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澳洲代写｜ECMT3150｜The Econometrics of Financial Markets金融市场计量经济学悉尼大学

Posted on 2023年10月12日2023年10月12日 by statistics-lab

statistics-lab^TM为您悉尼大学（英语：The University of Sydney）澳洲代写｜ECMT3150｜The Econometrics of Financial Markets金融市场计量经济学悉尼大学澳洲代写代考和辅导服务！

课程介绍：

This unit provides students with an understanding of the economic foundations of financial theory and the economic framework upon which that theory is based. Much of the work covered is an application of both microeconomic and macroeconomic theory to the special problems encountered in the study of the financial side of an economy. The relevance of these foundations is illustrated with empirical research using Australian and international data.

澳洲代写｜ECMT3150｜The Econometrics of Financial Markets金融市场计量经济学悉尼大学

Attribute	Detail
Course Code	ECMT3150
Course Title	The Econometrics of Financial Markets
Academic Unit	Economics
Session	Semester 1, 2023
Number of Units	6
Pre-Requisites	ECMT2130
Course Coordinator	Not explicitly mentioned in the provided text
Lecturer	Not explicitly mentioned in the provided text

Linear time series analysis线性时间序列分析的问题

问题 1.

Let $\mathbf{V}$ be an $n \times n$ positive definite matrix, and let $\mathbf{U}$ and $\mathbf{T}$ be $n \times k$ and $n \times(n-k)$ matrices, respectively, such that, if $\mathbf{W}=[\mathbf{U}, \mathbf{T}]$, then $\mathbf{W}^{\prime} \mathbf{W}=\mathbf{W} \mathbf{W}^{\prime}=\mathbf{I}_n$. Then
$$
\mathbf{V}^{-1}-\mathbf{V}^{-1} \mathbf{U}\left(\mathbf{U}^{\prime} \mathbf{V}^{-1} \mathbf{U}\right)^{-1} \mathbf{U}^{\prime} \mathbf{V}^{-1}=\mathbf{T}\left(\mathbf{T}^{\prime} \mathbf{V T}\right)^{-1} \mathbf{T}^{\prime}
$$

Let $\mathbf{P}=\mathbf{P}{\mathbf{X}}$ be the usual projection matrix on the column space of $\mathbf{X}$ from let $\mathbf{M}=\mathbf{I}_T-\mathbf{P}$, and let $\mathbf{G}$ and $\mathbf{H}$ be matrices such that $\mathbf{M}=\mathbf{G}^{\prime} \mathbf{G}$ and $\mathbf{P}=\mathbf{H}^{\prime} \mathbf{H}$, in which case $\mathbf{W}=\left[\mathbf{H}^{\prime}, \mathbf{G}^{\prime}\right]$ satisfies $\mathbf{W}^{\prime} \mathbf{W}=\mathbf{W} \mathbf{W}^{\prime}=\mathbf{I}_T$. Theorem 1.8 For the regression model given with $\widehat{\epsilon}=\mathbf{M}{\mathbf{\Sigma}} \mathbf{Y}$ from (1.62),
$$
\widehat{\epsilon}^{\prime} \Sigma^{-1} \widehat{\epsilon}=\epsilon^{\prime} \mathbf{G}^{\prime}\left(\mathbf{G} \Sigma \mathbf{G}^{\prime}\right)^{-1} \mathbf{G} \epsilon \text {. }
$$
Proof: $\mathbf{T}=\mathbf{G}^{\prime}, \mathbf{U}=\mathbf{H}^{\prime}$, and $\mathbf{V}=\mathbf{\Sigma}$, and the fact that $\mathbf{H}^{\prime}$ can be written as $\mathbf{X K}$, where $\mathbf{K}$ is a $k \times k$ full rank transformation matrix, we have
$$
\begin{aligned}
& \epsilon^{\prime} \mathbf{G}^{\prime}\left(\mathbf{G} \boldsymbol{\Sigma} \mathbf{G}^{\prime}\right)^{-1} \mathbf{G} \boldsymbol{\epsilon}=\mathbf{U}^{\prime}\left(\mathbf{\Sigma}^{-1}-\mathbf{\Sigma}^{-1} \mathbf{H}^{\prime}\left(\mathbf{H} \boldsymbol{\Sigma}^{-1} \mathbf{H}^{\prime}\right)^{-1} \mathbf{H} \boldsymbol{\Sigma}^{-1}\right) \mathbf{U} \
& =\mathbf{U}^{\prime}\left(\mathbf{\Sigma}^{-1}-\boldsymbol{\Sigma}^{-1} \mathbf{X K}\left(\mathbf{K}^{\prime} \mathbf{X}^{\prime} \mathbf{\Sigma}^{-1} \mathbf{X K}\right)^{-1} \mathbf{K}^{\prime} \mathbf{X}^{\prime} \mathbf{\Sigma}^{-1}\right) \mathbf{U} \
& =\mathbf{U}^{\prime}\left(\boldsymbol{\Sigma}^{-1}-\boldsymbol{\Sigma}^{-1} \mathbf{X}\left(\mathbf{X}^{\prime} \boldsymbol{\Sigma}^{-1} \mathbf{X}\right)^{-1} \mathbf{X}^{\prime} \boldsymbol{\Sigma}^{-1}\right) \mathbf{U}=\widehat{\boldsymbol{\epsilon}}^{\prime} \boldsymbol{\Sigma}^{-1} \widehat{\boldsymbol{\epsilon}}, \
&
\end{aligned}
$$

问题 2.

Consider the simple linear regression model $Y_t=\beta_1+\beta_2 X_t+\epsilon_t, t=1, \ldots, T$.
a) By setting $\partial \mathrm{S}(\boldsymbol{\beta}) / \partial \beta_1$ to zero, show that $\hat{\beta}1=\bar{Y}-\widehat{\beta}_2 \bar{X}$. Using this with $0=\partial \mathrm{S}(\boldsymbol{\beta}) / \partial \beta_2$, show that $\hat{\beta}_2=\hat{\sigma}{X, Y} / \hat{\sigma}X^2$, where $\hat{\sigma}{X, Y}$ denotes the sample covariance between $X$ and $Y$,
$$
\hat{\sigma}{X, Y}:=\frac{1}{T-1} \sum{t=1}^T\left(X_t-\bar{X}\right)\left(Y_t-\bar{Y}\right),
$$
and $\hat{\sigma}X^2:=\hat{\sigma}{X, X}$.
b) Show that $\widehat{Y}t-\bar{Y}=\widehat{\beta}_2\left(X_t-\bar{X}\right)$. c) Define the standardized variables $x_t=\left(X_t-\bar{X}\right) / \hat{\sigma}_X$ and $y_t=\left(Y_t-\bar{Y}\right) / \hat{\sigma}_Y$, and consider the regression $y_t=\alpha_1+\alpha_2 x_t+\varepsilon_t$. Show that $\hat{\alpha}_1=0$ and $\hat{\alpha}_2=\hat{\rho}$, where $\hat{\rho}=\hat{\rho}{X, Y}$ is the sample correlation between $X$ and $Y$, with $|\hat{\rho}| \leqslant 1$. Thus, we can write
$$
\hat{Y}_t=\hat{\alpha}_1+\hat{\alpha}_2 x_t=\hat{\rho} x_t,
$$
and squaring and summing both sides yields $\hat{\rho}^2=\sum \widehat{Y}_t^2 / \sum x_t^2$. Show that the $R^2$ statistics for the two regression models are the same, namely $\hat{\rho}^2$.

Continuous time models, option pricing连续时间模型，期权定价定义

Brownian Motion
A Brownian motion is a real-valued stochastic process with continuous trajectories that have independent and stationary increments. The trajectories are denoted by $t \rightarrow W_t$ and for the standard Brownian motion:

$W_0=0$;
for any $0<t_1<\cdots<t_n$, the random variables $\left(W_{t_1}, W_{t_2}-W_{t_1}, \cdots, W_{t_n}-W_{t_{n-1}}\right)$ are independent;
for any $0 \leq s<t$, the increment $W_t-W_s$ is a centered (mean-zero) normal random variable with variance $\mathbb{E}\left{\left(W_t-W_s\right)^2\right}=t-s$. In particular $W_t$ is $\mathcal{N}(0, t)$-distributed.
$\mathcal{F}t$ denotes the $\sigma$-algebra generated by $\left(W_s\right){s \leq t}$, the information on $W$ up to time $t$.

Conditional characteristic functions
For $0 \leq s<t$ and $u \in \mathbb{R}$
$$
\mathbf{E}\left{\mathrm{e}^{\mathrm{i} \mathbf{u}\left(\mathrm{W}{\mathrm{t}}-\mathrm{W}{\mathrm{s}}\right)} \mid \mathcal{F}_{\mathrm{s}}\right}=\mathrm{e}^{-\frac{\mathrm{u}^2(\mathrm{t}-\mathrm{s})}{2}} .
$$
If $W$ is a Brownian motion, by independence of the increment $W_t-W_s$ from the past $\mathcal{F}_s$, the left-hand side of (3) is simply
$$
\mathbb{E}\left{e^{i u\left(W_t-W_s\right)}\right}
$$
which is the characteristic function of a centered normal random variable with variance $t-s$, and is equal to the right-hand side.
Conversely, if (3) holds, then the continuous process $\left(W_t\right)$ is a standard Brownian motion.

Gaussian white noise
This independence of increments makes the Brownian motion an ideal candidate to define a complete family of independent infinitesimal increments $\mathrm{dW}{\mathbf{t}}$, which are $\mathcal{N}(\mathbf{0}, \mathbf{d t})$-distributed (centered, normally distributed with variance $d t$ ) and which will serve as a model of (Gaussian white) noise. The drawback is that the trajectories of $\left(W_t\right)$ are not of bounded variation. Let $t_0=0{i-1}=t / n$. The quantity
$$
\mathbb{E}\left{\sum_{i=1}^n\left|W_{t_i}-W_{t_{i-1}}\right|\right}=n \mathbb{E}\left{\left|W_{\frac{t}{n}}\right|\right}=n \sqrt{\frac{t}{n}} \mathbb{E}\left{\left|W_1\right|\right},
$$
goes to $+\infty$ as $n \nearrow+\infty$.
The integral with respect to $d W_t$ cannot be defined in the usual way “trajectory by trajectory”.

