### 统计代写|主成分分析代写Principal Component Analysis代考|ENVX2001

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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|Explained Variance

Finally, we introduce bounds for the explained variance $E V(X)$. Two results are obtained. The first is general and applicable to any basis $X$, not limited to sparse ones. The second is tailored to SPCArt.

Theorem 12 Let rank-r SVD of $A \in \mathbb{R}^{n \times p}$ be $U \Sigma V^{T}, \Sigma \in \mathbb{R}^{r \times r}$. Given $X \in \mathbb{R}^{p \times r}$, assume the $S V D$ of $X^{T} V$ to be $W D Q^{T}, D \in \mathbb{R}^{r \times r}, d_{\min }=\min {i} D{i i}$, then
$$d_{\min }^{2} \cdot E V(V) \leq E V(X),$$
and $E V(V)=\sum_{i} \Sigma_{i i^{\circ}}^{2}$
The theorem can be interpreted as follows. If $X$ is a basis that approximates the rotated PCA loadings well, then $d_{\min }$ will be close to one, and so the variance explained by $X$ is close to that explained by PCA. Note that the variance explained by PCA loadings is the largest value that is possible to be achieved by an orthonormal basis. Conversely, if $X$ deviates greatly from the rotated PCA loadings, then $d_{\min }$ tends to zero, so the variance explained by $X$ is not guaranteed to be large. Thus, the less the sparse loadings deviate from the rotated PCA loadings, the more variance they explain.

When SPCArt converges, i.e., $X_{i}=T_{\lambda}\left(Z_{i}\right) /\left|T_{\lambda}\left(Z_{i}\right)\right|_{2}$, where $Z=V R^{T}$, and $R=\operatorname{Polar}\left(X^{T} V\right)$ hold simultaneously, there is another estimation (mainly valid for $\mathrm{T}$-en).

Theorem 13 Let $C=Z^{T} X$, i.e., $C_{i j}=\cos \left(\theta\left(Z_{i}, X_{j}\right)\right)$, and let $\bar{C}$ be the diagonalremoved version. Assume $\forall i, \theta\left(Z_{i}, X_{i}\right)=\theta$ and $\sum_{j}^{r} C_{i j}^{2} \leq 1$, then
$$\left(\cos ^{2}(\theta)-\sqrt{r-1} \sin (2 \theta)\right) \cdot E V(V) \leq E V(X) .$$
When $\theta$ is sufficiently small,
$$\left(\cos ^{2}(\theta)-O(\theta)\right) \cdot E V(V) \leq E V(X) .$$
Since the sparse loadings are obtained by truncating small entries of the rotated PCA loadings, and $\theta$ is the deviation angle, the theorem implies that if the deviation

is small then the explained variance is close to that of $\mathrm{PCA}$, as $\cos ^{2}(\theta) \approx 1$. For example, if the truncated energy $|\bar{z}|_{2}^{2}=\sin ^{2}(\theta)$ is approximately $0.05$, then $95 \%$ $\mathrm{EV}$ of $\mathrm{PCA}$ loadings is guaranteed.

The assumptions $\theta\left(Z_{i}, X_{i}\right)=\theta$ and $\sum_{j}^{r} C_{i j}^{2} \leq 1, \forall i$, are broadly satisfied by T-en using small $\lambda$. Uniform deviation $\theta\left(Z_{i}, X_{i}\right)=\theta \forall i$ can be achieved by T-en, as indicated by Proposition 11. $\sum_{j}^{r} C_{i j}^{2} \leq 1$ means the sum of projected length is less than 1 when $Z_{i}$ is projected onto each $X_{j}$. It is satisfied if $X$ is exactly orthogonal, whereas it is likely satisfied if $X$ is nearly orthogonal (note $Z_{i}$ may not lie in the subspace spanned by $X$ ), which can be achieved by setting small $\lambda$ according to Proposition 6. In this case, about $(1-\lambda) E V(V)$ is guaranteed.

In practice, we prefer CPEV [21] to EV. CPEV measures the variance explained by subspace rather than basis. Since it is also the projected length of $A$ onto the subspace spanned by $X$, the higher CPEV, the better $X$ represents the data. If $X$ is not an orthogonal basis, EV may overestimate or underestimate the variance. However, if $X$ is nearly orthogonal, the difference is small, and it is nearly proportional to CPEV.

