### 数学代写|概率论代写Probability theory代考|STAT7203

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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

## 数学代写|概率论代写Probability theory代考|Determination of the Adjustment Parameter

Based on the existing studies of the scaling parameter in the AP model $[28,29]$, the adjustment parameter $\beta$ in this study is assumed related to the water content and matric suction of soil sample (1). In addition, the initial void ratios of the two soil samples are added to consider the effect of the initial void ratio on the relationship of matric suctions. For simplicity, a linear predictive model is built for $\beta$. Due to the large difference between the orders of magnitude of $\beta$ over the whole SWCC, the

natural logarithm of $\beta$ is adopted as a model output. A full predictive model of $\ln \beta$ is established as a linear function of the possible forms of three important factors: $S^{(1)}, e^{(2)} / e^{(1)}$ and $\varphi^{(1)}$. This model is as follows:
\begin{aligned} \ln \beta=& b_{1}+b_{2} \cdot S^{(1)}+b_{3} \cdot\left(S^{(1)}\right)^{2}+b_{4} \cdot \ln S^{(1)}+b_{5} \cdot \exp \left(S^{(1)}\right)+b_{6} \cdot e^{(2)} / e^{(1)} \ &+b_{7} \cdot \ln \varphi^{(1)}+b_{8} \cdot e^{(2)} / e^{(1)} \cdot \ln \varphi^{(1)} \end{aligned}
where $e^{(2)} / e^{(1)}$ is the initial void ratios of the two soil samples with the same texture. $S^{(1)}$ is the dimensionless volumetric water content of sample (1) and $\varphi^{(1)}$ is the corresponding matric suction. $b_{1}, b_{2}, \ldots, b_{8}$ are the regression coefficients that must be estimated. Measured data of $\ln \beta$ are obtained for coarse- and fine-grained soils based on the method presented in the previous section.

## 数学代写|概率论代写Probability theory代考|Established of Predictive Model

As briefly introduced in Sect. 1.2, the Bayesian approach can not only update the model parameters and characterize the uncertainties using their posterior probability distribution functions (PDFs) but also find the most plausible model for predicting the responses using Bayesian model class selection. In this section, the Bayesian model class selection is applied to establish the predictive model for adjustment parameter $\beta$.
A linear predictive model class $C$ can be written as follows:
$$\ln \beta(\mathbf{x} ; \mathbf{b}, C)=\sum_{l=1}^{N_{b}} b_{l} x_{l}$$
where $x_{1}, x_{2}, \ldots$, and $x_{N_{b}}$ are the corresponding data, that is, variables such as $S^{(1)},\left(S^{(1)}\right)^{2}, \ln S^{(1)}, \exp \left(S^{(1)}\right), e^{(2)} / e^{(1)}$ and $\ln \varphi^{(1)} . b_{1}, b_{2}, \ldots$, and $b_{N_{b}}$ are uncertain coefficients required for identification. $N_{b}$ is the number of uncertain coefficients in the predictive formula.
The measurement of is denoted by $y$ and is modeled as follows:
$$y=\ln \beta(\mathbf{x} ; \mathbf{b}, C)+\varepsilon$$
where $\varepsilon$ is the predictive error. Additionally, it is a zero-mean normal random variable with prediction-error variance $\sigma_{\varepsilon}^{2}$. Therefore, all the uncertain parameters in $\theta=$ $\left[\mathbf{b}^{\mathrm{T}}, \sigma_{\varepsilon}^{2}\right]^{\mathrm{T}}$, and the number of components $N_{j}$ is $N_{b}+1$. Their uncertainties can be evaluated using the posterior PDFs, and the expression of the posterior PDF for data $D$ is written as follows:
$$P(\theta \mid D, C)=c_{0} p(\theta \mid C) p(D \mid \theta, C)$$

where $c_{0}=1 / p(D \mid C)$ is a normalizing constant; $p(D \mid C)$ is the evidence of model class $C ; p(\theta \mid C)$ is the prior PDF of the uncertain parameters in $\theta$, which is based on the previous knowledge or user’s judgment; and $p(D \mid \theta, C)$ is the likelihood function expressing the level of data-fitting. If the prediction errors in different measured data are statistically independent, the likelihood function can be computed as follows:
$$\mathrm{p}(D \mid \theta, C)=\left(2 \pi \sigma_{\varepsilon}^{2}\right)^{-\frac{N}{2}} \exp \left[-\frac{N}{2 \sigma_{\varepsilon}^{2}} J_{\mathrm{g}}(\mathbf{b} ; D, C)\right]$$

## 数学代写|概率论代写Probability theory代考|Model Class Selection

A number of model class candidates can be generated by including different terms on the right side of Eq. (3.15). In all candidate model classes, the constant $b_{1}$ is retained as an intercept term. Therefore, there are 127 candidates in total. The most suitable predictive model class among the model class candidates can be obtained by Bayesian model class selection. The plausibility of a predictive model class $C_{j}$ given the data $D$ can be obtained by using Eq. (1.9). The evidence of the model class $C_{j}$ can be approximated as follows:
$$p\left(D \mid C_{j}\right) \approx p\left(\theta^{} \mid C_{j}\right) \exp \left(-\frac{N}{2}\right) \frac{\pi^{\frac{N_{j}-N}{2}}\left(\sqrt{2} \sigma_{\varepsilon}^{}\right)^{N_{j}+1-N}}{\sqrt{N^{N_{j}}|\mathbf{A}|}}$$
where $\theta^{}$ is the optimal parameter vector for a given model class $C_{j}$, that is, $\theta^{}=$ $\left[\mathbf{b}^{* \mathrm{~T}}, \sigma_{\varepsilon}^{2 }\right]^{\mathrm{T}}$. The Ockham factor $p\left(\theta^{} \mid C_{j}\right) / p\left(\theta^{*} \mid D, C_{j}\right)$ provides a measurement of the robustness of the model class, and its value decreases exponentially with the number of uncertain parameters in the model class. Therefore, given the same plausibility for all model class candidates, the model class with the largest value given by Eq. ( $3.24$ ) is regarded to have the best tradeoff between the data-fitting capability and the robustness to model error. A MATLAB code for the linear model class selection is presented in Appendix A.

## 数学代写|概率论代写Probability theory代考|Determination of the Adjustment Parameter

ln⁡b=b1+b2⋅小号(1)+b3⋅(小号(1))2+b4⋅ln⁡小号(1)+b5⋅经验⁡(小号(1))+b6⋅和(2)/和(1) +b7⋅ln⁡披(1)+b8⋅和(2)/和(1)⋅ln⁡披(1)

## 数学代写|概率论代写Probability theory代考|Established of Predictive Model

ln⁡b(X;b,C)=∑l=1ñbblXl

p(D∣θ,C)=(2圆周率σe2)−ñ2经验⁡[−ñ2σe2ĴG(b;D,C)]

## 数学代写|概率论代写Probability theory代考|Model Class Selection

p(D∣Cj)≈p(θ∣Cj)经验⁡(−ñ2)圆周率ñj−ñ2(2σe)ñj+1−ñññj|一个|

## 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。