统计代写|时间序列分析代写Time-Series Analysis代考|DSC425

2022年9月24日

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• 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
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统计代写|时间序列分析代写Time-Series Analysis代考|NEURAL NETWORKS

11.24 Neural networks (NNs) refer to a broad class of nonparametric models which have gained a good deal of popularity in recent years across a wide range of disciplines, including computer science, psychology, biology, linguistics, and pattern recognition (for a textbook treatment, see, for example, Haykin, 1999). These models originate from research in the cognitive sciences on emulating the structure and behavior of the human brain.

One of the most common types of $\mathrm{NN}$ is the multi-layered perceptron (MLP), which can be used for nonparametric regression and classification. These models are organized in three basic layers: the input layer of independent variables, the output layer of dependent variables, and one or more hidden layers in-between. An activation function regulates the dependencies between the elements of each layer. A univariate autoregressive MLP model with a single hidden layer can be represented as:
$$x_t=\sum_{i=1}^p \phi_i x_{t-i}+\sum_{j=1}^q \beta_j G\left(\sum_{i=1}^p \varphi_i x_{t-i}\right)+\varepsilon_t$$
$G(\cdot)$ is the activation function and is a bounded nonlinear function that operates in an analogous manner to that of the transition functions used in STAR models. Several activation functions are employed in practice, with the most common being the hyperbolic tangent and the logistic. The second term in (11.13) refers to the hidden layer in the MLP. Obviously, (11.13) collapses to a standard $\mathrm{AR}(p)$ model when the activation function is linear. The residual term $\varepsilon_t$ is usually assumed to be a white noise random variable.
11.25 The high flexibility, rich parameterization and nonlinear nature of NNs renders estimation particularly difficult (see White, 2006). One of the main problems is that NNs are highly susceptible to overfitting. Consequently, the estimation strategy of $\mathrm{NNs}$ is rather different to traditional linear model estimation in that it typically involves two steps: in-sample optimization (training or learning) with recurrent testing (crossvalidation), and out-of-sample testing. The in-sample optimization is usually terminated prior to reaching the maximum possible performance, when the performance of the model in the cross-validation sample starts to deteriorate. In this way overfitting is avoided and a good forecasting performance in the testing sample is more likely. The estimation (training) algorithms used vary considerably and typically involve adjusting the direction of the negative gradient of some error criterion (e.g., mean squared or absolute error).

统计代写|时间序列分析代写Time-Series Analysis代考|NONLINEAR DYNAMICS AND CHAOS

11.28 So far, all the processes introduced in this chapter have the common aim of modeling stochastic nonlinearities in time series. This would seem the natural approach to take when dealing with stochastic time series processes, but a literature has also developed that considers the question of deterministic laws of motion.
11.29 Research in the general area of nonlinear dynamics is concerned with the behavior of deterministic and stochastic nonlinear systems. Both applied and theoretical research has flourished over the past four decades across a variety of disciplines and an extensive overview of the research on nonlinear dynamics, albeit with a bias toward the natural sciences, is given by Hilborn (1997). The meaning of the term “nonlinear dynamics” seems to vary considerably across scientific disciplines and eras. For example, a popular interpretation, since the early $1980 \mathrm{~s}$, associates nonlinear dynamics with deterministic nonlinear systems and a specific dynamic behavior called chaos, although this term has itself been given several different interpretations.

11.30 This diversity of meanings is mainly a consequence of there being no formal and complete mathematical definition of a chaotic system (see, for example. Berliner, 1992). Broadly speaking. chaos is the mathematical condition whereby a simple (low-dimensional), nonlinear, dynamical system produces complex (infinite-dimensional or random-like) behavior. Even though these systems are deterministic, they are completely unpredictable in the long-run, due to “sensitive dependence on initial conditions,” also known as Lyapunov instability. Chaotic systems also invariably have “fractal” or “self-similar” pictorial representations.
11.31 An example of a chaotic process is one that is generated by a deterministic difference equation
$$x_t=f\left(x_{t-1}, \ldots, x_{t-p}\right)$$
such that $x_t$ does not tend to a constant or a (limit) cycle and has estimated covariances that are extremely small or zero. A simple example is provided by Brock (1986), where a formal development of deterministic chaos models is provided. Consider the difference equation,
$$x_t=f\left(x_{t-1}\right), \quad x_0 \in[0,1]$$
where
$$f(x)= \begin{cases}x / \alpha & x \in[0, \alpha] \ (1-x) /(1-\alpha) & x \in[\alpha, 1] \quad 0<\alpha<1\end{cases}$$

时间序列分析代考

统计代写|时间序列分析代写时间序列分析代考|神经网络

$\mathrm{NN}$最常见的类型之一是多层感知器(MLP)，它可以用于非参数回归和分类。这些模型组织在三个基本层中:自变量的输入层、因变量的输出层和中间的一个或多个隐藏层。激活函数调节每层元素之间的依赖关系。具有单一隐含层的单变量自回归MLP模型可以表示为:
$$x_t=\sum_{i=1}^p \phi_i x_{t-i}+\sum_{j=1}^q \beta_j G\left(\sum_{i=1}^p \varphi_i x_{t-i}\right)+\varepsilon_t$$
$G(\cdot)$是激活函数，是一个有界非线性函数，其作用方式类似于STAR模型中使用的过渡函数。在实践中使用了几种激活函数，最常见的是双曲正切函数和逻辑函数。(11.13)中的第二个术语指的是MLP中的隐藏层。显然，当激活函数是线性的时候，(11.13)会崩溃为标准的$\mathrm{AR}(p)$模型。残差项$\varepsilon_t$通常被认为是一个白噪声随机变量。
11.25网络的高灵活性、丰富的参数化和非线性特性使得估计特别困难(见white, 2006)。其中一个主要问题是神经网络非常容易过拟合。因此，$\mathrm{NNs}$的估计策略与传统的线性模型估计相当不同，因为它通常涉及两个步骤:样本内优化(训练或学习)与重复测试(交叉验证)，以及样本外测试。当交叉验证样本中的模型的性能开始恶化时，样本内优化通常会在达到可能的最大性能之前终止。这种方法避免了过拟合，更有可能在测试样本中获得良好的预测性能。所用的估计(训练)算法差异很大，通常涉及调整某些误差准则的负梯度的方向(例如，均方或绝对误差)

统计代写|时间序列分析代写时间序列分析代考|非线性动力学与混沌

$$x_t=f\left(x_{t-1}, \ldots, x_{t-p}\right)$$

$$x_t=f\left(x_{t-1}\right), \quad x_0 \in[0,1]$$

$$f(x)= \begin{cases}x / \alpha & x \in[0, \alpha] \ (1-x) /(1-\alpha) & x \in[\alpha, 1] \quad 0<\alpha<1\end{cases}$$

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。