![Page 1: >> x=rand(2,10000); %uniform in square >> ix=find(x(1,:)> x=x(:,ix); >> plot(x(1,:),x(2,:),'*'); %scatter plot](https://reader030.vdocuments.site/reader030/viewer/2022020117/56649d555503460f94a3265f/html5/thumbnails/1.jpg)
>> 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
![Page 2: >> x=rand(2,10000); %uniform in square >> ix=find(x(1,:)> x=x(:,ix); >> plot(x(1,:),x(2,:),'*'); %scatter plot](https://reader030.vdocuments.site/reader030/viewer/2022020117/56649d555503460f94a3265f/html5/thumbnails/2.jpg)
>> mean (d)ans =1.3384>> median(d)ans =1.4239>> mean(k)ans =2.0085>> median(k)ans =2.0275
![Page 3: >> x=rand(2,10000); %uniform in square >> ix=find(x(1,:)> x=x(:,ix); >> plot(x(1,:),x(2,:),'*'); %scatter plot](https://reader030.vdocuments.site/reader030/viewer/2022020117/56649d555503460f94a3265f/html5/thumbnails/3.jpg)
Rejection sampling:Y-coordinates have linear density function
![Page 4: >> x=rand(2,10000); %uniform in square >> ix=find(x(1,:)> x=x(:,ix); >> plot(x(1,:),x(2,:),'*'); %scatter plot](https://reader030.vdocuments.site/reader030/viewer/2022020117/56649d555503460f94a3265f/html5/thumbnails/4.jpg)
Plot of cdf of d Plot of cdf of d^2
![Page 5: >> x=rand(2,10000); %uniform in square >> ix=find(x(1,:)> x=x(:,ix); >> plot(x(1,:),x(2,:),'*'); %scatter plot](https://reader030.vdocuments.site/reader030/viewer/2022020117/56649d555503460f94a3265f/html5/thumbnails/5.jpg)
Statistical Data models,Non-parametrics,
Dynamics
![Page 6: >> x=rand(2,10000); %uniform in square >> ix=find(x(1,:)> x=x(:,ix); >> plot(x(1,:),x(2,:),'*'); %scatter plot](https://reader030.vdocuments.site/reader030/viewer/2022020117/56649d555503460f94a3265f/html5/thumbnails/6.jpg)
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
![Page 7: >> x=rand(2,10000); %uniform in square >> ix=find(x(1,:)> x=x(:,ix); >> plot(x(1,:),x(2,:),'*'); %scatter plot](https://reader030.vdocuments.site/reader030/viewer/2022020117/56649d555503460f94a3265f/html5/thumbnails/7.jpg)
Dirichlet Distribution-prior for discrete distribution
![Page 8: >> x=rand(2,10000); %uniform in square >> ix=find(x(1,:)> x=x(:,ix); >> plot(x(1,:),x(2,:),'*'); %scatter plot](https://reader030.vdocuments.site/reader030/viewer/2022020117/56649d555503460f94a3265f/html5/thumbnails/8.jpg)
Mean of Dirichlet - Laplaces estimator
![Page 9: >> x=rand(2,10000); %uniform in square >> ix=find(x(1,:)> x=x(:,ix); >> plot(x(1,:),x(2,:),'*'); %scatter plot](https://reader030.vdocuments.site/reader030/viewer/2022020117/56649d555503460f94a3265f/html5/thumbnails/9.jpg)
Occurence table probability
![Page 10: >> x=rand(2,10000); %uniform in square >> ix=find(x(1,:)> x=x(:,ix); >> plot(x(1,:),x(2,:),'*'); %scatter plot](https://reader030.vdocuments.site/reader030/viewer/2022020117/56649d555503460f94a3265f/html5/thumbnails/10.jpg)
Occurence table probabilityUniform prior:
![Page 11: >> x=rand(2,10000); %uniform in square >> ix=find(x(1,:)> x=x(:,ix); >> plot(x(1,:),x(2,:),'*'); %scatter plot](https://reader030.vdocuments.site/reader030/viewer/2022020117/56649d555503460f94a3265f/html5/thumbnails/11.jpg)
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
![Page 12: >> x=rand(2,10000); %uniform in square >> ix=find(x(1,:)> x=x(:,ix); >> plot(x(1,:),x(2,:),'*'); %scatter plot](https://reader030.vdocuments.site/reader030/viewer/2022020117/56649d555503460f94a3265f/html5/thumbnails/12.jpg)