## 统计代写|主成分分析代写Principal Component Analysis代考|A Unified View to Some Prior Work

A series of methods: PCA [10], SCoTLASS [11], SPCA [29], GPower [13], $\mathrm{rSVD}[21]$, TPower [25], SPC [24], and SPCArt, although proposed independently and formulated in various forms, can be derived from the common source of Theorem 1, the Eckart-Young Theorem. Most of them can be seen as the problems of matrix approximation (1), with different sparsity penalties. Most of them have two matrix variables, and the solutions of them can usually be obtained by an alternating scheme: fixing one and solving the other. Similar to SPCArt, the two subproblems are a sparsity penalized/constrained regression problem and a Procrustes problem.
PCA [10]. Since $Y^{}=A X^{}$, substituting $Y=A X$ into (1) and optimizing $X$, the problem is equivalent to
$$\max {X} \operatorname{tr}\left(X^{T} A^{T} A X\right), \text { s.t. } X^{T} X=I .$$ By the Ky Fan theorem [7], $X^{}=V{1: r} R, \forall R^{T} R=I$. If $A$ is a mean-removed data matrix, the special solution $X^{}=V_{1 r}$ contains exactly the $r$ loadings obtained by PCA.
SCoTLASS [11]. Constraining $X$ to be sparse in (19), we get SCotLASS
$$\max {X} \operatorname{tr}\left(X^{T} A^{T} A X\right), \text { s.t. } X^{T} X=I, \forall i,\left|X{i}\right|_{1} \leq \lambda$$
Unfortunately, this problem is not easy to solve.

## 统计代写|主成分分析代写Principal Component Analysis代考|Principal Component Analysis (PCA) Based

Principal component analysis (PCA) $[10,12,27]$ is an orthogonal basis transformation with the advantage that the first few principal components preserve most of the variance of the data set. This method [27], initially, calculates the covariance matrix of the given data set, and then finds the eigenvalues and eigenvectors of this matrix. Next it selects a few eigenvectors whose eigenvalues are more to form the transformation matrix to reduce the dimensions of the data set.

Suppose, there are $D$ number of band images. So, a pixel has $D$ number of different responses over different wavelengths. As a consequences, a pixel may be treated as a pattern of $D$ attributes. The main target is to reduce the dimensionality from $D$ to $d(d \ll D)$ of hyperspectral image pixel.

Let, there be a set of pattern $x_{i}$, where $x_{i} \in \mathfrak{R}^{D}, i=1,2, \ldots, N$. Assume that the data are centered, i.e., $x_{i} \Longleftarrow x_{i}-E\left{x_{i}\right}$. Conventional PCA formulates the eigenvalue problem by
$$\lambda V=\Sigma_{x} V$$
where $\lambda$ is eigenvalue, $V$ is eigenvector, $\Sigma_{x}$ is the corresponding covariance matrix over data set $x$ which is calculated by the following equation
$$\Sigma_{x}=\frac{1}{N} \sum_{i=1}^{N} x_{i} x_{i}^{T}$$
The projection on the eigenvector $V^{k}$ is calculated as
$$x_{p c}^{k}=V^{k} . x .$$
The principal component based transformation is defined as
$$y_{i}=W^{T} x_{i}$$
where $W$ is the matrix of first $d$ normalized eigenvectors of highest eigenvalues of the image covariance matrix $\Sigma_{x} . T$ denotes the transpose operation.

Here, a pattern $x_{i}$ from original $D$-dimensional space is transformed into $y_{i}$, a pattern in reduced $d$-dimensional space by choosing only the first $d$ components (eigenvectors of highest $d$ eigenvalues).

The transformed data set has two main properties which are significant to the application here. The variance in the original data set has been rearranged and reordered so that first few components contain almost all of the variance in the original data, and the components in the new feature space are uncorrelated in nature [20].

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

(因2⁡(θ)−r−1罪⁡(2θ))⋅和在(在)≤和在(X).

(因2⁡(θ)−○(θ))⋅和在(在)≤和在(X).

## 统计代写|主成分分析代写Principal Component Analysis代考|Principal Component Analysis (PCA) Based

λ在=ΣX在

ΣX=1ñ∑一世=1ñX一世X一世吨

XpCķ=在ķ.X.

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