## matlab代写|time series analysisEMET3007/8012 Assignment 2

statistics-lab™ 为您的留学生涯保驾护航 在代写时间序列分析time series 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。我们的专家在代写时间序列分析time series analysis代写方面经验极为丰富，各种代写时间序列分析time series analysis相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## Instructions:

This assignment is worth either 20% or 25% of the final grade, and is worth a total of 75 points. All working must be shown for all questions. For questions which ask you to write a program, you must provide the code you used. If you have found code and then modified it, then the original source must be cited. The assignment is due by 5pm Friday 1st of October (Friday of Week 8), using Turnitin on Wattle. Late submissions will only be accepted with prior written approval. Good luck.

[10 marks] In this exercise we will consider four different specifications for forecasting monthly Australian total employed persons. The dataset (available on Wattle) AUSEmp 1oy 2022. csv contains three columns; the first column contains the date; the second contains the sales figures for that month (FRED data series LFEMTTTTAUM647N), and the third contains Australian GDP for that month.1] The data runs from January 1995 to January $2022 .$

Let $M_{i t}$ be a dummy variable that denotes the month of the year. Let $D_{i t}$ be a dummy variable which denotes the quarter of the year. The four specifications we consider are
\begin{aligned} &S_1: y_t=a_0+a_1 t+\alpha_4 D_{4 t}+\epsilon_t \ &S_2: y_t=a_1 t+\sum_{i=1}^4 \alpha_i D_{i t}+\epsilon_t \ &S_3: y_t=a_0+a_1 t+\beta_{12} M_{12, t}+\epsilon_t \ &S_4: y_t=a_1 t+\sum_{i=1}^{12} \beta_i M_{i t}+\epsilon_t \end{aligned}
where $\mathbb{E} \epsilon_t=0$ for all $t$.

a) For each specification, describe this specification in words.
b) For each specification, estimate the values of the parameters, and compute the MSE, $\mathrm{AIC}$, and BIC. If you make any changes to the csv file, please describe the changes you make. As always, you must include your code.
c) For each specification, compute the MSFE for the 1-step and 5-step ahead forecasts, with the out-of-sample forecasting exercise beginning at $T_0=50$.
d) For each specification, plot the out-of-sample forecasts and comment on the results.

[10 marks] Now add to Question 1 the additional assumption that $\epsilon_t \sim \mathcal{N}\left(0, \sigma^2\right)$. One estimator ${ }^2$ for $\sigma^2$ is
$$\hat{\sigma}^2=\frac{1}{T-k} \sum_{t=1}^T\left(y_t-\hat{y}_t\right)^2$$
where $\hat{y}_t$ is the estimated value of $y_t$ in the model and $k$ is the number of regressors in the specification.
a) For each specification $\left(S_1, \ldots, S_4\right)$, compute $\hat{\sigma}^2$.
b) For each specification, make a $95 \%$ probability forecast for the sales in June $2021 .$
c) For each specification, compute the probability that the total employed persons in June 2022 will be greater than $13.5$ million. According to the FRED series LFEMTTTTAUM647N, what was the actual employment level for that month.
d) Do you think the assumption that $\epsilon_t$ is iid is a reasonable assumption for this data series.

[10 marks] Here we investigate whether adding GDP $\mathrm{Gs}^3$ as a predictor can improve our forecasts. Consider the following modified specifications:
\begin{aligned} &S_1^{\prime}: y_t=a_0+a_1 t+\alpha_4 D_{4 t}+\gamma x_{t-h}+\epsilon_t \ &S_2^{\prime}: y_t=a_1 t+\sum_{i=1}^4 \alpha_i D_{i t}+\gamma x_{t-h}+\epsilon_t \ &S_3^{\prime}: y_t=a_0+a_1 t+\beta_{12} M_{12, t}+\gamma x_{t-h}+\epsilon_t \ &S_4^{\prime}: y_t=a_1 t+\sum_{i=1}^{12} \beta_i M_{i t}+\gamma x_{t-h}+\epsilon_t \end{aligned}
where $\mathbb{E} \epsilon_t=0$ for all $t$, and $x_{t-h}$ is GDP at time $t-h$. For each specification, compute the MSFE for the 1-step ahead, and the 5-step ahead forecasts, with the out-of-sample forecasting exercise beginning at $T_0=50$. For each specification, plot the out-of-sample forecasts and comment on the results.

[15 marks] Here we investigate whether Holt-Winters smoothing can improve our forecasts. Use a Holt-Winters smoothing method with seasonality, to produce 1-step ahead and 5-step ahead forecasts and compute the MSFE for these forecasts. You should use smoothing parameters $\alpha=\beta=\gamma=0.3$ and start the out-of-sample forecasting exercise at $T_0=50$. Plot these out-of-sample forecasts and comment on the results.
Additionally, estimate the values for $\alpha, \beta$, and $\gamma$ which minimise the MSFE. Find the MSFE for these parameter vales and compare it to the baseline $\alpha=\beta=\gamma=0.3$.

[5 marks] Questions 1, 3 and 4 each provided alternative models for forecasting Australian Total Employment. Compare the efficacy of these forecasts. Your comparison should include discussions of MSFE, but must also make qualitative observations (typically based on your graphs).

[10 marks] Develop another model, either based on material from class or otherwise, to forecast Australian Total Employment. Your new model should perform better (have a lower MSFE or MAFE) than all models from Questions 1,3, and 4. As part of your response to this question you must provide:
a) a brief written explanation of what your model is doing,
b) a brief statement on why you think your new model will perform better,
c) any relevant equations or mathematics/statistics to describe the model,
d) the code to run the model, and
e) the MSFE and/or MAFE error found by your model, and a brief discussion of how this compares to previous cases.

[15 marks] Consider the ARX(1) model
$$y_t=\mu+a t+\rho y_{t-1}+\epsilon_t$$
where the errors follow an $\mathrm{AR}(2)$ process
$$\epsilon_t=\phi_1 \epsilon_{t-1}+\phi_2 \epsilon_{t-2}+u_t, \quad \mathbf{u} \sim \mathcal{N}\left(0, \sigma^2 I\right)$$
for $t=1, \ldots, T$ and $e_{-1}=e_0=0$. Suppose $\phi_1, \phi_2$ are known. Find (analytically) the maximum likelihood estimators for $\mu, a, \rho$, and $\sigma^2$.

Hint: First write $y$ and $\epsilon$ in vector/matrix form. You may wish to use different looking forms for each. Find the distribution of $\epsilon$ and $y$. Then apply some appropriate calculus. You may want to let $H=I-\phi_1 L-\phi_2 L^2$, where $I$ is the $T \times T$ identity matrix, and $L$ is the lag matrix.

