### 统计代写|数据科学代写data science代考|Disjunct Regions

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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

## 统计代写|数据科学代写data science代考|Assumptions

The assumptions imposed on the nonlinearity test are summarized below [38].

1. The variables are mean-centered and scaled to unit variance with respect to disjunct regions for which the accuracy bounds are to be determined.
2. Each disjunct region has the same number of observations.
3. A $\mathrm{PCA}$ model is determined for one region where the the accuracy bounds describe the variation for the sum of the discarded eigenvalues in that region.
4. PCA models are determined for the remaining disjunct regions.
5. The PCA models for each region include the same number of retained principal components.

## 统计代写|数据科学代写data science代考|Disjunct Regions

Here, we investigate how to construct the disjunct regions and how many disjunct regions should be considered. In essence, dividing the operating range into the disjunct regions can be carried out through prior knowledge of the process or by directly analyzing the recorded data. Utilizing a priori knowledge into the construction of the disjunct regions, for example, entails the incorporation of knowledge about distinct operating regions of the process. A direct analysis, on the other hand, by applying scatter plots of the first few retained principal components could reveal patterns that are indicative of distinct operating conditions. Wold et al. $[80]$, page 46 , presented an example of this based on a set of 20 “natural” amino acids.

If the above analysis does not yield any distinctive features, however, the original operating region could be divided into two disjunct regions initially. The nonlinearity test can then be applied to these two initial disjunct regions. Then, the number of regions can be increased incrementally, followed by a subsequent application of the test. It should be noted, however, that increasing the number of disjunct regions is accompanied by a reduction in the number of obervations in each region. As outlined the next subsection, a sufficient number of observations are required in order to prevent large Type I and II

errors for testing the hypothesis of using a linear model against the alternative hypothesis of rejecting that a linear model can be used.

Next, we discuss which of the disjunct regions should be used to establish the accuracy bounds. Intuitively, one could consider the most centered region for this purpose or alternatively, a region that is at the margin of the original operating region. More practically, the region at which the process is known to operate most often could be selected. This, however, would require a priori knowledge of the process. However, a simpler approach relies on the incorporation of the cross-validation principle $[64,65]$ to automate this selection. In relation to $\mathrm{PCA}$, cross-validation has been proposed as a technique to determine the number of retained principal components by Wold [79] and Krzanowski [39].

Applied to the nonlinearity test, the cross-validation principle could be applied in the following manner. First, select one disjunct region and compute the accuracy bounds of that region. Then, benchmark the residual variance of the remaining PCA models against this set of bounds. The test is completed if accuracy bounds have been computed for each of the disjunct regions and the residual variances of the PCA models of the respective remaining disjunct regions have been benchmarked against these accuracy bounds. For example, if 3 disjunct regions are established, the PCA model of the first region is used to calculate accuracy bounds and the residual variances of the $3 \mathrm{PCA}$ models (one for each region) is benchmarked against this set of bounds. Then, the PCA model for the second region is used to determine accuracy bounds and again, the residual variances of the $3 \mathrm{PCA}$ models are benchmarked against the second set of bounds. Finally, accuracy bounds for the PCA model of the 3rd region are constructed and each residual variance is compared to this 3rd set of bounds. It is important to note that the PCA models will vary depending upon which region is currently used to compute accuracy bounds. This is a result of the normalization procedure, since the mean and variance of each variable may change from region to region.

## 统计代写|数据科学代写data science代考|Confidence Limits for Correlation Matrix

The data correlation matrix, which is symmetric and positive semidefinite, for a given set of $N$ variables has the following structure:
$$\mathbf{R}{Z Z}=\left[\begin{array}{cccc} 1 & r{12} & \cdots & r_{1 N} \ r_{21} & 1 & \cdots & r_{2 N} \ \vdots & \vdots & \ddots & \vdots \ r_{N 1} & r_{N 2} & \cdots & 1 \end{array}\right]$$
Given that the total number of disjunct regions is $m$ the number of observations used to construct any correlation matrix is $\widetilde{K}=K / m$, rounded to the nearest integer. Furthermore, the correlation matrix for constructing the

