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• 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

## 统计代写|随机过程代写stochastic process代考|Prediction

In many applications, rather than being interested in the parameters, we shall be more concerned with the prediction of future observations of the variable of interest. This is especially true in the case of stochastic processes, when we will typically be interested in predicting both the short- and long-term behavior of the process.

For prediction of future values, say $\mathbf{Y}$, of the phenomenon, we use the predictive distribution. To do this, given the current data $\mathbf{x}$, if we knew the value of $\boldsymbol{\theta}$, we would use the conditional predictive distribution $f(\mathbf{y} \mid \mathbf{x}, \boldsymbol{\theta})$. However, since there is uncertainty about $\boldsymbol{\theta}$, modeled through the posterior distribution, $f(\boldsymbol{\theta} \mid \mathbf{x})$, we can integrate this out to calculate the predictive density
$$f(\mathbf{y} \mid \mathbf{x})=\int f(\mathbf{y} \mid \boldsymbol{\theta}, \mathbf{x}) f(\boldsymbol{\theta} \mid \mathbf{x}) \mathrm{d} \boldsymbol{\theta} .$$
Note that in the case that the sampled values of the phenomenon are conditionally IID, the formula (2.3) simplifies to
$$f(\mathbf{y} \mid \mathbf{x})=\int f(\mathbf{y} \mid \boldsymbol{\theta}) f(\boldsymbol{\theta} \mid \mathbf{x}) \mathrm{d} \boldsymbol{\theta},$$
although, in general, to predict the future values of stochastic processes, this simplification will not be available. The predictive density may be used to provide point or set forecasts and test hypotheses about future observations, much as we did earlier.
Example 2.6: In the normal-normal example, to predict the next observation $Y=X_{n+1}$, we have that in the case when a uniform prior was applied, then $X_{n+1} \mid x \sim \mathrm{N}\left(\bar{x}, \frac{n+1}{n} \sigma^2\right)$. Then a predictive $100(1-\alpha) \%$ probability interval is $\left[\bar{x}-z_{\alpha / 2} \sigma \sqrt{(n+1) / n}, \bar{x}+z_{\alpha / 2} \sigma \sqrt{(n+1) / n}\right]$.
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## 统计代写|随机过程代写stochastic process代考|Sensitivity analysis and objective Bayesian methods

As mentioned earlier, prior information may often be elicited from one or more experts. In such cases, the postulated prior distribution will often be an approximation to the expert’s beliefs. In case that different experts disagree, there may be considerable uncertainty about the appropriate prior distribution to apply. In such cases, it is important to assess the sensitivity of any posterior results to changes in the prior distribution. This is typically done by considering appropriate classes of prior distributions, close to the postulated expert prior distribution, and then assessing how the posterior results vary over such classes.

Example 2.8: Assume that the gambler in the gambler’s ruin problem is not certain about her $\operatorname{Be}(5,5)$ prior and wishes to consider the sensitivity of the posterior predictive ruin probability over a reasonable class of alternatives. One possible class of priors that generalizes the gambler’s original prior is
$$G={f: f \sim \operatorname{Be}(c, c), c>0},$$
the class of symmetric beta priors. Then, over this class of priors, it can be shown that the gambler’s posterior predictive ruin probability varies between $0.231$, when $c \rightarrow 0$ and $0.8$, when $c \rightarrow \infty$. This shows that there is a large degree of variation of this predictive probability over this class of priors.

When little prior information is available, or in order to promote a more objective analysis, we may try to apply a prior distribution that provides little information and ‘lets the data speak for themselves’. In such cases, we may use a noninformative prior. When $\boldsymbol{\Theta}$ is discrete, a sensible noninformative prior is a uniform distribution. However, when $\boldsymbol{\Theta}$ is continuous, a uniform distribution is not necessarily the best choice. In the univariate case, the most common approach is to use the Jeffreys prior.

# 随机过程代考

## 统计代写|随机过程代写stochastic -process代考|Prediction

.

$$f(\mathbf{y} \mid \mathbf{x})=\int f(\mathbf{y} \mid \boldsymbol{\theta}, \mathbf{x}) f(\boldsymbol{\theta} \mid \mathbf{x}) \mathrm{d} \boldsymbol{\theta} .$$

$$f(\mathbf{y} \mid \mathbf{x})=\int f(\mathbf{y} \mid \boldsymbol{\theta}) f(\boldsymbol{\theta} \mid \mathbf{x}) \mathrm{d} \boldsymbol{\theta},$$
，尽管一般来说，要预测随机过程的未来值，这种简化是不可用的。预测密度可以用来提供点或设定预测，并对未来的观察结果进行假设检验，就像我们前面所做的那样。例2.6:在正常-正常的例子中，为了预测下一个观察结果$Y=X_{n+1}$，我们在应用统一先验的情况下，那么$X_{n+1} \mid x \sim \mathrm{N}\left(\bar{x}, \frac{n+1}{n} \sigma^2\right)$。那么预测的$100(1-\alpha) \%$概率区间是$\left[\bar{x}-z_{\alpha / 2} \sigma \sqrt{(n+1) / n}, \bar{x}+z_{\alpha / 2} \sigma \sqrt{(n+1) / n}\right]$ .
$\Delta$

## 统计代写|随机过程代写随机过程代考|灵敏度分析和客观贝叶斯方法

$$G={f: f \sim \operatorname{Be}(c, c), c>0},$$

## 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。

assignmentutor™您的专属作业导师
assignmentutor™您的专属作业导师