## EMET3007/8012代写

matlab代写请认准statistics-lab™. statistics-lab™为您的留学生涯保驾护航。

## 数学代写|随机过程统计代写Stochastic process statistics代考|STAT3021

statistics-lab™ 为您的留学生涯保驾护航 在代写随机过程统计Stochastic process statistics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。我们的专家在代写随机过程统计Stochastic process statistics代写方面经验极为丰富，各种代写随机过程统计Stochastic process statistics相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|随机过程统计代写Stochastic process statistics代考|Wiener’s construction

This is also a series approach, but Wiener used the trigonometric functions $\left(e^{i n \pi t}\right){n \in Z}$ as orthonormal basis for $L^2[0,1]$. In this case we obtain Brownian motion on $[0,1]$ as a Wiener-Fourier series $$W(t, \omega):=\sum{n=1}^{\infty} \frac{\sin (n \pi t)}{n} G_n(\omega),$$
where $\left(G_n\right){n \geqslant 0}$ are iid standard normal random variables. Lemma $3.1$ remains valid for (3.6) and shows that the series converges in $L^2$ and that the limit satisfies (B0)(R3); only the pronf that the limiting process is continunus, Theorem 3.3, needs some changes. Proof of the continuity of (3.6). Let $$W_N(t, \omega):=\sum{n=1}^N \frac{\sin (n \pi t)}{n} G_n(\omega) .$$
It is enough to show that $\left(W_{2^n}\right){n \geqslant 1}$ is a Cauchy sequence in $L^2(\mathbb{P})$ uniformly for all $t \in[0,1]$. Set $$\Delta_j(t):=W{2^{j+1}}(t)-W_{2^j}(t)$$

Using $|\operatorname{Im} z| \leqslant|z|$ for $z \in \mathbb{C}$, we see
$$\left|\Delta_j(t)\right|^2=\left(\sum_{k=2^j+1}^{2^{j+1}} \frac{\sin (k \pi t)}{k} G_k\right)^2 \leqslant\left|\sum_{k=2^j+1}^{2^{j+1}} \frac{e^{i k \pi t}}{k} G_k\right|^2,$$
and since $|z|^2-z \bar{z}$ we get
\begin{aligned} \left|\Delta_j(t)\right|^2 & \leqslant \sum_{k=2^j+1} \sum_{\ell=2^j+1}^{2^{j+1}} \frac{2^{i k \pi t} e^{-i \ell \pi t}}{k \ell} G_k G_{\ell} \ &=\sum_{k=2^j+1}^{2^{j+1}} \frac{G_k^2}{k^2}+2 \sum_{k=2^j+1} \sum_{\ell=2^j+1}^{2^{j+1}} \frac{e^{i k \pi t} e^{-i \ell \pi t}}{k \ell} G_k G_{\ell} \ & \stackrel{m=k-\ell}{=} \sum_{k=2^j+1}^{2^{j+1}} \frac{G_k^2}{k^2}+2 \sum_{m=1}^{2^j-1} \sum_{\ell=2^j+1}^{2^{j+1}-m} \frac{e^{i m \pi t}}{\ell(\ell+m)} G_{\ell} G_{\ell+m} \ & \leqslant \sum_{k=2^j+1}^{2^{j+1}} \frac{G_k^2}{k^2}+2 \sum_{m=1}^{2^j-1}\left|\sum_{\ell=2^j+1}^{2^{j+1}-m} \frac{G_{\ell} G_{\ell+m}}{\ell(\ell+m)}\right| . \end{aligned}

## 数学代写|随机过程统计代写Stochastic process statistics代考|Donsker’s construction

Donsker’s invariance theorem shows that Brownian motion is a limit of linearly interpolated random walks – pretty much in the way we have started the discussion in Chapter 1 . As before, the difficult point is to prove the sample continuity of the limiting process.

Let, on a probability space $(\Omega, \mathcal{A}, \mathbb{P}), \epsilon_n, n \geqslant 1$, be iid Bernoulli random variables such that $\mathbb{P}\left(\epsilon_1=1\right)=\mathbb{P}^2\left(\epsilon_1=-1\right)=\frac{1}{2}$. Then
$$S_n:=\epsilon_1+\cdots+\epsilon_n$$
is a simple random walk. Interpolate linearly and apply Gaussian scaling
$$S^n(t):=\frac{1}{\sqrt{n}}\left(S_{\lfloor n t\rfloor}-(n t-\lfloor n t\rfloor) \epsilon_{\lfloor n t\rfloor+1}\right), \quad t \in[0,1] .$$
In particular, $S^n\left(\frac{\dot{L}}{n}\right)=\frac{1}{\sqrt{n}} S_j$. If $j=j(n)$ and $j / n=s=$ const., the central limit theorem shows that $S^n\left(\frac{\dot{j}}{n}\right)=\sqrt{s} S_j / \sqrt{j} \stackrel{d}{\longrightarrow} \sqrt{s} G$ as $n \rightarrow \infty$ where $G$ is a standard normal random variable. Moreover, with $s=j / n$ and $t=k / n$, the increment $S^n(t)-S^n(s)=\left(S_k-S_j\right) / \sqrt{n}$ is independent of $\epsilon_1, \ldots, \epsilon_j$, and therefore of all earlier increments of the same form. Moreover,
$$\mathbb{E}\left(S^n(t)-S^n(s)\right)=0 \quad \text { and } \quad \mathbb{V}\left(S^n(t)-S^n(s)\right)=\frac{k-j}{n}=t-s$$
in the limit we get a Gaussian increment with mean zero and variance $t-s$. Since independence and stationarity of the increments are distributional properties, they are inherited by the limiting process – which we will denote by $\left(B_t\right){t \in[0,1]}$. We have seen that $\left(B_q\right){q \in[0,1] \cap Q}$ would have the properties $(\mathrm{B} 0)-(\mathrm{B} 3)$ and it qualifies as a candidate for Brownian motion. If it had continuous sample paths, (B0)-(B3) would hold not only for rational times but for all $t \geqslant 0$. That the limit exists and is uniform in $t$ is the essence of Donsker’s invariance principle.

