Markov chains
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Markov chains. Probability distributions. Exercise. Use the Matlab function nchoosek(n,k) to implement a generic function BinomialPMF(k,n,p) for calculating the Binomial PMF with k successes in n trials with probability p.

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Markov chains

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Markov chains


Probability distributions


Exercise

Use the Matlab function nchoosek(n,k) to implement a generic function BinomialPMF(k,n,p) for calculating the Binomial PMF with k successes in n trials with probability p.

Use the barplot function to plot the pmf values (k=0..10) for Bin(10,0.5) as below:


Markov chains


Markov chains


Example: A very simple weather model

(modified from Wikipedia page ”Examples of Markov chains”)

The probabilities of weather conditions, modeled as either sunny=0 or rainy=1, given the weather on the preceding day, can be represented by a transition matrix


Weather model example

  • The matrix P represents the weather model in which a sunny day is 90% likely to be followed by another sunny day, and a rainy day is 50% likely to be followed by another rainy day. The columns can be labelled ”sunny” and ”rainy” respectively, and the rows can be labelled in the same order.

  • Pij is the probability that, if a given day is of type i, it will be followed by a day of type j.

  • Note that the rows of P add up to 1:

    sum(P’)


Weather model example


Exercise

  • If the the weather on the first day has 50% probability of being sunny or rainy, then what is the probability of the ninth day being sunny? What about the 30th day?


Weather model example


Eigenvectors / Eigenvalues


Eigenvectors / Eigenvalues

We can calculate eigenvalues and eigenvectors in Matlab using the built-in function eig. The default behavior is for right eigenvalues/eigenvectors, but left eigenvalues/eigenvectors are easily obtained by transposing the matrix.

We will be using the form (from Matlab help for eig):

[V,D] = eig(A) produces matrices of eigenvalues (D) and eigenvectors (V) of matrix A, so that A*V = V*D. Matrix D is the canonical form of A a diagonal matrix with A’s eigenvalues on the main diagonal. Matrix V is the modal matrix - its columns are the eigenvectors of A.

Note that Matlab always returns eigenvectors with norm 1.


Exercise

  • Use the eig function to find out, in the long term, what is the percentage of sunny days in the weather example.


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