### 统计代写|数据科学代写data science代考|Generalization of Linear PCA

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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
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• 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|Generalization of Linear PCA

The generalization properties of NLPCA techniques is first investigated for neural network techniques, followed for principal curve techniques and finally kernel PCA. Prior to this analysis, however, we revisit the cost function for determining the $k$ th pair of the score and loading vectors for linear $\mathrm{PCA}$. This analysis is motivated by the fact that neural network approaches as well as principả curves and manifólds minimize thé rêsidual variancees. Rēformulating Equations $(1.9)$ and $(1.10)$ to minimize the residual variance for linear PCA gives rise to:
$$\mathbf{e}{k}=\mathbf{z}-t{k} \mathbf{p}{k}$$ which is equal to: $$J{k}=E\left{\mathbf{e}{k}^{T} \mathbf{e}{k}\right}=E\left{\left(\mathbf{z}-t_{k} \mathbf{p}{k}\right)^{T}\left(\mathbf{z}-t{k} \mathbf{p}_{k}\right)\right}$$

and subject to the following constraints
$$t_{k}^{2}-\mathbf{p}{k}^{T} \mathbf{z z}^{T} \mathbf{p}{k}=0 \quad \mathbf{p}{k}^{T} \mathbf{p}{k}-1=0 .$$
The above constraints follow from the fact that an orthogonal projection of an observation, $\mathbf{z}$, onto a line, defined by $\mathbf{p}{k}$ is given by $t{k}=\mathbf{p}{k}^{T} \mathbf{z}$ if $\mathbf{p}{k}$ is of unit length. In a similar fashion to the formulation proposed by Anderson [2] for determining the PCA loading vectors in (1.11), (1.69) and (1.70) can be combined to produce:
$$J_{k}=\arg \min {\mathbf{p}{k}}\left{E\left{\left(\mathbf{z}-t_{k} \mathbf{p}{k}\right)^{T}\left(\mathbf{z}-t{k} \mathbf{p}{k}\right)-\lambda{k}^{(1)}\left(t_{k}^{2}-\mathbf{p}{k}^{T} \mathbf{z z}^{T} \mathbf{p}{k}\right)\right}-\lambda_{k}^{(2)}\left(\mathbf{p}{k}^{T} \mathbf{p}{k}-1\right)\right} .$$
Carrying out the a differentiation of $J_{k}$ with respect to $\mathbf{p}{k}$ yields: $$E\left{2 t{k}^{2} \mathbf{p}{k}-2 t{k} \mathbf{z}+2 \lambda_{k}^{(1)} \mathbf{z z}^{T} \mathbf{p}{k}\right}-2 \lambda{k}^{(2)} \mathbf{p}{k}=\mathbf{0} .$$ A pre multiplication of (1.72) by $\mathbf{p}{k}^{T}$ now reveals
$$E{\underbrace{t_{k}^{2}-\mathbf{p}{k}^{T} \mathbf{z z}^{T} \mathbf{p}{k}}{=0}+\lambda{k}^{(1)} \underbrace{\mathbf{p}{k}^{T} \mathbf{z z}^{T} \mathbf{p}{k}}{=t{k}^{2}}-\lambda_{k}^{(2)}}=0 .$$
It follows from Equation (1.73) that
$$E\left{t_{k}^{2}\right}=\frac{\lambda_{k}^{(2)}}{\lambda_{k}^{(1)}} .$$
Substituting (1.74) into Equation (1.72) gives rise to
$$\frac{\lambda_{k}^{(2)}}{\lambda_{k}^{(1)}} \mathbf{p}{k}+E\left{\lambda{k}^{(1)} \mathbf{z z}^{T} \mathbf{p}{k}-\mathbf{z z}^{T} \mathbf{p}{k}\right}-\lambda_{k}^{(2)} \mathbf{p}{k}=\mathbf{0} .$$ Utilizing (1.5), the above equation can be simplified to $$\left(\lambda{k}^{(2)}-1\right) \mathbf{S}{Z Z \mathbf{p}{k}}+\left(\frac{\lambda_{k}^{(2)}}{\lambda_{k}^{(1)}}-\lambda_{k}^{(2)}\right) \mathbf{p}{k}=\mathbf{0},$$ and, hence, $$\left[\mathbf{S}{Z Z}+\frac{\lambda_{k}^{(2)}}{\lambda_{k}^{(1)}} \frac{1-\lambda_{k}^{(1)}}{\lambda_{k}^{(2)}-1} \mathbf{I}\right] \mathbf{p}{\mathbf{k}}=\left[\mathbf{S}{Z Z}-\lambda_{k} \mathbf{I}\right] \mathbf{p}{k}=\mathbf{0}$$ with $\lambda{k}=\frac{\lambda_{k}^{(2)}}{\lambda_{k}^{(1)}} \frac{1-\lambda_{k}^{(1)}}{\lambda_{k}^{(2)}-1}$. Since Equation (1.77) is identical to Equation (1.14), maximizing the variance of the score variables produces the same solution as

