## 数学代写|基础数据分析代写Elementary data Analysis代考|MAT106

2022年10月11日

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## 数学代写|基础数据分析代写Elementary data Analysis代考|Errors in Variables

It is often the case that the input features we can actually measure, $\vec{X}$, are distorted versions of some other variables $\vec{U}$ we wish we could measure, but can’t:
$$\vec{X}=\vec{U}+\vec{\eta}$$
with $\vec{\eta}$ being some sort of noise. Regressing $Y$ on $\vec{X}$ then gives us what’s called an errors-in-variables problem.

In one sense, the errors-in-variables problem is huge. We are often much more interested in the connections between actual variables in the real world, than with our imperfect, noisy measurements of them. Endless ink has been spilled, for instance, on what determines students’ examination scores. One thing commonly thrown into the regression $-$ a feature included in $\vec{X}-$ is the income of children’s families. But this is typically not measured with absolute precision ${ }^5$, so what we are really interested in – the relationship between actual income and school performance $-$ is not what we are estimating in our regression. Typically, adding noise to the input features makes them less predictive of the response $-$ in linear regression, it tends to push $\hat{\beta}$ closer to zero than it would be if we could regress $Y$ on $\vec{U}$.

On account of the error-in-variables problem, some people get very upset when they see imprecisely-measured features as inputs to a regression. Some of them, in fact, demand that the input variables be measured exactly, with no noise whatsoever.
This position, however, is crazy, and indeed there’s a sense in which errors-invariables isn’t a problem at all. Our earlier reasoning about how to find the optimal linear predictor of $Y$ from $\vec{X}$ remains valid whether something like Eq. $2.32$ is true or not. Similarly, the reasoning in Ch. 1 about the actual regression function being the over-all optimal predictor, etc., is unaffected. If in the future we will continue to have $\vec{X}$ rather than $\vec{U}$ available to us for prediction, then Eq. $2.32$ is irrelevant for prediction. Without better data, the relationship of $Y$ to $\vec{U}$ is just one of the unanswerable questions the world is full of, as much as “what song the sirens sang, or what name $\Lambda$ chilles took when he hid among the women”.

Now, if you are willing to assume that $\vec{\eta}$ is a very nicely behaved Gaussian and you know its variance, then there are standard solutions to the error-in-variables problem for linear regression – ways of estimating the coefficients you’d get if you could regress $Y$ on $\vec{U}$. I’m not going to go over them, partly because they’re in standard textbooks, but mostly because the assumptions are hopelessly demanding. ${ }^6$

## 数学代写|基础数据分析代写Elementary data Analysis代考|Transformation

Let’s look at a simple non-linear example, $Y \mid X \sim \mathscr{N}(\log X, 1)$. The problem with smoothing data from this source on to a straight line is that the true regression curve isn’t very straight, $\mathbb{E}[Y \mid X=x]=\log x$. (Figure 2.5.) This suggests replacing the variables we have with ones where the relationship is linear, and then undoing the transformation to get back to what we actually measure and care about.

We have two choices: we can transform the response $Y$, or the predictor $X$. Here transforming the response would mean regressing exp $Y$ on $X$, and transforming the predictor would mean regressing $Y$ on $\log X$. Both kinds of transformations can be worth trying, but transforming the predictors is, in my experience, often a better bet, for three reasons.

1. Mathematically, $\mathbb{E}[f(Y)] \neq f(\mathbb{E}[Y])$. A mean-squared optimal prediction of $f(Y)$ is not necessarily close to the transformation of an optimal prediction of $Y$. And $Y$ is, presumably, what we really want to predict. (Here, however, it works out.)
2. Imagine that $Y=\sqrt{X}+\log Z$. There’s not going to be any particularly nice transformation of $Y$ that makes everything linear; though there will be transformations of the features.
3. This generalizes to more complicated models with features built from multiple covariates.
4. Suppose that we are in luck and $Y=\mu(X)+\epsilon$, with $\epsilon$ independent of $X$, and Gaussian, so all the usual default calculations about statistical inference apply. Then it will generally not be the case that $f(Y)=s(X)+\eta$, with $\eta$ a Gaussian random variable independent of $X$. In other words, transforming $Y$ completely messes up the noise model. (Consider the simple case where we take the logarithm of $Y$. Gaussian noise after the transformation implies lognormal noise before the transformation. Conversely, Gaussian noise before the transformation implies a very weird, nameless noise distribution after the transformation.)

# 基础数据分析代考

## 数学代写|基础数据分析代写基本数据分析代考|变量中的错误

.

$$\vec{X}=\vec{U}+\vec{\eta}$$

## 数学代写|基础数据分析代写基本数据分析代考|转换

.

1. 数学上，$\mathbb{E}[f(Y)] \neq f(\mathbb{E}[Y])$。$f(Y)$的均方最优预测不一定接近于$Y$的最优预测的变换。而$Y$大概就是我们真正想要预测的。
2. 想象一下$Y=\sqrt{X}+\log Z$。$Y$不会有什么特别好的变换使一切都是线性的;虽然会有特征的变化。这可以推广到由多个协变量构建的更复杂的模型。
3. 假设我们很幸运，并且$Y=\mu(X)+\epsilon$，其中$\epsilon$独立于$X$，并且是高斯分布，那么所有通常默认的统计推断计算都适用。那么它通常不会是$f(Y)=s(X)+\eta$的情况，$\eta$是一个独立于$X$的高斯随机变量。换句话说，转换$Y$完全搞乱了噪声模型。(考虑一个简单的例子，我们取$Y$的对数。变换后的高斯噪声意味着变换前的对数正态噪声。相反，变换前的高斯噪声意味着变换后非常奇怪的、无名的噪声分布。)

## 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。