Reference no: EM131053847

1. For the following cases, is it appropriate to model their data using HMMs? Explain the reasons.

-Monthly precipitation data in Rochester

-A dataset of indoor images

-Handwriting recognition

-Daily S&P 500 stock market price

2. Follow the graph to decompose the joint probability P(x₁, x₂, x₃, x₄, x₅, x₆).

3. A DNA sequence is a series of components {A, C, G, T}. Assume the hidden variable X takes 2 possible state values {s₁, s₂}, and the parameters of the HMM are as follows:

Transition probabilities: P(s₁|s₂) = 0.8, P(s₁|s₂) = 0.2

                                   P(s₂|s₁) = 0.2, P(s₂|s₂) = 0.8

Emission probabilities: P(A|s₁) = 0.4, P(C|s₁) = 0.1, P(G|s₁) = 0.4, P(T|s₁) = 0.1,

                                 P(A|s₂) = 0.1, P(C|s₂) = 0.4, P(G|s₂) = 0.1, P(T/s₂) = 0.4,

Initial probabilities: P(s₁) = 0.5, P(s₂) = 0.5

The observed sequence is Y = CGTCAG.

-Calculate P(Y/M). (Hint: refer to the lecture slides to calculate forward messages recursively, each forward message is a vector containing the probabilities of the two hidden states at that time step)

-Calculate P(x₃ = s₁|Y). (Hint: refer to the lecture slides for the recursive derivation of the forward-backward algorithm).