## 数学代写|随机过程统计代写Stochastic process statistics代考|Wiener’s construction

$$W(t, \omega):=\sum n=1^{\infty} \frac{\sin (n \pi t)}{n} G_n(\omega),$$

$$W_N(t, \omega):=\sum n=1^N \frac{\sin (n \pi t)}{n} G_n(\omega) .$$

$$\Delta_j(t):=W 2^{j+1}(t)-W_{2^j}(t)$$

$$\left|\Delta_j(t)\right|^2=\left(\sum_{k=2^j+1}^{2^{j+1}} \frac{\sin (k \pi t)}{k} G_k\right)^2 \leqslant\left|\sum_{k=2^j+1}^{2^{j+1}} \frac{e^{i k \pi t}}{k} G_k\right|^2$$

$$\left|\Delta_j(t)\right|^2 \leqslant \sum_{k=2^j+1} \sum_{\ell=2^j+1}^{2^{j+1}} \frac{2^{i k \pi t} e^{-i \ell \pi t}}{k \ell} G_k G_{\ell}=\sum_{k=2^j+1}^{2^{j+1}} \frac{G_k^2}{k^2}+2 \sum_{k=2^j+1} \sum_{\ell=2^j+1}^{2^{j+1}} \frac{e^{i k \pi t} e^{-i \ell \pi t}}{k \ell} G_k G_{\ell} \stackrel{m=k-\ell}{=}$$

## 数学代写|随机过程统计代写Stochastic process statistics代考|Donsker’s construction

Donsker 的不变性定理表明，布朗运动是线性揷值随机游走的极限一一与我们在第 1 章开始讨论的方式非常相 似。和以前一样，难点是证明限制过程的样本连续性。

$$S_n:=\epsilon_1+\cdots+\epsilon_n$$

$$S^n(t):=\frac{1}{\sqrt{n}}\left(S_{\lfloor n t\rfloor}-(n t-\lfloor n t\rfloor) \epsilon_{\lfloor n t\rfloor+1}\right), \quad t \in[0,1] .$$

$$\mathbb{E}\left(S^n(t)-S^n(s)\right)=0 \quad \text { and } \quad \mathbb{V}\left(S^n(t)-S^n(s)\right)=\frac{k-j}{n}=t-s$$

## 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|随机过程统计代写Stochastic process statistics代考|STAT3921

statistics-lab™ 为您的留学生涯保驾护航 在代写随机过程统计Stochastic process statistics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。我们的专家在代写随机过程统计Stochastic process statistics代写方面经验极为丰富，各种代写随机过程统计Stochastic process statistics相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|随机过程统计代写Stochastic process statistics代考|Brownian Motion in Rd

We will now show that $B_t=\left(B_t^1, \ldots, B_t^d\right)$ is a BM ${ }^d$ if, and only if, its coordinate processes $B_t^j$ are independent one-dimensional Brownian motions. We call two stochastic processes $\left(X_t\right){t \geqslant 0}$ and $\left(Y_t\right){t \geqslant 0}$ (defined on the same probability space) independent, if the $\sigma$-algebras generated by these processes are independent:
$$\mathcal{F}{\infty}^X \Perp \mathcal{F}{\infty}^Y$$
where
$$\mathcal{F}{\infty}^X:=\sigma\left(\bigcup{n \geqslant 1} \bigcup_{0 \leqslant t_1<\cdots<t_n<\infty} \sigma\left(X\left(t_j\right), \ldots, X\left(t_n\right)\right)\right) .$$
Note that the family of sets $\bigcup_n \bigcup_{t_1, \ldots, t_n} \sigma\left(X\left(t_1\right), \ldots, X\left(t_n\right)\right)$ is stable under finite intersections. Therefore, (2.15) follows already if
$$\left(X\left(s_1\right), \ldots, X\left(s_n\right)\right) \Perp\left(Y\left(t_1\right), \ldots, Y\left(t_m\right)\right)$$
for all $m, n \geqslant 1, s_1<\cdots<s_m$ and $t_1<\cdots<t_n$. Without loss of generality we can even assume that $m=n$ and $s_j=t_j$ for all $j$. This follows easily if we take the common refinement of the $s_j$ and $t_j$.

The following simple characterization of $d$-dimensional Brownian motion will be very useful for our purposes.

## 数学代写|随机过程统计代写Stochastic process statistics代考|The Lévy–Ciesielski construction

This approach goes back to Lévy [120, pp. 492-494] but it got its definitive form in the hands of Ciesielski, cf. $[26,27]$. The idea is to write the paths $[0,1] \ni t \mapsto B_t(\omega)$ for (almost) every $\omega$ as a random series with respect to a complete orthonormal system (ONS) in the Hilbert space $L^2(d t)=L^2([0,1], d t)$ with canonical scalar product $\langle f, g\rangle_{L^2}=\int_0^1 f(t) g(t) d t$. Assume that $\left(\phi_n\right){n \geqslant 0}$ is any complete ONS and let $\left(G_n\right){n \geqslant 0}$ be a sequence of real-valued iid Gaussian $N(0,1)$-random variables on the probability space $(\Omega, \mathcal{A}, \mathbb{P})$. Set
\begin{aligned} W_N(t) &:=\sum_{n=0}^{N-1} G_n\left\langle\mathbb{1}{[0, t)}, \phi_n\right\rangle{L^2} \ &=\sum_{n=0}^{N-1} G_n \int_0^t \phi_n(s) d s . \end{aligned}
We want to show that $\lim {N \rightarrow \infty} W_N(t)$ defines a Brownian motion on $[0,1]$. 3.1 Lemma. The limit $W(t):=\lim {N \rightarrow \infty} W_N(t)$ exists for every $t \in[0,1]$ in $L^2(\mathbb{P})$ and the process $W(t)$ satisfies (B0)-(B3).

