mixture models are noted for their flexibility in modeling complex data and are widely used in the statistical literature ( see ) .such models provide a natural framework for the modeling of heterogeneity in a population .moreover , due to the large class of functions that can be approximated by mixture models , they are attractive for describing non - standard distributions and have been adopted in many areas , as genetics , ecology , computer science , economics , biostatistics and many others .for instance , as stated in , in genetics , location of quantitative traits on a chromosome and interpretation of microarrays are both related to mixtures , while , in computer science , spam filters and web context analysis start from a mixture assumption to distinguish spams from regular emails and group pages by topic , respectively .  statistical analysis of mixtures has not been straightforward and the bayesian paradigm has been particularly suited to their analysis . this framework allows the complicated structure of a mixture model to be decomposed into a set of simpler structures through the use of hidden or latent variables . according to , when the number of components is unknown , the bayesian paradigm is the only sensible approach to its estimation .then , the bayesian approach contributed to mixture models become increasingly popular in many areas . in real applications ,the number of components can arise important conclusions about the problem , so it has to be well specified or estimated , despite we usually have little theoretical guidance . on the other hand ,even if prior theory suggests a particular number of components we may not be able to reliably distinguish between some of the components . in some cases additional componentsmay simply reflect the presence of outliers in the data .  when the number of subpopulations is assumed known , markov chain monte carlo methods ( mcmc ) can be used for bayesian estimation of the subpopulation parameters .nevertheless , this method , as originally formulated , requires the posterior distribution to have a density with respect to some fixed measure .when the number of components is considered unknown , i.e. , the size of the parametric space is also a parameter , it appears a problem with variable dimension , thus mcmc can not be used alone in this case and more sophisticated methods are required to perform the bayesian analysis .one alternative in this case is the approach based on reversible jump mcmc ( rjmcmc ) , which was first proposed in and applied in univariate normal mixture models with unknown numbers of components by .the method basically consists of jumps between the parameter subspaces corresponding to different numbers of components in the mixture .  whilst mcmc provides a convenient way to draw inference from complicated statistical models ,there are still many , perhaps under appreciated , problems associated with the mcmc analysis of mixtures .the problems are mainly caused by the nonidentifiability of the components under symmetric priors , which leads to the so called label - switching in the mcmc output , discussed in .the term describes the invariance of the likelihood under relabelling of the mixture components , which can lead to the posterior distribution of the parameters being highly symmetric and multimodal , making it difficult to summarize .in particular , the usual practice of estimating parameters by their posterior mean , and summarizing joint posterior distributions by marginal distributions is often inappropriate .one frequent response to this problem is to remove the symmetry by using artificial identifiability constraints .this and other alternative classes of approaches to this problem are described by .the aim of this work is to review and discuss the application of mixture models , in particular the normal mixture models , to heterogeneous populations under the bayesian approach .the main purpose is to evaluate the model s performance in different settings of heterogeneity and considering the number of components known and unknown .we verify if the label - switching phenomena generally persists when the subpopulations are not well separated , then , if the more the population is heterogeneous , the more the model parameters should be better estimated .furhermore , we evaluate the label - switching assuming more informative prior distributions for the mixture weights .the paper is organized as follows .section [ model ] presents the general definition of a mixture model and discusses some aspects of the inference .a simulation study for assessing the estimation of model parameters under different levels of heterogeneity is presented in section [ simul_study ] .additionally , a prior sensitivity analysis of the mixture proportions is presented .it also discusses the model fit when the number of components is known and unknown . in section [ real_data ]the performance of the methodology is assessed through an application to a real data set .finally , section [ sec : concl ] presents some conclusions and suggestions for further research .the basic mixture model for independent scalar or vector observations , is a convex combination given by : where is a given parametric family of densities indexed by a scalar or a vector .in general , the objective of the analysis is to make inferences about the unknowns : the number of components , ; the parameters with being specific to component ; and the components weights , , , .let be the parametric vector of the model ( [ mixt ] ) .  for a random sample observed , the likelihood function of is given by : the likelihood function leads to terms , what brings a computational difficulty .a context in which the model ( [ mixt ] ) can arise and we are interested in this paper is when we postulate a heterogeneous population consisting of heterogeneous groups of sizes proportional to , from which a random sample is drawn .the label of the group from which each observation is drawn is unknown and it is natural to regard the group label , for the -th observation as a latent variable and rewrite ( [ mixt ] ) as the following hierarchical model : for ,   integrating out from ( [ mixt2 ] ) we return to model ( [ mixt ] ) .the formulation given by ( [ mixt2 ] ) is convenient for interpretation and calculation , decreasing the computational cost .a bayesian approach to inference requires the specification of a prior distribution for the parameters of the mixture model ( [ mixt ] ) .in particular , the prior elicitation is an important question .being fully non - informative and obtaining proper posterior distributions are not possible in a mixture content .an alternative on this case is to define weakly informative priors , which may or may not be data dependent.  the mixture model in ( [ mixt2 ] ) is invariant to permutation of the labels .some implications of this for likelihood analyses are discussed by .if we have no prior information that distinguishes between the components of the mixture , so the prior distribution is the same for all permutations of , then the posterior distribution will be similarly symmetric and , there will be symmetric modes of the posterior distribution .therefore , the component labels are mixed up and can not be distinguished from each other . as a result ,the marginal on the parameters for all components is identical and the posterior expectation for the parameters is identical too , and so estimating the parameters on the basis of the mcmc output is not straightforward .there are some suggested solutions to this problem , see for details .one common response to the label - switching problem is to impose an identifiability constraint on the parameter space .this breaks the symmetry of the prior and thus , of the posterior distribution of the parametric vector .for example , we can impose an ordering constraint on s , such as , if it is a scalar .however , for any given data set , many choices of identifiability constraint may be ineffective in removing the symmetry in the posterior distribution .as we are in a bayesian framework , the inference consists in obtain the posterior distribution of the parametric vector of model ( [ mixt2 ] ) . in general , this distribution can not be obtained in closed form .therefore , it is necessary to use some numerical approximation methods .one alternative , which is often used and is feasible to implement , is to generate samples from the marginal distributions of the parameters based on the mcmc algorithm .a comprehensive bayesian treatment using mcmc methods was presented in for finite mixture models .nevertheless , this method , as originally formulated , requires the posterior distribution to have a density with respect to some fixed measure .thus , in the mixture context , the method can only be applied when the number of components in the model ( [ mixt2 ] ) is considered known.  