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
![Page 13: >> x=rand(2,10000); %uniform in square >> ix=find(x(1,:)> x=x(:,ix); >> plot(x(1,:),x(2,:),'*'); %scatter plot](https://reader030.vdocuments.site/reader030/viewer/2022020117/56649d555503460f94a3265f/html5/thumbnails/13.jpg)
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
![Page 14: >> x=rand(2,10000); %uniform in square >> ix=find(x(1,:)> x=x(:,ix); >> plot(x(1,:),x(2,:),'*'); %scatter plot](https://reader030.vdocuments.site/reader030/viewer/2022020117/56649d555503460f94a3265f/html5/thumbnails/14.jpg)
Averging MCMC run, positionsand number of breakpoints
![Page 15: >> x=rand(2,10000); %uniform in square >> ix=find(x(1,:)> x=x(:,ix); >> plot(x(1,:),x(2,:),'*'); %scatter plot](https://reader030.vdocuments.site/reader030/viewer/2022020117/56649d555503460f94a3265f/html5/thumbnails/15.jpg)
Averging MCMC run, positionswith uniform test data
![Page 16: >> x=rand(2,10000); %uniform in square >> ix=find(x(1,:)> x=x(:,ix); >> plot(x(1,:),x(2,:),'*'); %scatter plot](https://reader030.vdocuments.site/reader030/viewer/2022020117/56649d555503460f94a3265f/html5/thumbnails/16.jpg)
Mixture of Normals
![Page 17: >> x=rand(2,10000); %uniform in square >> ix=find(x(1,:)> x=x(:,ix); >> plot(x(1,:),x(2,:),'*'); %scatter plot](https://reader030.vdocuments.site/reader030/viewer/2022020117/56649d555503460f94a3265f/html5/thumbnails/17.jpg)
Mixture of Normalselimination of nuisance parameters
![Page 18: >> x=rand(2,10000); %uniform in square >> ix=find(x(1,:)> x=x(:,ix); >> plot(x(1,:),x(2,:),'*'); %scatter plot](https://reader030.vdocuments.site/reader030/viewer/2022020117/56649d555503460f94a3265f/html5/thumbnails/18.jpg)
Mixture of Normalselimination of nuisance parameters
(integrate using normalization constant of Gaussian and Gamma distributions)
![Page 19: >> x=rand(2,10000); %uniform in square >> ix=find(x(1,:)> x=x(:,ix); >> plot(x(1,:),x(2,:),'*'); %scatter plot](https://reader030.vdocuments.site/reader030/viewer/2022020117/56649d555503460f94a3265f/html5/thumbnails/19.jpg)
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)
![Page 20: >> x=rand(2,10000); %uniform in square >> ix=find(x(1,:)> x=x(:,ix); >> plot(x(1,:),x(2,:),'*'); %scatter plot](https://reader030.vdocuments.site/reader030/viewer/2022020117/56649d555503460f94a3265f/html5/thumbnails/20.jpg)
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
![Page 21: >> x=rand(2,10000); %uniform in square >> ix=find(x(1,:)> x=x(:,ix); >> plot(x(1,:),x(2,:),'*'); %scatter plot](https://reader030.vdocuments.site/reader030/viewer/2022020117/56649d555503460f94a3265f/html5/thumbnails/21.jpg)
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)
![Page 22: >> x=rand(2,10000); %uniform in square >> ix=find(x(1,:)> x=x(:,ix); >> plot(x(1,:),x(2,:),'*'); %scatter plot](https://reader030.vdocuments.site/reader030/viewer/2022020117/56649d555503460f94a3265f/html5/thumbnails/22.jpg)
Matlab Mixture of Normals, MCMC
The three componentsand the jointempirical distr
![Page 23: >> x=rand(2,10000); %uniform in square >> ix=find(x(1,:)> x=x(:,ix); >> plot(x(1,:),x(2,:),'*'); %scatter plot](https://reader030.vdocuments.site/reader030/viewer/2022020117/56649d555503460f94a3265f/html5/thumbnails/23.jpg)
Matlab Mixture of Normals, MCMC Putting them
together makesthe identificationseem harder.