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。

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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，工具箱允许您学习和应用专业技术。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。

R语言代写	问卷设计与分析代写
PYTHON代写	回归分析与线性模型代写
MATLAB代写	方差分析与试验设计代写
STATA代写	机器学习/统计学习代写
SPSS代写	计量经济学代写
EVIEWS代写	时间序列分析代写
EXCEL代写	深度学习代写
SQL代写	各种数据建模与可视化代写

澳洲代写｜MATH3404｜Optimisation Theory最优化理论昆士兰大学

Posted on 2023年10月11日2023年10月11日 by statistics-lab

statistics-lab^TM为您提供昆士兰大学（The University of Queensland）Optimisation Theory最优化理论澳洲代写代考和辅导服务！

课程介绍：

Calculus of variations: critical points; Euler equations; transversality; corner conditions; Hamilton equations; Jacobi equations; Legendre sufficient condition; Weierstrass E-function. Control theory: Lagrange, Mayer & Bolza problems; Pontryagin maximal principle, legendre transformations, augmented Hamiltonians, transversality, bang-bang control, linear systems.

澳洲代写｜MATH3404｜Optimisation Theory最优化理论昆士兰大学

Attribute	Detail
Course Code	MATH3404
Course Title	Optimisation Theory
Academic Unit	School of Mathematics and Physics
Session	Semester 2, 2023
Number of Units	2
Pre-Requisites	MATH2000 or MATH2001
Course Coordinator	Associate Professor Min-Chun Hong
Lecturer	Associate Professor Min-Chun Hong
Tutors	Mr Rhuaidi Burke, Mr Mitchell Jones, Mr Jacob Westerhout

Optimisation Theory最优化理论案例1

One of the fundamental operations in signal processing is filtering the data vector $x=\gamma s+n$, to remove the noise component $n$, while leaving the signal component $s$ relatively unaltered [53]. This can be done either to estimate $\gamma$, the amount of the signal vector $s$ present, or to detect if the signal is present at all, that is, to decide if $\gamma=0$ or not. The noise is typically known only through its covariance matrix $Q$, which is the positivedefinite, symmetric matrix having for its entries $Q_{j k}=E\left(n_j n_k\right)$. The filter usually is linear and takes the form of an estimate of $\gamma$ :
$$
\hat{\gamma}=b^T x
$$
We want $\left|b^T s\right|^2$ large, and, on average, $\left|b^T n\right|^2$ small; that is, we want $E\left(\left|b^T n\right|^2\right)=b^T E\left(n n^T\right) b=b^T Q b$ small. The best choice is the vector $b$ that maximizes the gain of the filter, that is, the ratio
$$
\left|b^T s\right|^2 / b^T Q b .
$$
We can solve this problem using the Cauchy Inequality.
Definition 2.2 Let $S$ be a square matrix. A non-zero vector $u$ is an eigenvector of $S$ if there is a scalar $\lambda$ such that $S u=\lambda u$. Then the scalar $\lambda$ is said to be an eigenvalue of $S$ associated with the eigenvector $u$.

Definition 2.3 The transpose, $B=A^T$, of an $M$ by $N$ matrix $A$ is the $N$ by $M$ matrix having the entries $B_{n, m}=A_{m, n}$.
Definition 2.4 A square matrix $S$ is symmetric if $S^T=S$.
A basic theorem in linear algebra is that, for any symmetric $N$ by $N$ matrix $S, \mathbb{R}^N$ has an orthonormal basis consisting of mutually orthogonal, norm-one eigenvectors of $S$. We then define $U$ to be the matrix whose columns are these orthonormal eigenvectors $u^n$ and $L$ the diagonal matrix with the associated eigenvalues $\lambda_n$ on the diagonal, we can easily see that $U$ is an orthogonal matrix, that is, $U^T U=I$. We can then write

S=U L U^T ;

this is the eigenvalue/eigenvector decomposition of $S$. The eigenvalues of a symmetric $S$ are always real numbers.

Definition 2.5 $A J$ by $J$ symmetric matrix $Q$ is non-negative definite if, for every $x$ in $\mathbb{R}^J$, we have $x^T Q x \geq 0$. If $x^T Q x>0$ whenever $x$ is not the zero vector, then $Q$ is said to be positive definite.

We leave it to the reader to show, in Exercise 2.13, that the eigenvalues of a non-negative (positive) definite matrix are always non-negative (positive).
A covariance matrix $Q$ is always non-negative definite, since
$$
x^T Q x=E\left(\left|\sum_{j=1}^J x_j n_j\right|^2\right) .
$$
Therefore, its eigenvalues are non-negative; typically, they are actually positive, as we shall assume now. We then let $C=U \sqrt{L} U^T$, which is called the symmetric square root of $Q$ since $Q=C^2=C^T C$. The Cauchy Inequality then tells us that
$$
\left|b^T s\right|^2=\left|b^T C C^{-1} s\right|^2 \leq\left[b^T C C^T b\right]\left[s^T\left(C^{-1}\right)^T C^{-1} s\right],
$$
with equality if and only if the vectors $C^T b$ and $C^{-1} s$ are parallel. It follows that
$$
b=\alpha\left(C C^T\right)^{-1} s=\alpha Q^{-1} s,
$$
for any constant $\alpha$. It is standard practice to select $\alpha$ so that $b^T s=1$, therefore $\alpha=1 / s^T Q^{-1} s$ and the optimal filter $b$ is
$$
b=\frac{1}{s^T Q^{-1} s} Q^{-1} s .
$$

Calculus of variations变分法知识点

Invariance of the Euler-Lagrange Equation
The principles in physics that lead to variational formulations do not depend on coördinate systems. Geometrical problems such as the determination of geodesics are likewise “coördinate free” in character. The path of a particle, for instance, does not depend on the coördinate system the observer uses to describe it; a geodesic does not depend on a particular parametrization of the surface. These types of problems can be framed in terms of maximizing functionals and ultimately lead to solutions to an Euler-Lagrange equation. On physical (and geometrical) grounds one thus expects the Euler-Lagrange equation to also be invariant with respect to coördinate transformations. In this section we take an informal but practical look at the invariance of the Euler-Lagrange equation.
A coördinate transformation
$$
x=x(u, v), \quad y=y(u, v),
$$
is called smooth if the functions $x$ and $y$ have continuous partial derivatives with respect to $u$ and $v$. A smooth transformation is called nonsingular if the Jacobian
$$
\frac{\partial(x, y)}{\partial(u, v)}=\operatorname{det}\left(\begin{array}{ll}
x_u & y_u \
x_v & y_v
\end{array}\right)
$$
satisfies the condition
$$
\frac{\partial(x, y)}{\partial(u, v)} \neq 0 .
$$
Here we use the notation $x_u=\partial x / \partial u$ etc. for succinctness. Note that condition (2.29) implies that the transformation is invertible: to every pair $(x, y)$ there corresponds a unique pair $(u, v)$ satisfying equation $(2.28) .{ }^7$ We assume that the coördinate transformation defined by equation (2.28) is smooth and nonsingular.
Let $J$ be a functional of the form
$$
J(y)=\int_{x_0}^{x_1} f\left(x, y, y^{\prime}\right) d x,
$$
and let $S$ be defined by

$$
S=\left{y \in C^2\left[x_0, x_1\right]: y\left(x_0\right)=y_0 \text { and } y\left(x_1\right)=y_1\right},
$$
where $y_0$ and $y_1$ are given numbers. Suppose now that we write the functional in terms of the $(u, v)$ coördinates and, for definiteness, let us regard $v$ as a function of $u$. Then,
$$
\frac{d y}{d x}=\frac{d y / d u}{d x / d u}=\frac{y_u+y_v \dot{v}}{x_u+x_v \dot{v}},
$$
and
$$
d x=\frac{d x}{d u} d u=\left(x_u+x_v \dot{v}\right) d u,
$$
where $\dot{v}$ denotes $d v / d u$. The functional defined by equation (2.30) thus becomes
$$
\begin{aligned}
J(y) & =\int_{x_0}^{x_1} f\left(x, y, y^{\prime}\right) d x \
& =\int_{u_0}^{u_1} f\left(x(u, v), y(u, v), \frac{y_u+y_v \dot{v}}{x_u+x_v \dot{v}}\right)\left(x_u+x_v \dot{v}\right) d u \
& =\int_{u_0}^{u_1} F(u, v, \dot{v}) d u .
\end{aligned}
$$
Here, the numbers $u_0, u_1$ and the new boundary values $v\left(u_0\right)=v_0, v\left(u_1\right)=v_1$ are the unique solutions to the equations
$$
\begin{aligned}
& x_0=x\left(u_0, v_0\right), x_1=x\left(u_1, v_1\right), \
& y_0=y\left(u_0, v_0\right), y_1=y\left(u_1, v_1\right) .
\end{aligned}
$$
For clarity, let
$$
K(v)=\int_{u_0}^{u_1} F(u, v, \dot{v}) d u,
$$
and let $T$ be the set defined by
$$
T=\left{v \in C^2\left[u_0, u_1\right]: v\left(u_0\right)=v_0 \text { and } v\left(u_1\right)=v_1\right} .
$$
Given a curve in the $x y$-plane described by a function $y=y(x)$, the transformation (2.28) defines the curve in the $u v$-plane described by some function $v=v(u)$. The essence of the invariant question is: if $v \in T$ is an extremal for $K$, is $y \in S$ and extremal for $J$ (and vice versa)? The next theorem resolves this question.

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

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。

有限元方法代写

随机分析代写

时间序列分析代写

回归分析代写

MATLAB代写

R语言代写	问卷设计与分析代写
PYTHON代写	回归分析与线性模型代写
MATLAB代写	方差分析与试验设计代写
STATA代写	机器学习/统计学习代写
SPSS代写	计量经济学代写
EVIEWS代写	时间序列分析代写
EXCEL代写	深度学习代写
SQL代写	各种数据建模与可视化代写

澳洲代写｜ECON6021｜Financial Economics金融经济学悉尼大学

Posted on 2023年10月11日2023年10月11日 by statistics-lab

statistics-lab^TM为您悉尼大学（英语：The University of Sydney）Financial Economics金融经济学澳洲代写代考和辅导服务！

课程介绍：

澳洲代写｜ECON6021｜Financial Economics金融经济学悉尼大学

Attribute	Detail
Course Code	ECON6021
Course Title	Financial Economics
Academic Unit	Economics
Session	Semester 1, 2023
Attendance Mode	Normal day
Location	Camperdown/Darlington, Sydney
Number of Units	6
Course Coordinators/Examiner	Information not provided in the given text
Pre-Requisites	Information not provided in the given text

Contingent claim economy或有债权经济的问题

Basic Characteristics of Option Prices
The owner of a call option has the right, but not the obligation, to buy a given asset in the future at a pre-agreed price, known as the exercise price, or strike price. Similarly, the owner of a put option has the right, but not the obligation,

to sell a given asset in the future at a preagreed price. For each owner (buyer) of an option, there is an option seller, also referred to as the option writer. If the owner of a call (put) option chooses to exercise, the seller must deliver (receive) the underlying asset or commodity in return for receiving (paying) the pre-agreed exercise price. Since an option always has a non-negative payoff to the owner, this buyer of the option must make an initial payment, called the option’s premium, to the seller of the option. ${ }^{10}$

Options can have different features regarding which future date(s) that exercise can occur. A European option can be exercised only at the maturity of the option contract, while an American option can be exercised at any time prior to the maturity of the contract.

