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

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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|Nonlinear PCA Extensions

This section reviews nonlinear PCA extensions that have been proposed over the past two decades. Hastie and Stuetzle [25] proposed bending the loading vectors to produce curves that approximate the nonlinear relationship between

a set of two variables. Such curves, defined as principal curves, are discussed in the next subeection, including their multidimensional extensions to produce principal surfaces or principal manifolds.

Another paradigm, which has been proposed by Kramer [37], is related to the construction of an artificial neural network to represent a nonlinear version of (1.2). Such networks that map the variable set $\mathbf{z}$ to itself by defining a reduced dimensional bottleneck layer, describing nonlinear principal components, are defined as autoassociative neural networks and are revisited in Subsect. 4.2.

A more recently proposed NLPCA technique relates to the definition of nonlinear mapping functions to define a feature space, where the variable space $\mathbf{z}$ is assumed to be a nonlinear transformation of this feature space. By carefully selecting these transformation using Kernel functions, such as radial basis functions, polynomial or sigmoid kernels, conceptually and computationally efficient NLPCA algorithms can be constructed. This approach, referred to as Kermel $P C A$, is reviewed in Subsect. $4.3 .$

## 统计代写|数据科学代写data science代考|Introduction to principal curves

Principal Curves (PCs), presented by Hastie and Stuetzle $[24,25]$, are smooth one-dimensional curves passing through the middle of a cloud representing a data set. Utilizing probability distribution, a principal curve satisfies the self-consistent property, which implies that any point on the curve is the average of all data points projected onto it. As a nonlinear generalization of principal component analysis, $\mathrm{PCs}$ can be also regarded as a one-dimensional manifold embedded in high dimeneional data space. In addition to the sta tistical property inherited from linear principal components, $\mathrm{PCs}$ also reflect the geometrical structure of data due. More precisely, the natural parameter arc-length is regarded as a projection index for each sample in a similar fashion to the score variable that represents the distance of the projected data point from the origin. In this respect, a one-dimensional nonlinear topological relationehip between two variables can be eetimated by a principal curve [85].

## 统计代写|数据科学代写data science代考|From a weight vector to a principal curve

Inherited from the basic paradigm of $\mathrm{PCA}, \mathrm{PCs}$ assume that the intrinsic middle structure of data is a curve rather than a straight line. In relation to

the total least squares concept $[71]$, the cost function of $\mathrm{PCA}$ is to minimize the sum of projection distances from data points to a line. This produces the same solution as that presented in Sect. 2. Eq. (1.14). Geometrically, eigenvectors and their corresponding eigenvalues of $\mathbf{S}_{Z Z}$ reflect the principal directions and the variance along the principal directions of data, respectively. Applying the above analysis to the first principal component, the following properties can be established [5]:

1. Maximize the variance of the projection location of data in the principal directions.
2. Minimize the squared distance of the data points from their projections onto the lst principal component.
3. Each point of the first principal component is the conditional mean of all data points projected into it.

Assuming the underlying interrelationships between the recorded variables are governed by:
$$\mathbf{z}=\mathbf{A t}+\mathbf{e},$$
where $\mathbf{z} \in \mathbb{R}^{N}, \mathbf{t} \in \mathbb{R}^{n}$ is the latent variable (or projection index for the $\mathrm{PCs}$ ), $\mathbf{A} \in \mathbb{R}^{N \times n}$ is a matrix describing the linear interrelationships between data $\mathbf{z}$ and latent variables $\mathbf{t}$, and e represent statistically independent noise, i.e. $E{\mathbf{e}}=\mathbf{0}, E{\mathbf{e e}}=\delta \mathbf{I}, E\left{\mathbf{e t}^{T}\right}=\mathbf{0}$ with $\delta$ being the noise variance. $\mathrm{PCA}$, in this context, uses the above principles of the first principal component to extract the $n$ latent variables $\mathbf{t}$ from a recorded data set $\mathbf{Z}$.

