### 数学代写|数学建模代写math modelling代考|MATH1013

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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|数学建模代写math modelling代考|TRANSLATION INTO MATHEMATICS

How do gull bones grow? A bit of thought will convince you that this question cannot be translated into the language of mathematics, because it is too vague. What does “grow” mean? We might define an object to be “growing” if and only if its size is changing over time. (This definition for growth includes shrinking as well as expanding.) But what does “size” mean? Are we interested in length, diameter, volume, or what? And which kind of bone are we talking about? Humerus? Ulna?
You can see that the very first step in the modeling process, which is the translation into mathematics, typically requires careful thought. This step can be quite fruitful in and of itself, even if you never go any further in the modeling cycle, because the act of translating forces you to clarify concepts and sharpen questions. Translation into mathematics can help you ask whether your scientific question makes sense and whether it can be expected to have a solution. This is important, because some apparently meaningful questions are actually nonsensical (Exercise 3). Indeed, some of the burning scientific questions of history have simply “gone away” because they were discovered to be meaningless.

Let’s pose our problem precisely. How does the length of the humerus change in time over the life of a gull? That is, how does the length of the humerus change as a function of age? Consider a single “average” gull. Let
\begin{aligned} x & =\text { Age in days } \ f(x) & =\text { Length of humerus in } \mathrm{cm} . \end{aligned}
Mathematically, the question becomes: How does $f(x)$ depend on $x$ ?

## 数学代写|数学建模代写math modelling代考|The Stochastic Model

We wish to explicitly model the stochasticity in the system based on assumption (A2) about the source of the noise. Let $F(x)$ be a random variable denoting the measurement of the humerus length in a chick of known age $x$. We can think of the random variable $F(x)$ as the deterministic prediction $f(x)$ plus a random perturbation (noise):
$$F(x)=f(x)+\text { noise. }$$
Think of equation (2.3) as the deterministic skeleton $f(x)$ “clothed” with noise. The deterministic skeleton of a stochastic model is the part of the model that would remain if all the noise could be tuned to zero.
Typically, however, noise is not additive as in equation (2.3). Usually, one must first transform the observational data and the deterministic predictions with a variance-stabilizing transformation $\phi$ under which noise becomes additive:
$$\phi(F(x))=\phi(f(x))+\text { noise. }$$
Here, we mention an important point from statistical theory: Demographic noise is approximately additive on the square root scale, whereas environmental noise is approximately additive on the log scale (Cushing et al. 2003). That is, if demographic noise is dominant, then $\phi(\cdot)=\sqrt{\cdot}$, and if environmental noise is dominant, then $\phi(\cdot)=\ln (\cdot)$.
Thus, in our current example, under assumption (A2), equation (2.4) becomes
$$\sqrt{F(x)}=\sqrt{f(x)}+\sigma \varepsilon,$$
where $\sigma>0$ is a parameter representing the standard deviation of the noise and $\varepsilon$ is a standard normal random variable (a normal random variable with mean zero and standard deviation one). Equation (2.5) is the stochastic model for our current example.

## 数学代写|数学建模代写math modelling代考|TRANSLATION INTO MATHEMATICS

$x=$ Age in days $f(x) \quad=$ Length of humerus in $\mathrm{cm}$.

## 数学代写|数学建模代写math modelling代考|The Stochastic Model

$$F(x)=f(x)+\text { noise. }$$

$$\phi(F(x))=\phi(f(x))+\text { noise. }$$

$$\sqrt{F(x)}=\sqrt{f(x)}+\sigma \varepsilon,$$

## 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。