however , rarely the number is known , and fix it on an incorrect value can bring important consequences to the posterior distribution .other times , the target of the study is exactly the estimation of .the approach based on rjmcmc is an alternative in this case , which was proposed on this context by .it operates on the augmented parameter space , where the allocation variables are included as unknown parameters .the method basically consists of jumps between the parameter subspaces corresponding to different numbers of components in the mixture , after updating them .if the current model is a mixture with components , then it is usual to reduce the searching strategy to moves that either preserve the number of components , or lead to a mixture with or components .the idea is then to supplement each of the spaces with adequate artificial spaces in order to create a bijection between them , most often by augmenting the space of the smaller model .jumps are achieved by adding new components , deleting existing components , and splitting or merging these .these moves are randomly chosen and after being drawn makes necessary corresponding changes to .with respect to the label - switching problem , during mcmc computation , the sampler should switch from modes to modes between the iterations , however the failure to visit the identical posterior expectations reveals that the mcmc sampler has not converged and the posterior distribution surface is multimodal . if the value of is very large , it would be hard for the regular mcmc sampler to thoroughly and explore the high multimodality .thus , the mcmc samples might be trapped into local modes and it would require an enormous number of iterations to escape from it and the label switch would cause very poor estimates and the results might be very different from different runs of the mcmc .in this work we are particularly interested in the univariate normal case presented in , then in ( [ mixt ] ) becomes a vector with expectation and variance parameters .the model is stated below : for and ,   assuming that the parameters in are prior independent and identically distributed and that is unknown , the prior distribution is given by : for and , where generically denotes the symmetric dirichlet distribution with parameter .the symmetric dirichlet distributions are often used , since there typically is no prior knowledge favoring one component over another . since all elements of the parameter vectorhave the same value , the distribution alternatively is parametrized by a single scalar value . represents the gamma distribution with mean and variance and is the uniform distribution defined on the integers .moreover , for identifiability , we can use for example that the are in increasing numerical order , thus the joint prior distribution of is times the product of their marginal prior distributions .  in this paper , as treated in considered the bayesian estimation in the set - up where we do not have strong prior information on the mixture parameters . on the other hand , being fully non - informative and obtaining proper posterior distributions are not possible in a mixture content .an alternative on this case is to keep the simple independence and define weakly informative priors , which may or may not be data dependent .therefore , the default hyperparameter choices can be viewed with further details in .furthermore , in a normal mixture model the posterior distribution of the means for example could overlap , but the extent of the overlap depends on its separation and the sample size .when the means are well separated , labels of the realizations from the posterior by ordering their means generally coincide with the population ones . as the separation reduces, label - switching may occur .this problem can be also minimized by choosing to order other parameters of the mixture components , for example , the variance , weights or some combination of all three parameters .  in this workthe unique labeling will be achieved by imposing a restriction on .we will use that in which the are in increasing numerical order ; thus the joint prior distribution of the parameters is times the prior density , restricted to the set assess the convergence of the mcmc and rjmcmc estimation , we generated some samples under different settings .we obtained samples from the posterior distributions of the model parameters , supposing known and estimating it .the population estimates were then compared with the true values to evaluate the model s performance .the aim is to evaluate the performance of the normal mixture model varying the level of heterogeneity and the prior information elicited for the mixture proportions .furthermore , we also compared the results obtained under each simulation method considered , mcmc and rjmcmc .to check the convergence of the rjmcmc and mcmc estimations , we generated one sample with observations under two levels of heterogeneity , the first one with groups well separated , which we call as the more heterogeneous sample and the other with groups less well separated , which represents the more homogeneous one . on both scenarios we fixed , and . with this value fixed for we expected to have groups with a reasonable number of observations , thus we did not consider scenarios with groups outliers .the heterogeneous scenario was obtained fixing and the homogeneous fixing .figure [ data1 ] presents the distribution of both data generated .the aim of this study is to verify if the level of heterogeneity of the population affects the results , mainly the label - switching problem .cc  the prior distribution considered are described in ( [ prior ] ) , and we elicited the prior for and using the same idea of weakly informative prior suggested by .first , we assumed unknown and in its prior distribution presented in ( [ prior ] ) we assumed , thus rjmcmc was used to obtain samples from the posterior distribution .we also did a brief prior sensitivity analysis assuming two values on the dirichlet prior distribution for each data : for the heterogeneous case and for the homogeneous case .to assume is equivalent to a uniform distribution over all points in its support .on the other hand , the parameter value above gives some information that all sample proportions in subpopulations are similar to each other .for the rjmcmc simulations , we generated , respectively for the homogeneous sample and the heterogeneous one , 350,000 and 70,000 samples from the posterior distribution , discarded the first 10,000 and 20,000 , and then thinned the chain by taking every 10th sample value .figure [ kpost1 ] displays the histogram with the posterior densities of for some values of .it shoud be noted that for the heterogeneous case the parameter is well estimated , but when the estimate is more accurate . on the other hand , is underestimated when assuming with the homogeneous sample .the same happen with or any value less than .in this case , when the value of is well estimated .cc +  figure [ cadeia1 ] shows the trace plot with the posterior distribution of parameters conditional on the posterior samples , whose estimated value of is the one with higher posterior probability . here, we also considered the value of known and fixed it on the true value used to generate the samples , so mcmc was also used to generate samples from the posterior distribution . for the mcmc simulations , we generated 70,000 samples from the posterior distribution , discarded the first 20,000 , and then thinned the chain by taking every 10th sample value , for both data generated .all the results were obtained for each scenario and value of considered .the black trace represents the posterior density fixing , the blue trace when in both scenarios and is represented by the red trace in the homogeneous case .the gray line represents the true value of each .note that on the homogeneous case , when rjmcmc is used , there is only a red trace for , that is because the posterior for favors the value 5 , only when .when analyzing figure [ cadeia1 ] it can be seen the effects of label - switching in the sampled values of the component means for many cases , even in the heterogeneous case .however this behavior improves when giving some prior information about the mixture proportions .it is also possible to observe that on the homogeneous case to obtain better results it is needed to increase even more the value of , this is , to give more prior information that the proportion observed in groups are similar .l +  +  +  +  the results obtained indicate that the rjmcmc and mcmc chains have converged for some cases , but for others the label - switching phenomena appears significantly , and so estimating the means on the basis of the rjmcmc and mcmc output is not straightforward .however , as the value of increases this behavior improves .if the number of iterations increases and then , the lag of the chain , the convergence may also improve , but it would require a high computational effort , thus we suggest here the carefully elicitation of the prior distribution to have better estimates .almost all the mean parameters are well estimated when and in the heterogeneous and homogeneous case , respectively .traces and density estimates for the mixture proportions and variances present this same behavior .in general , mcmc and rjmcmc present a similar behavior , mainly for the heterogeneous sample generated .a more interesting comparison between both approaches is presented in the next subsection .figure [ summary1 ] shows summary statistics of the posterior distributions of