minimizing the residual variance by orthogonally projecting the observations onto the $k$ th weight vector. It is interesting to note that a closer analysis of Equation (1.74) yields that $E\left{t_{k}^{2}\right}=\frac{\lambda_{k}^{(2)}}{\lambda_{k}^{(1)}}=\lambda_{k}$, according to Equation (1.9), and hence, $\lambda_{k}^{(1)}=\frac{2}{1+\lambda_{k}}$ and $\lambda_{k}^{(2)}=2 \frac{\lambda_{k}}{1+\lambda_{k}}$, which implies that $\lambda_{k}^{(2)} \neq 1$ and $\frac{\frac{x_{k}^{(2)}}{x_{k}^{(1)}}-\lambda_{k}^{(2)}}{\lambda_{k}^{(2)}-1}=\lambda_{k}>0$.

More precisely, minimizing residual variance of the projected observations and maximizing the score variance are equivalent formulations. This implies that determining a NLPCA model using a minimizing of the residual variance would produce an equivalent linear model if the nonlinear functions are simplified to be linear. This is clearly the case for principal curves and manifolds as well as the netral network approaches. In contrast, the kernel PCA approach computes a linear PCA analysis using nonlinearly transformed variables and directly addresses the variance maximization and residual minimization as per the discussion above.

## 统计代写|数据科学代写data science代考|Neural network approaches

It should also be noted, however, that residual variance minimization alone is a necessary but not a sufficient condition. This follows from the analysis of the ANN topology proposed by Kramer [37] in Fig. 1.6. The nonlinear scores, which can extracted from the bottleneck layer, do not adhere to the fundamental principle that the first component is asseciated with the largest variance, the second component with the second largest variance etc. However, utilizing the sequential training of the ANN, detailed in Fig.1.7, provides an improvement, such that the first nonlinear score variables minimizes the residual variance $e_{1}=\mathbf{z}-\widehat{\mathbf{z}}$ and so on. However, given that the network weights and bias terms are not subject to a length restriction as it is the case for linear PCA, this approach does also not guarantee that the first score variables possesses a maximum variance.

The same holds true for the IT network algorithm by Tan and Mavrovouniotis [68], the computed score variables do not adhere to the principal that the first one has a maximum variance. Although score variables may not be extracted that maximize a variance criterion, the computed scores can certainly be useful for feature extraction $[15,62]$. Another problem of the technique by Tan and Mavrovouniotis is its application as a condition monitoring tool. Assuming the data describe a fault condition the score variables are obtained by an optimization routine to best reconstruct the fault data. It therefore follows that certain fault conditions may not be noticed. This can be illustrated using the following linear example
$$\mathbf{z}{f}-\mathbf{z}+\mathbf{f} \longrightarrow \mathbf{P}\left(\mathbf{z}{0}+\mathbf{f}\right),$$

where $f$ represents a step type fault superimposed on the original variable set $\mathbf{z}$ to produce the recorded fault variables $\mathbf{z}{f}$. Separating the above equation produces by incorporating thẻ statistical first order móment: $$E\left{\mathbf{z}{0}+\mathbf{f}{0}\right}+\mathbf{P}{0}^{-T} \mathbf{P}{1}^{T} E\left{\mathbf{z}{1}+\mathbf{f}{1}\right}=\mathbf{P}{0}^{-T} \mathbf{t},$$
where the subscript $-T$ is the transpose of an inverse matrix, respectively, $\mathbf{P}^{T}=\left[\mathbf{P}{0}^{T} \mathbf{P}{1}^{T}\right], \mathbf{z}^{T}=\left(\mathbf{z}{0} \mathbf{z}{1}\right), \mathbf{f}^{T}=\left(\mathbf{f}{0} \mathbf{f}{1}\right), \mathbf{P}{0} \in \mathbb{R}^{n \times n}, \mathbf{P}{1} \in \mathbb{R}^{N-n \times n}$, $\mathbf{z}{0}$ and $\mathbf{f}{0} \in \mathbb{R}^{N}$, and $\mathbf{z}{1}$ and $\mathbf{f}{1} \in \mathbb{R}^{N-n}$. Since the expectation of the original variables are zero, Equation (1.79) becomes:
$$\mathbf{f}{0}+\mathbf{P}{0}^{-T} \mathbf{P}{1}^{T} \mathbf{f}{1}=\mathbf{0}$$
which implies that if the fault vector $\mathbf{f}$ is such that $\mathbf{P}{0}^{-T} \mathbf{P}{1}^{T} \mathbf{f}{1}=-\mathbf{f}{0}$ the fault condition cannot be detected using the computed score variables. However, under the assumption that the fault condition is a step type fault but the variance of $\mathbf{z}$ remains unchanged, the first order moment of the residuals would clearly be affected since
$$E{\mathbf{e}}=E{\mathbf{z}+\mathbf{f}-\mathbf{P t}}=\mathbf{f} .$$
However, this might not hold true for an NLPCA model, where the PCA model plane, constructed from the retained loading vectors, becomes a surface. In this circumstances, it is possible to construct incipient fault conditions that remain unnoticed given that the optimization routine determines scores from the faulty observations and the IT network that minimize the mismatch between the recorded and predicted observations.

