Homework Help is Here – Start Your Trial Now!
Math
Statistics
where K2 is the number of clusters you think are in the data, and ₁,...,K ERP are unknown cluster means. Fitting this model to the data you obtain MLES 1,..., K. (a) Simplifying your expressions as far as possible, give the condition under which this model-based clustering approach would tell you to assign X; to cluster k. Which other clustering method from the module is this most similar to? You may assume that all X₁,..., X, and p₁,..., K are distinct. (b) Give an expression for ÎK in this model, the maximised value of the likelihood. (c) Suppose that the model is correct, with K being the correct number of clusters, and that mink k k k is large. What will be the approximate value of -2 log(LK)? You do not need to give rigorous mathematical proofs, but you should explain your reasoning. - (d) Show that for any y₁,...,Ym ERP we have im 1 2 m i,j=1 m |y₁ - y₁|² = m =mΣlyi-yml², i=1

where K2 is the number of clusters you think are in the data, and ₁,...,K ERP are unknown cluster means. Fitting this model to the data you obtain MLES 1,..., K. (a) Simplifying your expressions as far as possible, give the condition under which this model-based clustering approach would tell you to assign X; to cluster k. Which other clustering method from the module is this most similar to? You may assume that all X₁,..., X, and p₁,..., K are distinct. (b) Give an expression for ÎK in this model, the maximised value of the likelihood. (c) Suppose that the model is correct, with K being the correct number of clusters, and that mink k k k is large. What will be the approximate value of -2 log(LK)? You do not need to give rigorous mathematical proofs, but you should explain your reasoning. - (d) Show that for any y₁,...,Ym ERP we have im 1 2 m i,j=1 m |y₁ - y₁|² = m =mΣlyi-yml², i=1

MATLAB: An Introduction with Applications
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
See similar textbooks
icon
Related questions
Question
Q3. Suppose that we have observations X₁,..., Xn in RP and wish to carry out a cluster analysis. Since you have reason to believe that the clusters are equally sized and each cluster is spherically symmetric, you consider the model Xi | Zik~ Np (μk, Ip), P(Z₁ = k) = 1/K, where K2 is the number of clusters you think are in the data, and ₁,...,K ERP are unknown cluster means. Fitting this model to the data you obtain MLEs 1,..., K. (a) Simplifying your expressions as far as possible, give the condition under which this model-based clustering approach would tell you to assign X, to cluster k. Which other clustering method from the module is this most similar to? You may assume that all X₁,..., Xn and p1,..., K are distinct. (b) Give an expression for ÎK in this model, the maximised value of the likelihood. (c) Suppose that the model is correct, with K being the correct number of clusters, and that minkkMkMk is large. What will be the approximate value of -2 log(LK)? You do not need to give rigorous mathematical proofs, but you should explain your reasoning. (d) Show that for any y₁,..., Ym ERP we have 2 -1 m Σlyi-y₁²: i,j=1 = m 2 m Σyi-m², m m where ym= Zi1 Yi. Starting from this equality and (c) and giving brief heuristic justification, using the AIC or BIC to choose K is similar to which other method for choosing K from the notes on clustering? i=1
Transcribed Image Text:Q3. Suppose that we have observations X₁,..., Xn in RP and wish to carry out a cluster analysis. Since you have reason to believe that the clusters are equally sized and each cluster is spherically symmetric, you consider the model Xi | Zik~ Np (μk, Ip), P(Z₁ = k) = 1/K, where K2 is the number of clusters you think are in the data, and ₁,...,K ERP are unknown cluster means. Fitting this model to the data you obtain MLEs 1,..., K. (a) Simplifying your expressions as far as possible, give the condition under which this model-based clustering approach would tell you to assign X, to cluster k. Which other clustering method from the module is this most similar to? You may assume that all X₁,..., Xn and p1,..., K are distinct. (b) Give an expression for ÎK in this model, the maximised value of the likelihood. (c) Suppose that the model is correct, with K being the correct number of clusters, and that minkkMkMk is large. What will be the approximate value of -2 log(LK)? You do not need to give rigorous mathematical proofs, but you should explain your reasoning. (d) Show that for any y₁,..., Ym ERP we have 2 -1 m Σlyi-y₁²: i,j=1 = m 2 m Σyi-m², m m where ym= Zi1 Yi. Starting from this equality and (c) and giving brief heuristic justification, using the AIC or BIC to choose K is similar to which other method for choosing K from the notes on clustering? i=1
Expert Solution
steps

Step by step

Solved in 2 steps

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
Recommended textbooks for you
MATLAB: An Introduction with Applications
MATLAB: An Introduction with Applications
Statistics
ISBN:
9781119256830
Author:
Amos Gilat
Publisher:
John Wiley & Sons Inc
Probability and Statistics for Engineering and th…
Probability and Statistics for Engineering and th…
Statistics
ISBN:
9781305251809
Author:
Jay L. Devore
Publisher:
Cengage Learning
Statistics for The Behavioral Sciences (MindTap C…
Statistics for The Behavioral Sciences (MindTap C…
Statistics
ISBN:
9781305504912
Author:
Frederick J Gravetter, Larry B. Wallnau
Publisher:
Cengage Learning
Elementary Statistics: Picturing the World (7th E…
Elementary Statistics: Picturing the World (7th E…
Statistics
ISBN:
9780134683416
Author:
Ron Larson, Betsy Farber
Publisher:
PEARSON
The Basic Practice of Statistics
The Basic Practice of Statistics
Statistics
ISBN:
9781319042578
Author:
David S. Moore, William I. Notz, Michael A. Fligner
Publisher:
W. H. Freeman
Introduction to the Practice of Statistics
Introduction to the Practice of Statistics
Statistics
ISBN:
9781319013387
Author:
David S. Moore, George P. McCabe, Bruce A. Craig
Publisher:
W. H. Freeman
SEE MORE TEXTBOOKS