the mean parameters after reaching the supposed convergence for each of the scenarios and prior assumed , when assuming unknown .the crosses represent the true value , the lines the 95% credibility interval and the points are the posterior mean .also , the results in black are obtained assuming , the blue one when and the red when . in almost all the casesthe intervals contain the true value .it is possible to observe the impact of the label - switching , which difficults the parameters estimation , but also the improvement of the results when assuming a more informative prior to .cc  therefore , we conclude that in those cases considered here the identifiability problem can be minimized under more informative priors and it was not necessary to use other alternative class of approaches to deal with the identifiability problem , as those described in .the prior distribution of seems to have strong impact on the posterior distribution , improving the results , even in a homogeneous case .furthermore , as the degree of heterogeneity increases , the mixture model performs considerably better even under less prior information .the same conclusions were attained when estimating the value of or considering it known .additionally , figure [ pred1 ] shows the predictive densities for the two data sets generated , for all the prior distributions considered for , represented by the solid ( ) , dashed ( ) and dotted ( ) lines , respectively .the predictive densities in black are those obtained when the value of is estimated , so rjmcmc was used , and the red ones are obtained when the value of is fixed on the true value , so mcmc was used . the densities obtained under rjmcmc are conditional on the posterior samples whose sampled value of is equal to the value with high posterior probability among all the samples .in contrast with the above results , an estimate of the predictive density based on the rjmcmc and mcmc outputs is unaffected by the label - switching problem , since it does not depend on how the components are labeled . it should be noted that the predictive density is better estimated on the heterogeneous sample than the homogeneous one and that the prior distribution does not affect the estimates .moreover , the results obtained in the estimation considering unknown and fixed are very similar to each other .cc   to examine the performance of the bayes estimators obtained under each simulation method , we generated two artificial samples of size , fixing into two different values , and , in order to also evaluate the results when varying the value of . then , we obtained samples from the posterior distribution of the parametric vector , supposing known ( mcmc ) and estimating it ( rjmcmc ) . in the mcmc simulationwe particularly fixed for each case in three different values : we assumed it and for the first sample and and for the second one .we assumed here the same prior distribution used in section [ sens ] .thus , we are interested in evaluate the method s performance when we fix on its true value , on a smaller and a greater value than the true one and when it is estimated . for the rjmcmc and mcmc simulations , we generated 350,000 samples from the posterior distribution , discarded the first 10,000 , then thinned the chain by taking every 50th sample value , and the convergence was supposed achieved .figure [ preditivacomp ] presents the posterior distribution of obtained under the rjmcmc simulation and the predictive densities obtained for each sample generated .it should be noted that is well estimated and all predictive densities are very similar , except when we fixed in a value lower than the true one .moreover , fixing in a greater value than the truth does not affect the results .+  table [ dic ] presents the deviance information criterion ( dic ) , introduced by , for each approach considered in this study .dic evaluates the goodness of fit of the model , thus the model with the smallest dic should be the one that would best fit .the model with known seems to fit the data better than its counterparts .however , the results are very similar , even when is estimated , increasing the size of the parametric vector , except when is fixed in a smaller value than the truth .[ cols="^,^,^,^,^,^,^,^,^,^,^",options="header " , ]   the model fit estimating and fixing it in results a dic of 3692.59 and 3693.99 , respectively .thus , as dic increases with the number of parameters , it is expected that rjmcmc presents a higher dic .however , since both dics were very similar , it is possible to conclude that both methods are efficient in this case .we have considered the problem of the fit of mixture models for heterogeneous populations under different levels of heterogeneity .we have discussed the improvement of the convergence when assigning a weakly informative prior distribution for the mixture proportions even for more homogeneous populations .finally , we have also evaluated the inference for the model when the number of components is unknown ( rjmcmc ) and when it is fixed in a known value ( mcmc ) .we have concluded that when the number of mixture components is unknown , the rjmcmc is a feasible alternative , achieving similar results when this number is fixed in the true value .nevertheless , it requires slightly bigger computational effort than mcmc . on the other hand ,if we are not interesting in estimating this number , fixing it in a smaller value than the truth will generate poor estimates , however , similar results are obtained when fixing it in the true value or greater than this .spiegelhalter , d. j. , best , n. g. , carlin , b. p. and van der linde , a. ( 2002 ) bayesian measures of model complexity and fit .journal of the royal statistical society : series b ( statistical methodology ) , * 64 * , 583 - 639 .

Abstract: mixture models provide a flexible representation of heterogeneity in a finite number of latent classes . from the bayesian point of view , markov chain monte carlo methods provide a current way to draw inference from these models . in particular , when the number of subpopulations is considered unknown , more sophisticated methods are required to perform the bayesian analysis . the reversible jump markov chain monte carlo is an alternative method for computing the posterior distribution by simulation in this case . some problems associated with the bayesian analysis of these class of models are frequent , as the so - called label - switching " . however , as the level of the heterogeneity in the population increases , it is expected that these problems become less frequent and the model s performance improves . thus , the aim of this work is to evaluate the normal mixture model fit using simulated data under different settings of heterogeneity and prior information about the mixture proportions . a simulation study was also carried out to evaluate the model s performance considering the number of components known and estimating it . finally , the model is applied to a real data set that consists of antibody levels of cytomegalovirus in individuals .  * keywords * : identifiability , sensitivity analysis , subpopulations , frequentist properties , nhanes


in the last thirty years , a large number of diverse visual tracking methods has been proposed  .the methods differ in the formulation of the problem , assumptions made about the observed motion , in optimization techniques , the features used , in the processing speed , and in the application domain .some methods focus on specific challenges like tracking of articulated or deformable objects  , occlusion handling  , abrupt motion  or long - term tracking  .three observations motivate the presented research .first , most trackers perform poorly if run outside the scenario they were designed for .second , some trackers make different and complementary assumptions and their failures are not highly correlated ( called complementary trackers in the paper ) . andfinally , even fairly complex well performing trackers run at frame rate or faster on standard hardware , opening the possibility for multiple trackers to run concurrently and yet in or near real - time .we propose a novel methodology that exploits a hidden markov model ( hmm ) for fusion of non - uniform observables and pose prediction of multiple complementary trackers using an on - line learned high - precision detector .the non - uniform observables , in this sense , means that each tracker can produce its own type of confidence estimate which may not be directly comparable between each other .  the hmm , trained in an unsupervised manner , estimates the state of the trackers failed , operates correctly and outputs the pose of the tracked object taking into account the past performance and observations of the trackers and the detector .the hmm treats the detector output as correct if it is not in contradiction with its current most probable state in which the majority of trackers are correct .this limits the cases where the hmm would be wrongly updated by a false detection . for the potentially many frames where reliable detector output is not available, it combines the trackers .the detector is trained on the first image and interacts with the learning of the hmm by partially annotating the sequence of hmm states in the time of verified detections .the recall of the detector is not critical but it affects the learning rate of the hmm and the long - term properties of the hmmtxd method , i.e. its ability to reinitialize trackers after occlusions or object disappearance .