, but that’s another story.) Similarly, financial risk managers will talk about big market moves as being three-sigma events or six-sigma events. Other areas of risk management can be valuable sources of techniques and terminology for financial risk management.

Within this broader field of risk management, though, how do we determine what is and is not financial risk management? One approach would be to define risk in terms of organizations, to say that financial risk management concerns itself with the risk of financial firms. By this definition, assessing the risks faced by Goldman Sachs or a hedge fund is financial risk management, whereas assessing the risks managed by the Army Corps of Engineers or NASA is not. A clear advantage to this approach is that it saves us from having to create a long list of activities that are the proper focus of financial risk management. The assignment is unambiguous. If a task is being performed by a financial firm, it is within the scope of financial risk management. This definition is future proof as well. If HSBC, one of the world’s largest financial institutions, starts a new business line tomorrow, we do not have to ask ourselves if this new business line falls under the purview of financial risk management. Because HSBC is a financial firm, any risk associated with the new business line would be considered financial risk.

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 金融代写|量化风险管理代写Quantitative Risk Management代考|FINC6023

statistics-lab™ 为您的留学生涯保驾护航 在代写量化风险管理Quantitative Risk Management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。我们的专家在代写量化风险管理Quantitative Risk Management代写方面经验极为丰富，各种代写量化风险管理Quantitative Risk Management相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• Advanced Probability Theory 高等概率论
• Advanced Mathematical Statistics 高等数理统计学
• (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

## 金融代写|量化风险管理代写Quantitative Risk Management代考|OVERVIEW OF FINANCIAL RISK MANAGEMENT

Imagine you are a chef at a restaurant. You’ve just finished preparing eggs benedict for a customer. The eggs are cooked perfectly, the hollandaise sauce has just the right mix of ingredients, and it all sits perfectly on the plate. The presentation is perfect! You’re so proud of the way this has turned out that you decide to deliver the dish to the customer yourself. You place the plate in front of the customer, and she replies, “This looks great, but I ordered a filet mignon, and you forgot my drink.”

Arguably, the greatest strength of modern financial risk management is that it is highly objective. It takes a scientific approach, using math and statistics to measure and evaluate financial products and portfolios. While these mathematical tools can be very powerful, they are simply that-tools. If we make unwarranted assumptions, apply models incorrectly, or present results poorly – or if our findings are ignored – then the most elegant mathematical models in the world will not help us. The eggs might be perfect, but that’s irrelevant if the customer ordered a steak.

This is not a new idea, Vitruvius, a famous Roman architect wrote, “Neque enim ingenium sine disciplina aut disciplina sine ingenio perfectum artificem potest efficere”, which roughly translates to “Neither genius without knowledge, nor knowledge without genius, will make a perfect artist.” Applying this to risk management, we might say, “Neither math without knowledge of financial markets, nor knowledge of financial markets without math, will make a perfect risk manager.”

Before we get to the math and statistics, then, we should take a step back and look at risk management more broadly. Before delving into the models, we explore the following questions: What is risk management? What is the proper role for a risk manager within a financial organization? What do risk managers actually do on a day-to-day basis?

We end this chapter with a brief history of risk management. As you will see, risk management has made many positive contributions to finance, but it is far from being a solved problem.

## 金融代写|量化风险管理代写Quantitative Risk Management代考|WHAT IS RISK

Before we can begin to describe what financial risk managers do, we need to understand what financial risk $i$. In finance, risk arises from uncertainty surrounding future profits or returns. There are many ways to define risk, and we may change the definition slightly, depending on the task at hand.

In everyday speech, the word risk is associated with the possibility of negative outcomes. For something to be risky, the final outcome must be uncertain and there must be some possibility that the final outcome will have negative consequences. While this may seem obvious, some popular risk measures treat positive and negative outcomes equally, while others focus only negative outcomes. For this reason, in order to avoid any ambiguity when dealing specifically with negative outcomes, risk managers will often talk about downside risk.
Risk is often defined relative to expectations. If we have one investment with a $50 / 50$ chance of earning $\$ 0$or$\$200$, and a second investment with a $50 / 50$ chance of earning $\$ 400$or$\$600$, are both equally risky? The first investment earns $\$ 100$on average, and the second$\$500$, but both have a $50 / 50$ chance of being $\$ 100$above or below this expected value. Because the deviations from expectations are equal, many risk managers would consider the two investments to be equally risky. By this logic, the second investment is more attractive because it has a higher expected return, not because it is less risky. It is also important to note that risk is about possible deviations from expectations. If we expect an investment to make$\$1$ and it does make $\$ 1$, the investment was not necessarily risk free. If there were any possibility that the outcome could have been something other than$\$1$, then the investment was risky.

