>> x=rand(2,10000); %uniform in square>> ix=find(x(1,:)<x(2,:));% below diagonal: linear density>> x=x(:,ix);>> plot(x(1,:),x(2,:),'*'); %scatter plot>> d=x(2,:)*2; %distribution of sphere %random point distances >> d=sort(d);>> plot(d);>> k=d.^2;>> plot(k);

HW2- linear density and squares

>> mean (d)ans =1.3384>> median(d)ans =1.4239>> mean(k)ans =2.0085>> median(k)ans =2.0275

Rejection sampling:Y-coordinates have linear density function

Plot of cdf of d Plot of cdf of d^2

Statistical Data models,Non-parametrics,

Dynamics

Non-informative, proper and improper priors

• For real quantity bounded to interval,standard prior is uniform distribution

• For real quantity, unbounded, standard is uniform - but with what density?

• For real quantity on half-open interval, standard prior is f(s)=1/s - but integral diverges!

• Divergent priors are called improper -they can only be used with convergent likelihoods

Dirichlet Distribution-prior for discrete distribution

Mean of Dirichlet - Laplaces estimator

Occurence table probability

Occurence table probabilityUniform prior:

Non-parametric inference

• How to perform inference about a distribution without assuming a distribution family?

• A distribution over reals can be approximated by a piecewise uniform distribution a mixture of real distributions

• But how many parts? This is non-parametric inference

Non-parametric inferenceChange-points, Rao-Blackwell

• Given times for events (eg coal-mining disasters)Infer a piecewise constant intensity function(change-point problem)

• State is set of change-points with intensities inbetween• But how many pieces? This is non-parametric inference• MCMC: Given current state, propose change in segment

bounadry or intensity• But it is possible to integrate out intensities proposed

Probability ratio in MCMC

For a proposed merge of intervals j and j+1, with sizesproportional to (,1-), were the counts and obtained by tossing a ‘coin’ with success probability or not? Compute model probability ratio as in HW1.

Also, the total number of breakpoints has prior distributionPoisson with parameter (average) . Probability ratio in favor of split :

€

n j

€

n j+1

€

€

λ

Averging MCMC run, positionsand number of breakpoints

Averging MCMC run, positionswith uniform test data

Mixture of Normals

Mixture of Normalselimination of nuisance parameters

Mixture of Normalselimination of nuisance parameters

(integrate using normalization constant of Gaussian and Gamma distributions)

Matlab Mixture of Normals, MCMC (AutoClass method)

function [lh,lab,trlpost,trm,trstd,trlab,trct,nbounc]= mmnonu1(x,N,k,labi,NN);%[lh,lab,trlpost,trm,trstd,trlab,trct,nbounc]=% MMNONU1(x,N,k,labi,NN);%inputs% 1D MCMC mixture modelling,% x - 1D data column vector% N - MCMC iterations.% k - number of components%lab,labi - component labelling of data vector)% NN - thinning (optional)

Matlab Mixture of Normals, MCMC

function [lab,trlh,trm,trstd,trlab,trct,nbounc]= mmnonu1(x,N,k,labi,NN);%[lh,lab,trlpost,trm,trstd,trlab,trct,nbounc]=% MMNONU1(x,N,k,labi,NN);%outputs%trlh - thinned trace of log probability (optional)%trm - thinned trace of means vector (optional)%trstd - thinned vector of standard deviations (optional)%trlab - thinned trace of labels vector (size(x,1) by N/NN (optional)%trct - thinned trace of mixing proportions

Matlab Mixture of Normals, MCMC

N=10000;NN=100;x=[randn(100,1)-1;randn(100,1)*3;randn(100,1)+1];% 3 components synthetic datak=2; labi=ceil(rand(size(x))*2);[llhc,lab2,trl,trm,trstd,trlab,trct,nbounc]= … mmnonu1(x,N,k,labi,NN);[llhc2,lab2,trl2,trm2,trstd2,trlab2,trct2,nbounc]=… mmnonu1(x,N,k,lab2,NN); … (k=3, 4, 5)

Matlab Mixture of Normals, MCMC

The three componentsand the jointempirical distr

Matlab Mixture of Normals, MCMC Putting them

together makesthe identificationseem harder.