* related work . *the most closely related approaches include santner et al . , where three tracking methods with different rates of appearance adaptation are combined to prevent drift due to incorrect model updates .the approach uses simple , hard - coded rules for tracker selection .kalal et al . combine a tracking - by - detection method with a short - term tracker that generates so called p - n events to learn new object appearance .the output is defined either by the detector or the tracker based on visual similarity to the learned object model .both these methods employ pre - defined rules to make decisions about object pose and use one type of measurement , a certain form of similarity between the object and the estimated location . in contrary, hmmtxd learns continuously and causally the performance statistics of individual parts of the systems and fuses multiple `` confidence '' measurements in the form of probability densities of observables in the hmm .zhang et al . use a pool of multiple classifiers learned from different time spans and choose the one that maximize an entropy - based cost function .this method addresses the problem of model drifting due to wrong model updates , but the failure modes inherent to the classifier itself remains the same .this is unlike the proposed method which allows to combine diverse tracking methods with different inherent failure modes and with different learning strategies to balance their weaknesses .similarly to the proposed method , wang et al . and bailer et at . fuse different out - of - the box tracking methods .bailer et al .combine offline the _ outputs _ of multiple tracking algorithms .there is no interaction between trackers , which for instance implies that the method avoids failure only if one method correctly tracks the whole sequence .wang et al .use a factorial hidden markov model and a bayesian approach .the state space of their factorial hmm is the set of potential object positions , therefore it is very large .the model contains a probability description of the object motion based on a particle filter .trackers interact by reinitializing those with low reliability to the pose of the most confident one .the yuan et al . using hmm in the same setup , but rather than merging multiple tracking method , they focus on modeling the temporal change of the target appearance in the hmm framework by introducing a observational dependencies . in contrast , the hmmtxd method is online with tracker interaction via a high precision object detector that supervises tracker reinitializations which happen on the fly .the appearance modeling is performed inside of each tracker and the hmmtxd capture the relation of the confidence provided by tracker and its performance , validated by the object detector , by the observable distributions . moreover , the hmmtxd confidence estimation is motion - model free and this prevents biases towards support of trackers with a particular motion model .yoon et al . combines multiple trackers in a particle filter framework .this approach models observables and transition behavior of individual trackers , but the trackers are self - adapting which makes it prone to wrong model updates .the adaptation of hmmtxd model is supervised by a detector method set to a specific mode of operation near precision alleviating the incorrect update problem .the contributions of the paper are : a novel method for fusion of multiple trackers based on hmms using non - uniform observables , a simple , and so far unused , unsupervised method for hmms training in the context of tracking , tunable feature - based detector with very low false positive rate , and the creation of a tracking system that shows state - of - the - art performance .hmmtxd uses a hidden markov model ( hmm ) to integrate pose and observational confidence of different trackers and a detector , and updates its own confidence estimates that in turn define the pose that it outputs . in the hmm , each tracker is modeled as working correctly ( 1 ) or incorrectly ( 0 ) .the hmm poses no constraints on the definition of tracker correctness , we adopted target overlap above a threshold .having at our disposal trackers , the set of all possible states is and the initial state .note that the trackers are not assumed to be independent , because an independence of tracker correctness is not a realistic assumption .for example , if the tracking problem is relatively easy , all trackers tend to be correct and in the case of occlusion all tend to be incorrect ( see the analysis in  ) .the number of states grows exponentially with the number of trackers .however , we do not consider this a significant issue due to real - time requirements of tracking , the need to combine more than a small number of trackers , say , is unlikely .the hmmtxd method overview is illustrated in fig .[ fig : method ] .each tracker provides an estimate of the object pose ( ) and a vector of observables ( ) , which may contain a similarity measure to some model ( such as normalized cross - correlation to the initial image patch , distance of template and current histograms at given position , etc . ) or any other estimates of the tracker performance .the serve as observables to relate the tracker current confidence to the hmm .each individual observable depends only on one particular tracker and its correctness , hence , they are assumed to be conditionally independent conditioned on the state of the hmm ( which encodes the tracker correctness ) .   and the detector outputs either the verified object pose or nothing .if detector fires , hmm is updated and trackers are reinitialized and the final output is the , otherwise , hmm estimate the most probable state and outputs an average bounding box of trackers that are correct in the estimated state .,scaledwidth=85.0% ]  in general , there are no constraints on observable values , however , in the proposed hmm the observable values are required to be normalized to the interval .the observables are modeled as beta - distributed random variables ( eq .[ eq : beta ] ) and its parameters are estimated online .the beta distribution was chosen for its versatility , where practically any kind of unimodal random variable on can be modeled by the beta distribution , i.e. for any choice of any lower and upper quantiles , a beta distribution exists satisfying the given quantile constraint  .learning the parameters of the beta distributions online is crucial for the adaptability to particular tracking scenes , where the observable values from a different trackers may be biased due to scene properties , or to adapt to a different types of observables of trackers and their correlations to the real tracker performance .for example , taking correlation with the initial target patch as an observable for one tracker and color histogram distance to a initial target for a second tracker , the correlation between their values and the performance of the tracker may differ depending on object rigidity and color distribution of object and background.  the hmm is parameterized by the pair , where are the probabilities of state transition and are the beta distributions of observables with shape parameters and density defined for     since the goal is real - time tracking without any specific pre - processing , learning of hmm parameters has to be done online . towards this goal ,the object detector , which is set to operating mode with low false positive rate , is utilized to partially annotate the sequence of hidden states .in contrast to classical hmm , where only a sequence of observations is available , we are in a semi - supervised setting and have a time sequence of observed states of a markov chain , and markov chain starting again in state , all trackers correct , at any time , since there are reinitialized to common object pose .this information is provided by the detector , where is a sequence of detection times .the hmm parameters are learned by a modified baum - welch algorithm run on the observations and the annotated sequence of states .the partial annotation and hmm parameter estimation update is done strictly online .the output of the hmmtxd is an average bounding box of correct trackers of the current most probable state .for the forward - backward procedure  for hmm is used to calculate probability of each state at time ( see eq .[ eq : a1]-[eq : a7 ] ) and the state is the state for which    is maximal .this equation is computed using eq .[ eq : app_chain ] and maximized w.r.t . for the eq .[ eq : state - prob ] holds with .this ensures that the algorithm outputs a pose for each frame which is required by most benchmark protocols .illustration of the tracking process and hmm insight is shown in fig .