### 统计代写|主成分分析代写Principal Component Analysis代考|MATH5855

statistics-lab™ 为您的留学生涯保驾护航 在代写主成分分析Principal Component Analysis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。我们的专家在代写主成分分析Principal Component Analysis代写方面经验极为丰富，各种代写主成分分析Principal Component Analysis相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• Advanced Probability Theory 高等概率论
• Advanced Mathematical Statistics 高等数理统计学
• (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|主成分分析代写Principal Component Analysis代考|Segmented Principal Component Analysis

In hyperspectral images, the correlations between neighboring spectral bands are generally higher than for bands further apart. If conventional PCA based method is modified so that the transformation is carried out by avoiding the low correlations between the highly correlated blocks, the efficiency of PCA will be improved. Also, the computational load is a major consideration in the case of hyperspectral data transformation, i.e., it is inefficient to transform the complete data set. So, a segmented principal component analysis comes into picture.

In this scheme [20], the complete data set is first partitioned into several subgroups, depending on the correlations of neighboring features of hyperspectral images. Highly correlated features are selected as subgroups. Then, PCA based transformation is conducted separately on each subgroup of data.

At the onset, the $D$ number of bands of a hyperspectral images is partitioned into a few number of contiguous intervals with constant intensities (i.e., $K$ subgroups). Highly correlated bands should be in a subgroup. Let $I_{1}, I_{2}, \ldots, I_{k}$, be the number of bands in the $1 \mathrm{st}, 2 \mathrm{nd}$, and $K$ th group, correspondingly. The purpose is to obtain a set of $\mathrm{K}$ breakpoints $P=\left{p_{1}, p_{2}, \ldots, p_{K}\right}$, which defines the contiguous intervals $I_{k}=\left[p_{k}, p_{k+1}\right)$. The partition should follow the principle that each band should be inside one block.

Let $\Gamma$ be a correlation matrix of size $D \times D$, where $D$ is the number of bands present in a hyperspectral image. Each element of $\Gamma$ is $\gamma_{i j}$, where $\gamma_{i j}$ represents the correlation between band images $B_{i}$ and $B_{j}$. Let the size of each band image be $M \times N$. The correlation coefficient between $B_{i}$ and $B_{j}$ is defined as
$$\gamma_{i, j}=\frac{\Sigma_{x=1}^{M} \Sigma_{y=1}^{N}\left|B_{i}(x, y)-\mu_{i}\right|\left|B_{j}(x, y)-\mu_{j}\right|}{\sqrt{\left(\Sigma_{x=1}^{M} \Sigma_{y=1}^{N}\left[B_{i}(x, y)-\mu_{i}\right]^{2}\right)\left(\Sigma_{x=1}^{M} \Sigma_{y=1}^{N}\left[B_{j}(x, y)-\mu_{j}\right]^{2}\right)}}$$
where $\mu_{i}$ and $\mu_{j}$ are the mean of band images $B_{i}$ and $B_{j}$, respectively. $\left|B_{i}(x, y)-\mu_{i}\right|$ measures the difference between the reflectance value of pixel $(x, y)$ from the mean value of the total image.

It is observed that the correlation between neighboring spectral bands are generally higher than for bands further apart. Partitioning is performed based on the results obtained by first considering only correlations whose absolute value exceeds a given threshold, and simultaneously searching for edges in the “image” of the correlation matrix [20]. Each value of the correlation matrix is compared with a threshold (correlation). If the magnitude is greater than the threshold value (i.e., denoted by $\Theta$ ), then replace it by 1 ; otherwise by 0 . The value of $\Theta$ has been determined depending on the value of average correlation $\left(\mu_{c a r r}\right)$ and standard deviation $\left(\sigma_{c a r r}\right)$ of correlation matrix $\Gamma$ as
$$\Theta=\mu_{\text {corr }}+\sigma_{\text {corr }}$$

## 统计代写|主成分分析代写Principal Component Analysis代考|Kernel Principal Component Analysis

PCA, basically, rotates the original axes, so that the new coordinate system aligns with the orientation of maximum variability of data. Rotation is a linear transformation and the new coordinate axes are then a linear combination of the original axes. So, PCA as a linear algorithm is inadequate to extract the non linear structures of the data. Also, PCA only considers variance between patterns which is a second order statistics, that may limit the effectiveness of the method. So, a non-linear version of PCA is considered, which is called kernel PCA (KPCA). It is capable of capturing

a part of higher order statistics. So it is useful for representing the information from the original data set which is more useful to discriminate among themselves.