PCA model for the $h$ th disjunct region, which is utilized to determine of the accuracy bound, is further defined by $\mathbf{R}{Z Z}^{(h)}$. Whilst the diagonal elements of this matrix are equal to one, the nondiagonal elements represent correlation coefficients for which confidence limits can be determined as follows: $$r{i j}^{(h)}=\frac{\exp \left(2 \varsigma_{i j}^{(h)}\right)-1}{\exp \left(2 \varsigma_{i j}^{(h)}\right)+1} \text { if } i \neq j$$
where $\varsigma_{i j}^{(h)}=\varsigma_{i j}^{(h)^{}} \pm \varepsilon, \varsigma_{i j}^{(h)^{}}=\ln \left(1+r_{i j}^{(h)^{}} / 1-r_{i j}^{(h)^{}}\right) / 2, r_{i j}^{(h)^{*}}$ is the sample correlation coefficient between the $i$ th and $j$ th process variable, $\varepsilon=\mathrm{c}{\alpha} / \sqrt{\overparen{K}-3}$ and $c{\alpha}$ is the critical value of a normal distribution with zero mean, unit variance and a significance level $\alpha$. This produces two confidence limits for each of the nondiagonal elements of $\mathbf{R}{Z Z}^{(h)}$, which implies that the estimate nondiagonal elements with a significance level of $\alpha$, is between $\mathbf{R}{Z Z}^{(h)}=\left[\begin{array}{c|c|c|c}1 & r_{12 L}^{(h)} \leq r_{12}^{(h)} \leq r_{12 U}^{(h)} & \cdots & r_{1 N_{L}}^{(h)} \leq r_{1 N}^{(h)} \leq r_{1 N_{U}}^{(h)} \ \hline r_{21_{L}}^{(h)} \leq r_{21}^{(h)} \leq r_{21_{U}}^{(h)} & 1 & \cdots & r_{2 N_{L}}^{(h)} \leq r_{2 N}^{(h)} \leq r_{2 N_{\nu}}^{(h)} \ \vdots & \vdots & \ddots & \vdots \ \hline r_{N 1_{L}}^{(h)} \leq r_{N 1}^{(h)} \leq r_{N 1_{U}}^{(h)} & r_{N 22_{L}}^{(h)} \leq r_{N 2}^{(h)} \leq r_{N 2 U}^{(h)} & \cdots & 1\end{array}\right]$
where the indices $U$ and $L$ refer to the upper and lower confidence limit, that is $r_{i j L}^{(h)}=\frac{\exp \left(2\left(c_{i j}^{(h)}-\varepsilon\right)\right)-1}{\exp \left(2\left(c_{i j}^{(h)}-\varepsilon\right)\right)+1}$ and $r_{i j u}^{(h)}=\frac{\exp \left(2\left(c_{i j}^{(h)}+\varepsilon\right)\right)-1}{\exp \left(2\left(c_{i j}^{(h)}+\varepsilon\right)\right)+1}$. A simplified version of Equation (1.18) is shown below
$$\mathbf{R}{Z Z{L}}^{(h)} \leq \mathbf{R}{Z Z}^{(h)} \leq \mathbf{R}{Z Z_{U}}^{(h)}$$
which is valid elementwise. Here, $\mathbf{R}{Z Z{L}}^{(h)}$ and $\mathbf{R}{Z Z{U}}^{(h)}$ are matrices storing the lower confidence limits and the upper confidence limits of the nondiagonal elements, respectively.

It should be noted that the confidence limits for each correlation coefficient is dependent upon the number of observations contained in each disjunct region, $\tilde{K}$. More precisely, if $\tilde{K}$ reduces the confidence region widens according to (1.17). This, in turn, undermines the sensitivity of this test. It is therefore important to record a sufficiently large reference set from the analyzed process in order to (i) guarantee that the number of observations in each disjunct region does not produce excessively wide confidence regions for each correlation coefficient, (ii) produce enough disjunct regions for the test and (iii) extract information encapsulated in the recorded observations.

## 统计代写|数据科学代写data science代考|Assumptions

1. 这些变量以均值为中心，并相对于要确定准确度界限的分离区域缩放为单位方差。
2. 每个分离区域具有相同数量的观察值。
3. 一种磷C一种模型是为一个区域确定的，其中精度界限描述了该区域中丢弃的特征值之和的变化。
4. 为剩余的分离区域确定 PCA 模型。
5. 每个区域的 PCA 模型包括相同数量的保留主成分。

## 统计代写|数据科学代写data science代考|Confidence Limits for Correlation Matrix

R从从=[1r12⋯r1ñ r211⋯r2ñ ⋮⋮⋱⋮ rñ1rñ2⋯1]

PCA 模型H用于确定精度界限的分离区域进一步定义为R从从(H). 虽然该矩阵的对角元素等于 1，但非对角元素表示相关系数，其置信限可按如下方式确定：r一世j(H)=经验⁡(2ε一世j(H))−1经验⁡(2ε一世j(H))+1 如果 一世≠j

R从从大号(H)≤R从从(H)≤R从从在(H)

## 广义线性模型代考

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