[ fig : insight ] .theoretically the parameters of hmm could be updated after each frame . however , in our implementation , learning takes place only at frames where the detector positively detects the object , i.e. the sequence of states starting and ending with observed state inferred by the detector .the detector is used only if the detection pose is not in contradiction with the pose of the current most probable state in which the majority of trackers are correct .this ensure that even when the detector makes a mistake , the hmm is not wrongly updated .when we are in the state that one or none of the trackers are correct , the detector get precedence .   with colorencoded correct trackers ( 1 ) are displayed .the final output is defined by the state and the bounding box is highlighted by white color .best viewed zoomed in color.,scaledwidth=85.0% ]for learning of the parameters of the hmm a mle inference is employed , however maximizing the likelihood function is a complicated task that can not be solved analytically . in the proposed method ,the baum - welch algorithm  is adapted .the baum - welch algorithm is a widespread iterative procedure for estimating parameters of hmm where each iteration increases the likelihood function but , in general , the convergence to the global maximum is not guaranteed .the baum - welch algorithm is in fact an application of the em ( expectation - maximization ) algorithm  .let us assume the hmm with possible states , the matrix of state transition probabilities , the vector of initial state probabilities , the initial state , a sequence of observations and the system of conditional probability densities of observations conditioned on     where are random variables representing the state at time , and is denoting the parameter set of the model .let us denote , \label{eg : q_baum}\ ] ] where is a set of all possible t - tuples of states and + is one sequence of states . according to theorem 2.1 .in  and the equality holds if and only if . the classical baum - welch algorithm repeats the following steps until convergence :  1 .compute 2 .set .we propose the modified baum - welch algorithm that exploits the partially annotated sequence of states , where the known states are inferred from the detector output .let be a sequence of detection times , be observed states of markov chain , marked by the detector , and for .so the sequence of observations of the hmm is divided into independent subsequences , each with a fixed initial state , the first subsequences with a known terminal state defined by the detector and the last subsequence with an unknown terminal state .the following equations are obtained by employing the modification to the baum - welch algorithm , =\sum_{t=1}^{t-1}\log a_{\mathfrak{s}_{t}\mathfrak{s}_{t+1 } } + \sum_{t=1}^{t } \log f_{\mathfrak{s}_{t}}(x_{t}),\ ] ]   the maximization of the can be separated to maximization w.r.t .transition probability matrix by maximizing the first term and w.r.t .observable densities by maximizing the second term .  the maximization of eq .[ eq : q ] w.r.t . constrained by is obtained by re - estimating the parameters as follows :      this equation is computed using modified forward and backward variables of the baum - welch algorithm to reflect the partially annotated states . for the exact derivation of formulas for computation of see the [ app : fb ] .   the maximization of eq .[ eq : q ] w.r.t . depends on assumptions on the system of probability densities .it is usually assumed ( e.g. in  ) that is a system of probability distributions of the same type and differ only in their parameters . in the hmmtxdthe -dimensional observed random variables + are assumed conditionally independent and to have the beta - distribution , so are products of one - dimensional beta distributions with parameters of shape . in this casemaximization of the second term of the eq .[ eq : q ] is an iterative procedure using inverse digamma function which is very computationally expensive  .we propose to estimate the shape parameters of the beta distributions with a generalized method of moments .the classical method of moments is based on the fact that sample moments of independent observations converge to its theoretical ones due to the law of large numbers for independent random variables . in the hmmtxd observations are not independent .the generalized method of moments is based on the fact that is a sequence of martingale differences for which the law of large numbers also holds .using the generalized method of moments gives estimates of the parameters of shape    and where and   let us denote the system of probability densities with re - estimated parameters as .the generalized method of moments is described in detail in the [ app : moments ] .the complete modified baum - welch algorithm is summarized in alg .[ alg : mbw ] , where after each iteration and we repeat these steps until convergence .note that is a maximum likelihood estimate of therefore always increases ( shown in  ) but is estimated by the method of moments so the test on likelihood increase is required ( if statement in the alg .[ alg : mbw ] ) .in fact , this algorithm structure match to the generalized em algorithm ( gem ) introduced in  .the requirements for the detector are : adjustable operation mode ( e.g. set for high precision but possibly low recall ) , ( near ) real - time performance and the ability to model pose transformations up to at least similarity ( translation , rotation , isotropic scaling ) .basically , any detector - like approach can be used and it may vary based on application .we choose to adapt a feature - based detector which has been shown to perform well in image retrieval , object detection and object tracking  tasks .there are many possible combinations of features and their descriptors with different advantages and drawbacks .we exploit multiple feature types : specifically , hessian keypoints with the sift  descriptor , orb  with brisk and orb with freak  .each feature type is handled separately , up to the point where point correspondences are established .a weight is assigned to each feature type and is set to be inversely proportional to the number of features on the reference template , to balance the disparity in individual feature numbers .the detector works as follows . in the initialization step ,features are extracted from the inside and the outside of the region specifying the tracked object .descriptors of the features outside of the region are stored as the background model .usually , the input region is not occupied by the target ; therefore , fast color segmentation  attempts to delineate the object more precisely than the axis - aligned bounding box to remove the features that are most likely not on the target .the step is not critical for the function of the detector , since the bounding box is a fall - back option .we assume that at least of the bounding box is filled with pixels that belong to the target , if the segmentation fails ( returns a region containing less than of area of the bounding box ) , all features in the initial bounding box are used .additionally , for each target feature , we use a normal distribution to model the similarity of the feature to other features .the parameters and are estimated in the first frame by randomly sampling 100 features , other than , and computing distances to the feature , from which the mean and variation are computed .this allows defining the quality of correspondence matches in a probabilistic manner for each feature , thus getting rid of global static threshold for the acceptable correspondence distance .  in the detection phase, features are detected and described in the whole image . for each feature from the image the nearest neighbour ( in euclidean space or in hamming distance metric space , depending on the feature type ) feature from the background model and the nearest neighbour feature from the foreground model are computed .a tentative correspondence is formed if the feature match passes the second nearest neighbour test and a probability that the correspondence distance belongs to the outlier distribution is lower than a predefined significance set to .so    where is a c.d.f . of the normal distribution with parameters and of a distance distribution of features not corresponding to .the significance corresponds to the threshold . finally , ransac estimates the target current pose using a sum of weighted inliers as a cost function for model support } , \ ] ] which takes into account the different numbers of features per feature type on the target .the decision whether the detected pose is considered correct depends on the number of weighted inliers that supports the ransac - selected transformation and it controls the trade - of between precision and recall of the method .this threshold is automatically computed in the first frame of the sequence as . the threshold interval ( 5,10 ) and the feature multiplier ( 0.03 )were set experimentally to have the false positive rate close to zero for the most of the testing sequences .furthermore , majority voting is used to verify that the detection is not in contradiction to the estimated hmm state , i.e. if we are in the state where two or more ( majority ) trackers are correct and the detector is not consistent with them , the detection is not used .this mitigates the false positive detections , therefore hmm updates , when the trackers works correctly .the true and false positives for 77 sequences are shown in fig .