Let us define the following notation, similar to that used to describe a forward contract. Let $S_0$ denote the current date 0 price per share of the underlying asset, and let this asset’s price at the maturity date of the option contract, $\tau$, be denoted as $S_\tau$. We let $X$ be the exercise price of the option and denote the date $t$ price of European call and put options as $c_t$ and $p_t$, respectively. Then based on our description of the payoffs of call and put options, we can write the maturity values of European call and put options as
$$
\begin{aligned}
c_\tau & =\max \left[S_\tau-X, 0\right] \
p_\tau & =\max \left[X-S_\tau, 0\right]
\end{aligned}
$$
Now we recall that the payoffs to the long and short parties of a forward contract are $S_\tau-F_{0 \tau}$ and $F_{0 \tau}-S_\tau$, respectively. If we interpret the pre-agreed forward price, $F_0$, as analogous to an option’s preagreed exercise price, $X$, then we see that a call option’s payoff equals that of the long forward payoff whenever thelong forward payoff is positive, and it equals 0 when the long forward payoff is negative. Similarly, the payoff of the put option equals the short forward payoff when this payoff is positive, and it equals 0 when the short forward payoff is negative. Hence, assuming $X=F_{0 \tau}$, we see that the payoff of a call option weakly dominates that of a long forward position, while the payoff of a put option weakly dominates that of a short forward position. ${ }^{11}$ This is due to the consequence of option payoffs always being nonnegative whereas forward contract payoffs can be of either sign.

期权价格的基本特征
看涨期权的所有者有权利（但没有义务）在未来以预先约定的价格（称为执行价格或执行价格）购买特定资产。同样，看跌期权的所有者有权利，但没有义务，

将来以预先约定的价格出售特定资产。对于每个期权的所有者（买方）来说，都有一个期权卖方，也称为期权卖方。如果看涨（看跌）期权的所有者选择行权，卖方必须交付（接收）标的资产或商品，以换取接收（支付）预先约定的行权价格。由于期权总是给所有者带来非负的回报，因此期权的买方必须向期权的卖方支付一笔初始付款，称为期权费。 ${}^{10}$

关于行使的未来日期，期权可以具有不同的特征。欧式期权只能在期权合约到期时行权，而美式期权则可以在合约到期前的任何时间行权。

让我们定义以下符号，类似于用于描述远期合约的符号。令 $S_0$ 表示标的资产的当前日期 0 每股价格，并令该资产在期权合约到期日的价格 $\tau$ 表示为 $S_\tau$。我们让 $X$ 为期权的执行价格，并将欧式看涨期权和看跌期权的日期 $t$ 价格分别表示为 $c_t$ 和 $p_t$。然后根据我们对看涨期权和看跌期权收益的描述，我们可以将欧式看涨期权和看跌期权的到期价值写为$$
\begin{aligned}
c_\tau & =\max \left[S_\tau-X, 0\right] \
p_\tau & =\max \left[X-S_\tau, 0\right]
\end{aligned}
$$

现在我们回想一下，远期合约多头和空头双方的收益分别为 $S_\tau-F_{0 \tau}$ 和 $F_{0 \tau}-S_\tau$。如果我们将预先商定的远期价格 $F_0$ 解释为类似于期权的预先商定的执行价格 $X$，那么我们会看到，只要长期远期收益为正，看涨期权的收益就等于长期远期收益的收益，并且当长期远期收益为负时，它等于 0。同样，当收益为正时，看跌期权的收益等于空头远期收益，当空头远期收益为负时，它等于0。因此，假设 $X=F_{0 \tau}$，我们看到看涨期权的收益微弱地支配多头远期头寸的收益，而看跌期权的收益微弱地支配空头远期头寸的收益。 ${ }^{11}$ 这是因为期权收益总是非负的，而远期合约收益可以是任一符号。

Risk Allocation Principles风险分配原则定义

Principles for the management of operational risk

Operational risk ${ }^5$ is inherent in all banking products, activities, processes and systems, and the effective management of operational risk has always been a fundamental element of a bank’s risk management programme. As a result, sound operational risk management is a reflection of the effectiveness of the board and senior management in administering its portfolio of products, activities, processes, and systems. The Committee, through the publication of this paper, desires to promote and enhance the effectiveness of operational risk management throughout the banking system.

Risk management generally encompasses the process of identifying risks to the bank, measuring exposures to those risks (where possible), ensuring that an effective capital planning and monitoring programme is in place, monitoring risk exposures and corresponding capital needs on an ongoing basis, taking steps to control or mitigate risk exposures and reporting to senior management and the board on the bank’s risk exposures and capital positions. Internal controls are typically embedded in a bank’s day-to-day business and are designed to ensure, to the extent possible, that bank activities are efficient and effective, information is reliable, timely and complete and the bank is compliant with applicable laws and regulation. In practice, the two notions are in fact closely related and the distinction between both is less important than achieving the objectives of each.

Sound internal governance forms the foundation of an effective operational risk management Framework. Although internal governance issues related to the management of operational risk are not unlike those encountered in the management of credit or market risk operational risk management challenges may differ from those in other risk areas.

The Committee is seeing sound operational risk governance practices adopted in an increasing number of banks. Common industry practice for sound operational risk governance often relies on three lines of defence – (i) business line management, (ii) an independent corporate operational risk management function and (iii) an independent review. ${ }^6$ Depending on the bank’s nature, size and complexity, and the risk profile of a bank’s activities, the degree of formality of how these three lines of defence are implemented will vary. In all cases, however, a bank’s operational risk

操作风险${ }^5$是所有银行产品、活动、流程和系统所固有的，操作风险的有效管理一直是银行风险管理计划的基本要素。因此，健全的操作风险管理反映了董事会和高级管理层在管理其产品、活动、流程和系统组合方面的有效性。委员会希望通过发布本文件，促进和提高整个银行体系操作风险管理的有效性。

风险管理通常包括识别银行面临的风险、衡量这些风险的暴露（如果可能）、确保有效的资本规划和监控计划到位、持续监控风险暴露和相应的资本需求、采取措施的过程控制或减轻风险敞口，并向高级管理层和董事会报告银行的风险敞口和资本状况。内部控制通常嵌入银行的日常业务中，旨在尽可能确保银行活动高效、有效，信息可靠、及时和完整，以及银行遵守适用的法律和法规。在实践中，这两个概念实际上密切相关，两者之间的区别并不重要，重要的是实现各自的目标。

健全的内部治理是有效操作风险管理框架的基础。尽管与操作风险管理相关的内部治理问题与信用或市场风险管理中遇到的问题没有什么不同，操作风险管理挑战可能与其他风险领域的挑战不同。

委员会发现越来越多的银行采用了良好的操作风险治理实践。健全的操作风险治理的常见行业惯例通常依赖于三道防线 – (i) 业务线管理，(ii) 独立的公司操作风险管理职能，以及 (iii) 独立审查。 ${ }^6$ 根据银行的性质、规模和复杂性以及银行活动的风险状况，这三道防线实施的正式程度会有所不同。然而，在所有情况下，银行的操作风险

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

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。

有限元方法代写

随机分析代写

时间序列分析代写

回归分析代写

MATLAB代写

R语言代写	问卷设计与分析代写
PYTHON代写	回归分析与线性模型代写
MATLAB代写	方差分析与试验设计代写
STATA代写	机器学习/统计学习代写
SPSS代写	计量经济学代写
EVIEWS代写	时间序列分析代写
EXCEL代写	深度学习代写
SQL代写	各种数据建模与可视化代写

澳洲代写｜MATH2988｜Number Theory and Cryptography Adv 进阶数论与密码学悉尼大学

Posted on 2023年10月11日2023年10月11日 by statistics-lab

statistics-lab^TM为您悉尼大学（英语：The University of Sydney）Applied Financial Econometrics应用金融计量经济学澳洲代写代考和辅导服务！

课程介绍：

This unit of study is an advanced version of MATH2088, sharing the same lectures but with more advanced topics introduced in the tutorials and computer laboratory sessions.

澳洲代写｜MATH2988｜Number Theory and Cryptography Adv 进阶数论与密码学悉尼大学

Attribute	Detail
Course Code	MATH2988
Course Title	Number Theory and Cryptography Adv
Academic Unit	Mathematics and Statistics Academic Operations
Session	Semester 2, 2023
Attendance Mode	Normal day
Location	Camperdown/Darlington, Sydney
Number of Units	6
Course Coordinators	Information not provided in the given text
Tutors	Information not provided in the given text
Pre-Requisites	This unit of study is an advanced version of MATH2088

Computational complexity计算复杂度的问题

Modeling computation and efficiency
We start with an informal description of computation. Let $f$ be a function that takes a string of bits (i.e., a member of the set ${0,1}^$ ) and outputs, say, either 0 or 1 . Informally speaking, an algorithm for computing $f$ is a set of mechanical rules, such that by following them we can compute $f(x)$ given any input $x \in{0,1}^$. The set of rules being followed is finite (i.e., the same set must work for all infinitely many inputs) though each rule in this set may be applied arbitrarily many times. Each rule must involve one of the following “elementary” operations:

Read a bit of the input.
Read a bit (or possibly a symbol from a slightly larger alphabet, say a digit in the set ${0, \ldots, 9})$ from the “scratch pad” or working space we allow the algorithm to use.
Write a bit/symbol to the scratch pad.
Stop and output either 0 or 1 .
Decide which of the above operations to apply based on the values that were just read.

Finally, the running time is the number of these basic operations performed.
Below, we formalize all of these notions.