## 金融代写|量化风险管理代写Quantitative Risk Management代考|Absolute, Relative, and Conditional Risk

There may be no better way to understand the limits of financial risk management-why and where it may fail or succeed – than to understand the difference between absolute, relative, and conditional risk.

Financial risk managers are often asked to assign probabilities to various financial outcomes. What is the probability that a bond will default? What is the probability that an equity index will decline by more than $10 \%$ over the course of a year? These types of predictions, where risk managers are asked to assess the total or absolute risk of an investment, are incredibly difficult to make. As we will see, assessing the accuracy of these types of predictions, even over the course of many years, can be extremely difficult.

It is often much easier to determine relative risk than to measure risk in isolation. Bond ratings are a good example. Bond ratings can be used to assess absolute risk, but they are on much surer footing when used to assess relative risk. The number of defaults in a bond portfolio might be much higher or lower next year depending on the state of the economy and financial markets. No matter what happens, though, a portfolio consisting of a large number of AAA-rated bonds will almost certainly have fewer defaults than a portfolio consisting of a large number of C-rated bonds. Similarly, it is much easier to say that emerging market equities are riskier than U.S. equities, or that one hedge fund is riskier than another hedge fund.
What is the probability that the S\&P 500 will be down more than $10 \%$ next year? What is the probability that a particular U.S. large-cap equity mutual fund will be down more than $8 \%$ next year? Both are very difficult questions. What is the probability that this same mutual fund will be down more than $8 \%$, if the S\&P 500 is down more than $10 \%$ ? This last question is actually much easier to answer. What’s more, these types of conditional risk forecasts immediately suggest ways to hedge and otherwise mitigate risk.

Given the difficulty of measuring absolute risk, risk managers are likely to be more successful if they limit themselves to relative and conditional forecasts, when possible. Likewise, when there is any ambiguity about how a risk measure can be interpreted —as with bond ratings – encouraging a relative or conditional interpretation is likely to be in a risk manager’s best interest.

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 数学代写|信息论代写information theory代考|ECE4042

statistics-lab™ 为您的留学生涯保驾护航 在代写信息论information theory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。我们的专家在代写信息论information theory代写方面经验极为丰富，各种代写信息论information theory相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• Advanced Probability Theory 高等概率论
• Advanced Mathematical Statistics 高等数理统计学
• (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|信息论代写information theory代考|Definition of entropy of a continuous random variable

Up to now we have assumed that a random variable $\xi$, with entropy $H_{\xi}$, can take values from some discrete space consisting of either a finite or a countable number of elements, for instance, messages, symbols, etc. However, continuous variables are also widespread in engineering, i.e. variables (scalar or vector), which can take values from a continuous space $X$, most often from the space of real numbers. Such a random variable $\xi$ is described by the probability density function $p(\xi)$ that assigns the probability
$$\Delta P=\int_{\xi \varepsilon \Delta X} p(\xi) d \xi \approx p(A) \Delta V \quad(A \in \Delta X)$$
of $\xi$ appearing in region $\Delta X$ of the specified space $X$ with volume $\Delta V(d \xi=d V$ is a differential of the volume).

How can we define entropy $H_{\xi}$ for such a random variable? One of many possible formal ways is the following: In the formula
$$H_{\xi}=-\sum_{\xi} P \xi \ln P(\xi)=-\mathbb{E}[\ln P(\xi)]$$
appropriate for a discrete variable we formally replace probabilities $P(\xi)$ in the argument of the logarithm by the probability density and, thereby, consider the expression
$$H_{\xi}=-\mathbb{E}[\ln p(\xi)]=-\int_{x} p(\xi) \ln p(\xi) d \xi .$$
This way of defining entropy is not well justified. It remains unclear how to define entropy in the combined case, when a continuous distribution in a continuous space coexists with concentrations of probability at single points, i.e. the probability density contains delta-shaped singularities. Entropy (1.6.2) also suffers from the drawback that it is not invariant, i.e. it changes under a non-degenerate transformation of variables $\eta=f(\xi)$ in contrast to entropy (1.6.1), which remains invariant under such transformations.