Proof. Using the independence of the $G_n \sim \mathrm{N}(0,1)$ and Parseval’s identity we get for every $t \in[0.1]$
\begin{aligned} \mathbb{E}\left(W_N(t)^2\right) &=\mathbb{E}\left[\sum_{m, n=0}^{N-1} G_n G_m\left\langle\mathbb{1}{[0, t)}, \phi_m\right\rangle{L^2}\left\langle\mathbb{1}{[0, t)}, \phi_n\right\rangle{L^2}\right] \ &=\sum_{m, n=1}^{N-1} \underbrace{\mathbb{E}\left(G_n G_m\right)}{=0(n \neq m), \text { or }=1(n=m)}\left\langle\mathbb{1}{[0, t)}, \phi_m\right\rangle_{L^2}\left\langle\mathbb{1}{[0, t)}, \phi_n\right\rangle{L^2} \ &=\sum_{n=1}^{N-1}\left\langle\mathbb{1}{[0, t)}, \phi_n\right\rangle{L^2}^2 \underset{N \rightarrow \infty}{ }\left\langle\mathbb{1}{[0, t)}, \mathbb{1}{[0, t)}\right\rangle_{L^2}=t . \end{aligned}
This shows that $W(t)=L^2-\lim {N \rightarrow \infty} W_N(t)$ exists. An analogous calculation yields for s{n=0}^{\infty}\left\langle\mathbb{1}{[0, t)}-\mathbb{1}{[0, s)}, \phi_n\right\rangle_{L^2}\left\langle\mathbb{1}{[0, v)}-\mathbb{1}{[0, u)}, \phi_n\right\rangle_{L^2} \ &=\left\langle\mathbb{1}{[s, t)}, \mathbb{1}{[u, v)}\right\rangle_{L^2}= \begin{cases}t-s, & {[s, t)=[u, v) ;} \ 0, & {[s, t) \cap[u, v)=\emptyset} \ (v \wedge t-u \vee s)^{+}, & \text {in general. }\end{cases} \end{aligned} $$## 随机过程统计代考 ## 数学代写|随机过程统计代写Stochastic process statistics代考|Brownian Motion in Rd 我们现在将证明 B_t=\left(B_t^1, \ldots, B_t^d\right) 是一个BM { }^d 当且仅当其协调过程 B_t^j 是独立的一维布朗运动。我们称两个随 机过程 \left(X_t\right) t \geqslant 0 和 \left(Y_t\right) t \geqslant 0 (在相同的概率空间上定义) 独立的，如果 \sigma-这些过程生成的代数是独立的:$$ \mathcal{F} \infty^X \backslash \operatorname{Perp} \mathcal{F} \infty^Y $$在哪里$$ \mathcal{F} \infty^X:=\sigma\left(\bigcup n \geqslant 1 \bigcup_{0 \leqslant t_1<\cdots<t_n<\infty} \sigma\left(X\left(t_j\right), \ldots, X\left(t_n\right)\right)\right) $$注意集合族 \bigcup_n \bigcup_{t_1, \ldots, t_n} \sigma\left(X\left(t_1\right), \ldots, X\left(t_n\right)\right) 在有限的交点下是稳定的。因此，(2.15) 已经成立，如果 \left(X\left(s_1\right), \ldots, X\left(s_n\right)\right) \backslash \operatorname{Perp}\left(Y\left(t_1\right), \ldots, Y\left(t_m\right)\right) 对所有人 m, n \geqslant 1, s_1<\cdots<s_m 和 t_1<\cdots<t_n. 不失一般性，我们甚至可以假设 m=n 和 s_j=t_j 对所 有人j. 如果我们对 s_j 和 t_j. 以下简单表征 d 维布朗运动对我们的目的非常有用。 ## 数学代写|随机过程统计代写Stochastic process statistics代考|The Lévy–Ciesielski construction 这种方法可以追溯到 Lévy [120, pp. 492-494]，但它在 Ciesielski 手中得到了确定的形式，参见。[26, 27]. 这个 想法是写路径 [0,1] \ni t \mapsto B_t(\omega) 对于 (几平) 每个 \omega 作为关于希尔伯特空间中完整正交系统 (ONS) 的随机序列 L^2(d t)=L^2([0,1], d t) 具有规范标量积 \langle f, g\rangle_{L^2}=\int_0^1 f(t) g(t) d t. 假使，假设 \left(\phi_n\right) n \geqslant 0 是任何完整的 ONS 并且让 \left(G_n\right) n \geqslant 0 是一个实值独立同分布高斯序列 N(0,1)-概率空间上的随机变量 (\Omega, \mathcal{A}, \mathbb{P}). 放$$ W_N(t):=\sum_{n=0}^{N-1} G_n\left\langle 1[0, t), \phi_n\right\rangle L^2=\sum_{n=0}^{N-1} G_n \int_0^t \phi_n(s) d s . $$我们想证明 \lim N \rightarrow \infty W_N(t) 定义了一个布朗运动 [0,1] .3 .1 引理。极限 W(t):=\lim N \rightarrow \infty W_N(t) 存在 于每个 t \in[0,1] 在 L^2(\mathbb{P}) 和过程 W(t) 满足 (B0)-(B3)。 证明。利用独立性 G_n \sim \mathrm{N}(0,1) 和 Parseval 的身份，我们得到每个 t \in[0.1] 这表明 W(t)=L^2-\lim N \rightarrow \infty W_N(t) 存在。类似的计算产生 \$$ 统计代写请认准statistics-lab™. statistics-lab™为您的留学生涯保驾护航。 ## 金融工程代写 金融工程是使用数学技术来解决金融问题。金融工程使用计算机科学、统计学、经济学和应用数学领域的工具和知识来解决当前的金融问题，以及设计新的和创新的金融产品。 ## 非参数统计代写 非参数统计指的是一种统计方法，其中不假设数据来自于由少数参数决定的规定模型；这种模型的例子包括正态分布模型和线性回归模型。 ## 广义线性模型代考 广义线性模型（GLM）归属统计学领域，是一种应用灵活的线性回归模型。该模型允许因变量的偏差分布有除了正态分布之外的其它分布。 术语 广义线性模型（GLM）通常是指给定连续和/或分类预测因素的连续响应变量的常规线性回归模型。它包括多元线性回归，以及方差分析和方差分析（仅含固定效应）。 ## 有限元方法代写 有限元方法（FEM）是一种流行的方法，用于数值解决工程和数学建模中出现的微分方程。典型的问题领域包括结构分析、传热、流体流动、质量运输和电磁势等传统领域。 有限元是一种通用的数值方法，用于解决两个或三个空间变量的偏微分方程（即一些边界值问题）。为了解决一个问题，有限元将一个大系统细分为更小、更简单的部分，称为有限元。这是通过在空间维度上的特定空间离散化来实现的，它是通过构建对象的网格来实现的：用于求解的数值域，它有有限数量的点。边界值问题的有限元方法表述最终导致一个代数方程组。该方法在域上对未知函数进行逼近。[1] 然后将模拟这些有限元的简单方程组合成一个更大的方程系统，以模拟整个问题。然后，有限元通过变化微积分使相关的误差函数最小化来逼近一个解决方案。 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。 ## 随机分析代写 随机微积分是数学的一个分支，对随机过程进行操作。它允许为随机过程的积分定义一个关于随机过程的一致的积分理论。这个领域是由日本数学家伊藤清在第二次世界大战期间创建并开始的。 ## 时间序列分析代写 随机过程，是依赖于参数的一组随机变量的全体，参数通常是时间。 随机变量是随机现象的数量表现，其时间序列是一组按照时间发生先后顺序进行排列的数据点序列。通常一组时间序列的时间间隔为一恒定值（如1秒，5分钟，12小时，7天，1年），因此时间序列可以作为离散时间数据进行分析处理。研究时间序列数据的意义在于现实中，往往需要研究某个事物其随时间发展变化的规律。这就需要通过研究该事物过去发展的历史记录，以得到其自身发展的规律。 ## 回归分析代写 多元回归分析渐进（Multiple Regression Analysis Asymptotics）属于计量经济学领域，主要是一种数学上的统计分析方法，可以分析复杂情况下各影响因素的数学关系，在自然科学、社会和经济学等多个领域内应用广泛。 ## 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|随机过程统计代写Stochastic process statistics代考|STAT4061 如果你也在 怎样代写随机过程统计Stochastic process statistics这个学科遇到相关的难题，请随时右上角联系我们的24/7代写客服。 随机过程 用于表示在时间上发展的统计现象以及在处理这些现象时出现的理论模型，由于这些现象在许多领域都会遇到，因此这篇文章具有广泛的实际意义。 statistics-lab™ 为您的留学生涯保驾护航 在代写随机过程统计Stochastic process statistics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。