建模计算和效率
我们从计算的非正式描述开始。令 $f$ 为一个函数，它接受一串位（即集合 ${0,1}^$ 的成员）并输出（例如 0 或 1 ）。通俗地说，计算 $f$ 的算法是一组机械规则，通过遵循这些规则，我们可以在给定任何输入 $x \in{0,1}^$ 的情况下计算 $f(x)$。所遵循的规则集是有限的（即，同一组必须适用于所有无限多个输入），尽管该组中的每个规则可以任意应用多次。每条规则必须涉及以下“基本”操作之一：

读取一点输入。
从我们允许算法使用的“便笺本”或工作空间中读取一点（或者可能是稍大的字母表中的符号，例如集合 ${0, \ldots, 9}）$ 中的一个数字。
将位/符号写入暂存器。
停止并输出 0 或 1 。
根据刚刚读取的值决定应用上述哪一个操作。

最后，运行时间是执行这些基本操作的次数。
下面，我们将所有这些概念形式化。

Theorem 1.6 (EFFicient Universal Turing Machine)
There exists a $T M \mathcal{U}$ and a representation scheme of TM’s satisfying:

Every string $\alpha \in{0,1}^*$ is a representation of some $T M M_\alpha$.
Every TM $M$ is represented by infinitely many strings $\alpha \in{0,1}^*$.
For every $t \in \mathbb{N}$ and $x, \alpha \in{0,1}^*$, if on input $x$, the machine $M_\alpha$ outputs a string $y$ within at most $t$ steps, then $\mathcal{U}(t, x, \alpha)=y$.
On every triple $\langle t, x, \alpha\rangle$, the machine $\mathcal{U}$ runs for at most $C t \log t$ steps, where $C$ is a constant depending on $M_\alpha$ ‘s alphabet and number of states and tapes but independent of $|\alpha|,|x|,|t|$.

To simulate a single computational step of $M$, the machine $\mathcal{U}$ performs the following actions:

Scan the main work tape (the one containing the contents of $M$ ‘s $k$ tapes), and copy into its scratch tape the $k$ symbols that follow the $\star$ symbol of each tape.
Scan the transition function of $M$ to find out how $M$ behaves on these symbols (what $M$ writes on its tapes, how it changes its state register, and how it moves the head). Then write down this information on the scratch pad.
Scan the main work tape and update it (both symbols written and head locations) according to the scratch pad.
Update the tape containing $M$ ‘s state according to the new state.
Use the same head movement and write instructions of $M$ on the input and output tape.
Decrease the counter by 1 , check if it has reached 0 and if so halt.
Now let’s count how many computational steps $\mathcal{U}$ performs to simulate a single step of $M: \mathcal{U}$ ‘s main tape contains at most $k t$ symbols, and so scanning it takes $O(t)$ steps (as $k$ is a constant depending only on $M$ ). Decreasing the counter takes $O(\log t)$ steps. The transition function, the current state, and the scratch pad only require a constant number of bits to store (where this constant depends on $M$ ‘s alphabet size, and number of tapes and states) and so only require a constant number of operations to read and update. Thus, simulating a single step of $M$ takes $O(t+\log t)=O(t)$ operations, and simulating $M$ for $t$ steps takes $O\left(t^2\right)$ operations.
为了模拟 $M$ 的单个计算步骤，机器 $\mathcal{U}$ 执行以下操作：
扫描主工作磁带（包含 $M$ 的 $k$ 磁带内容的磁带），并将每个磁带 $\star$ 符号后面的 $k$ 符号复制到其暂存磁带中。
扫描 $M$ 的转换函数，找出 $M$ 在这些符号上的行为（$M$ 在其磁带上写入什么，它如何更改其状态寄存器，以及它如何移动磁头）。然后将这些信息写在便签本上。
扫描主要工作磁带并根据暂存器更新它（写入的符号和磁头位置）。
根据新状态更新包含 $M$ 状态的磁带。
使用相同的磁头运动并在输入和输出磁带上写入$M$指令。
将计数器减 1，检查它是否已达到 0，如果是则停止。
现在让我们计算一下 $\mathcal{U}$ 执行多少个计算步骤来模拟 $M 的单个步骤： \mathcal{U}$ 的主磁带最多包含 $k t$ 个符号，因此扫描它需要 $O( t)$ 步骤（因为 $k$ 是仅取决于 $M$ 的常数）。减少计数器需要 $O(\log t)$ 步。转换函数、当前状态和暂存器仅需要恒定数量的位来存储（其中该常数取决于 $M$ 的字母大小以及磁带和状态的数量），因此仅需要恒定数量的读取和更新操作。因此，模拟 $M$ 的单个步骤需要 $O(t+\log t)=O(t)$ 操作，而模拟 $M$ 的 $t$ 个步骤需要 $O\left(t^2\right)$ 操作运营。

Euler–Fermat Theorem欧拉-费马定理定义

Euler’s $\phi$-function
Definition :
Let $\mathrm{m} \geq 1$. The Euler $\phi$-function is the function $\phi$-with domain $\mathrm{N}$, the set of positive integers defined as follows :
$\phi(\mathrm{m})$ equals the number of integers in $\mathrm{Z}{\mathrm{m}}$ that are relatively prime to $\mathrm{m}$, e.g., $\phi(1)=1, \phi(2)=1, \phi(3)=2, \phi(4)=2, \phi(5)=4, \phi(6)=2$ and so on. Thus $\phi$ is not a one-to-one function and in case $\mathrm{p}$ is a prime $$ \phi(p)=p-1 . $$ Definition : The residue class $\overline{\mathrm{r}}$ modulo $\mathrm{m}$ is prime to $\mathrm{m}$ if $(\mathrm{r}, \mathrm{m})=1$. Definition A reduced set of residues modulo $m$ is a set of positive integers $\left{r_1, r_2, \ldots . . r{\phi(m)}\right}$ such that exactly one of them lies in each residue class prime to $\mathrm{m}$. If each $r_{\mathrm{i}}$ satisifies $0 \leq \mathrm{r}{\mathrm{i}}<\mathrm{m}$, we called $\left{r_1, r_2, \ldots . r{\phi(m)}\right}$ the reduced set of least positive residues modulo $m$.

For an example, take $\mathrm{m}=6$. Then ${0,1,2,3,4,5}$ is a complete set of residues modulo 6 and ${1,5}$ is a reduced set of least positive residues modulo 6 . In addition ${6,19$, $20,39,16,17}$ is also a complete set of residues, while ${13,17}$ is a reduced set of residues modulo 6 .

We also denote complete set of residues modulo $\mathrm{m}$ by CRS (Mod $\mathrm{m}$ ) and reduced set of least positive residues modulo $\mathrm{m}$ by RRS $(\bmod \mathrm{m})$
Thus CRS $(\bmod 5)={5,13,27,31,34}$
and $\operatorname{RRS}(\bmod 12)={1,5,7,11}$

澳洲代写｜ECMT6006｜Applied Financial Econometrics应用金融计量经济学悉尼大学

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

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。

有限元方法代写

随机分析代写

时间序列分析代写

回归分析代写

MATLAB代写

R语言代写	问卷设计与分析代写
PYTHON代写	回归分析与线性模型代写
MATLAB代写	方差分析与试验设计代写
STATA代写	机器学习/统计学习代写
SPSS代写	计量经济学代写
EVIEWS代写	时间序列分析代写
EXCEL代写	深度学习代写
SQL代写	各种数据建模与可视化代写

澳洲代写｜ECMT6006｜Applied Financial Econometrics应用金融计量经济学悉尼大学

Posted on 2023年10月11日2023年10月11日 by statistics-lab

statistics-lab^TM为您悉尼大学（英语：The University of Sydney）Applied Financial Econometrics应用金融计量经济学澳洲代写代考和辅导服务！

课程介绍：

This unit provides an introduction to some of the widely used econometric models designed for the analysis of financial data, and the procedures used to estimate them. Special emphasis is placed upon empirical work and applied analysis of real market data. The unit deals with topics such as: the statistical nature of financial data; the specification, estimation and testing of assets pricing models; the analysis of high frequency financial data; and the modelling of volatility in financial returns. Throughout the unit, students are encouraged (especially in assignments) to familiarise themselves with financial data and learn how to apply the models to these data.

Attribute	Detail
Course Code	ECMT6006
Course Title	Applied Financial Econometrics
Academic Unit	Economics
Session	Semester 1, 2023
Attendance Mode	Normal day
Location	Camperdown/Darlington, Sydney
Number of Units	6
Course Coordinators	Information not provided in the given text
Tutors	Information not provided in the given text
Pre-Requisites	Information not provided in the given text

Applied Financial Econometrics应用金融计量经济学的问题

Which type of the errors are the worst: Type I or II”?
(a) Type I errors
(b) Type II errors
(c) Both Type I and II: It depends on the context
(d) None
Stochastic specification $(\varepsilon)$ in the model is also known as
(a) Error terms
(b) Noise terms
(c) “disturbance terms” or “residuals terms”
(d) All of the above
Panel data often referred as the
(a) Longitudinal data
(b) Cross-sectional data
(c) Time series data
(d) All of the above
Daily mean excess return series of the stock ” $\mathrm{X}$ ” for five years is an example of
(a) Panel data
(b) Pooled data
(c) Time series data
(d) Cross-sectional data
Which of the following is an example of open source software?
(a) EViews
(b) SPSS(c) SAS(d) R programming
Which among the following is not a logical operator?
(a)! “NOT”
(b) \&\& “AND”
(c) || “OR”
(d) $==$ “Equal”
Descriptive statistics estimates of a variable includes:
(a) Mean
(b) Median
(c) Skewness \& Kurtosis
(d) All of the above
Which among the following is not used for testing the unit root ?
(a) Augmented Dickey-Fuller Test
(b) Phillips-Perron test
(c) KPSS test
(d) BDS independence test
Which among the following test is/are used for correlation analysis?
(a) Kendall’s tau
(b) Spearman’s rank correlation
(c) Pearson correlation
(d) All of the above
Which among the following test is not used for the time series normality test?
(a) Shapiro-Wilk test
(b) Jarque-Bera test
(c) Phillips-Perron test
(d) Anderson-Darling test

Applied Financial Econometrics应用金融计量经济学问题2

A professor ask his/her student to do a basic comparative analysis of the COVID-19 first and second phase impact on the Asian stock markets ? Assume yourself as the student and perform the task. Then prepare a detailed report of the analysis to be submitted to the professor.
A researcher want to conduct elementary analyse on the performance of cryptocurrencies during the noble Coronavirus pandemic. Assume yourself as the researcher: perform the mentioned task in details and develop the analysis report.
42 M. MAITI
A Financial analyst want to study the elementary performances of the world indices before, during, and after the first phase of COVID19. Help him/her to perform the said analysis and prepare the report.
A student need to perform descriptive examination on the major three currency pairs exchange rate for the last three years as his/her project dissertation. Help the student to complete his/her project dissertation successfully and satisfactorily.
A senior manager of the Reserve bank of India (RBI) asked an intern working under him/her to prepare a detailed report on the total amount of paperless transactions done in India through the various medium before and during the COVID-19 first phase. Help the intern in analysing and developing the final report for timely submission to the senior RBI manager.

