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

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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|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

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## MATLAB代写

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