Given two data points in 2 dimensions:
\displaystyle \displaystyle \mathbf{x}^{(1)} \displaystyle = \displaystyle (x^{(1)}, y^{(1)})= (0,1)
\displaystyle \mathbf{x}^{(2)} \displaystyle = \displaystyle (x^{(2)}, y^{(2)})=(0,-1)
Notice that this sample is already centered, with sample mean 0 in both x- and y- coordinates.
Without computation, find the direction of largest variance. (This is the first principal component \text {PC1}.
(1,0)
(0,1)
(1,1)
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Let us now compute the first principal component and check if it matches with the above. Find the sample covarance matrix \mathbf{S} for the given data.
Recall: Sample covarance matrix
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(Enter as a matrix, e.g. type [[1,2],[3,4]] for the matrix \begin{pmatrix} 1& 2\\ 3& 4\end{pmatrix}.)
\mathbf{S}=\quad
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What are the eigenvalues \lambda _1,\lambda _2 of the covariance matrix \mathbf{S}? In other words, find pairs of scalar \lambda and \mathbf v_\lambda such that \mathbf{S}\mathbf v_\lambda =\lambda \mathbf v_\lambda, i.e. the matrix \mathbf{S} acts on a vector \mathbf v_\lambda by rescaling it.
Review: eigenvalues and eigenvectors
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(Enter the eigenvalues in decreasing order, i.e. \lambda _1> \lambda _2. )
\lambda _1=\quad
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\lambda _2=\quad
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Find an eigenvector corresponding to the largest eigenvalue \lambda _1. The span of this vector is called the first principal component (PC1) .
(All non-zero scalar multiples will be accepted as correct.)
\mathbf v_{\text {PC1}}=\quad
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{::}
(Is your answer for the first principal component the same as your answer in the first part is this question, the direction of largest variance? )
Yes, the direction of the largest variance is the same as the first principal component.
Yes, the answer for the first principal component is the same as the direction of largest variance. The first principal component represents the direction along which the data has the largest spread or variance.
To find the direction of the largest variance without computation, we can visualize the given data points on a graph.
The first data point is (0, 1) and the second data point is (0, -1). Since the mean of both x and y coordinates is 0, these data points are already centered.
If we plot these points on a graph, we can see that they lie on a vertical line through the origin. Therefore, the direction of largest variance is the y-axis, which is represented by the answer choice (0, 1).