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

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。

有限元方法代写

随机分析代写

时间序列分析代写

回归分析代写

MATLAB代写

R语言代写	问卷设计与分析代写
PYTHON代写	回归分析与线性模型代写
MATLAB代写	方差分析与试验设计代写
STATA代写	机器学习/统计学习代写
SPSS代写	计量经济学代写
EVIEWS代写	时间序列分析代写
EXCEL代写	深度学习代写
SQL代写	各种数据建模与可视化代写

澳洲代写｜STAT3006｜Statistical Learning统计学习昆士兰大学

Posted on 2023年10月11日2023年10月11日 by statistics-lab

statistics-lab^TM为您提供昆士兰大学（The University of Queensland）Statistical Learning统计学习澳洲代写代考和辅导服务！

课程介绍：

Statistics is a key building block for numerous modern machine learning algorithms that have found success in many important applications. The course will cover key ideas behind these algorithms from a statistical perspective, and provide an in-depth knowledge that will enable students to apply the methods with awareness of their strengths and limitations. The topics covered will include probabilistic and analytic foundations, multivariate statistical analysis and machine learning, with a particular focus on clustering, classification, model selection and high-dimensional statistical analysis. Students will gain a theoretical understanding of these techniques and develop practical skills by using the R statistical programming language to solve problems involving various real and simulated datasets.

澳洲代写｜STAT3006｜Statistical Learning统计学习昆士兰大学

Statistical Learning统计学习案例1

Least Angle Regression: Lasso Modification.
If a non-zero coefficient hits zero, drop its variable from the active set of variables and recompute the current joint least squares direction.
The LAR(lasso) algorithm is extremely efficient, requiring the same order of computation as that of a single least squares fit using the $p$ predictors. Least angle regression always takes $p$ steps to get to the full least squares estimates. The lasso path can have more than $p$ steps, although the two are often quite similar. Algorithm 3.2 with the lasso modification $3.2 \mathrm{a}$ is an efficient way of computing the solution to any lasso problem, especially when $p \gg N$. Osborne et al. (2000a) also discovered a piecewise-linear path for computing the lasso, which they called a homotopy algorithm.

We now give a heuristic argument for why these procedures are so similar. Although the LAR algorithm is stated in terms of correlations, if the input features are standardized, it is equivalent and easier to work with innerproducts. Suppose $\mathcal{A}$ is the active set of variables at some stage in the algorithm, tied in their absolute inner-product with the current residuals $\mathbf{y}-\mathbf{X} \beta$. We can express this as
$$
\mathbf{x}_j^T(\mathbf{y}-\mathbf{X} \beta)=\gamma \cdot s_j, \forall j \in \mathcal{A}
$$
where $s_j \in{-1,1}$ indicates the sign of the inner-product, and $\gamma$ is the common value. Also $\left|\mathbf{x}_k^T(\mathbf{y}-\mathbf{X} \beta)\right| \leq \gamma \forall k \notin \mathcal{A}$. Now consider the lasso criterion (3.52), which we write in vector form
$$
R(\beta)=\frac{1}{2}|\mathbf{y}-\mathbf{X} \beta|_2^2+\lambda|\beta|_1 .
$$
Let $\mathcal{B}$ be the active set of variables in the solution for a given value of $\lambda$. For these variables $R(\beta)$ is differentiable, and the stationarity conditions give
$$
\mathbf{x}_j^T(\mathbf{y}-\mathbf{X} \beta)=\lambda \cdot \operatorname{sign}\left(\beta_j\right), \forall j \in \mathcal{B}
$$

最小角度回归：套索修改。
如果非零系数为零，则从活动变量集中删除其变量并重新计算当前联合最小二乘方向。
LAR(lasso) 算法非常高效，需要与使用 $p$ 预测器的单个最小二乘拟合相同的计算顺序。最小角度回归总是需要 $p$ 步才能获得完整的最小二乘估计。套索路径可以有超过 $p$ 步，尽管两者通常非常相似。带有套索修改的算法 3.2 $3.2 \mathrm{a}$ 是计算任何套索问题的解决方案的有效方法，特别是当 $p \gg N$ 时。奥斯本等人。 (2000a) 还发现了一种用于计算套索的分段线性路径，他们将其称为同伦算法。

我们现在给出一个启发式的论证来解释为什么这些过程如此相似。尽管 LAR 算法是用相关性来表述的，但如果输入特征是标准化的，那么使用内积是等效的并且更容易使用。假设 $\mathcal{A}$ 是算法中某个阶段的活跃变量集，其绝对内积与当前残差 $\mathbf{y}-\mathbf{X} \beta$ 相关联。我们可以将其表示为

$$
\mathbf{x}_j^T(\mathbf{y}-\mathbf{X} \beta)=\gamma \cdot s_j, \forall j \in \mathcal{A}
$$

$$
其中$s_j \in{-1,1}$表示内积符号，$\gamma$是公共值。还有 $\left|\mathbf{x}_k^T(\mathbf{y}-\mathbf{X} \beta)\right| \leq \gamma \forall k \notin \mathcal{A}$。现在考虑套索准则 (3.52)，我们将其写成向量形式
$$

$$
R(\beta)=\frac{1}{2}|\mathbf{y}-\mathbf{X} \beta|_2^2+\lambda|\beta|_1 .
$$

令$\mathcal{B}$ 为给定$\lambda$ 解中的活跃变量集。对于这些变量 $R(\beta)$ 是可微的，平稳性条件给出
$$

$$
\mathbf{x}_j^T(\mathbf{y}-\mathbf{X} \beta)=\lambda \cdot \operatorname{sign}\left(\beta_j\right), \forall j \in \mathcal{B}
$$

偏最小二乘法 Partial Least Squares知识点

Partial Least Squares
This technique also constructs a set of linear combinations of the inputs for regression, but unlike principal components regression it uses $\mathbf{y}$ (in addition to $\mathbf{X}$ ) for this construction. Like principal component regression, partial least squares (PLS) is not scale invariant, so we assume that each $\mathbf{x}j$ is standardized to have mean 0 and variance 1 . PLS begins by computing $\hat{\varphi}{1 j}=\left\langle\mathbf{x}j, \mathbf{y}\right\rangle$ for each $j$. From this we construct the derived input $\mathbf{z}_1=\sum_j \hat{\varphi}{1 j} \mathbf{x}_j$, which is the first partial least squares direction. Hence in the construction of each $\mathbf{z}_m$, the inputs are weighted by the strength of their univariate effect on $\mathbf{y}^3$. The outcome $\mathbf{y}$ is regressed on $\mathbf{z}_1$ giving coefficient $\hat{\theta}_1$, and then we orthogonalize $\mathbf{x}_1, \ldots, \mathbf{x}_p$ with respect to $\mathbf{z}_1$. We continue this process, until $M \leq p$ directions have been obtained. In this manner, partial least squares produces a sequence of derived, orthogonal inputs or directions $\mathbf{z}_1, \mathbf{z}_2, \ldots, \mathbf{z}_M$. As with principal-component regression, if we were to construct all $M=p$ directions, we would get back a solution equivalent to the usual least squares estimates; using $M<p$ directions produces a reduced regression.

In the prostate cancer example, cross-validation chose $M=2$ PLS directions in Figure 3.7. This produced the model given in the rightmost column of Table 3.3 .

What optimization problem is partial least squares solving? Since it uses the response $\mathbf{y}$ to construct its directions, its solution path is a nonlinear function of $\mathbf{y}$. It can be shown that partial least squares seeks directions that have high variance and have high correlation with the response, in contrast to principal components regression which keys only on high variance (Stone and Brooks 1990, Frank and Friedman 1993). In particular, the $m$ th principal component direction $v_m$ solves:
$$
\begin{gathered}
\max \alpha \operatorname{Var}(\mathbf{X} \alpha) \ \text { subject to }|\alpha|=1, \alpha^T \mathbf{S} v{\ell}=0, \ell=1, \ldots, m-1,
\end{gathered}
$$
where $\mathbf{S}$ is the sample covariance matrix of the $\mathbf{x}j$. The conditions $\alpha^T \mathbf{S} v{\ell}=$ 0 ensures that $\mathbf{z}m=\mathbf{X} \alpha$ is uncorrelated with all the previous linear combinations $\mathbf{z}{\ell}=\mathbf{X} v_{\ell}$. The $m$ th PLS direction $\hat{\varphi}m$ solves: $$ \begin{gathered} \max \alpha \operatorname{Corr}^2(\mathbf{y}, \mathbf{X} \alpha) \operatorname{Var}(\mathbf{X} \alpha) \
\text { subject to }|\alpha|=1, \hat{\varphi}_{\ell}^T \mathbf{S} \alpha=0, \ell=1, \ldots, m-1 .
\end{gathered}
$$