Following from this linear analysis, a general nonlinear form of (1.28) is as follows:
$$\mathbf{z}=\mathbf{f}(\mathbf{t})+\mathbf{e},$$
where $\mathbf{f}(\mathbf{t})$ is a nonlinear function and represents the interrelationships between the latent variables $\mathbf{t}$ and the original data $\mathbf{z}$. Reducing $\mathbf{f}(\cdot)$ to be a linear function, Equation (1.29) clearly becomes (1.28), that is a special case of Equation (1.29).

To uncover the intrinsic latent variables, the following cost function, defined as
$$R=\sum_{i=1}^{K}\left|\mathbf{z}{i}-\mathbf{f}\left(\mathbf{t}{i}\right)\right|_{2}^{2},$$
where $K$ is the number available observations, can be used.
With respect to (1.30), linear $\mathrm{PCA}$ calculates a vector $\mathbf{p}{1}$ for obtaining the largest projection index $t{i}$ of Equation (1.28), that is the diagonal elements of $E\left{t^{2}\right}$ represent a maximum. Given that $\mathbf{p}{1}$ is of unit length, the location of the projection of $\mathbf{z}{i}$ onto the first principal direction is given by $\mathbf{p}{1} t{i}$. Incorporating a total of $n$ principal directions and utilizing (1.28), Equation (1.30) can be rewritten as follows:
$$R=\sum_{i=1}^{K}\left|\mathbf{z}{i}-\mathbf{P} \mathbf{t}{i}\right|_{2}^{2}=\operatorname{trace}\left{\mathbf{Z Z}^{T}-\mathbf{Z}^{T} \mathbf{A}\left[\mathbf{A}^{T} \mathbf{A}\right]^{-1} \mathbf{A} \mathbf{Z}^{T}\right}$$

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

Kramer [37] 提出的另一种范式与构建人工神经网络有关，以表示 (1.2) 的非线性版本。映射变量集的此类网络和通过定义一个降维瓶颈层，描述非线性主成分，将其定义为自关联神经网络，并在 Subsect 中重新讨论。4.2.

## 统计代写|数据科学代写data science代考|Introduction to principal curves

Hastie 和 Stuetzle 提出的主曲线 (PC)[24,25], 是通过表示数据集的云中间的平滑一维曲线。利用概率分布，主曲线满足自洽性质，这意味着曲线上的任何点都是投影到其上的所有数据点的平均值。作为主成分分析的非线性推广，磷Cs也可以看作是嵌入在高维数据空间中的一维流形。除了继承自线性主成分的统计特性外，磷Cs也反映了数据的几何结构所致。更准确地说，自然参数 arc-length 被视为每个样本的投影索引，其方式类似于表示投影数据点与原点的距离的分数变量。在这方面，两个变量之间的一维非线性拓扑关系可以通过主曲线来估计[85]。

## 统计代写|数据科学代写data science代考|From a weight vector to a principal curve

1. 最大化数据在主方向上的投影位置的方差。
2. 最小化数据点从它们投影到第一个主成分的平方距离。
3. 第一个主成分的每个点都是投影到其中的所有数据点的条件平均值。

R=∑一世=1ķ|和一世−F(吨一世)|22,

R=\sum_{i=1}^{K}\left|\mathbf{z}{i}-\mathbf{P} \mathbf{t}{i}\right|_{2}^{2}= \operatorname{trace}\left{\mathbf{Z Z}^{T}-\mathbf{Z}^{T} \mathbf{A}\left[\mathbf{A}^{T} \mathbf{A}\对]^{-1} \mathbf{A} \mathbf{Z}^{T}\right}R=\sum_{i=1}^{K}\left|\mathbf{z}{i}-\mathbf{P} \mathbf{t}{i}\right|_{2}^{2}= \operatorname{trace}\left{\mathbf{Z Z}^{T}-\mathbf{Z}^{T} \mathbf{A}\left[\mathbf{A}^{T} \mathbf{A}\对]^{-1} \mathbf{A} \mathbf{Z}^{T}\right}

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