我们的专家在代写随机过程统计Stochastic process statistics代写方面经验极为丰富，各种代写随机过程统计Stochastic process statistics相关的作业也就用不着说。 我们提供的随机过程统计Stochastic process statistics及其相关学科的代写，服务范围广, 其中包括但不限于: • 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|随机过程统计代写Stochastic process statistics代考|Brownian motion as a Gaussian process Recall that a one-dimensional random variable\Gamma$is Gaussian if it has the characteristic function $$\mathbb{E} e^{i \xi \Gamma}=e^{i m \xi-\frac{1}{2} \sigma^2 \xi^2}$$ for some real numbers$m \in \mathbb{R}$and$\sigma \geqslant 0$. If we differentiate (2.1) two times with respect to$\xi$and set$\xi=0$, we see that $$m=\mathbb{E} \Gamma \quad \text { and } \quad \sigma^2=\mathbb{V} \Gamma .$$ A random vector$\Gamma=\left(\Gamma_1, \ldots, \Gamma_n\right) \in \mathbb{R}^n$is Gaussian, if$\langle\ell, \Gamma\rangle$is for every$\ell \in \mathbb{R}^n$a one-dimensional Gaussian random variable. This is the same as to say that $$\mathbb{E} e^{i\langle\xi, \Gamma\rangle}=e^{i \mathbb{E}(\xi, \Gamma)-\frac{1}{2} \mathbb{V}(\xi, \Gamma)} .$$ Setting$m=\left(m_1, \ldots, m_n\right) \in \mathbb{R}^n$and$\Sigma=\left(\sigma_{j k}\right){j, k=1 \ldots, n} \in \mathbb{R}^{n \times n}$where $$m_j:=\mathbb{E} \Gamma_j \quad \text { and } \quad \sigma{j k}:=\mathbb{E}\left(\Gamma_j-m_j\right)\left(\Gamma_k-m_k\right)=\operatorname{Cov}\left(\Gamma_j, \Gamma_k\right),$$ we can rewrite (2.3) in the following form $$\mathbb{E} e^{i(\xi, \Gamma\rangle}=e^{i\langle\xi, m\rangle-\frac{1}{2}(\xi, \Sigma \xi)} .$$ We call$m$the mean vector and$\Sigma$the covariance matrix of$\Gamma$. ## 数学代写|随机过程统计代写Stochastic process statistics代考|Invariance properties of Brownian motion The fact that a stochastic process is a Brownian motion is preserved under various operations at the level of the sample paths. Throughout this section$\left(B_t\right){t \geqslant 0}$denotes a$d$-dimensional Brownian motion. 2.8 Reflection. If$\left(B_t\right){t \geqslant 0}$is a$\mathrm{BM}^d$, so is$\left(-B_t\right){t \geqslant 0}$. 2.9 Renewal. Let$(B(t)){t \geqslant 0}$be a Brownian motion and fix some time$a>0$. Then$(W(t)){t \geqslant 0}, W(t):=B(t+a)-B(a)$, is again a$\mathrm{BM}^d$. The properties (B0) and (B4) are obvious for$W(t)$. For all$s \leqslant t\begin{aligned} W(t)-W(s) &=B(t+a)-B(a)-(B(s+a)-B(a)) \ &=B(t+a)-B(s+a) \ & \stackrel{(\mathrm{B} 3)}{\sim} \mathrm{N}(0, t-s) \end{aligned} which proves (B3) and (B2) for the processW$. Finally, if$t_0=0{l-1}\right)=B\left(t_l+a\right)-B\left(t_{l-1}+a\right) \text { for all } j=1, \ldots, n
$$i. e. the independence of the W-increments follows from (B1) for B at the times t_j+a, j=1, \ldots, d A consequence of the independent increments property is that a Brownian motion has no memory. This is the essence of the next lemma. 2.10 Lemma (Markov property of BM). Let (B(t)){t \geqslant 0} be a \mathrm{BM}^d and denote by W(t):=B(t+a)-B(a) the shifted Brownian motion constructed in Paragraph 2.9. Then (B(t)){0 \leqslant t \leqslant a} and (W(t)){t \geqslant 0} are independent, i.e. the \sigma-algebras generated by these processes are independent:$$ \sigma(B(t): 0 \leqslant t \leqslant a)=: \mathcal{F}_a^B \Perp \mathcal{F}{\infty}^W:=\sigma(W(t): 0 \leqslant t<\infty) .
$$In particular, B(t)-B(s) \Perp \mathcal{F}s^B for all 0 \leqslant s{j-1}: j=1, \ldots, n\right) .$$
Since $X_0$ and $X_j-X_{j-1}$ are $\sigma\left(X_j: j=0, \ldots, n\right)$ measurable, we see the inclusion ‘ $\supset$ ‘. For the converse we observe that $X_k=\sum_{j=1}^k\left(X_j-X_{j-1}\right)+X_0, k=0, \ldots, n$. Let $0=s_0<s_1<\cdots<s_m=a=t_0<t_1<\cdots<t_n$. By (B1) the random variables
$$B\left(s_1\right)-B\left(s_0\right), \ldots, B\left(s_m\right)-B\left(s_{m-1}\right), B\left(t_1\right)-B\left(t_0\right), \ldots, B\left(t_n\right)-B\left(t_{n-1}\right)$$
are independent, thus
$$\sigma\left(B\left(s_j\right)-B\left(s_{j-1}\right): j=1, \ldots, m\right) \Perp \sigma\left(B\left(t_k\right)-B\left(t_{k-1}\right): k=1, \ldots, n\right) .$$
Using $W\left(t_k-t_0\right)-W\left(t_{k-1}-t_0\right)=B\left(t_k\right)-B\left(t_{k-1}\right)$ and $B(0)=W(0)=0$, we can apply (2.14) to get
$$\sigma\left(B\left(s_j\right): j=1, \ldots, m\right) \Perp \sigma\left(W\left(t_k-t_0\right): k=1, \ldots, n\right)$$
and
$$\bigcup_{\substack{0<s_i<\cdots<s_m \leqslant a \ m \geqslant 1}} \sigma\left(B\left(s_j\right): j=1, \ldots, m\right) \Perp \bigcup_{\substack{0<u_i<\cdots<<u_n \ n \geqslant 1}} \sigma\left(W\left(u_k\right): k=1, \ldots, n\right) .$$
The families on the left and right-hand side are $\cap$-stable generators of $\mathcal{F}a^B$ and $\mathcal{F}{\infty}^W$, respectively, thus $\mathcal{F}a^B \Perp \mathcal{F}{\infty}^W$.

