Markov PCA

Question 1

Markov Chain

Answer 1

• Finite number of states
• Transition matrix dictates probability of variable to change.
• “Ability to forget the past”
• x-order Markov Chain considers previous x states (including itself)

Question 2

Markov Chain 1-step

Answer 2

P(x_ = P(x1|x0) * P(x0)

Question 3

Hidden Markov Model

Answer 3

• Current state hidden
• Emits a symbol when at a state with probability
• Exact state never know, only guess from output
• Underlying states of HMM are in a Markov Chain

Question 4

HMM 1 step sequence

Answer 4

P(X) = E_s P(x|S)P(S)

Question 5

Maximum Likelihood

Answer 5

Estimate T by maximising probability of data P(D;T) as a function of matrix T.
Good estimate, asymptotically consistent and efficient.

Question 6

Max Likelihood equation

Answer 6

P(D;T)=P(x1) . nij Ti->j, n(i->j)

n(i->j) is number of transitions from i->j

Question 7

PCA Description

Answer 7

Reduce dimensions of the matrix by keeping only principle data. Key principle: Maintain as much variance in data as possible whilst reducing dimensionality.

Question 8

PCA Method:

Answer 8

Centre data around origin.
Calculate Covariance matric
Take eigenvectors of the largest eigenvalue in the covariance matrix to capture most data.
Take highest d eigvecs such that sum of d eigvecs is 0.95x total variance.

Question 9

Eigenvector EigenValue

Answer 9

EigVec of a matrix is a vector that when multiplied by original matrix comes out parallel to direction is went in (scale factor - the eigenvalue)