Matlab Mixture of Normals, MCMC

K=2:

std

mean

Matlab Mixture of Normals, MCMC

K=3:

std

mean

Burn inprogressing

Matlab Mixture of Normals, MCMC

K=3:

std

mean

Burnt in

Matlab Mixture of Normals, MCMC

K=4: Low prob

std

mean

No focus-No interpretationas 4 clusters

Matlab Mixture of Normals, MCMC

K=5: Low prob

std

mean

Matlab Mixture of Normals, MCMC

X sample: 1-100 : (-1 1) 101:200: (0 3) 201:300: (1 1)

Trace of state labels

Unsorted sample label trace sorted

Dynamic Systems,time series

• An abundance of linear prediction models exists

• For non-linear and Chaotic systems, method was developed in 1990:s (Santa Fe)

• Gershenfeld, Weigend: The Future of Time Series

• Online/offline: prediction/retrodiction

QuickTime™ and aTIFF (LZW) decompressor

are needed to see this picture.

Berry and Linoff have eloquently stated their preferences with the often quoted sentence:

"Neural networks are a good choice for most classification problemswhen the results of the model are more important than understandinghow the model works".

“Neural networks typically give the right answer”

Dynamic Systems and Taken’s Theorem

• Lag vectors (xi,x(i-1),…x(i-T), for all i,occupy a submanifold of E^T, if T is large enough

• This manifold is ‘diffeomorphic’ to original state space and can be used to create a good dynamic model

• Taken’s theorem assumes no noise and must be empirically verified.

Dynamic Systems and Taken’s Theorem

Santa Fe 1992 Competition

Unstable Laser

Intensive Care Unit Data,Apnea

Exchange rate Data

Synthetic series with drift

White Dwarf Star Data

Bach’s unfinished Fugue

Stereoscopic 3D view of statespace manifold, series A (Laser)The points seem to lie on asurface, which means that alag-vector of 3 gives goodprediction of the time series.

QuickTime™ and aTIFF (LZW) decompressor

are needed to see this picture.

Variational Bayes

QuickTime™ and a decompressor

are needed to see this picture.

True trajectory in state space

QuickTime™ and a decompressor

are needed to see this picture.

Reconstructed trajectory in inferred state space

Hidden Markov Models

• Given a sequence of discrete signals xi

• Is there a model likely to have produced xi from a sequence of states si of a Finite Markov Chain?

• P(.|s) - transition probability in state s

• S(.|s) - signal probability in state s

• Speech Recognition, Bioinformatics, …

Hidden Markov Models

function [Pn,Sn,stn,trP,trS,trst,tll]=… hmmsim(A,N,n,s,prop,Po,So,sto,NN);%[Pn,Sn,stn,trP,trS,trst]=HMMSIM(A,N,n,s,prop,Po,So,sto,NN);% Compute trace of posterior for hmm parameters% A - the sequence of signals% N - the length of trace% n - number of states in Markov chain% s - number of signal values % prop - proposal stepsize% optional inputs:% Po - starting transition matrix (each of n columns a discrete pdf% in n-vector% So - starting signal matrix (each of n columns a discrete pdf

Hidden Markov Models

function [Pn,Sn,stn,trP,trS,trst,tll]=… hmmsim(A,N,n,s,prop,Po,So,sto,NN);% in s-vector% sto - starting state sequence (congruent to vector A)% NN - thining of trace, default 10% outputs% Pn - last transition matrix in trace% Sn - last signal emission matrix% stn - last hidden state vector (congruent to A)% trP - trace of transition matrices% trS - trace of signal matrices% trace of hidden state vectors

Hidden Markov Models

Hidden Markov Models

Hidden Markov Models

Hidden Markov ModelsOver 100000 iterations, burnin is visible2 states, 2 signalsP-transition matrix S-signaling

Chapman Kolmogorov version of Bayes’ rule

f (λt |Dt) ∝ f(dt |λt)∫ f (λt |λt−1) f (λt−1 |Dt−1)dλt−1

Chapman Kolmogorov version of Bayes’ rule

f (λt |Dt) ∝ f(dt |λt)∫ f (λt |λt−1) f (λt−1 |Dt−1)dλt−1

Observation and video based particle filter tracking

Defence: tracking with heterogeneousobservations

Crowd analysis: tracking from video

Cycle in Particle filter

Importance (weighted)sampleResampled ordinary sample

Diffused sample

Weighted by likelihood

X- state Z - Observation

Time step cycle

Particle filter-general tracking