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# 统计代写|线性回归代写Linear Regression代考|Logarithms

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## 统计代写|线性回归代写Linear Regression代考|Logarithms

$$\mathrm{E}(Y \mid X=x)=\beta_0+\beta_1 x$$
a useful way to interpret the coefficient $\beta_1$ is as the first derivative of the mean function with respect to $x$,
$$\frac{d \mathrm{E}(Y \mid X=x)}{d x}=\beta_1$$
We recall from elementary geometry that the first derivative is the rate of change, or the slope of the tangent to a curve, at a point. Since the mean function for simple regression is a straight line, the slope of the tangent is the same value $\beta_1$ for any value of $x$, and $\beta_1$ completely characterizes the change in the mean when the predictor is changed for any value of $x$.

When the predictor is replaced by $\log (x)$, the mean function as a function of $x$
$$\mathrm{E}(Y \mid X=x)=\beta_0+\beta_1 \log (x)$$
is no longer a straight line, but rather it is a curve. The tangent at the point $x>0$ is
$$\frac{d \mathrm{E}(Y \mid X=x)}{d x}=\frac{\beta_1}{x}$$
The slope of the tangent is different for each $x$ and the effect of changing $x$ on $\mathrm{E}(Y \mid X=x)$ is largest for small values of $x$ and gets smaller as $x$ is increased.
When the response is in log scale, we can get similar approximate results by exponentiating both sides of the equation:
$$\begin{gathered} \mathrm{E}(\log (Y) \mid X=x)=\beta_0+\beta_1 x \ \mathrm{E}(Y \mid X=x) \approx e^{\beta_0} e^{\beta_1 x} \end{gathered}$$
Differentiating this second equation gives
$$\frac{d \mathrm{E}(Y \mid X=x)}{d x}=\beta_1 \mathrm{E}(Y \mid X=x)$$

## 统计代写|线性回归代写Linear Regression代考|EXPERIMENTATION VERSUS OBSERVATION

There are fundamentally two types of predictors that are used in a regression analysis, experimental and observational. Experimental predictors have values that are under the control of the experimenter, while for observational predictors, the values are observed rather than set. Consider, for example, a hypothetical study of factors determining the yield of a certain crop. Experimental variables might include the amount and type of fertilizers used, the spacing of plants, and the amount of irrigation, since each of these can be assigned by the investigator to the units, which are plots of land. Observational predictors might include characteristics of the plots in the study, such as drainage, exposure, soil fertility, and weather variables. All of these are beyond the control of the experimenter, yet may have important effects on the observed yields.

The primary difference between experimental and observational predictors is in the inferences we can make. From experimental data, we can often infer causation.

If we assign the level of fertilizer to plots, usually on the basis of a randomization scheme, and observe differences due to levels of fertilizer, we can infer that the fertilizer is causing the differences. Observational predictors allow weaker inferences. We might say that weather variables are associated with yield, but the causal link is not available for variables that are not under the experimenter’s control. Some experimental designs, including those that use randomization, are constructed so that the effects of observational factors can be ignored or used in analysis of covariance (see, e.g., Cox, 1958; Oehlert, 2000).

Purely observational studies that are not under the control of the analyst can only be used to predict or model the events that were observed in the data, as in the fuel consumption example. To apply observational results to predict future values, additional assumptions about the behavior of future values compared to the behavior of the existing data must be made. From a purely observational study, we cannot infer a causal relationship without additional information external to the observational study.

## 统计代写|线性回归代写Linear Regression代考|Logarithms

$$\mathrm{E}(Y \mid X=x)=\beta_0+\beta_1 x$$

$$\frac{d \mathrm{E}(Y \mid X=x)}{d x}=\beta_1$$

$$\mathrm{E}(Y \mid X=x)=\beta_0+\beta_1 \log (x)$$

$$\frac{d \mathrm{E}(Y \mid X=x)}{d x}=\frac{\beta_1}{x}$$

$$\begin{gathered} \mathrm{E}(\log (Y) \mid X=x)=\beta_0+\beta_1 x \ \mathrm{E}(Y \mid X=x) \approx e^{\beta_0} e^{\beta_1 x} \end{gathered}$$

$$\frac{d \mathrm{E}(Y \mid X=x)}{d x}=\beta_1 \mathrm{E}(Y \mid X=x)$$

## 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。