### 统计代写|主成分分析代写Principal Component Analysis代考|OLET5610

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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|Performance Measures

In this section, four feature evaluation indices namely, class separability ( $S$ ) [8], overall classification accuracy $(O A)[32]$, kappa coefficient ( $\kappa)$ [32] and entropy $(E)$ [33], have been described which are considered for evaluating the effectiveness of the extracted features. The first three measuring indices need class label information of the samples while the last one does not require the same. The details of the evaluation indices used in this thesis, are given below.
Overall Accuracy $(O A):$
Overall accuracy [32] represents the ratio between the number of samples correctly recognized by the classification algorithm and the total number of test samples. To measure the overall accuracy, initially, confusion matrix is determined. The confusion matrix is a square matrix of size $C \times C$, where $C$ represents the number of classes of the given data set. The element $n_{i j}$ of the matrix denotes the number of samples of the $j$ th $(j=1,2, \ldots, C)$ category which are classified into $i$ th $(i=1,2, \ldots, C)$ category. Let $N$ be the total number of samples; where $N=\sum_{i=1}^{C} \sum_{j=1}^{C} n_{i j}$. The

overall accuracy $(O A)$ is defined as
$$O A=\frac{\sum_{i=1}^{C} n_{i i}}{N}$$
Kappa Coefficient $(\kappa)$ :
The kappa coefficient $(\kappa)[32]$ is a measure defined on the difference between the actual agreement in the confusion matrix and the chance agreement, which is indicated by row and column totals of the confusion matrix. The kappa coefficient is widely adopted, as it also takes into consideration the off-diagonal elements of the confusion matrix and compensates for chance agreement. The value of $\kappa$ lies in the range $[-1,+1]$. Closer the value of $\kappa$ to $+1$, better is the classification.

Let, in the confusion matrix, the sum of the elements of $i$ th row be denoted as $n_{i+}$ (where, $n_{i+}=\sum_{j=1}^{C} n_{i j}$ ) and the sum of the elements of column $j$ be $n_{+j}$ (where $n_{+j}=\sum_{i=1}^{C} n_{i j}$. The kappa coefficient is then defined as
$$\kappa=\frac{N \sum_{i=1}^{C} n_{i i}-\sum_{i=1}^{C} n_{i+} n_{+i}}{N^{2}-\sum_{i=1}^{C} n_{i+} n_{+i}}$$
where $N$ denotes the total number of samples and $C$ denotes the number of classes of the given data set.

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

Experiments are conducted on three hyperspectral data sets, namely, Indian Pine, $\mathrm{KSC}$ and Botswana. Details about the data sets are given in Sect. 6.1. As already mentioned in the previous section, the clustering oriented KPCA based method first perform DBSCAN clustering technique on pixels to choose $N$ representative patterns and then perform KPCA based transformation on the data set to reduce the dimensionality.

DBSCAN clustering algorithm uses two parameters, namely, minimum distance with respect to a point for which neighborhood is calculated (denoted as Eps) and the minimum number of points in an Eps-neighborhood of that point (denoted by MinPts). Ester et al. [29] suggested to use MinPts equal to 4 and used a method which considers the variation of the number of points with respect to their 4th nearest neighbor distance to calculate the value of Eps. Although higher values for MinPts have also been tested, it did not produce better results. The value of Eps is taken to be the location of the first valley of this graph. In the clustering oriented KPCA based strategy, MinPts and Eps are calculated in accordance to Ester et al. [29]. For Indian Pine data set, Eps value is 110 , which is the 4th nearest neighbor distance of the first valley of the graph described at Ester et al. [29] with MinPts equal to 4. There are about 19 clusters of pixels and few isolated pixels which do not belong to any cluster. It is better to discard the isolated pixels and not consider them in formation of representative patterns, because KPCA is susceptible to noise. Generally, the principle for selecting representative patterns from each cluster is discussed in the proposed method section. But the percentage of total patterns which are selected for representative patterns, is needed to determine. Here, $2-12 \%$ of total patterns are selected for representative patterns for calculating kernel matrix of KPCA and the performance of the clustering oriented KPCA based method in terms of overall accuracy for 18 number of extracted features for Indian Pine data is depicted in Table 4. From the table, it is observed that $8-10 \%$ data patterns are sufficient for calculating kernel matrix. Similar observations are also found for the other data sets. So, $10 \%$ data from each cluster are selected for making representative patterns. So in the set of representative patterns, a small cluster has less number of pixels and vice verse. For example, the number of representative patterns for Indian Pine data is about 850 .