## 数学代写|信息论代写information theory代考|Properties of entropy in the generalized version

Entropy (1.6.13), (1.6.16) defined in the previous section possesses a set of properties, which are analogous to the properties of an entropy of a discrete random variable considered earlier. Such an analogy is quite natural if we take into account the interpretation of entropy (1.6.13) (provided in Section 1.6) as an asymptotic case (for large $N$ ) of entropy (1.6.1) of a discrete random variable.

The non-negativity property of entropy, which was discussed in Theorem $1.1$, is not always satisfied for entropy (1.6.13), (1.6.16) but holds true for sufficiently large $N$. The constraint
$$H_{\xi}^{P / Q} \leqslant \ln N$$
results in non-negativity of entropy $H_{\xi}$.
Now we move on to Theorem $1.2$, which considered the maximum value of entropy. In the case of entropy (1.6.13), when comparing different distributions $P$ we need to keep measure $v$ fixed. As it was mentioned, quantity (1.6.17) is non-negative and, thus, (1.6.16) entails the inequality
$$H_{\xi} \leqslant \ln N .$$
At the same time, if we suppose $P=Q$, then, evidently, we will have
$$H_{\xi}=\ln N .$$
This proves the following statement that is an analog of Theorem $1.2$.

## 数学代写|信息论代写information theory代考|Encoding of discrete information

The definition of the amount of information, given in Chapter 1, is justified when we deal with a transformation of information from one kind into another, i.e. when considering encoding of information. It is essential that the law of conservation of information amount holds under such a transformation. It is very useful to draw an analogy with the law of conservation of energy. The latter is the main argument for introducing the notion of energy. Of course, the law of conservation of information is more complex than the law of conservation of energy in two respects. The law of conservation of energy establishes an exact equality of energies, when one type of energy is transformed into another. However, in transforming information we have a more complex relation, namely ‘not greater’ $(\leqslant)$, i.e. the amount of information cannot increase. The equality sign corresponds to optimal encoding. Thus, when formulating the law of conservation of information, we have to point out that there possibly exists such an encoding, for which the equality of the amounts of information occurs.

The second complication is that the equality is not exact. It is approximate, asymptotic, valid for complex (large) messages and for composite random variables. The larger a system of messages is, the more exact such a relation becomes. The exact equality sign takes place only in the limiting case. In this respect, there is an analogy with the laws of statistical thermodynamics, which are valid for large thermodynamic systems consisting of a large number (of the order of the Avogadro number) of molecules.

When conducting encoding, we assume that a long sequence of messages $\xi_{1}, \xi_{2}$, … is given together with their probabilities, i.e. a sequence of random variables. Therefore, the amount of information (entropy $H$ ) corresponding to this sequence can be calculated. This information can be recorded and transmitted by different realizations of the sequence. If $M$ is the number of such realizations, then the law of conservation of information can be expressed by the equality $H=\ln M$, which is complicated by the two above-mentioned factors (i.e. actually. $H \leqslant \ln M$ ).

Two different approaches may be used for solving the encoding problem. One can perform encoding of an infinite sequence of messages, i.e. online (or ‘sliding’) encoding. The inverse procedure, i.e. decoding, will be performed analogously.

## 数学代写|信息论代写information theory代考|Definition of entropy of a continuous random variable

$$\Delta P=\int_{\xi \varepsilon \Delta X} p(\xi) d \xi \approx p(A) \Delta V \quad(A \in \Delta X)$$

$$H_{\xi}=-\sum_{\xi} P \xi \ln P(\xi)=-\mathbb{E}[\ln P(\xi)]$$

$$H_{\xi}=-\mathbb{E}[\ln p(\xi)]=-\int_{x} p(\xi) \ln p(\xi) d \xi$$

## 数学代写|信息论代写information theory代考|Properties of entropy in the generalized version

$$H_{\xi}^{P / Q} \leqslant \ln N$$

$$H_{\xi} \leqslant \ln N \text {. }$$

$$H_{\xi}=\ln N .$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。