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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

## 物理代写|光学代写Optics代考|Chi-square test

We know that a value of $\chi_{\text {red }}^{2}$ significantly greater than one implies that the fit-function is likely to be inconsistent with the data. However, we’d prefer to make a quantitative statement about the probability that the fit-function is consistent with the data based on the actual value of $\chi_{\text {red }}^{2}$. Under the assumption that the errors are Gaussian, and assuming our fit function is correct, it’s possible to construct a function $P_{\chi}(X, v)$ that will tell us the probability of getting a value of $\chi_{\text {red }}^{2}$ greater than or equal to $X$ due to random-error fluctuations alone. $P_{\chi}(X, v)$ takes both the value $X$ of the reduced chi-square and the number of degrees of freedom $v=N-m$. Finding the probability $P_{\chi}(X, v)$ based on the $\chi_{\text {red }}^{2}$ and $v$ for a particular fit to 102 data points is known as applying the chi-square test. For example, if we perform a two-parameter fit and find that the value of the reduced chi-squared is $1.243$, then $P_{\chi}(1.243,100)=0.05$. The result of this chi-square test is that there is only a $5 \%$ chance that the fit-function is consistent with the data.

The calculation of $P_{X}(X, v)$ is described in statistics books and the values are tabulated. For example, you can use Table C.4 in Bevington (2003). (Online calculators also exist but make sure you find the right one.) To use a chi-square test table, you’ll need the number of degrees of freedom, $v=N-m$ and the value of $\chi_{\text {red }}^{2}$. The table will then give $P_{\chi}(X, v) .^{11}$
So far we’ve mostly concentrated on $\chi_{\text {red }}^{2}$ values above one because these indicate that the fit is too far from the data points. If your $\chi_{\text {red }}^{2}$ is significantly less than one, it generally indicates that you’ve over-estimated your errors. Your fit is “too good” given the uncertainties you’re assigning. This doesn’t serve to rule out your model but indicates that your data is not varying as much as one would assume given the assigned uncertainties.

## 物理代写|光学代写Optics代考|Finding the Uncertainty in the Fit Parameters

Since the underlying data $\left(x_{i}, y_{i}\right)$ have uncertainties, the best-fit values $\left(a_{1}^{\prime}, a_{2}^{\prime}, \ldots a_{m}^{\prime}\right)$ of the parameters will have corresponding uncertainties $\left(\Delta a_{i}^{\prime}, \Delta a_{2}^{\prime}, \ldots a_{m}^{\prime}\right)$. To estimate the uncertainty in the fit parameters, we need the chi-square, Eq. (2.12), rather than the reduced chi-square. A plot of $\chi^{2}$ versus any one of the parameters $a_{i}$ while fixing the rest at their best-fit values should always show a minimum at the best-fit point, $a_{i}^{\prime}$. This is the overall minimum of the chi-square surface and the minimum value is $\chi_{\min }^{2}=\chi^{2}\left(a_{1}^{\prime}, a_{2}^{\prime} \ldots a_{m}^{\prime}\right)$ at this point. The uncertainty in $a_{i}$ is the amount by which you must vary $a_{i}$ from its best fit value in order for the $\chi^{2}$ to increase by one. Figure $2.15$ shows an example of this. The upper panel shows the chi-square surface corresponding to a two-parameter, straightline fit similar to the fit shown in Figure 2.14. The dashed white contour in the upper panel indicates the locations $\left(a_{1}, a_{2}\right)$ at which $\chi^{2}\left(a_{1}, a_{2}\right)=\chi_{\text {min }}^{2}+1$. In other words, the contour shows the distance from the minimum at which the chi-square has risen by 1 . The horizontal distance from the minimum to the contour is the uncertainty $\Delta a_{1}^{\prime}$ in the best fit value $a_{1}^{\prime}$. Similarly, $\Delta a_{2}^{\prime}$ is the vertical distance from the minimum to the contour. The lower panel shows the cut along the long-dashed, horizontal line that corresponds to allowing $a_{1}$ to vary while keeping $a_{2}$ at its best-fit value, $a_{2}^{\prime}$. The cut shows the minimum chi-square at the best fit values of the parameters and the corresponding uncertainty $\Delta a_{1}^{\prime}$ is easy to read directly from the graph.

As demonstrated by Figure 2.15b, it’s not necessary to calculate the full chi-square surface to find the uncertainties. One only needs to calculate the chi-square cuts. Generating these cuts is a fairly easy way of obtaining the uncertainty in the parameters. It’s a good way to get a handle on fitting uncertainties, especially to begin with. However, if you do a lot of fitting, generating chi-square cuts soon gets old. Since the cuts are through a minimum of the chi-square surface, they can almost always be approximated by tangent parabolas. The uncertainties in the fit parameters are then estimated by calculating where these tangent parabolas have increased by 1 . The tangent parabolas can be found from the curvature of the chi-square cuts at the minimum. Some fit routines will use this information to estimate the uncertainties in the parameters for you. Other fit routines may return the curvature of the cuts at the minimum or some related quantity from which you can find the parameter uncertainties. The code in Appendix B.3 shows an example how to make a direct estimate of the parameter uncertainties based on information about the chi-square curvature.

# 光学代考

## 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
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