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

The cumulative eigenvalues of PCA, KPCA and clustering oriented KPCA based methods are depicted in Table 5 in percentage for Indian Pine data set. The cumulative eigenvalues represent the cumulative variance of the data $[22,34]$. It shows that ninety five percent of cumulative variance of $\mathrm{PCA}$ is retained by the first six components, while KPCA and clustering oriented KPCA based methods need 14 to 18 components. In PCA most of the information content is retained in the first few features, where as, KPCA and clustering oriented KPCA based methods require more number of components.

The obtained OA and $\kappa$ for Indian Pine data after applying fuzzy $k$-NN classifier over the transformed set of features by PCA, segmented PCA (SPCA), kernel PCA (KPCA) and clustering oriented KPCA based methods are given in Table 6. For PCA based method, OA becomes saturated when the number of transformed feature is 10 and after that it is stabilized. For KPCA and clustering oriented KPCA based methods, OA saturated at 18 and 16 number of features, respectively. It is due to the fact that the number of principal components for PCA, KPCA and clustering oriented KPCA methods, for containing most of the variance of data, are 10,18 and 16, respectively (shown in Table 5). It is noticed from Table 6 that Kernel PCA based methods (i.e., KPCA and clustering oriented KPCA) give better results than PCA and segmented PCA based methods. From Table 6, it is also observed that clustering oriented KPCA method achieves better results in terms of OA and $\kappa$ for different number of transformed features. The reason behind this finding is that all the four methods transform the original set of features into a new set of features considering the maximum variance of data. Moreover, KPCA based methods incorporate the non linearity in transformation. The clustering oriented KPCA method gives better results than KPCA, because the representative patterns, for calculating kernel matrix for KPCA, are not selected randomly (like KPCA). The DBSCAN clustering technique is used to select the representative patterns so that it properly represents all the clusters of the data set, as well as, discard noisy pattern.

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

○一个=∑一世=1Cn一世一世ñ

kappa 系数(ķ)[32]是对混淆矩阵中的实际一致性与机会一致性之间的差异定义的度量，由混淆矩阵的行和列总计表示。kappa 系数被广泛采用，因为它还考虑了混淆矩阵的非对角元素并补偿了机会一致性。的价值ķ位于范围内[−1,+1]. 更接近的价值ķ至+1，更好的是分类。

ķ=ñ∑一世=1Cn一世一世−∑一世=1Cn一世+n+一世ñ2−∑一世=1Cn一世+n+一世

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

DBSCAN 聚类算法使用两个参数，即相对于计算邻域的点的最小距离（表示为 Eps）和该点的 Eps 邻域中的最小点数（表示为 MinPts）。酯等。[29] 建议使用等于 4 的 MinPts，并使用一种考虑点数相对于其第 4 个最近邻距离的变化的方法来计算 Eps 的值。尽管还测试了更高的 MinPts 值，但它并没有产生更好的结果。Eps 的值被视为该图的第一个谷的位置。在基于聚类的 KPCA 策略中，MinPts 和 Eps 是根据 Ester 等人的方法计算的。[29]。对于 Indian Pine 数据集，Eps 值为 110 ，这是 Ester 等人描述的图的第一个谷的第四个最近邻距离。[29] MinPts 等于 4。大约有 19 个像素簇和少数不属于任何簇的孤立像素。最好丢弃孤立的像素，而不是在形成代表性图案时考虑它们，因为 KPCA 容易受到噪声的影响。通常，在建议的方法部分中讨论了从每个集群中选择代表性模式的原则。但需要确定为代表模式选择的总模式的百分比。这里，最好丢弃孤立的像素，而不是在形成代表性图案时考虑它们，因为 KPCA 容易受到噪声的影响。通常，在建议的方法部分中讨论了从每个集群中选择代表性模式的原则。但需要确定为代表模式选择的总模式的百分比。这里，最好丢弃孤立的像素，而不是在形成代表性图案时考虑它们，因为 KPCA 容易受到噪声的影响。通常，在建议的方法部分中讨论了从每个集群中选择代表性模式的原则。但需要确定为代表模式选择的总模式的百分比。这里，2−12%为计算 KPCA 的核矩阵选择了总模式的代表模式，表 4 描述了基于聚类的 KPCA 方法在印度松数据的 18 个提取特征的总体准确度方面的性能。从表中，它观察到8−10%数据模式足以计算内核矩阵。对于其他数据集也发现了类似的观察结果。所以，10%选择来自每个集群的数据来制作具有代表性的模式。因此，在一组代表性图案中，一个小簇的像素数较少，反之亦然。例如，Indian Pine 数据的代表模式数量约为 850 。

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## MATLAB代写

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