Finally, taking $a=s$, we see that $B(t)-B(s)=W(t-s)$ which is $\mathcal{F}_{\infty}^W$ measurable and therefore independent of $\mathcal{F}_s^B$.

## 数学代写|随机过程统计代写Stochastic process statistics代考|Brownian motion as a Gaussian process

$$\mathbb{E} e^{i \xi \Gamma}=e^{i m \xi-\frac{1}{2} \sigma^2 \xi^2}$$

$$m=\mathbb{E} \Gamma \quad \text { and } \quad \sigma^2=\mathbb{V} \Gamma .$$

$$\mathbb{E} e^{i\langle\xi, \Gamma\rangle}=e^{i \mathbb{E}(\xi, \Gamma)-\frac{1}{2} \mathbb{V}(\xi, \Gamma)} .$$

$$m_j:=\mathbb{E} \Gamma_j \quad \text { and } \quad \sigma j k:=\mathbb{E}\left(\Gamma_j-m_j\right)\left(\Gamma_k-m_k\right)=\operatorname{Cov}\left(\Gamma_j, \Gamma_k\right),$$

$$\mathbb{E} e^{i(\xi, \Gamma\rangle}=e^{i\langle\xi, m\rangle-\frac{1}{2}(\xi, \Sigma \xi)}$$

## 数学代写|随机过程统计代写Stochastic process statistics代考|Invariance properties of Brownian motion

$$W(t)-W(s)=B(t+a)-B(a)-(B(s+a)-B(a)) \quad=B(t+a)-B(s+a) \stackrel{(\mathrm{B} 3)}{\sim} \mathrm{N}(0, t-s)$$

\left(\lim i \rightarrow \infty f_i\right)(\omega) \equiv \lim i \rightarrow \infty f_i(\omega)

## 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|概率论代写Probability theory代考|STAT4061

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

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|概率论代写Probability theory代考|Natural Numbers

We start with the natural numbers as known in elementary schools. All mathematical objects are constructed from natural numbers, and every theorem is ultimately a calculation on the natural numbers. From natural numbers are constructed the integers and the rational numbers, along with the arithmetical operations, in the manner taught in elementary schools.

We claim to have a natural number only when we have provided a finite method to calculate it, i.e., to find its decimal representation. This is the fundamental difference from classical mathematics, which requires no such finite method; an infinite procedure in a proof is considered just as good in classical mathematics.
The notion of a finite natural number is so simple and so immediate that no attempt is needed to define it in even simpler terms. A few examples would suffice as clarification: 1,2 , and 3 are natural numbers. So are $9^9$ and $9^{9^9}$; the multiplication method will give, at least in principle, their decimal expansion in a finite number of steps. In contrast, the “truth value” of a particular mathematical statement is a natural number only if a finite method has been supplied that, when carried out, would prove or disprove the statement.

## 数学代写|概率论代写Probability theory代考|Calculation and Theorem

An algorithm or a calculation means any finite, step-by-step procedure. A mathematical object is defined when we specify the calculations that need to be done to produce this object. We say that we have proved a theorem if we have provided a step-by-step method that translates the calculations doable in the hypothesis to a calculation in the conclusion of the theorem. The statement of the theorem is merely a summary of the algorithm contained in the proof.

Although we do not, for good reasons, write mathematical proofs in a computer language, the reader would do well to compare constructive mathematics to the development of a large computer software library, with successive objects and library functions being built from previous ones, each with a guarantee to finish in a finite number of steps.

There is a trivial form of proof by contradiction that is valid and useful in constructive mathematics. Suppose we have already proved that one of two given alternatives, $A$ and $B$, must hold, meaning that we have given a finite method, that, when unfolded, gives either a proof for $A$ or a proof for $B$. Suppose subsequently we also prove that $A$ is impossible. Then we can conclude that we have a proof of $B$; we need only exercise said finite method, and see that the resulting proof is for $B$.

## 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|线性规划作业代写Linear Programming代考|MATH7232

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

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|线性规划作业代写Linear Programming代考|Basic Feasible Solutions

Consider the system of equalities
$$\mathbf{A x}=\mathbf{b},$$
where $\mathbf{x}$ is an $n$-vector, $\mathbf{b}$ is an $m$-vector, and $\mathbf{A}$ is an $m \times n$ matrix. Suppose that from the $n$ columns of $\mathbf{A}$ we select a set of $m$ linearly independent columns (such a set exists if the rank of $\mathbf{A}$ is $m$ ). For notational simplicity assume that we select the first $m$ columns of $\mathbf{A}$ and denote the $m \times m$ matrix determined by these columns by B. The matrix $\mathbf{B}$ is then nonsingular and we may uniquely solve the equation.
$$\mathbf{B x}{\mathbf{B}}=\mathbf{b} \quad \text { or } \quad \mathbf{x}{\mathbf{B}}=\mathbf{B}^{-1} \mathbf{b}$$
for the $m$-vector $\mathbf{x}{\mathbf{B}}$ whose components are associated with the columns of submatrix $\mathbf{B}$ according to the same index order. By putting $\mathbf{x}=\left(\mathbf{x}{\mathbf{B}}, \mathbf{0}\right)$ (that is, setting the first $m$ components of $\mathbf{x}$ equal to those of $\mathbf{x}{\mathbf{B}}$ and the remaining components equal to zero), we obtain a solution to $\mathbf{A x}=\mathbf{b}$. This leads to the following definition. Definition Given the set of $m$ simultaneous linear equations in $n$ unknowns (2.10), let $\mathbf{B}$ be any nonsingular $m \times m$ submatrix made up of columns of $\mathbf{A}$. Then, if all $n-m$ components of $\mathbf{x}$ not associated with columns of $\mathbf{B}$ are set equal to zero, the solution to the resulting set of equations is said to be a basic solution to (2.10) with respect to basis $\mathbf{B}$. The components of $\mathbf{x}$ associated with the columns of $\mathbf{B}$. denoted by subvector $\mathbf{X}{\mathbf{R}}$ according to the same column index order in $\mathbf{B}$ throughout this book, are called basic variables.
In the above definition we refer to $\mathbf{B}$ as a basis, since $\mathbf{B}$ consists of $m$ linearly independent columns that can be regarded as a basis for the space $E^{m}$. The basic solution corresponds to an expression for the vector $\mathbf{b}$ as a linear combination of these basis vectors. This interpretation is discussed further in the next section.

