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The squared distance from any sample point to the origin has a x² distribution with mean d. Consider a prediction point x₁ drawn from this distribution, and let a = Xo/|xo| be an associated unit vector. Let zi aTx; be the projection of each of the training points on this - direction. (a). Show that the z; are distributed N(0, 1) with expected squared distance from the origin 1, while the target point has expected squared distance d from the origin. (b). For d = 10 show that the expected distance of a test point from the centre of the training data is 3.1 standard deviations, while all the training points have expected distance 0.80 along direction a. So most prediction points see themselves as lying on the edge of the training set. Note: for this question you need to use a result for the expected value of a squared root of a chi-squared distribution. Either find such a result, or obtain your answer by simulation.

The squared distance from any sample point to the origin has a x² distribution with mean d. Consider a prediction point x₁ drawn from this distribution, and let a = Xo/|xo| be an associated unit vector. Let zi aTx; be the projection of each of the training points on this - direction. (a). Show that the z; are distributed N(0, 1) with expected squared distance from the origin 1, while the target point has expected squared distance d from the origin. (b). For d = 10 show that the expected distance of a test point from the centre of the training data is 3.1 standard deviations, while all the training points have expected distance 0.80 along direction a. So most prediction points see themselves as lying on the edge of the training set. Note: for this question you need to use a result for the expected value of a squared root of a chi-squared distribution. Either find such a result, or obtain your answer by simulation.

Calculus For The Life Sciences
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter13: Probability And Calculus
Section13.2: Expected Value And Variance Of Continuous Random Variables
Problem 10E
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The problem with KNN is that in high dimensions, most points tend to lie on the boundary of the data space. Consider explanatory variables drawn from a spherical multinormal distribution x ~ N(0, I), where x is a random d-vector, and I is a d x d identity matrix.

The squared distance from any sample point to the origin has a x² distribution with mean d. Consider a prediction point xo drawn from this distribution, and let a = Xo/||xo|| be an associated unit vector. Let z; = aTx; be the projection of each of the training points on this direction. (a). Show that the z; are distributed N(0, 1) with expected squared distance from the origin 1, while the target point has expected squared distance d from the origin. (b). For d = 10 show that the expected distance of a test point from the centre of the training data is 3.1 standard deviations, while all the training points have expected distance 0.80 along direction a. So most prediction points see themselves as lying on the edge of the training set. Note: for this question you need to use a result for the expected value of a squared root of a chi-squared distribution. Either find such a result, or obtain your answer by simulation.
Transcribed Image Text:The squared distance from any sample point to the origin has a x² distribution with mean d. Consider a prediction point xo drawn from this distribution, and let a = Xo/||xo|| be an associated unit vector. Let z; = aTx; be the projection of each of the training points on this direction. (a). Show that the z; are distributed N(0, 1) with expected squared distance from the origin 1, while the target point has expected squared distance d from the origin. (b). For d = 10 show that the expected distance of a test point from the centre of the training data is 3.1 standard deviations, while all the training points have expected distance 0.80 along direction a. So most prediction points see themselves as lying on the edge of the training set. Note: for this question you need to use a result for the expected value of a squared root of a chi-squared distribution. Either find such a result, or obtain your answer by simulation.
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