In this task the idea is to study the behaviour of extreme distances and relative contrast...

  

Transcribed Image Text:

In this task the idea is to study the behaviour of extreme distances and relative contrast with different Lp norm based distance measures when the dimensionality of data increases. The relative contrast for data vector x; is defined as Ci = 2 Dmax — Dmin Dmin 2 where Dmax and Dmin are the maximum and minimum distances from vector x; to any other vector in the data, calculated with a given distance function. In the experiment, vary the dimensionality as d = 1, 2, 3, 4, 5, 10, 20,..., 100. For each d, simulate random data with n = 100 data points x; E Rd, where each component has uniform distribution in the range [0, 1]. For each x; € {x₁,...,Xn}, find the minimum, average and maximum distances from it to all other vectors. Then calculate the difference between the maximum and minimum distances and the relative contrast value. Finally compute the averages of all these five measures over all xi. Plot the average values and their logarithms as functions of increasing d. (Place all distance values in one plot and the contrast values in another plot. Similarly, do not mix the original and the logarithmic values in the same plot. You should thus have in total four plots.) Repeat the experiment four times with the following Lp measures as the distance function: L0.5, L₁, L2, and L3. a) What happens to each one of the distance values and the difference between the maxi- mum and minimum distances when d increases? Explain, how you use the original and logarithmic plots to draw these conclusions. b) What happens to the relative contrast measure when d increases? How can this finding be explained based on the distance plots? c) What kind of an effect does the p value have on the bahavior of the relative contrast measure as a function of d? d) How do you interpret the results with respect to so called "curse of dimensionality"? In this task the idea is to study the behaviour of extreme distances and relative contrast with different Lp norm based distance measures when the dimensionality of data increases. The relative contrast for data vector x; is defined as Ci = 2 Dmax — Dmin Dmin 2 where Dmax and Dmin are the maximum and minimum distances from vector x; to any other vector in the data, calculated with a given distance function. In the experiment, vary the dimensionality as d = 1, 2, 3, 4, 5, 10, 20,..., 100. For each d, simulate random data with n = 100 data points x; E Rd, where each component has uniform distribution in the range [0, 1]. For each x; € {x₁,...,Xn}, find the minimum, average and maximum distances from it to all other vectors. Then calculate the difference between the maximum and minimum distances and the relative contrast value. Finally compute the averages of all these five measures over all xi. Plot the average values and their logarithms as functions of increasing d. (Place all distance values in one plot and the contrast values in another plot. Similarly, do not mix the original and the logarithmic values in the same plot. You should thus have in total four plots.) Repeat the experiment four times with the following Lp measures as the distance function: L0.5, L₁, L2, and L3. a) What happens to each one of the distance values and the difference between the maxi- mum and minimum distances when d increases? Explain, how you use the original and logarithmic plots to draw these conclusions. b) What happens to the relative contrast measure when d increases? How can this finding be explained based on the distance plots? c) What kind of an effect does the p value have on the bahavior of the relative contrast measure as a function of d? d) How do you interpret the results with respect to so called "curse of dimensionality"?

Answer rating: 100% (QA)

The image contains text describing a task for studying the behavior of extreme distances and relative contrast with increasing dimensionality of data The task involves conducting experiments with vary... View the full answer