9. The first principle direction is the direction in the input space along which the projections...
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9. The first principle direction is the direction in the input space along which the projections of the data points have the largest variance. We use W₁ to denote the first principle direction (w₁ has unit length i.e. L2 norm is 1), 2₁ to represent the first/largest eigenvalue of the covariance matrix, u to represent the sample mean, and x to represent a data point. The deviation of x from the mean µ is x − µ. We do dimensionality reduction: y = PCA(x) with "whiten-True" in sk-learn, and the number of components/elements of y is less than the the number of components/elements of x (1) what is the vector-projection of x in the direction of w₁ ? (2) what is the vector-projection of the deviation x - u in the direction of w₁? (3) what is the first component of y? (whiten=True: something divided by √₁ (4) assuming y only has one component, then we do inverse transform to recover/approximate the input x = PCA-¹(y) compute x using μ, y, λ₁ and w₁: x = ??? (5) what is x x if x and y have the same number of elements? 9. The first principle direction is the direction in the input space along which the projections of the data points have the largest variance. We use W₁ to denote the first principle direction (w₁ has unit length i.e. L2 norm is 1), 2₁ to represent the first/largest eigenvalue of the covariance matrix, u to represent the sample mean, and x to represent a data point. The deviation of x from the mean µ is x − µ. We do dimensionality reduction: y = PCA(x) with "whiten-True" in sk-learn, and the number of components/elements of y is less than the the number of components/elements of x (1) what is the vector-projection of x in the direction of w₁ ? (2) what is the vector-projection of the deviation x - u in the direction of w₁? (3) what is the first component of y? (whiten=True: something divided by √₁ (4) assuming y only has one component, then we do inverse transform to recover/approximate the input x = PCA-¹(y) compute x using μ, y, λ₁ and w₁: x = ??? (5) what is x x if x and y have the same number of elements?
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