![Page 24: >> x=rand(2,10000); %uniform in square >> ix=find(x(1,:)> x=x(:,ix); >> plot(x(1,:),x(2,:),'*'); %scatter plot](https://reader030.vdocuments.site/reader030/viewer/2022020117/56649d555503460f94a3265f/html5/thumbnails/24.jpg)
Matlab Mixture of Normals, MCMC
K=2:
std
mean
![Page 25: >> x=rand(2,10000); %uniform in square >> ix=find(x(1,:)> x=x(:,ix); >> plot(x(1,:),x(2,:),'*'); %scatter plot](https://reader030.vdocuments.site/reader030/viewer/2022020117/56649d555503460f94a3265f/html5/thumbnails/25.jpg)
Matlab Mixture of Normals, MCMC
K=3:
std
mean
Burn inprogressing
![Page 26: >> x=rand(2,10000); %uniform in square >> ix=find(x(1,:)> x=x(:,ix); >> plot(x(1,:),x(2,:),'*'); %scatter plot](https://reader030.vdocuments.site/reader030/viewer/2022020117/56649d555503460f94a3265f/html5/thumbnails/26.jpg)
Matlab Mixture of Normals, MCMC
K=3:
std
mean
Burnt in
![Page 27: >> x=rand(2,10000); %uniform in square >> ix=find(x(1,:)> x=x(:,ix); >> plot(x(1,:),x(2,:),'*'); %scatter plot](https://reader030.vdocuments.site/reader030/viewer/2022020117/56649d555503460f94a3265f/html5/thumbnails/27.jpg)
Matlab Mixture of Normals, MCMC
K=4: Low prob
std
mean
No focus-No interpretationas 4 clusters
![Page 28: >> x=rand(2,10000); %uniform in square >> ix=find(x(1,:)> x=x(:,ix); >> plot(x(1,:),x(2,:),'*'); %scatter plot](https://reader030.vdocuments.site/reader030/viewer/2022020117/56649d555503460f94a3265f/html5/thumbnails/28.jpg)
Matlab Mixture of Normals, MCMC
K=5: Low prob
std
mean
![Page 29: >> x=rand(2,10000); %uniform in square >> ix=find(x(1,:)> x=x(:,ix); >> plot(x(1,:),x(2,:),'*'); %scatter plot](https://reader030.vdocuments.site/reader030/viewer/2022020117/56649d555503460f94a3265f/html5/thumbnails/29.jpg)
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
![Page 30: >> x=rand(2,10000); %uniform in square >> ix=find(x(1,:)> x=x(:,ix); >> plot(x(1,:),x(2,:),'*'); %scatter plot](https://reader030.vdocuments.site/reader030/viewer/2022020117/56649d555503460f94a3265f/html5/thumbnails/30.jpg)
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
![Page 31: >> x=rand(2,10000); %uniform in square >> ix=find(x(1,:)> x=x(:,ix); >> plot(x(1,:),x(2,:),'*'); %scatter plot](https://reader030.vdocuments.site/reader030/viewer/2022020117/56649d555503460f94a3265f/html5/thumbnails/31.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
![Page 32: >> x=rand(2,10000); %uniform in square >> ix=find(x(1,:)> x=x(:,ix); >> plot(x(1,:),x(2,:),'*'); %scatter plot](https://reader030.vdocuments.site/reader030/viewer/2022020117/56649d555503460f94a3265f/html5/thumbnails/32.jpg)
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”
![Page 33: >> x=rand(2,10000); %uniform in square >> ix=find(x(1,:)> x=x(:,ix); >> plot(x(1,:),x(2,:),'*'); %scatter plot](https://reader030.vdocuments.site/reader030/viewer/2022020117/56649d555503460f94a3265f/html5/thumbnails/33.jpg)
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.