Kernel principal component analysis [22], a nonlinear version of the PCA is capable of capturing a part of higher order statistics, which may represent the information in a better way from the original data set to reduced data set [25]. This technique is used for reducing the dimensionality of hyperspectral images. Here, the data of the input space $\Re^{D}$ is mapped into another space, called feature space $F$, to capture higher-order statistics. A non-linear mapping function $\Phi$ is used to transfer the data from input feature space to a new feature space by
$$\begin{gathered} \Phi: \mathfrak{R}^{D} \rightarrow F \ x \rightarrow \Phi(x) \end{gathered}$$
The non-linear function $\Phi$ transforms a pattern $x$ from $D$-dimensional input space to another feature space $F$. The covariance matrix in this feature space is calculated as
$$\Sigma_{\Phi(x)}=\frac{1}{N} \sum_{i=1}^{N} \Phi\left(x_{i}\right) \Phi\left(x_{i}\right)^{T}$$
The principal components are then computed by solving the eigenvalue problem
$$\lambda V=\Sigma_{\Phi(x)} V=\frac{1}{N} \sum_{i=1}^{N}\left(\Phi\left(x_{i}\right) \cdot V\right) \Phi\left(x_{i}\right) .$$

## 统计代写|主成分分析代写Principal Component Analysis代考|Clustering Oriented Kernel Principal Component

The clustering oriented KPCA based feature extraction method [26] performs kernel principal component analysis to transform the original data set of dimension $D$ into $d$ dimensional space. The KPCA is non linear in nature and uses higher order statistics of data set to discriminate the classes. The most important thing is to select the proper training set for calculating kernel matrix for KPCA. A randomly selected training pattern may not represent the overall data set properly. Also, it should not be too large so that the method becomes computationally prohibitive. So, a proper subset of original hyperspectral data set which can represent the total data set properly should be selected and this training set should not contain any noisy data. DBSCAN clustering technique is used for choosing the proper representative training set. In this section, selection of $N$ representative patterns using DBSCAN clustering technique is described and then discuss about the KPCA based transformation using these data.
KPCA shares the same properties as the PCA, but in a different space. Both PCA and KPCA need to solve eigenvalue problem, but the dimensions of the problem are different, $D \times D$ for PCA and $N \times N$ for KPCA, where $D$ is the dimensions of data set and $N$ is number of representative patterns required to calculate kernel matrix $\Psi$. The size of the matrix becomes problematic for large $N$. Number of pixel points $(N)$ in hyperspectral images is huge, so it is difficult to perform KPCA by taking all the pixels. If some percentage of total pixels are selected randomly, then the selected pixels may not represent the characteristics of total data. So, it is better to make small group of pixels according to their similarity, and then take some representative pixels from each group to make the representative pattern set for KPCA.

## 统计代写|主成分分析代写Principal Component Analysis代考|Segmented Principal Component Analysis

C一世,j=ΣX=1米Σ是=1ñ|乙一世(X,是)−μ一世||乙j(X,是)−μj|(ΣX=1米Σ是=1ñ[乙一世(X,是)−μ一世]2)(ΣX=1米Σ是=1ñ[乙j(X,是)−μj]2)

θ=μ更正 +σ更正

## 统计代写|主成分分析代写Principal Component Analysis代考|Kernel Principal Component Analysis

PCA 基本上旋转原始轴，以便新坐标系与数据最大可变性的方向对齐。旋转是一种线性变换，新的坐标轴是原始坐标轴的线性组合。因此，PCA 作为一种线性算法不足以提取数据的非线性结构。此外，PCA 仅考虑模式之间的方差，这是二阶统计量，这可能会限制方法的有效性。因此，考虑了 PCA 的非线性版本，称为内核 PCA (KPCA)。它能够捕捉

Σ披(X)=1ñ∑一世=1ñ披(X一世)披(X一世)吨

λ在=Σ披(X)在=1ñ∑一世=1ñ(披(X一世)⋅在)披(X一世).

## 统计代写|主成分分析代写Principal Component Analysis代考|Clustering Oriented Kernel Principal Component

KPCA 与 PCA 具有相同的属性，但在不同的空间中。PCA和KPCA都需要求解特征值问题，但是问题的维度不同，D×D对于 PCA 和ñ×ñ对于 KPCA，其中D是数据集的维度和ñ是计算核矩阵所需的代表性模式数Ψ. 矩阵的大小对于大型ñ. 像素点数(ñ)在高光谱图像中是巨大的，因此很难通过取所有像素来执行 KPCA。如果随机选择总像素的某个百分比，则选择的像素可能不代表总数据的特征。因此，最好根据它们的相似性制作一小组像素，然后从每组中取出一些具有代表性的像素来制作KPCA的代表模式集。

## 有限元方法代写

tatistics-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。