[ fig : det_perf ] , where the detector works on almost all sequences with zero false positive rate ( average false positive rate on the dataset ) and recall rate .the failure cases of this feature - based detector are mostly caused by the imprecise initial bounding box , which contains large portion of structured background ( i.e. background where the detector finds features ) and due to the presence of similar object in the scene , e.g. sequences _hand2 , basketball , singer2_.to demonstrate the performance of the proposed framework , a pair and a triplet of published short - term trackers were plugged into the framework to show the performance gain by combination of a different number of trackers .as bailer et al . pointed out , not all trackers when combined can improve the overall performance ( i.e. adding tracking method with similar failure mode will not benefit ) .we therefore choose methods that have a different designs and work with different assumptions ( e.g. rigid global motion vs. color mean - shift estimation vs. maximum correlation response ) .these trackers are the flock of trackers ( fot )  , scale adaptive mean - shift tracker ( asms )  and kernelized correlation filters ( kcf )  .this choice shows that superior performance can be achieved by using simple , fast trackers ( above 100fps ) that may not represent the state - of - the - art .the trackers can be arbitrarily replaced depending on the user application or requirements .the flock of trackers ( fot )  evenly covers the object with patches and establishes frame - to - frame correspondence by the lucas - kanade method  .the global motion of the target is estimated by ransac .the second tracker is a scale adaptive mean - shift tracker ( asms )  where the object pose is estimated by minimizing the distance between rgb histograms of the reference and the candidate bounding box .the kcf  tracker learns a correlation filter by ridge regression to have high response to target object and low response on background .the correlation is done in the fourier domain which is very efficient .these three trackers have been selected since they are complementary by design .fot enforces a global motion constrain and works best for rigid object with texture . on the other hand, asms does not enforce object rigidity and is well suited for articulated or deformable objects assuming their color distribution is discriminative w.r.t .the background .kcf can be viewed as a tracking - by detection approach using sliding window like scanning .  for each tracker position ,two global observable measurements are computed , namely the hellinger distance between the target template histogram and the histogram of the current position and normalized cross - correlation score of the current patch and the target model patch .these target models are initialized in the first frame and then updated exponentially with factor of during each positive detection of the detector part .additionally , each tracker produces its own estimate of performance . for fotit is the number of predicted correspondences ( for details please see  ) that support the global model . for asmsit is the hellinger distance between its histogram model and current neighbourhood background ( i.e. color similarity of the object and background ) and for kcf it is a correlation response of the tracking procedure .0.49   0.49the hmmtxd was compared with state - of - the - art methods on two standard benchmarks and on a dataset tv77 containing 77 public video sequences collected from tracking - related publications .the dataset exhibits wider diversity of content and variability of conditions than the benchmarks .parameters of the method were fixed for all the experiments . in the hmm ,the initial beta distribution shape parameters were set to for correct state ( 1 ) and for fail state ( 0 ) for all observations and the transition matrix was set to prefer staying in the current state .the transition matrix has on diagonal , in fist column , in last column , in last row and otherwise .the matrix is normalized so that rows sum to one .states in the matrix are binary encoded starting from the left column which corresponds to the state . the number of iteration for baum - welch alg . was set to  .  the processing speed on the vot2015 dataset is ( in frames per second ) minimum 1.03 , maximum 33.72 and average 10.83 measured on a standard notebook with intel core - i7 processor .this speed is mostly affected by the number of features detected in the images which correlates to the resolution of the image ( in the dataset the range is from 320x180 to 1280x720 ) .first , we compare the performance of individual parts of the hmmtxd framework ( i.e. kcf , asms , fot trackers ) and their combination via hmm as proposed in this paper .two variants of hmmtxd are evaluated 2-hmmtxd refers to combination of fot and asms trackers and the 3-hmmtxd to combination of all mentioned trackers .we also show the benefit of the proposed detector when simply combined with the individual trackers in such way that if detector fires the tracker is reinitialized .the figure [ fig : bench_parts ] shows the benefit gained from the detector and further consistent improvement achieved by the combination of the trackers .more detailed per sequence analysis on the tv77 dataset ( fig .[ fig : hmm_parts_single ] and fig .[ fig : hmm_parts_det ] ) shows more clearly the efficiency of learning tracker performance online . in almost all sequencesthe hmmtxd is able to identify and learn which trackers works correctly and achieve the performance of at least the best tracker or higher ( e.g. _ motocross1 , skating1(low ) , volkswagen , singer1 , pedestrian3 , surfer _ ) .most notable failure cases are caused by the detector failure , e.g. in sequences _singer2 , woman , skating1 , basketball , girl_mov_.  in all other experiments , the abbreviation hmmtxd refers to the combination of all 3 trackers .* evaluation on the cvpr2013 benchmark *  that contains 50 video sequences .results on the benchmark have been published for about 30 trackers .the benchmark defines three types of experiments : ( i ) one - pass evaluation ( ope ) a tracker initialized in the first frame is run to the end of the sequence , ( ii ) temporal robustness evaluation ( tre ) the tracker is initialized and starts at a random frame , and ( iii ) spatial robustness evaluation ( sre ) the initialization is perturbed spatially .performance is measured by precision ( spatial accuracy , i.e. center distance of ground truth and reported bounding box ) and success rate ( the number of frames where overlap with the ground truth was higher than a threshold ) .the results are visualized in fig .[ fig : cvpr2013bench ] where only results of the 10 top performing trackers are plotted .together with the tracker from this benchmark , we also added the meem  tracker , which is a recent state - of - the - art tracker .the proposed hmmtxd outperforms all trackers in the success rate in all three experiments .its precision is comparable to meem  the top performing tracker in terms of precision .hmmtxd outperforms significantly the ope results reported in wang et al . , where 5 top performing trackers from this particular benchmark were used for combination ( other experiments were not reported in the paper ) .0.32   0.32   0.32   0.32   0.32   0.32   * vot2013 benchmark *  evaluates trackers on a collection containing 16 sequences carefully selected from a large pool by a semi- automatic clustering method . for comparison ,results of 27 tracking methods are available and the added meem tracker was evaluated by us using default setting from the publicly available source code .the performance is measured by accuracy , average overlap with the ground truth , and robustness , the number of re - initialization of the tracker so that it is able to track the whole sequence .average rank of trackers is used as an overall performance indicator .0.49  interval ) . for both plotsthe top right corner is the best performance.,title="fig : " ]  0.49  interval ) . for both plotsthe top right corner is the best performance.,title="fig : " ]  in this benchmark , the proposed hmmtxd achieves clearly the best accuracy ( fig .[ fig : vot2013bench ] ) . with less than one re - initialization per sequenceit performs slightly worse in terms of robustness due to two reasons .firstly , the hmm recognizes a tracker problem with a delay and switching to other tracker ( here even one frame where the overlap with ground truth is zero leads to penalization ) and secondly the vot evaluation protocol , which require re - initialization after failure and to forget all previously learned models ( the vot2013 refer to this as causal tracking ) , therefore the learned performance of the trackers is forgotten and has to be learned from scratch .the results for the baseline and region - noise experiments are shown in fig .