![Page 34: >> x=rand(2,10000); %uniform in square >> ix=find(x(1,:)> x=x(:,ix); >> plot(x(1,:),x(2,:),'*'); %scatter plot](https://reader030.vdocuments.site/reader030/viewer/2022020117/56649d555503460f94a3265f/html5/thumbnails/34.jpg)
Dynamic Systems and Taken’s Theorem
![Page 35: >> x=rand(2,10000); %uniform in square >> ix=find(x(1,:)> x=x(:,ix); >> plot(x(1,:),x(2,:),'*'); %scatter plot](https://reader030.vdocuments.site/reader030/viewer/2022020117/56649d555503460f94a3265f/html5/thumbnails/35.jpg)
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
![Page 36: >> x=rand(2,10000); %uniform in square >> ix=find(x(1,:)> x=x(:,ix); >> plot(x(1,:),x(2,:),'*'); %scatter plot](https://reader030.vdocuments.site/reader030/viewer/2022020117/56649d555503460f94a3265f/html5/thumbnails/36.jpg)
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.
![Page 37: >> x=rand(2,10000); %uniform in square >> ix=find(x(1,:)> x=x(:,ix); >> plot(x(1,:),x(2,:),'*'); %scatter plot](https://reader030.vdocuments.site/reader030/viewer/2022020117/56649d555503460f94a3265f/html5/thumbnails/37.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
![Page 38: >> x=rand(2,10000); %uniform in square >> ix=find(x(1,:)> x=x(:,ix); >> plot(x(1,:),x(2,:),'*'); %scatter plot](https://reader030.vdocuments.site/reader030/viewer/2022020117/56649d555503460f94a3265f/html5/thumbnails/38.jpg)
Variational Bayes
![Page 39: >> x=rand(2,10000); %uniform in square >> ix=find(x(1,:)> x=x(:,ix); >> plot(x(1,:),x(2,:),'*'); %scatter plot](https://reader030.vdocuments.site/reader030/viewer/2022020117/56649d555503460f94a3265f/html5/thumbnails/39.jpg)
QuickTime™ and a decompressor
are needed to see this picture.
True trajectory in state space
![Page 40: >> x=rand(2,10000); %uniform in square >> ix=find(x(1,:)> x=x(:,ix); >> plot(x(1,:),x(2,:),'*'); %scatter plot](https://reader030.vdocuments.site/reader030/viewer/2022020117/56649d555503460f94a3265f/html5/thumbnails/40.jpg)
QuickTime™ and a decompressor
are needed to see this picture.
