### 统计代写|数据科学代写data science代考|Standard Nonlinear 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|Standard Nonlinear PCA

Nonlinear PCA (NLPCA) is based on a multi-layer perceptron (MLP) with an autoassociative topology, also known as an autoencoder, replicator network, bottleneck or sandglass type network. An introduction to multi-layer perceptrons can be found in [28].

The autoassociative network performs an identity mapping. The output $\hat{\boldsymbol{x}}$ is enforced to equal the input $\boldsymbol{x}$ with high accuracy. It is achieved by minimising the squared reconstruction error $E=\frac{1}{2}|\hat{\boldsymbol{x}}-\boldsymbol{x}|^{2}$.

This is a nontrivial task, as there is a ‘bottleneck’ in the middle: a layer of fewer units than at the input or output layer. Thus, the data have to be projected or compressed into a lower dimensional representation $Z$.

The network can be considered to consist of two parts: the first part represents the extraction function $\Phi_{\text {extr }}: \mathcal{X} \rightarrow \mathcal{Z}$, whereas the second part represents the inverse function, the generation or reconstruction function $\Phi_{g e n}: \mathcal{Z} \rightarrow \hat{\mathcal{X}}$. A hidden layer in each part enables the network to perform nonlinear mapping functions. Without these hidden layers, the network would only be able to perform linear PCA even with nonlinear units in the component layer, as shown by Bourlard and Kamp [29]. To regularise the network, a weight decay term is added $E_{\text {total }}=E+\nu \sum_{i} w_{i}^{2}$ in order to penalise large network weights $w$. In most experiments, $\nu=0.001$ was a reasonable choice.

In the following, we describe the applied network topology by the notation $l_{1}-l_{2}-l_{3} \ldots-l_{S}$ where $l_{s}$ is the number of units in layer $s$. For example, 3-4-1-4-3 specifies a network of five layers having three units in the input and output layer, four units in both hidden layers, and one unit in the component layer, as illustrated in Flg. $2.2$.

## 统计代写|数据科学代写data science代考|Hierarchical nonlinear PCA

In order to decompose data in a PCA related way, linearly or nonlinearly, it is important to distinguish applications of pure dimensionality reduction from applications where the identification and discrimination of unique and meaningful components is of primary interest, usually referred to as feature extraction. In applications of pure dimensionality reduction with clear emphasis on noise reduction and data compression, only a subspace with high descriptive capacity is requred. How the individual components form this subspace is not particularly constrained and hence does not need to be unique. The only requirement is that the subspace explains maximal information in the mean squared error sense. Since the individual components which span this subspace, are treated equally by the algorithm without any particular order or differential weighting, this is referred to as symmetric type of learning. This includes the nonlinear PCA performed by the standard autoassociative neural network which is therefore referred to as s-NLPCA.

By contrast, hierarchical nonlinear PCA $(h-N L P C A)$, as proposed by Scholz and Vigário [10], provides not only the optimal nonlinear subspace spanned by components, it also constrains the nonlinear components to have the same hierarchical order as the linear components in standard PCA.

Hierarchy, in this context, is explained by two important properties: scalability and stability. Scalability means that the first $n$ components explain the maximal variance that can be covered by a $n$-dimensional subspace. Stability means that the i-th component of an $n$ component solution is identical to the $i$-th component of an $m$ component solution.

## 统计代写|数据科学代写data science代考|The Hierarchical Error Function

$E_{1}$ and $E_{1,2}$ are the squared reconstruction errors when using only the first or both the first and the second component, respectively. In order to perform the h-NLPCA, we have to impose not only a small $E_{1,2}$ (as in s-NLPCA), but also a small $E_{1}$. This can be done by minimising the hierarchical error:
$$E_{H}=E_{1}+E_{1,2}$$

Fig. 2.3. Hierarchical NLPCA. The standard autoassociative network is hierarchically extended to perform the hierarchical NLPCA (h-NLPCA). In addition to the whole 3-4-2-4-3 network (grey+black), there is a 3-4-1-4-3 subnetwork (black) explicitly considered. The component layer in the middle has either one or two units which represent the tirst and second components, respectively. Both the error $E_{1}$ of the subnotwork with one componont and the srror of the total network with two components are estimated in each iteration. The network weights are then adapted at once with regard to the total hierarchic error $E=E_{1}+E_{1,2}$

To find the optimal network weights for a minimal error in the h-NLPCA as well as in the standard symmetric approach, the conjugate gradient descent algorithm [31] is used. At each iteration, the single error terms $E_{1}$ and $E_{1,2}$ have to be calculated separately. This is performed in the standard s-NLP $\overline{\mathrm{C} A}$ way by a network either with one or with two units in the component layer. Here, one network is the subnetwork of the other, as illustrated in Fig. 2.3. The gradient $\nabla E_{H}$ is the sum of the individual gradients $\nabla E_{H}=\nabla E_{1}+\nabla E_{1,2}$. If a weight $w_{i}$ does not exist in the subnetwork, $\frac{\partial E_{1}}{\partial w_{i}}$ is set to zero.

To achieve more robust results, the network weights are set such that the sigmoidal nonlinearities work in the linear range, corresponding to initialise the network with the simple linear PCA solution.

The hierarchical error function (2.1) can be easily extended to $k$ components $(k \leq d)$ :
$$E_{H}=E_{1}+E_{1,2}+E_{1,2,3}+\cdots+E_{1,2,3, \ldots, k} .$$
The hierarchical condition as given by $E_{H}$ can then be interpreted as follows: we search for a $k$-dimensional subspace of minimal mean square error (MSE) under the constraint that the $(k-1)$-dimensional subspace is also of minimal MSE. This is successively extended such that all $1, \ldots, k$ dimensional subspaces are of minimal MSE. Hence, each subspace represents the data with regard to its dimensionalities best. Hierarchical nonlinear PCA can therefore be seen as a true and natural nonlinear extension of standard linear PCA.

## 广义线性模型代考

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## MATLAB代写

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