偏最小二乘法
该技术还构造了一组用于回归的输入的线性组合，但与主成分回归不同，它使用 $\mathbf{y}$ （除了 $\mathbf{X}$ 之外）进行此构造。与主成分回归一样，偏最小二乘法 (PLS) 不是尺度不变的，因此我们假设每个 $\mathbf{x}j$ 被标准化为均值 0 和方差 1 。 PLS 首先计算每个 $j$ 的 $\hat{\varphi}{1 j}=\left\langle\mathbf{x}j, \mathbf{y}\right\rangle$。由此我们构造导出输入 $\mathbf{z}_1=\sum_j \hat{\varphi}{1 j} \mathbf{x}_j$，这是第一个偏最小二乘方向。因此，在构建每个 $\mathbf{z}_m$ 时，输入按其对 $\mathbf{y}^3$ 的单变量影响的强度进行加权。结果 $\mathbf{y}$ 在 $\mathbf{z}_1$ 上回归，给出系数 $\hat{\theta}_1$，然后我们正交化 $\mathbf{x}_1, \ldots, \mathbf{ x}_p$ 相对于 $\mathbf{z}_1$。我们继续这个过程，直到获得 $M \leq p$ 方向。以这种方式，偏最小二乘产生一系列导出的正交输入或方向$\mathbf{z}_1，\mathbf{z}_2，\ldots，\mathbf{z}_M$。与主成分回归一样，如果我们要构造所有 $M=p$ 方向，我们将得到相当于通常最小二乘估计的解；使用 $M<p$ 方向会产生减少的回归。偏最小二乘法解决什么优化问题？由于它使用响应 $\mathbf{y}$ 来构造其方向，因此其解路径是 $\mathbf{y}$ 的非线性函数。可以证明，偏最小二乘寻求具有高方差且与响应具有高相关性的方向，这与仅依赖于高方差的主成分回归相反（Stone 和 Brooks 1990，Frank 和 Friedman 1993）。特别是，第 $m$ 个主成分方向 $v_m$ 求解：

$$
\begin{gathered}
\max \alpha \operatorname{Var}(\mathbf{X} \alpha) \ \text { subject to }|\alpha|=1, \alpha^T \mathbf{S} v{\ell}=0, \ell=1, \ldots, m-1,
\end{gathered}
$$

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

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。

有限元方法代写

随机分析代写

时间序列分析代写

回归分析代写

MATLAB代写

R语言代写	问卷设计与分析代写
PYTHON代写	回归分析与线性模型代写
MATLAB代写	方差分析与试验设计代写
STATA代写	机器学习/统计学习代写
SPSS代写	计量经济学代写
EVIEWS代写	时间序列分析代写
EXCEL代写	深度学习代写
SQL代写	各种数据建模与可视化代写

澳洲代写｜MATH4090｜Computation in Financial Mathematics金融数学计算昆士兰大学

Posted on 2023年10月11日2023年10月11日 by statistics-lab

statistics-lab^TM为您提供昆士兰大学（The University of Queensland）Computation in Financial Mathematics金融数学计算澳洲代写代考和辅导服务！

课程介绍：

The past few decades have witnessed an explosion in the trading of sophisticated financial instruments, known as derivatives. Financial derivative securities, such as options, can be viewed as a form of insurance or used for speculation purposes. These instruments are routinely used by large corporations to hedge currency fluctuations, uncertain energy costs and commodity price volatility.

澳洲代写｜MATH4090｜Computation in Financial Mathematics金融数学计算昆士兰大学

Attribute	Detail
Course Code	MATH4090
Course Title	Computation in Financial Mathematics
Coordinating Unit	School of Mathematics and Physics
Semester	Semester 2, 2023
Mode	In Person
Delivery Location	St Lucia
Number of Units	2
Pre-Requisites	MATH3090 or MN391
Recommended Pre-Requisites	MATH4091or MS479
Course Coordinator	Dr Duy-Minh Dang
Lecturers	Dr Duy-Minh Dang, Dr Kazutoshi Yamazaki
Tutors	Mr Nhat Pham

Computation in Financial Mathematics金融数学计算案例

问题 1.

$\Lambda$ house was sold for $\$ 17,500$. The buyer paid $\$ 6,000$ eash and the remainder in 24 equal scmiannual payments, the first one 6 months after date of purehase. Find the amount of each of these payments if they also provide for interest on the outstunding debt at $(.05, m=4)$.

\begin{aligned}
11,500 & =R \cdot{ }^{(4)} a_{12 \mid .05^{\circ}}^{(2)} \
\frac{R}{2} & =11,500() a_{4 \times 1.0125}^{-1} \times s_{2] .0125} . \
\log \frac{R}{2} & =4.0606978+8.4445249-10+.3037359=2.8089586 . \
\frac{R}{2} & =\$ 644.11 .
\end{aligned}

问题 2.

Example 2. What sum must a rompany set aside each month, if arcumulated at $(.03, m=4)$, in order to provide a depreciation fund of $\$ 15,000$ at the end of 15 years?

Solution. (f) the different final forms in which the solution might be expressed, the following is probably preferable:
$$
\begin{aligned}
R \cdot{ }^{(1)} s_{181.03}^{(12)} & =15,000 . \
\frac{R}{12} & =\frac{5000}{s_{\text {sio } / .0075} \cdot s_{1 \mid .075}^{(3)}} . \
\log \frac{R}{12} & =3.69 \times 9700-1.8775104-.0010826=1.8203770 . \
\frac{R}{12} & =\$ 66.13 .
\end{aligned}
$$

问题 3.

Example 1. A must borrow $\$ 2,400$ for building purposes. He can borrow from a building and loan association which charges $6 \%$, payable monthly in advance. A $\$ 100$ share, requiring a payment of $\$ 1.42$ per month, will mature at the end of 5 years with a payment of $\$ .43$. A can secure a private loan at only $\mathbf{5 \%}$, payable semiannually, and can provide for retirement of the debt at the end of 5 years by semiannual payments into a sinking fund which allows $(.03, m=2)$. Which of the two sources is the cheaper?

Solution. If $\mathbf{A}$ accepts the services of the building and loan association, he will pay at the beginning of each month dues amounting to $\$ 34.08$, plus $\$ 12$ interest. Amortization of the $\$ 2,400$ debt by monthly payments of $\$ 34.08+\$ 12$, or $\$ 46.08$, would be equivalent to amortizing the loan at $(.0617, m=12)$, as found by interpolation from the equation
$$
46.08 a_{50, j / 12}+24(.43)\left(1+\frac{j}{12}\right)^{-00}=2400-46.08 \text {. }
$$
Acceptance of the private loan would entail a semiannual expense of $\$ 60$ interest and a sinking-fund payment of $\$ 224.24$; that is, $\left(\frac{R}{2}=2400 \cdot s_{i 01.015}^{-1}\right)$. The corresponding amortization rate from the equation $284.24 \cdot a_{\overline{10} / / 2}=2400$ is found to be $(.0641, n=2)$. The building and loan association rate is obviously the better rate. (The student should check this last statement.)

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

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。

有限元方法代写

随机分析代写

时间序列分析代写

回归分析代写

MATLAB代写

R语言代写	问卷设计与分析代写
PYTHON代写	回归分析与线性模型代写
MATLAB代写	方差分析与试验设计代写
STATA代写	机器学习/统计学习代写
SPSS代写	计量经济学代写
EVIEWS代写	时间序列分析代写
EXCEL代写	深度学习代写
SQL代写	各种数据建模与可视化代写

澳洲代写｜MATH3075｜Financial Derivatives 金融衍生工具悉尼大学

Posted on 2023年10月10日2023年10月10日 by statistics-lab

statistics-lab^TM为您悉尼大学（英语：The University of Sydney）Financial Derivatives 金融衍生工具澳洲代写代考和辅导服务！

课程介绍：

This unit will introduce you to the mathematical theory of modern finance with the special emphasis on the valuation and hedging of financial derivatives, such as: forward contracts and options of European and American style. You will learn about the concept of arbitrage and how to model risk-free and risky securities. Topics covered by this unit include: notions of a martingale and a martingale measure, the fundamental theorems of asset pricing, complete and incomplete markets, the binomial options pricing model, discrete random walks and the Brownian motion, the Black-Scholes options pricing model and the valuation and hedging of exotic options. Students completing this unit have been highly sought by the finance industry, which continues to need graduates with quantitative skills. Lectures in the mainstream unit are held concurrently with those of the corresponding advanced unit.

澳洲代写｜MATH3075｜Financial Derivatives 金融衍生工具悉尼大学

Attribute	Detail
Course Code	MATH3075
Course Title	Financial Derivatives
Coordinating Unit	Mathematics and Statistics Academic Operations
Semester	Semester 2, 2023
Mode	Normal day
Delivery Location	Camperdown/Darlington, Sydney
Number of Units	6
Pre-Requisites	ECB1101, ECC1000, ECF1100, ECS1101, ECW1101
Lecturers	Associate Professor Paola Labrecciosa

Financial Derivatives 金融衍生工具的基础理论

“Derivative” includes

Security derived from a debt instrument, share, loan whether secured or unsecured, risk instrument or contract for differences or any other form of security.
A contract which derives its value from the prices, or index of prices of underlying securities.
The above definition conveys that
The derivatives are financial products.
Derivative is derived from another financial instrument/contract called the underlying. In the case of Nifty futures, Nifty index is the underlying. A derivative derives its value from the underlying assets. Accounting Standard SFAS133 defines a derivative as, ‘a derivative instrument is a financial derivative or other contract with all three of the following characteristics:
(i) It has (1) one or more underlings, and (2) one or more notional amount or payments provisions or both. Those terms determine the amount of the settlement or settlements.
(ii) It requires no initial net investment or an initial net investment that is smaller than would be required for other types of contract that would be expected to have a similar response to changes in market factors.
(iii) Its terms require or permit net settlement. It can be readily settled net by a means outside the contract or it provides for delivery of an asset that puts the recipients in a position not substantially different from net settlement

In general, from the aforementioned, derivatives refer to securities or to contracts that derive from another-whose value depends on another contract or assets. As such the financial derivatives are financial instruments whose prices or values are derived from the prices of other underlying financial instruments or financial assets. The underlying instruments may be a equity share, stock, bond, debenture, treasury bill, foreign currency or even another derivative asset. For example, a stock option’s value depends upon the value of a stock on which the option is written. Similarly, the value of a treasury bill of futures contracts or foreign currency forward contract will depend upon the price or value of the underlying assets, such as Treasury bill or foreign currency. In other words, the price of the derivative is not arbitrary rather it is linked or affected to the price of the underlying asset that will automatically affect the price of the financial derivative. Due to this reason, transactions in derivative markets are used to offset thein the underlying assets. In fact, the derivatives can be formed on almost any variable, for example, from the price of hogs to the amount of snow falling at a certain ski resort. risk of price changesin the underlying assets. In fact, the derivatives can be formed on almost any variable, for example, from the price of hogs to the amount of snow falling at a certain ski resort.