In general, of course, Eq. (2.10) may have no basic solutions. However, we may avoid trivialities and difficulties of a nonessential nature by making certain elementary assumptions regarding the structure of the matrix $\mathbf{A}$. First, we usually assume that $n>m$, that is, the number of variables $x_{j}$ exceeds the number of equality constraints. Second, we usually assume that the rows of $\mathbf{A}$ are linearly independent, corresponding to linear independence of the $m$ equations. A linear dependency among the rows of $\mathbf{A}$ would lead either to contradictory constraints and hence no solutions to $(2.10)$, or to a redundancy that could be eliminated. Formally, we explicitly make the following assumption in our development, unless noted otherwise.

## 数学代写|线性规划作业代写Linear Programming代考|The Fundamental Theorem of Linear Programming

In this section, through the fundamental theorem of linear programming, we establish the primary importance of basic feasible solutions in solving linear programs. The method of proof of the theorem is in many respects as important as the result itself, since it represents the beginning of the development of the simplex method. The theorem (due to Carathéodory) itself shows that it is necessary only to consider basic feasible solutions when seeking an optimal solution to a linear program because the optimal value is always achieved at such a solution.
Corresponding to a linear program in standard form
\begin{aligned} &\operatorname{minimize} \mathbf{c}^{T} \mathbf{x} \ &\text { subject to } \mathbf{A x}=\mathbf{b}, \mathbf{x} \geqslant \mathbf{0} \end{aligned}
a feasible solution to the constraints that achieves the minimum value of the objective function subject to those constraints is said to be an optimal feasible solution. If this solution is basic, it is an optimal basic feasible solution.
Fundamental Theorem of Linear Programming Given a linear program in standard form (2.13) where $\mathbf{A}$ is an $m \times n$ matrix of rank $m$,
i) if there is a feasible solution, there is a basic feasible solution;
ii) if there is an optimal feasible solution, there is an optimal basic feasible solution.
Proof of (i) Denote the columns of $\mathbf{A}$ by $\mathbf{a}{1}, \mathbf{a}{2}, \ldots, \mathbf{a}{n}$. Suppose $\mathbf{x}=$ $\left(x{1}, x_{2}, \ldots, x_{n}\right)$ is a feasible solution. Then, in terms of the columns of $\mathbf{A}$, this solution satisfies:
$$x_{1} \mathbf{a}{1}+x{2} \mathbf{a}{2}+\cdots+x{n} \mathbf{a}{n}=\mathbf{b} .$$ Assume that exactly $p$ of the variables $x{i}$ are greater than zero, and for convenience, that they are the first $p$ variables. Thus
$$x_{1} \mathbf{a}{1}+x{2} \mathbf{a}{2}+\cdots+x{p} \mathbf{a}_{p}=\mathbf{b}$$

## 数学代写|线性规划作业代写Linear Programming代考|Basic Feasible Solutions

$$\mathbf{A} \mathbf{x}=\mathbf{b},$$

$$\mathbf{B} \mathbf{x} \mathbf{B}=\mathbf{b} \quad \text { or } \quad \mathbf{x B}=\mathbf{B}^{-1} \mathbf{b}$$

## 数学代写|线性规划作业代写Linear Programming代考|The Fundamental Theorem of Linear Programming

minimize $\mathbf{c}^{T} \mathbf{x} \quad$ subject to $\mathbf{A} \mathbf{x}=\mathbf{b}, \mathbf{x} \geqslant \mathbf{0}$

i) 如果有可行的解决方案，则有一个基本的可行解决方案；
ii) 如果有最佳的可行解决方案，则有一个最佳的基本可行解决方案。
(i) 表示的证明 $\mathbf{A}$ 经过 $\mathbf{a} 1, \mathbf{a} 2, \ldots, \mathbf{a}$. 认为 $\mathbf{x}=\left(x 1, x_{2}, \ldots, x_{n}\right)$ 是一个可行的解决方案。然后，就 $\mathbf{A}$ ，该解 决方案满足:
$$x_{1} \mathbf{a} 1+x 2 \mathbf{a} 2+\cdots+x n \mathbf{a} n=\mathbf{b} .$$

$$x_{1} \mathbf{a} 1+x 2 \mathbf{a} 2+\cdots+x p \mathbf{a}_{p}=\mathbf{b}$$

## 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|线性规划作业代写Linear Programming代考|MA3212

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

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|线性规划作业代写Linear Programming代考|Iterative Algorithms and Convergence

The most important characteristic of a high-speed computer is its ability to perform repetitive operations efficiently, and in order to exploit this basic characteristic, most algorithms designed to solve large optimization problems are iterative in nature. Typically, in seeking a vector that solves the programming problem, an initial vector $\mathbf{x}{0}$ is selected and the algorithm generates an improved vector $\mathbf{x}{1}$. The process is repeated and a still better solution $\mathbf{x}{2}$ is found. Continuing in this fashion, a sequence of ever-improving points $\mathbf{x}{0}, \mathbf{x}{1}, \ldots, \mathbf{x}{k}, \ldots$, is found that approaches a solution point $\mathbf{x}^{*}$. For linear programming problems solved by the simplex method, the generated sequence is of finite length, reaching the solution point exactly after a finite (although initially unspecified) number of steps. For nonlinear programming problems or interior-point methods, the sequence generally does not ever exactly reach the solution point, but converges toward it. In operation, the process is terminated when a point sufficiently close to the solution point, say with at most a positive number $\epsilon(<1)$ error for practical purposes, is obtained (a solution with error $\epsilon=0$ is an exact solution).