[ fig : vot2013bench ] .note that the ranking of the methods differs from the original publication since two new methods ( hmmtxd and meem ) were added and the relative ranking of the methods changed .the top three performing trackers and their average ranks are hmmtxd ( ) , plt ( ) , lgtpp  ( ) .meem tracker ends up at the fifth place with average rank .the rankings were obtained by the toolkit provided by the vot in default settings for baseline and region noise experiments .the second best performing method on the vot2013 is the unpublished plt for which just a short description is available in .plt is a variation of structural svm that uses multiple features ( color , gradients ). struck  and meem  are similar method to the plt based on svm classification .we compared these method with hmmtxd on the diverse 77 videos along with the tld  which has a similar design as hmmtxd .hmmtxd outperforms all these methods by a large margin on average recall measured as number of frames where the tracker overlap with ground truth is higher than averaged over all sequences .results are shown in fig .[ fig : tv77bench ] .qualitative comparison of these state - of - the - art methods is shown in fig .[ fig : qualitative ] .even for sequences with lower recall ( e.g. _ bird_1 , skating2 _ ) , the hmmtxd is able to follow the object of interest .a novel method called hmmtxd for fusion of multiple trackers has been proposed .the method utilizes an on - line trained hmm to estimate the states of the individual trackers and to fuse a different types of observables provided by the trackers .the hmmtxd outperforms its constituent parts ( fot , asms , kcf , detector and its combinations ) by a large margin and shows the efficiency of the hmm with combination of three trackers .hmmtxd outperforms all methods included in the cvpr2013 benchmark and perform favorably against most recent state - of - the - art tracker .the hmmtxd also outperforms all method of the vot2013 benchmark in accuracy , while maintaining very good robustness , and ranking in the first place in overall ranking .experiments conducted on a diverse dataset tv77 show that the hmmtxd outperforms state - of - the - art meem , struck and tld methods , which are similar in design , by a large margin .the processing speed of the hmmtxd is frames per second on average , which is comparable with other complex tracking methods .the research was supported by the czech science foundation project gacr p103/12/g084 and by the technology agency of the czech republic project te01020415 ( v3c visual computing competence center ) .let us assume the hmm with possible states , the matrix of state transition probabilities , the vector of initial state probabilities , the initial state , a sequence of observations and the system of conditional probability densities of observations conditioned on .let be a sequence of detection times , be observed states of markov chain , marked by the detector , and for .the forward variable for the baum - welch algorithm is defined as follows .let and  and for  for the forward variable is in principle the same as above with .so  the backward variable for is where and for and for the backward variable is in principle the same as above where for .given the forward and backward variables , we get the following probabilities , that are used to update parameters of hmm . for       and for  final equation for the update of transition probabilities of hmm is as follows .for a simplification let us assume hmm with one - dimensional observed random variables .the sequence is a martingale difference series where      under the assumption that are uniformly bounded random variables i.e. for all , the strong law of large numbers for a sum of martingale differences can be used(see theorem 2.19 in  ) .so  = 0 \text {  almost surely}.\ ] ]  let us denote for and the estimate of based on the modified method of moments .the estimate is a solution of a following equation w.r.t .   having one equation for unknown variables it is necessary to add some constrains to get a unique solution .we propose to minimize    w.r.t . giving which satisfy the moment equation ( [ eq : app1 ] ) .the same way of reasoning can be used for higher moments of .for example using we get estimates for for ,    in the hmmtxd -dimensional observed random variables are assumed , each of them having beta- distribution and being conditionally independent .there are well - known relations for a mean value and a variance of a random variable having beta distribution and its shape parameters     and   using the modified method of moments gives and then and   if we assume in our model that for some then    and where and

Abstract: in this paper , we propose a novel method for visual object tracking called hmmtxd . the method fuses observations from complementary out - of - the box trackers and a detector by utilizing a hidden markov model whose latent states correspond to a binary vector expressing the failure of individual trackers . the markov model is trained in an unsupervised way , relying on an online learned detector to provide a source of tracker - independent information for a modified baum- welch algorithm that updates the model w.r.t . the partially annotated data .   we show the effectiveness of the proposed method on combination of two and three tracking algorithms . the performance of hmmtxd is evaluated on two standard benchmarks ( cvpr2013 and vot ) and on a rich collection of 77 publicly available sequences . the hmmtxd outperforms the state - of - the - art , often significantly , on all datasets in almost all criteria .   visual tracking , on - line learning , hidden markov model , object detection


more and more global optimization tasks are completed today using algorithms originating from mimicking the way the nature solves them .we have two branches of science , which describe the world around us , namely biology ( for living things ) and physics ( for the rest of it ) .the problem of global optimization ( either minimization or maximization , with or without constraints ) of the objective function of many variables still remains a challenge for practitioners .there is no single , universal and deterministic algorithm capable of solving all kinds of optimization problems : those involving smooth as well as non - smooth objective functions , mixed integer - real - boolean valued unknowns , etc .classical mathematical analysis was long the only tool for finding extrema of functions of many variables .unfortunately it is of very limited use in many practical applications , especially when the objective function is not differentiable at least once .on the other hand , we are ready to accept the solutions , which are not perfect in the mathematical sense ( often called `` -optimal '' ) , but are sufficiently close to them .  to overcome such difficulties , researchers in various fields of science and engineering turned to stochastic algorithms .there are several kinds of arguments for doing so .the first , and certainly not the most important one , is increasing availability of the computing power .we are able to examine , usually in a fraction of a second , many trial solutions of the optimization problem under study . loosely speaking ,this is the base of a rich family of monte - carlo - type optimization algorithms .trying to mimic nature s actions is another justification for rich variety of optimization algorithms , just because nature seems very successful .let s put aside algorithms of genetic type , grown on biological grounds , and concentrate instead on those , which are based on behavior of purely physical systems .various physical phenomena were taken into account , mostly from classical mechanics of a single particle ( deterministic algorithms of `` gradient type '' ) and thermodynamics ( simulated annealing ) as models for optimization procedures .  in this paperwe propose the diffusion processes and quantum tunneling as a base for a class of stochastic optimization routines .consider the simplest version of familiar monte - carlo optimization procedure .its operation may be summarized as a random walk in the search space ( bounded or not ) and sampling the values of objective function in visited points .the trajectory of such a random walker is very similar to the trajectory of physical particle subjected to brownian motion .if we consider many random walkers at the same time ( multiple start point monte - carlo optimization algorithm ) , then emerging set of trajectories resembles closely the diffusion process . in the derivation of the law of brownianmotion one assumes that the lengths of individual `` jumps '' are not equal to each other but are distributed normally , as a result of a huge number ( estimated as ) of independent `` kicks '' from surrounding molecules .this is practical manifestation of the law of large numbers , known also as the central limit theorem . under those assumptionsit may be shown ( einstein , smoluchowski ) that the average distance , , of a random walker from starting point is a function of time , , and can be expressed as where is the diffusion constant .the formula ( [ brown ] ) was later confirmed experimentally for small particles suspended in liquids .this , in turn , made possible to estimate the value of very important physical constant , the avogadro s number , , thus finally confirming the atomic structure of matter .it is worth noting that the early value of , obtained this way , differs less than % from the one known today , almost years later .extensive investigations of diffusion processes revealed , that at least some of them must be governed by other mechanisms , different from familiar brownian motion .they were classified as _ subdiffusion _( , see eq .