“衍生产品 “包括

由债务工具、股票、有担保或无担保贷款、风险工具或差价合约或任何其他形式的证券衍生而来的证券。

从基础证券的价格或价格指数中获取价值的合约。
上述定义表明

衍生产品是金融产品。

衍生产品来源于另一种称为标的物的金融工具/合约。就 Nifty 期货而言，Nifty 指数就是标的物。衍生产品的价值来源于标的资产。会计准则 SFAS133 对衍生工具的定义是：”衍生工具是一种金融衍生工具或具有以下三个特征的其他合同：
(i) 它有：(1) 一个或多个标的；(2) 一个或多个名义金额或支付条款，或两者兼而有之。这些条款决定结算的金额。
(ii) 不需要初始净投资，或初始净投资少于对市场因素变化有类似反应的其他类型的合同。
(iii) 其条款要求或允许净额结算。它可以很容易地通过合同以外的方式进行净结算，或者它规定了一种资产的交割，使接受者处于与净结算没有本质区别的地位。

一般来说，从上述情况来看，衍生工具指的是证券或衍生于另一种合同的合同，其价值取决于另一种合同或资产。因此，金融衍生品是其价格或价值源自其他基础金融工具或金融资产价格的金融工具。基础工具可以是股票、股票、债券、债权、国库券、外币，甚至是另一种衍生资产。例如，股票期权的价值取决于作为期权标的的股票的价值。同样，期货合约的国库券或外币远期合约的价值取决于相关资产（如国库券或外币）的价格或价值。换句话说，衍生产品的价格不是随意确定的，而是与相关资产的价格相关联或受其影响，而相关资产的价格会自动影响金融衍生产品的价格。正因为如此，衍生品市场的交易被用来抵消基础资产的价格。事实上，衍生产品几乎可以针对任何变量，例如，从生猪价格到某个滑雪胜地的降雪量。事实上，衍生工具几乎可以根据任何变量形成，例如，从生猪价格到某个滑雪胜地的降雪量。

Fundamental theorem of asset pricing资产定价基本定理定义相关

The previously constructed “maximal” sequence of semi-martingales $X^n \in \mathcal{X}^1$ converges in a pathwise uniform way in probability, i.e. $\left|X^n-X\right|_1^* \rightarrow 0$ in probability for some càdlàg process $X$.
It is now the goal to show that indeed $X^n \rightarrow X$ in the Emery topology, an apparently much stronger statement. Convergence in the Emery topology can be shown with respect to any equivalent measure $Q \sim P$. since this notion of convergence only depends on the equivalence class of probability measures.
By the basic convergence result (1) we know that $\xi:=\sup _n\left|X^n\right|_1^* \in L^0$. We can therefore find a measure $Q \sim P$ (take, e.g., $d Q / d P=c \exp (-\xi)$ ) such that $X^n \in L^2(Q)$, hence we can continue the analysis with $L^2$-methods, in order to prove Emery-convergence with respect to $Q$. Now the proof starts!
$42 / 24$
先前构建的半马尔廷态 $X^n \ in \mathcal{X}^1$ 的 “最大 “序列在概率上以路径均匀的方式收敛，即对于某个 càdlàg 过程 $X$ ，$\left|X^n-X\right|_1^* \rightarrow 0$ 在概率上收敛。
现在，我们的目标是证明在埃默里拓扑中，$X^n \rightarrow X$ 确实是一个明显更强的陈述。埃默里拓扑中的收敛可以用任何等价度量 $Q \sim P$ 来证明，因为这个收敛的概念只取决于概率度量的等价类。
根据基本收敛结果（1），我们知道 $\xi:=\sup _n\left|X^n\right|_1^* \in L^0$。因此，我们可以找到一个度量 $Q \sim P$（例如，取 $d Q / d P=c \exp (-\xi)$ ），使得 $X^n \ in L^2(Q)$，因此我们可以继续用 $L^2$ 方法进行分析，以证明关于 $Q$ 的埃默里收敛性。现在开始证明
$42 / 24$

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

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。

有限元方法代写

随机分析代写

时间序列分析代写

回归分析代写

MATLAB代写

R语言代写	问卷设计与分析代写
PYTHON代写	回归分析与线性模型代写
MATLAB代写	方差分析与试验设计代写
STATA代写	机器学习/统计学习代写
SPSS代写	计量经济学代写
EVIEWS代写	时间序列分析代写
EXCEL代写	深度学习代写
SQL代写	各种数据建模与可视化代写

澳洲代写｜OLET2610｜Foundations of Quantum Computing量子计算的基础悉尼大学

Posted on 2023年10月9日2023年10月9日 by statistics-lab

statistics-lab^TM为您悉尼大学（英语：The University of Sydney）Foundations of Quantum Computing量子计算的基础澳洲代写代考和辅导服务！

课程介绍：

This OLE will provide a general introduction to the research field of quantum computing, covering hardware, software, and potential societal impact. The circuit model of quantum computing and example algorithms will be introduced, then building on this knowledge you will code and execute simple algorithms in a quantum software environment. Emphasis will be given to comparing quantum vs ‘classical’ performance of key algorithms. For hardware, the research challenges in developing quantum computer technology will be introduced, and you will undertake a critical analysis of specific hardware platforms (advantages and challenges). The potential societal impact of quantum computers, and quantum technologies more broadly, will be surveyed. On completion of this OLE, you will have gained an informed appreciation of this new technology and its potential impact, as well as generic skills that allow for the critical assessment and evaluation of potential new technologies.

澳洲代写｜OLET2610｜Foundations of Quantum Computing量子计算的基础悉尼大学

Attribute	Detail
Code	OLET2610
Academic Unit	Physics Academic Operations
Unit Name	Foundations of Quantum Computing
Credit Points	2
Pre-requisites	None mentioned in the provided text
Lecturers	Not explicitly mentioned in the text

Foundations of Quantum Computing量子计算的基础理论

1.4.3 Reduced Density Matrix
Consider a Hilbert space $\mathcal{H}=\mathcal{H}_A \oplus \mathcal{H}_B$. The reduced density matrix for system $A$ is defined as:
$$
\rho^A \equiv \operatorname{tr}_B\left(\rho^{A B}\right)
$$
$\operatorname{tr}_B$ is known as the partial trace of $\rho^{A B}$ over $B$. The partial trace is defined as :
$$
\operatorname{tr}_B\left(\left|a_1\right\rangle\left\langle a_1|| b_1\right\rangle\left\langle b_2\right|\right) \equiv\left|a_1\right\rangle\left\langle a_1\right| \operatorname{tr}\left(\left|b_1\right\rangle\left\langle b_2\right|\right)
$$
As we expect, if our state is just a tensor product a density matrix $\rho_A$ in $\mathcal{H}_A$ and a density matrix $\rho_B$ in $\mathcal{H}_B$ then
$$
\rho^A=\operatorname{tr}_B\left(\rho_A \rho_B\right)=\rho_A \operatorname{tr}\left(\rho_B\right)=\rho_A
$$
A less trivial example is the bell state.

1.4.6 Example: Infinite Square Well
As an example let us consider a particle sitting in a infinite square well with the state $\psi(x)=\sqrt{\frac{2}{\pi}} \sin (x), x \in[0, \pi]$. The density matrix is:
$$
\rho=\frac{2}{\pi} \sin x^{\prime} \sin x
$$
Now let us break the square well in half into two subspaces, $\mathcal{H}A \otimes \mathcal{H}_B$. Somewhat suprisingly, now we can calculate entropy and entanglement of these subspaces, even though the entire system is pure (and has zero entropy and entanglement). The key difference now is that in each subspace we no longer know if there is a particle or not. Thus, we must express each sub-Hilbert space in the basis $(|0\rangle+|1\rangle) \otimes \psi(x)$ where $|0\rangle$ represents no particle and $|1\rangle$ represents one particle. Equivalently, you can think of this as a way of breaking up the wave function into two functions. $$ \psi=|0,0\rangle \otimes|1, \psi(x)\rangle+|1, \psi(x)\rangle \otimes|0,0\rangle $$ Or, compressing notation and ignoring normalization for now, $$ \psi=|01\rangle+|10\rangle $$ Now we can find the reduced density matrix for $\mathcal{H}_A$. First we rewrite $\rho$ : $$ \begin{aligned} & \rho=(|01\rangle+|10\rangle)(\langle 01|+\langle 10|) \ & \rho=|01\rangle\langle 01|+| 10\rangle\langle 10|+| 10\rangle\langle 01|+| 01\rangle+\langle 10| \end{aligned} $$ The trace of $|10\rangle\langle 01|$ is zero since $\operatorname{tr}(|10\rangle\langle 01|)=\operatorname{tr}(\langle 01 \mid 10\rangle)=0$. We are left with $$ \begin{aligned} & \rho_A=\frac{2}{\pi} \sin x \sin y|1\rangle\left\langle 1\left|+\int{\pi / 2}^\pi \frac{2}{\pi} \sin (x)^2 d x\right| 0\right\rangle\langle 0| \
& \rho_A=\frac{2}{\pi} \sin x \sin y|1\rangle\left\langle 1\left|+\frac{1}{2}\right| 0\right\rangle\langle 0|
\end{aligned}
$$
To calculate the entropy of this subsystem we must know the eigenvalues of $\rho_A$. By inspection, it is clear that $|0\rangle$ is an eigenvector with eigenvalue $\frac{1}{2}$. We know the sum of the eigenvalues must be 1 , so the other eigenvalue is $\frac{1}{2}$. (Alternativly we could try $f(x)|1\rangle$ and work out the corresponding inner product, which is an integral.)
$$
S=-\operatorname{tr}(\rho \ln (\rho))=-\frac{1}{2} \ln \frac{1}{2}-\frac{1}{2} \ln \frac{1}{2}=\ln (2)
$$
This result could have been anticipated because there are two possibilities for measurement – we find the particle on the left or the right. Still, it is suprising that the entropy of a given subsystem can be non-zero while the entire system has zero entropy.