Reconstructed trajectory in inferred state space
![Page 41: >> x=rand(2,10000); %uniform in square >> ix=find(x(1,:)> x=x(:,ix); >> plot(x(1,:),x(2,:),'*'); %scatter plot](https://reader030.vdocuments.site/reader030/viewer/2022020117/56649d555503460f94a3265f/html5/thumbnails/41.jpg)
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, …
![Page 42: >> x=rand(2,10000); %uniform in square >> ix=find(x(1,:)> x=x(:,ix); >> plot(x(1,:),x(2,:),'*'); %scatter plot](https://reader030.vdocuments.site/reader030/viewer/2022020117/56649d555503460f94a3265f/html5/thumbnails/42.jpg)
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
![Page 43: >> x=rand(2,10000); %uniform in square >> ix=find(x(1,:)> x=x(:,ix); >> plot(x(1,:),x(2,:),'*'); %scatter plot](https://reader030.vdocuments.site/reader030/viewer/2022020117/56649d555503460f94a3265f/html5/thumbnails/43.jpg)
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
![Page 44: >> x=rand(2,10000); %uniform in square >> ix=find(x(1,:)> x=x(:,ix); >> plot(x(1,:),x(2,:),'*'); %scatter plot](https://reader030.vdocuments.site/reader030/viewer/2022020117/56649d555503460f94a3265f/html5/thumbnails/44.jpg)
Hidden Markov Models
![Page 45: >> x=rand(2,10000); %uniform in square >> ix=find(x(1,:)> x=x(:,ix); >> plot(x(1,:),x(2,:),'*'); %scatter plot](https://reader030.vdocuments.site/reader030/viewer/2022020117/56649d555503460f94a3265f/html5/thumbnails/45.jpg)
Hidden Markov Models
![Page 46: >> x=rand(2,10000); %uniform in square >> ix=find(x(1,:)> x=x(:,ix); >> plot(x(1,:),x(2,:),'*'); %scatter plot](https://reader030.vdocuments.site/reader030/viewer/2022020117/56649d555503460f94a3265f/html5/thumbnails/46.jpg)
Hidden Markov Models
![Page 47: >> x=rand(2,10000); %uniform in square >> ix=find(x(1,:)> x=x(:,ix); >> plot(x(1,:),x(2,:),'*'); %scatter plot](https://reader030.vdocuments.site/reader030/viewer/2022020117/56649d555503460f94a3265f/html5/thumbnails/47.jpg)
Hidden Markov ModelsOver 100000 iterations, burnin is visible2 states, 2 signalsP-transition matrix S-signaling
![Page 48: >> x=rand(2,10000); %uniform in square >> ix=find(x(1,:)> x=x(:,ix); >> plot(x(1,:),x(2,:),'*'); %scatter plot](https://reader030.vdocuments.site/reader030/viewer/2022020117/56649d555503460f94a3265f/html5/thumbnails/48.jpg)
Chapman Kolmogorov version of Bayes’ rule
f (λt |Dt) ∝ f(dt |λt)∫ f (λt |λt−1) f (λt−1 |Dt−1)dλt−1
![Page 49: >> x=rand(2,10000); %uniform in square >> ix=find(x(1,:)> x=x(:,ix); >> plot(x(1,:),x(2,:),'*'); %scatter plot](https://reader030.vdocuments.site/reader030/viewer/2022020117/56649d555503460f94a3265f/html5/thumbnails/49.jpg)
Chapman Kolmogorov version of Bayes’ rule
f (λt |Dt) ∝ f(dt |λt)∫ f (λt |λt−1) f (λt−1 |Dt−1)dλt−1
![Page 50: >> x=rand(2,10000); %uniform in square >> ix=find(x(1,:)> x=x(:,ix); >> plot(x(1,:),x(2,:),'*'); %scatter plot](https://reader030.vdocuments.site/reader030/viewer/2022020117/56649d555503460f94a3265f/html5/thumbnails/50.jpg)
Observation and video based particle filter tracking
Defence: tracking with heterogeneousobservations
Crowd analysis: tracking from video
![Page 51: >> x=rand(2,10000); %uniform in square >> ix=find(x(1,:)> x=x(:,ix); >> plot(x(1,:),x(2,:),'*'); %scatter plot](https://reader030.vdocuments.site/reader030/viewer/2022020117/56649d555503460f94a3265f/html5/thumbnails/51.jpg)
Cycle in Particle filter
Importance (weighted)sampleResampled ordinary sample
Diffused sample
Weighted by likelihood
X- state Z - Observation
Time step cycle
![Page 52: >> x=rand(2,10000); %uniform in square >> ix=find(x(1,:)> x=x(:,ix); >> plot(x(1,:),x(2,:),'*'); %scatter plot](https://reader030.vdocuments.site/reader030/viewer/2022020117/56649d555503460f94a3265f/html5/thumbnails/52.jpg)
Particle filter-general tracking