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# 电子代写|数字信号处理代写Digital Signal Processing代考|ECE538 Finding the Principal Components

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## 电子代写|数字信号处理代写Digital Signal Processing代考|Finding the Principal Components

To find the principal components, the unit vector of $v_1$ and $v_2$, return to section $1.3 .1 .2$ which illustrated the eigenvectors of different matrices. Notice that symmetric matrices stretch their data along different directions and these directions are characterized by the eigenvectors of that matrix. It is our goal to generate a symmetric, or Hermitian symmetric, matrix given the data set at hand. Fortunately, the covariance matrix of the data set provides exactly what we needed. If you are unfamiliar with concepts of variance and covariance, please jump ahead to section 1.4.2 and $1.5 .3$ to get a quick preview. However, we can easily define the variance and covariance calculations of a sample set as follows. Given a sequence of $N$ numbers, $x[n]$, where $n=0,1, \ldots$ $N-1$, the variance is computed using the following expression. Remember that the asterisk
$$\sigma_x^2=\frac{1}{N} \sum_{n=0}^{N-1}(x[n]-\bar{x}) \cdot(x[n]-\bar{x})^*$$
The covariance between two sequences, $x[n]$ and $y[n]$, is similarly defined as follows.
$$\sigma_{x y}=\frac{1}{N} \sum_{n=0}^{N-1}(x[n]-\bar{x}) \cdot(y[n]-\bar{y})^*$$
The covariance matrix is a hermitian symmetric matrix and obeys the following expression. The eigenvectors of this matrix provide us with the principle components, ProjA_vI and ProjA_v2 , that we are looking for. Note that the sequences $x[n]$ and $y[n]$ in our example above represent the voltage and current values of each observation, $n$.
$$C_{x y}=\left[\begin{array}{cc} \sigma_x^2 & \sigma_{x y} \ \sigma_{y x} & \sigma_y^2 \end{array}\right]$$

## 电子代写|数字信号处理代写Digital Signal Processing代考|Understanding Projection

To find the projections of the measurement along principle component vectors, we recall the geometric interpretation of the dot product of two vectors $A$ and $B$, which states the following.
$$\operatorname{Dot}(A, B)=a_1 v_1^+a_2 b_2^+a_3 b_3^+\ldots+a_N b_N^=|A| \cdot|V| \cdot \cos (\theta)$$
Observe the figure below that illustrates the geometric interpretation of the dot product using the vectors A (one of our measurements), B (one of the two principle component vectors), and ProjA_B $\cdot B_{\text {unit }}$ (A projected onto B). Simple algebra proves that the length ProjA_B is equal to the ||$A|| \cdot \cos (\theta)$, which leads us to the equation for the projection of $A$ onto $B$. Note that the unit vector of $B=B / \mid B |$, where $|B|$ represents the length of norm of $B$.

The projection of the measurement, $\mathrm{A}$, onto the principle component vector, $\mathrm{B}$, has the following form, which is valid for both real and complex vectors.
\begin{aligned} \text { ProjA_B }_{\text {Unit }} &=|A| \cdot \cos (\theta) \cdot \frac{B}{|B|} \ &=|A| \cdot \cos (\theta) \cdot \frac{|B|}{|B|} \cdot \frac{B}{|B|} \ &=\frac{|A| \cdot|B| \cdot \cos (\theta)}{|B| \cdot|B|} \cdot B \ &=\frac{\operatorname{Dot}(A, B)}{\operatorname{Dot}(B, B)} \cdot B \end{aligned}

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