The theory of iterative algorithms can be divided into two aspects. The first is concerned with the creation of the algorithms themselves. Algorithms are not conceived arbitrarily, but are based on a creative examination of the programming problem, its inherent structure, and the efficiencies of digital computers. The second aspect is the verification that a given algorithm will in fact generate a sequence that converges to a solution point. This aspect is referred to as global convergence, since it addresses the important question of whether the point sequence generated by an algorithm, when initiated far from the solution point, will eventually converge to it, and at what speed the sequence converges to the solution. One cannot regard a problem as solved simply because an algorithm is known which will converge to the solution, since it may require an exorbitant amount of time to reduce the error to an acceptable tolerance. It is essential when prescribing algorithms that some estimate of the time required is available. It is the convergence-rate aspect of the theory that allows some quantitative evaluation and comparison of different algorithms, and at least crudely, assigns a measure of tractability to a problem, as discussed in Sect.

## 数学代写|线性规划作业代写Linear Programming代考|Examples of Linear Programming Problems

Linear programming has long proved its merit as a significant model of numerous allocation problems and economic phenomena. The continuously expanding literature of applications repeatedly demonstrates the importance of linear programming as a general framework for problem formulation. In this section we present some classic examples of situations that have natural formulations.

Example 1 (The Diet Problem) How can we determine the most economical diet that satisfies the basic minimum nutritional requirements for good health? Such a problem might, for example, be faced by the dietitian of a large army. We assume that there are available at the market $n$ different foods and that the $j$ th food sells at a price $c_{j}$ per unit. In addition there are $m$ basic nutritional ingredients and, to achieve a balanced diet, each individual must receive at least $b_{i}$ units of the $i$ th nutrient per day. Finally, we assume that each unit of food $j$ contains $a_{i j}$ units of the $i$ th nutrient.

If we denote by $x_{j}$ the number of units of food $j$ in the diet, the problem then is to select the $x_{j}$ ‘s to minimize the total cost
$$c_{1} x_{1}+c_{2} x_{2}+\cdots+c_{n} x_{n}$$
subject to the nutritional constraints
$$a_{i 1} x_{1}+a_{i 2} x_{2}+\cdots+a_{i n} x_{n} \geqslant b_{i}, i=1, \ldots, m,$$
and the nonnegativity constraints
$$x_{1} \geqslant 0, x_{2} \geqslant 0, \ldots, x_{n} \geqslant 0$$
on the food quantities.
This problem can be converted to standard form by subtracting a nonnegative surplus variable from the left side of each of the $m$ linear inequalities. The diet problem is discussed further in Chap. $3 .$

## 数学代写|线性规划作业代写Linear Programming代考|Examples of Linear Programming Problems

$$c_{1} x_{1}+c_{2} x_{2}+\cdots+c_{n} x_{n}$$
subject to the nutritional constraints
$$a_{i 1} x_{1}+a_{i 2} x_{2}+\cdots+a_{i n} x_{n} \geqslant b_{i}, i=1, \ldots, m,$$
and the nonnegativity constraints
$$x_{1} \geqslant 0, x_{2} \geqslant 0, \ldots, x_{n} \geqslant 0$$

## 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|线性规划作业代写Linear Programming代考|MAT2200

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

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|线性规划作业代写Linear Programming代考|Unconstrained Problems

It may seem that unconstrained optimization problems are so devoid of structural properties as to preclude their applicability as useful models of meaningful problems. Quite the contrary is true for two reasons. First, it can be argued, quite convincingly, that if the scope of a problem is broadened to the consideration of all relevant decision variables, there may then be no constraints-or put another way, constraints represent artificial delimitations of scope, and when the scope is broadened the constraints vanish. Thus, for example, it may be argued that a budget constraint is not characteristic of a meaningful problem formulation; since by borrowing at some interest rate it is always possible to obtain additional funds, and hence rather than introducing a budget constraint, a term reflecting the cost of funds should be incorporated into the objective. A similar argument applies to constraints describing the availability of other resources which at some cost (however great) could be supplemented.

The second reason that many important problems can be regarded as having no constraints is that constrained problems are sometimes easily converted to unconstrained problems. For instance, the sole effect of equality constraints is simply to limit the degrees of freedom, by essentially making some variables functions of others. These dependencies can sometimes be explicitly characterized, and a new problem having its number of variables equal to the true degree of freedom can be determined. As a simple specific example, a constraint of the form $x_{1}+x_{2}=B$ can be eliminated by substituting $x_{2}=B-x_{1}$ everywhere else that $x_{2}$ appears in the problem.

Aside from representing a significant class of practical problems, the study of unconstrained problems, of course, provides a stepping stone toward the more general case of constrained problems. Many aspects of both theory and algorithms are most naturally motivated and verified for the unconstrained case before progressing to the constrained case.

## 数学代写|线性规划作业代写Linear Programming代考|Constrained Problems

In spite of the arguments given above, many problems met in practice are formulated as constrained problems. This is because in most instances a complex problem such as, for example, the detailed production policy of a giant corporation, the planning of a large government agency, or even the design of a complex device cannot be directly treated in its entirety accounting for all possible choices, but instead must be decomposed into separate subproblems-each subproblem having constraints that are imposed to restrict its scope. Thus, in a planning problem, budget constraints are commonly imposed in order to decouple that one problem from a more global one. Therefore, one frequently encounters general nonlinear constrained mathematical programming problems.
The general mathematical programming problem can be stated as
In this formulation, $\mathbf{x}$ is an $n$-dimensional vector of unknowns, $\mathbf{x}=\left(x_{1}, x_{2}, \ldots\right.$, $\left.x_{n}\right)$, and $f, h_{i}, i=1,2, \ldots, m$, and $g_{j}, j=1,2, \ldots, p$, are real-valued functions of the variables $x_{1}, x_{2}, \ldots, x_{n}$. The set $S$ is a subset of $n$-dimensional space. The function $f$ is the objective function of the problem and the equations, inequalities, and set restrictions are constraints.

Generally, in this book, additional assumptions are introduced in order to make the problem smooth in some suitable sense. For example, the functions in the problem are usually required to be continuous, or perhaps to have continuous derivatives. This ensures that small changes in $\mathbf{x}$ lead to small changes in other values associated with the problem. Also, the set $S$ is not allowed to be arbitrary but usually is required to be a connected region of $n$-dimensional space, rather than, for example, a set of distinct isolated points. This ensures that small changes in $\mathbf{x}$ can be made. Indeed, in a majority of problems treated, the set $S$ is taken to be the entire space; there is no set restriction.

In view of these smoothness assumptions, one might characterize the problems treated in this book as continuous variable programming, since we generally discuss problems where all variables and function values can be varied continuously. In fact, this assumption forms the basis of many of the algorithms discussed, which operate essentially by making a series of small movements in the unknown $\mathbf{x}$ vector.

## 有限元方法代写

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## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。