## 统计代写|数据科学代写data science代考|Nonlinear subspace identification

Subspace identification has been extensively studied over the past decade. This technique enables the identification of a linear state space model using input/output observations of the process. Nonlinear extensions of subspace identification have been proposed in references $[23,41,43,74,76]$ mainly employ Hammerstein or Wiener models to represent a nonlinear steady state transformation of the process outputs. As this is restrictive, kernel PCA may be considered to determine nonlinear filters to efficiently determine this nonlinear transformation.

## 统计代写|数据科学代写data science代考|Generalization of Linear PCA

NLPCA 技术的泛化特性首先针对神经网络技术进行了研究，然后是主曲线技术，最后是核 PCA。然而，在此分析之前，我们重新审视成本函数以确定ķ线性的第 th 对分数和加载向量磷C一种. 这种分析的动机是神经网络方法以及原理曲线和流形最小化残差方差。重新制定方程(1.9)和(1.10)最小化线性 PCA 的残差导致：

J_{k}=\arg \min {\mathbf{p}{k}}\left{E\left{\left(\mathbf{z}-t_{k} \mathbf{p}{k}\right) ^{T}\left(\mathbf{z}-t{k} \mathbf{p}{k}\right)-\lambda{k}^{(1)}\left(t_{k}^{2 }-\mathbf{p}{k}^{T} \mathbf{z z}^{T} \mathbf{p}{k}\right)\right}-\lambda_{k}^{(2)}\左(\mathbf{p}{k}^{T} \mathbf{p}{k}-1\right)\right} 。J_{k}=\arg \min {\mathbf{p}{k}}\left{E\left{\left(\mathbf{z}-t_{k} \mathbf{p}{k}\right) ^{T}\left(\mathbf{z}-t{k} \mathbf{p}{k}\right)-\lambda{k}^{(1)}\left(t_{k}^{2 }-\mathbf{p}{k}^{T} \mathbf{z z}^{T} \mathbf{p}{k}\right)\right}-\lambda_{k}^{(2)}\左(\mathbf{p}{k}^{T} \mathbf{p}{k}-1\right)\right} 。

E\left{t_{k}^{2}\right}=\frac{\lambda_{k}^{(2)}}{\lambda_{k}^{(1)}} 。E\left{t_{k}^{2}\right}=\frac{\lambda_{k}^{(2)}}{\lambda_{k}^{(1)}} 。

\frac{\lambda_{k}^{(2)}}{\lambda_{k}^{(1)}} \mathbf{p}{k}+E\left{\lambda{k}^{(1 )} \mathbf{z z}^{T} \mathbf{p}{k}-\mathbf{z z}^{T} \mathbf{p}{k}\right}-\lambda_{k}^{(2 )} \mathbf{p}{k}=\mathbf{0} 。\frac{\lambda_{k}^{(2)}}{\lambda_{k}^{(1)}} \mathbf{p}{k}+E\left{\lambda{k}^{(1 )} \mathbf{z z}^{T} \mathbf{p}{k}-\mathbf{z z}^{T} \mathbf{p}{k}\right}-\lambda_{k}^{(2 )} \mathbf{p}{k}=\mathbf{0} 。利用 (1.5)，上式可以简化为(λķ(2)−1)小号从从pķ+(λķ(2)λķ(1)−λķ(2))pķ=0,因此，[小号从从+λķ(2)λķ(1)1−λķ(1)λķ(2)−1一世]pķ=[小号从从−λķ一世]pķ=0和λķ=λķ(2)λķ(1)1−λķ(1)λķ(2)−1. 由于方程 (1.77) 与方程 (1.14) 相同，因此最大化分数变量的方差会产生与

## 统计代写|数据科学代写data science代考|Neural network approaches

Tan 和 Mavrovouniotis [68] 的 IT 网络算法也是如此，计算的分数变量不遵守第一个具有最大方差的原则。尽管可能无法提取使方差标准最大化的分数变量，但计算出的分数肯定可用于特征提取[15,62]. Tan 和 Mavrovouniotis 提出的技术的另一个问题是其作为状态监测工具的应用。假设数据描述了故障条件，则通过优化程序获得分数变量以最好地重建故障数据。因此，可能不会注意到某些故障情况。这可以使用以下线性示例来说明

F0+磷0−吨磷1吨F1=0

## 广义线性模型代考

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## MATLAB代写

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