[ brown ] ) and _ enhanced diffusion _ ( ) .it is often said that the enhanced diffusion is governed by _ lvy flights _ , which will be explained below ._ lvy flights _  paul lvy ( 18861971 ) , the french mathematician , considered in thirties ( xx century )the following problem :  _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ what , if any , should be the probability density of independent , identically distributed random variables ( ) , , , to satisfy the requirement that the probability density of their sum has the same functional form ? _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _  today we could say , that lvy tried to find a class of self - similar objects , known as _fractals _ since benoit mandelbrot had invented them , much later , in 1968 .well known answer to lvy s problem was based on famous _ central limit theorem _ , which , in most widely known version , states that the sum of random variables has normal probability density .we can even drop the requirement of identical distributions ( but _ not _ the independence ! ) and still have the same result .there is a catch , however : the individual distributions have to be _ narrow _ , i.e. their first and second moment ( lindberg ) , and in lyapunov version also the third moment , have to be finite .taking into considerations also other , non - gaussian distributions , lvy obtained the following condition for the fourier transform of probability density of the sum of  random variables : where the normalization constant was dropped , and .going back to the searched distribution , not its fourier transform , is not trivial , and the analytical form of the result is known only for few special cases .generally it may be expressed as  and is known as _ symmetrical lvy stable distribution of index _ ( ) _ and scale factor _ ( ) . for simplicity one usually sets .the special cases mentioned earlier are :  * cauchy distribution ( among physicists known also as lorentzian shape ) : * gauss normal distribution , when .the integral ( [ calka ] ) can be written in a form of ( truncated ) power series  with of order and the leading term is proportional to .looking at the above series and original lvy s result ( [ orig ] ) , one can see that the searched probability density should behave as now we understand , why the index _ must _ belong to the interval ,2\left.\right] ] , as largely unexplored until now .values result in slower spreading of random walkers in search space , what may very significantly affect their ability to find desired extremum , when the problem is defined on unbounded domain . on the other hand , the case may be considered as a computer imitation of the phenomenon known from quantum mechanics , namely * quantum tunneling * , see fig .[ l0.67 ] .it is interesting to note , that we have obtained this behavior without even mentioning the quantum mechanical methods .= 1.05 = 0.408  one may wonder , whether such an optimization procedure can still be classified as an evolutionary algorithm .our answer is _ yes _ , because one can always identify long jumps in search space with _ mutations _ , but remember that some researchers are simply ruling out mutations from algorithms regarded as evolutionary , reserving them instead for as they call it genetic - type routines .we have to mention here , that similar behavior , i.e. occasional long jumps , was introduced earlier , rather heuristically , by galar and kopciuch under the amusing name of _ impatience operator _ and interpreted there mainly in context of social sciences .here we present the random number generator , implemented in fortran 77 , which produces sequences of numbers distributed according to eq .( [ easy ] ) .it is intentionally not optimized and works by inverting the distributive function , which is given by where is uniformly distributed on  ] , random number generator taking x ( of type double precision ) as a dummy argument .there is no check , whether .the subroutine will work even for , however one should not expect to obtain normally distributed random numbers in such case .in this paper we have described two distinct , but closely related classes of stochastic optimization algorithms , based on a single mathematical model and being the computer counterparts of two different physical effects : classical diffusion and quantum tunneling .they received unified mathematical background and may be distinguished according to the properties of random walkers , as summarized below :  [ cols="<,<,<",options="header " , ]   the unexpected similarities between classical diffusion processes and quantum tunneling have their roots probably in properties ( similarities ) of the corresponding partial differential equations describing them . both equations , i.e. the diffusion equation ( fick s law ) and schrdinger equation relate first partial derivatives of the unknown function with respect to time with its second spatial derivatives .the important difference is the explicit presence of imaginary unit , , in schrdinger s equation .the algorithm we describe here is able to mimic the properties of both types of solutions .it can be easily switched from one type of behavior to another one by merely changing the value of a single control parameter , i.e. lvy index .the question arises , whether the familiar evolutionary algorithms should be immediately thrown away in favor of lvy flights based ones .even , if we stick to the orthodox definition of evolutionary algorithms as the ones , which accept only small , gradual changes in position within the searched domain then the answer is _no_. the main problem with evolutionary algorithms is their poor ability to escape from unwanted , suboptimal extrema . indeed ,if evolutionary random walkers make steps with lengths distributed uniformly on $ ] , then they are unable to escape from extremum , which is wider than .using normally distributed numbers as step lengths good in theory does nt help much in practice . and here is why : gaussian generators of poor quality _ never _ produce random numbers exceeding few ( say ) in magnitude . on the other hand ,even perfect normal generators produce _ very rarely _ steps longer than that .so it is a pure illusion , that it is possible to find the global optimum in reasonable time with one of such algorithms even if the appropriate theorem states so .this may only happen , when our first approximation to the solution is already quite good .quite different situation occurs , when our goal is to keep track and adapt to the varying environment , i.e. when objective function changes smoothly and slowly enough while the optimization procedure is in progress ( for example satellite or missile tracking ) . in such cases , providing the starting point is already well known i.e. is located closer to the global optimum than to any other one the evolutionary algorithm , without any mutations , is indispensable .this work was done as part of author s statutory activity in institute of physics , polish academy of sciences .exact form of lvy distribution is rarely quoted in literature , we have found it in : rosario n. mantegna and h. eugene stanley _stochastic process with ultraslow convergence to a gaussian : the truncated lvy flight _. lett . * 73*(22 ) , 29462947 , 1994   franois bardou _ cooling gases with lvy flights : using the generalized central limit theorem in physics _ ,in mini proceedings of conference on _ lvy processes : theory and applications _ , aarhus , january 1822 , 1999 , maphysto publication ( miscellanea no . 11 , issn 1398 - 5957 ) , o. barndorff - nielsen , s.e . craversen and t. mikosch ( eds . ) , also available at url : http://arxiv.org/abs/physics/0012049 .  roman galar , ryszard kopciuch _ zniecierpliwienie i polaryzacja w procesach ewolucyjnych_ ii krajowa konferencja _ algorytmy ewolucyjne i optymalizacja globalna _ , potok zoty , 2528 maja 1999 , str .115122 ( _ impatience and polarization in evolutionary processes _ , ii domestic conference _evolutionary algorithms and global optimization _ , potok zoty ( poland ) , may 2528 , 1999 , conference materials , page 115 - 122 , in polish ) .

Abstract: in this paper we propose and advocate the use of the so called _ lvy flights _ as a driving mechanism for a class of stochastic optimization computations . this proposal , for some reasons overlooked until now , is in author s opinion very appropriate to satisfy the need for algorithm , which is capable of generating trial steps of very different length in the search space . the required balance between short and long steps can be easily and fully controlled . a simple example of approximated lvy distribution , implemented in fortran 77 , is given . we also discuss the physical grounds of presented methods .   lvy flights , diffusive processes , brownian motion , quantum tunneling , evolutionary computations , evolutionary algorithms , lvy distributions , random number generators