>> 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
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>> x=rand(2,10000); %uniform in square >> ix=find(x(1,:)
HW2- linear density and squares. >> x=rand(2,10000); %uniform in square >> ix=find(x(1,:)> x=x(:,ix); >> plot(x(1,:),x(2,:),'*'); %scatter plot >> d=x(2,:)*2; %distribution of sphere %random point distances - PowerPoint PPT Presentation
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>> 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
• 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
• 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
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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.
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are needed to see this picture.
Variational Bayes
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True trajectory in state space
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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