### 统计代写|数据科学代写data science代考|Nonlinear Principal Component Analysis

statistics-lab™ 为您的留学生涯保驾护航 在代写数据科学data science方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写数据科学data science方面经验极为丰富，各种代写数据科学data science相关的作业也就用不着说。

• 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|Neural Network Models and Applications

Many natural phenomena behave in a nonlinear way meaning that the observed data describe a curve or curved subspace in the original data space. Identifying such nonlinear manifolds becomes more and more important in the field of molecular biology. In general, molecular data are of very high dimensionality because of thousands of molecules that are simultaneously measured at a time. Since the data are usually located within a low-dimensional subspace, they can be well described by a single or low number of components. Experimental time course data are usually located within a curved subspace which requires a nonlinear dimensionality reduction as illustrated in Fig. 2.1.
Visualising the data is nne aspert of molecnlar data analysis, another important aspẹct is to módel the mapping from original epace to component

Fig. 2.1. Nonlinear dimensionality reduction. Illustrated are threedimensional samples that are located on a one-dimensional subspace, and hence can be described without loes of informution by a vingle variable (the component). The transformation is given by the two functions $\Phi_{\text {extr }}$ and $\Phi_{\text {gen. }}$. The extraction funccomponent value (right). The inverse mapping is given by the generation function $\Phi_{g e n}$ which transforms any scalar component value back into the original data space. Such helical trajectory over time is not uncommon in molecular data. The horizontal axes may represent molecule concentrations driven by a circadian rhythm, whereas the vertical axis might represent a molecule with an increase in concentration
space in order to interpret the impact of observed variables on the subspace (component space). Both the component values (scores) and the mapping function is provided by the neural network approach for nonlinear PCA.
Three important extensions of nonlinear $\mathrm{PCA}$ are discussed in this chapter: the hierarchical NLPCA, the circular PCA, and the inverse NLPCA. All of them can be used in combination. Hierarchical NLPCA means to enforce the nonlinear components to have the same hierarchical order as the linear components of standard PCA. This hierarchical condition yields a higher meaning of individual components. Circular $P C A$ enables nonlinear PCA to extract circular components which describe a closed curve instead of the standard curve with an open interval. This is very useful for analysing data from cyclic or oscillatory phenomena. Inverse $N L P C A$ defines nonlinear $\mathrm{PCA}$ as an inverse problem, where only the assumed data generation process is modelled, which has the advantage that more complex curves can be identified and NLPCA becomes applicable to incomplete data sets.

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

Nonlinear PCA based on autoassociative neural networks was investigated in several studies $[1-4]$. Kirby and Miranda [5] constrained network units to work in a circular manner resulting in a circular $P C A$ whose components are closed curves. In the fields of atmospheric and oceanic sciences, circular PCA is applied to oscillatory geophysical phenomena, for example, the oceanatmosphere El Niño-Southern oscillation [6] or the tidal cycle at the German North Sea coast [7]. There are also applications in the field of robotics in

order to analyse and control periodic movements [8]. In molecular biology, circular PCA is used for gene expression analysis of the reproductive cycle of the malaria parasite Plasmodium falciparum in red blood cells $[9]$. Scholz and Vigário [10] proposed a hienanchical nonlinear $P C A$ which achieves a hierarchical order of nonlinear components similar to standard linear $\mathrm{PCA}$. This hierarchical NLPCA was applied to spectral data of stars and to electromyographic (EMG) recordings of muscle activities. Neural network models for inverse $N L P C A$ were first studied in $[11,12]$. A more general Bayesian framework based on such inverse network architecture was proposed by Valpola and Honkela ${13,14}$ for a nonlinear factor analysis (NFA) and a nonlinear independent factor analysis (NIFA). In $[15]$, such inverse NLPCA model was adapted to handle missing data in order to use it for molecular data analysis. It was applied to metabolite data of a cold stress experiment with the model plant A rabidopsis thaliana. Hinton and Salakhutdinov [16] have demonstrated the use of the autoassociative network architecture for visualisation and dimensionality reduction by using a special initialisation technique.

Even though the term nonlinear PCA (NLPCA) is commonly referred to as the autoassociative approach, there are many other methods which visualise data and extract components in a nonlinear manner. Locally linear embedding (LLE) $[17,18]$ and Isomap [19] were developed to visualise high dimensional data by projecting (embedding) them into a two or low-dimensional space, but the mapping function is not explicitly given. Principal curves [20] and self organising maps (SOM) [21] are useful for detecting nonlinear curves and two-dimensional nonlinear planes. Practically both methods are limited in the number of extracted components, usually two, due to high computational costs. Kernel $P C A[22]$ is useful for visualisation and noise reduction [23].
Several efforts are made to extend independent component analysis (ICA) into a nonlinear ICA. However, the nonlinear extension of ICA is not only very challenging, but also intractable or non-unique in the absence of any a priori knowledge of the nonlinear mixing process. Therefore, special nonlinear ICA models simplify the problem to particular applications in which some information about the mixing system and the factors (source signals) is available, e.g., by using sequence information [24]. A discussion of nonlinear approaches to $\mathrm{ICA}$ can be found in $[25,26]$. This chapter focuses on the less difficult task of nonlinear PCA. A perfect nonlinear PCA should, in principle, be able to remove all nonlinearities in the data such that a standard linear ICA can be applied subsequently to achieve, in total, a nonlinear ICA. This chapter is mainly based on $[9,10,15,27]$.

## 统计代写|数据科学代写data science代考|Data generation and component extraction

To extract components, linear as well as nonlinear, we assume that the data are determined by a number of factors and hence can be considered as being generated from them. Since the number of varied factors is often smaller than

the number of observed variables, the data are located within a subspace of the given data space. The aim is to represent these factors by components which together describe this subspace. Nonlinear PCA is not limited to linear components, the subspace can be curved, as illustrated in Fig. 2.1.

Suppose we have a data space $\mathcal{X}$ given by the observed variables and a component space $Z$ which is a subspace of $\mathcal{X}$. Nonlinear PCA aims to provide both the subspace $\mathcal{Z}$ and the mapping between $\mathcal{X}$ and $\mathcal{Z}$. The mapping is given by nonlinear functions $\Phi_{\text {extr }}$ and $\Phi_{g e n}$. The extruction function $\Phi_{\text {extr }}: \mathcal{X} \rightarrow \mathcal{Z}$ transforms the sample coordinates $x=\left(x_{1}, x_{2}, \ldots, x_{d}\right)^{T}$ of the $d$-dimensional data space $\mathcal{X}$ into the corresponding coordinates $z=\left(z_{1}, z_{2}, \ldots, z_{k}\right)^{T}$ of the component space $\mathcal{Z}$ of usually lower dimensionality $k$. The generation function $\bar{\Phi}{g e n}: \mathcal{Z} \rightarrow \hat{\mathcal{X}}$ is the inverse mapping which reconstructs the original sample vector $x$ from their lower-dimensional component representation $z$. Thus, $\Phi{g e n}$ approximates the assumed data generation process.

## 广义线性模型代考

statistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。