Quantum Algorithms 量子算法定义

the two possible states of the electron in classical physics. Many of the most counterintuitive aspects of quantum physics arise from the superposition principle which states that if a quantum system can be in one of two states, then it can also be in any linear superposition of those two states. For instance, the state of the electron could well be $\frac{1}{\sqrt{2}}|0\rangle+\frac{1}{\sqrt{2}}|1\rangle$ or $\frac{1}{\sqrt{2}}|0\rangle-\frac{1}{\sqrt{2}}|1\rangle$; or an infinite number of other combinations of the form $\alpha_0|0\rangle+\alpha_1|1\rangle$. The coefficient $\alpha_0$ is called the amplitude of state $|0\rangle$, and similarly with $\alpha_1$. And-if things aren’t already strange enough – the $\alpha$ ‘s can be complex numbers, as long as they are normalized so that $\left|\alpha_0\right|^2+\left|\alpha_1\right|^2=1$. For example, $\frac{1}{\sqrt{5}}|0\rangle+\frac{2 i}{\sqrt{5}}|1\rangle$ (where $i$ is the imaginary unit, $\sqrt{-1}$ ) is a perfectly valid quantum state! Such a superposition, $\alpha_0|0\rangle+\alpha_1|1\rangle$, is the basic unit of encoded information in quantum computers (Figure 10.1). It is called a qubit (pronounced “cubit”).
The whole concept of a superposition suggests that the electron does not make up its mind about whether it is in the ground or excited state, and the amplitude $\alpha_0$ is a measure of its inclination toward the ground state. Continuing along this line of thought, it is tempting to think of $\alpha_0$ as the probability that the electron is in the ground state. But then how are we to make sense of the fact that $\alpha_0$ can be negative, or even worse, imaginary? This is one of the most mysterious aspects of quantum physics, one that seems to extend beyond our intuitions about the physical world.

This linear superposition, however, is the private world of the electron. For us to get a glimpse of the electron’s state we must make a measurement, and when we do so, we get a single bit of information -0 or 1 . If the state of the electron is $\alpha_0|0\rangle+\alpha_1|1\rangle$, then the outcome of the measurement is 0 with probability $\left|\alpha_0\right|^2$ and 1 with probability $\left|\alpha_1\right|^2$ (luckily we normalized so $\left|\alpha_0\right|^2+\left|\alpha_1\right|^2=1$ ). Moreover, the act of measurement causes the system to change its state: if the outcome of the measurement is 0 , then the new state of the system is $|0\rangle$ (the ground state), and if the outcome is 1 , the new state is $|1\rangle$ (the excited state). Thisfeature of quantum physics, that a measurement disturbs the system and forces it to choose (in this case ground or excited state), is another strange phenomenon with no classical analog.

The superposition principle holds not just for 2-level systems like the one we just described, but in general for $k$-level systems. For example, in reality the electron in the hydrogen atom can be in one of many energy levels, starting with the ground state, the first excited state, the second excited state, and so on. So we could consider a $k$-level system consisting of the ground state and the first $k-1$ excited states, and we could denote these by $|0\rangle,|1\rangle,|2\rangle, \ldots,|k-1\rangle$. The superposition principle would then say that the general quantum state of the system is $\alpha_0|0\rangle+\alpha_1|1\rangle+\cdots+\alpha_{k-1}|k-1\rangle$, where $\sum_{j=0}^{k-1}\left|\alpha_j\right|^2=1$. Measuring the state of the system would now reveal a number between 0 and $k-1$, and outcome $j$ would occur with probability $\left|\alpha_j\right|^2$. As before, the measurement would disturb the system, and the new state would actually become $|j\rangle$ or the $j$ th excited state.

How do we encode $n$ bits of information? We could choose $k=2^n$ levels of the hydrogen atom. But a more promising option is to use $n$ qubits.

Let us start by considering the case of two qubits, that is, the state of the electrons of two hydrogen atoms. Since each electron can be in either the ground or excited state, in classical physics the two electrons have a total of four possible states-00, 01, 10, or 11-and are therefore suitable for storing 2 bits of information. But in quantum physics, the superposition principle tells us that the quantum state of the two electrons is a linear combination of the four classical states,
$$
|\boldsymbol{\alpha}\rangle=\alpha_{00}|00\rangle+\alpha_{01}|01\rangle+\alpha_{10}|10\rangle+\alpha_{11}|11\rangle,
$$

澳洲代写｜ECOS3022｜The Economics of Financial Markets金融市场经济学悉尼大学

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

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。

有限元方法代写

随机分析代写

时间序列分析代写

回归分析代写

MATLAB代写

R语言代写	问卷设计与分析代写
PYTHON代写	回归分析与线性模型代写
MATLAB代写	方差分析与试验设计代写
STATA代写	机器学习/统计学习代写
SPSS代写	计量经济学代写
EVIEWS代写	时间序列分析代写
EXCEL代写	深度学习代写
SQL代写	各种数据建模与可视化代写

澳洲代写｜COMP5328｜Advanced Machine Learning高级机器学习悉尼大学

Posted on 2023年10月9日2023年10月9日 by statistics-lab

statistics-lab^TM为您悉尼大学（英语：The University of Sydney）Advanced Machine Learning高级机器学习澳洲代写代考和辅导服务！

课程介绍：

Machine learning models explain and generalise data. This course introduces some fundamental machine learning concepts, learning problems and algorithms to provide understanding and simple answers to many questions arising from data explanation and generalisation. For example, why do different machine learning models work? How to further improve them? How to adapt them to different purposes?

澳洲代写｜COMP5328｜Advanced Machine Learning高级机器学习悉尼大学

Attribute	Detail
Year/Session	Semester 2, 2020
Code	COMP5328
Academic Unit	Computer Science
Unit Name	Advanced Machine Learning
Credit Points	6
Pre-requisites	None
Lecturers	Not explicitly mentioned in the text

Advanced Machine Learning高级机器学习理论

Definition: $\operatorname{AR}(p)$ model is a linear generative model based on the $p$ th order Markov assumption:
$$
\forall t, Y_t=\sum_{i=1}^p a_i Y_{t-i}+\epsilon_t
$$
where

$\epsilon_t \mathrm{~s}$ are zero mean uncorrelated random variables with variance $\sigma$.

$a_1, \ldots, a_p$ are autoregressive coefficients.

$Y_t$ is observed stochastic process.

Definition: the generalized autoregressive conditional heteroscedasticity $\operatorname{GARCH}(p, q)$ model is an $\operatorname{ARMA}(p, q)$ model for the variance $\sigma_t$ of the noise term $\epsilon_t$ :

$$\forall t, \sigma_{t+1}^2=\omega+\sum_{i=0}^{p-1} \alpha_i \sigma_{t-i}^2+\sum_{j=0}^{q-1} \beta_j \epsilon_{t-j}^2$$

where

$\epsilon_t \mathrm{~S}$ are zero mean Gaussian random variables with variance $\sigma_t$ conditioned on $\left{Y_{t-1}, Y_{t-2}, \ldots\right}$.

$\omega>0$ is the mean parameter.

Continuous state space version of Hidden Markov Models:

$$\begin{aligned}\mathbf{X}_{t+1} & =\mathbf{B X}_t+\mathbf{U}_t, \Y_t & =\mathbf{A} \mathbf{X}_t+\epsilon_t\end{aligned}$$where$\mathbf{X}_t$ is an $n$-dimensional state vector.$Y_t$ is an observed stochastic process.$\mathbf{A}$ and $\mathbf{B}$ are model parameters.$\mathbf{U}_t$ and $\epsilon_t$ are noise terms.

Different methods for estimating model parameters:

Maximum likelihood estimation:
Requires further parametric assumptions on the noise distribution (e.g. Gaussian).
Method of moments (Yule-Walker estimator).
Conditional and unconditional least square estimation.
Restricted to certain models.

Complexity and Generalisation复杂性和概括性定义

Given a space $Z$ and a fixed distribution $\left.D\right|Z$, let $S=\left{z_1, \ldots, z_m\right}$ be a set of examples drawn i.i.d. from $\left.D\right|_Z$. Furthermore, let $\mathcal{F}$ be a class of functions $f: Z \rightarrow \mathbb{R}$. Definition. The empirical Rademacher complexity of $\mathcal{F}$ is defined to be $$ \hat{R}_m(\mathcal{F})=\mathrm{E}\sigma\left[\sup {f \in \mathcal{F}}\left(\frac{1}{m} \sum{i=1}^m \sigma_i f\left(z_i\right)\right)\right]
$$
where $\sigma_1, \ldots, \sigma_m$ are independent random variables uniformly chosen from ${-1,1}$. We will refer to such random variables as Rademacher variables.
Definition. The Rademacher complexity of $\mathcal{F}$ is defined as
$$
R_m(\mathcal{F})=\mathrm{E}_D\left[\hat{R}_m(\mathcal{F})\right]
$$

Intuitively, the supremum measures, for a given set $S$ and Rademacher vector $\sigma$, the maximum correlation between $f\left(z_i\right)$ and $\sigma_i$ over all $f \in \mathcal{F}$. Taking the expectation over $\sigma$, we can then say that the empirical Rademacher complexity of $\mathcal{F}$ measures the ability of functions from $\mathcal{F}$ (when applied to a fixed set $S$ ) to fit random noise. The Rademacher complexity of $\mathcal{F}$ then measures the expected noise-fitting-ability of $\mathcal{F}$ over all data sets $S \in Z^m$ that could be drawn according to the distribution $\left.D\right|_Z$.

We note that Rademacher complexity can be defined even more generally on sets $A \subseteq \mathbb{R}^m$ by making the supremum over $a \in A$ (instead of $f \in \mathcal{F}$ ) and replacing each $f\left(z_i\right)$ with $a_i$. Taking $A=\mathcal{F}(S)={f(z) \mid f \in \mathcal{F}, z \in S}$ recovers the definition above. It will sometimes be convenient to use the more general definition.

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

R语言代写	问卷设计与分析代写
PYTHON代写	回归分析与线性模型代写
MATLAB代写	方差分析与试验设计代写
STATA代写	机器学习/统计学习代写
SPSS代写	计量经济学代写
EVIEWS代写	时间序列分析代写
EXCEL代写	深度学习代写
SQL代写	各种数据建模与可视化代写

Linear time series analysis线性时间序列分析的问题

Continuous time models, option pricing连续时间模型，期权定价定义

Optimisation Theory最优化理论案例1

Calculus of variations变分法知识点

Contingent claim economy或有债权经济的问题

Risk Allocation Principles风险分配原则定义

Computational complexity计算复杂度的问题

Euler–Fermat Theorem欧拉-费马定理定义

Applied Financial Econometrics应用金融计量经济学的问题

Applied Financial Econometrics应用金融计量经济学问题2

Statistical Learning统计学习案例1

偏最小二乘法 Partial Least Squares知识点

Computation in Financial Mathematics金融数学计算案例

Financial Derivatives 金融衍生工具的基础理论

Fundamental theorem of asset pricing资产定价基本定理定义相关

Foundations of Quantum Computing量子计算的基础理论

Quantum Algorithms 量子算法定义

Advanced Machine Learning高级机器学习理论

Complexity and Generalisation复杂性和概括性 定义

Complexity and Generalisation复杂性和概括性定义