Instatistics, amixture model is aprobabilistic model for representing the presence ofsubpopulations within an overall population, without requiring that an observed data set should identify the sub-population to which an individual observation belongs. Formally a mixture model corresponds to themixture distribution that represents theprobability distribution of observations in the overall population. However, while problems associated with "mixture distributions" relate to deriving the properties of the overall population from those of the sub-populations, "mixture models" are used to makestatistical inferences about the properties of the sub-populations given only observations on the pooled population, without sub-population identity information. Mixture models are used for clustering, under the namemodel-based clustering, and also fordensity estimation.

Mixture models should not be confused with models forcompositional data, i.e., data whose components are constrained to sum to a constant value (1, 100%, etc.). However, compositional models can be thought of as mixture models, where members of the population are sampled at random. Conversely, mixture models can be thought of as compositional models, where thetotal size reading population has been normalized to 1.

A typical finite-dimensional mixture model is ahierarchical model consisting of the following components:

• N random variables that are observed, each distributed according to a mixture ofK components, with the components belonging to the sameparametric family of distributions (e.g., allnormal, allZipfian, etc.) but with different parameters. However, it is also possible to have a finite mixture model where each component belongs to a differentparametric family of distributions,^[1] for example, a mixture of amultivariate normal distribution and ageneralized hyperbolic distribution.

• N randomlatent variables specifying the identity of the mixture component of each observation, each distributed according to aK-dimensionalcategorical distribution
• A set ofK mixture weights, which are probabilities that sum to 1.
• A set ofK parameters, each specifying the parameter of the corresponding mixture component. In many cases, each "parameter" is actually a set of parameters. For example, if the mixture components areGaussian distributions, there will be amean andvariance for each component. If the mixture components arecategorical distributions (e.g., when each observation is a token from a finite alphabet of sizeV), there will be a vector ofV probabilities summing to 1.

In addition, in aBayesian setting, the mixture weights and parameters will themselves be random variables, andprior distributions will be placed over the variables. In such a case, the weights are typically viewed as aK-dimensional random vector drawn from aDirichlet distribution (theconjugate prior of the categorical distribution), and the parameters will be distributed according to their respective conjugate priors.

Mathematically, a basic parametric mixture model can be described as follows:

In a Bayesian setting, all parameters are associated with random variables, as follows:

This characterization usesF andH to describe arbitrary distributions over observations and parameters, respectively. TypicallyH will be theconjugate prior ofF. The two most common choices ofF areGaussian aka "normal" (for real-valued observations) andcategorical (for discrete observations). Other common possibilities for the distribution of the mixture components are:

• Binomial distribution, for the number of "positive occurrences" (e.g., successes, yes votes, etc.) given a fixed number of total occurrences
• Multinomial distribution, similar to the binomial distribution, but for counts of multi-way occurrences (e.g., yes/no/maybe in a survey)
• Negative binomial distribution, for binomial-type observations but where the quantity of interest is the number of failures before a given number of successes occurs
• Poisson distribution, for the number of occurrences of an event in a given period of time, for an event that is characterized by a fixed rate of occurrence
• Exponential distribution, for the time before the next event occurs, for an event that is characterized by a fixed rate of occurrence
• Log-normal distribution, for positive real numbers that are assumed to grow exponentially, such as incomes or prices
• Multivariate normal distribution (aka multivariate Gaussian distribution), for vectors of correlated outcomes that are individually Gaussian-distributed
• Multivariate Student'st-distribution, for vectors of heavy-tailed correlated outcomes^[2]
• A vector ofBernoulli-distributed values, corresponding, e.g., to a black-and-white image, with each value representing a pixel; see the handwriting-recognition example below

A typical non-BayesianGaussian mixture model looks like this:

A Bayesian version of aGaussian mixture model is as follows:

${\displaystyle }$

A Bayesian Gaussian mixture model is commonly extended to fit a vector of unknown parameters (denoted in bold), or multivariate normal distributions. In a multivariate distribution (i.e. one modelling a vector${\displaystyle \mathit{x}}$ withN random variables) one may model a vector of parameters (such as several observations of a signal or patches within an image) using a Gaussian mixture model prior distribution on the vector of estimates given by$${\displaystyle p(\mathit{\theta })=\sum _{i=1}^K{\varphi }_i\mathcal{N}({\mathit{\mu }}_i,{\mathbf{\Sigma }}_i)}$$where thei^th vector component is characterized by normal distributions with weights${\displaystyle {\varphi }_i}$, means${\displaystyle {\mathit{\mu }}_i}$ and covariance matrices${\displaystyle {\mathbf{\Sigma }}_i}$. To incorporate this prior into a Bayesian estimation, the prior is multiplied with the known distribution${\displaystyle p(\mathit{x}|\mathit{\theta })}$ of the data${\displaystyle \mathit{x}}$ conditioned on the parameters${\displaystyle \mathit{\theta }}$ to be estimated. With this formulation, theposterior distribution${\displaystyle p(\mathit{\theta }|\mathit{x})}$ isalso a Gaussian mixture model of the form$${\displaystyle p(\mathit{\theta }|\mathit{x})=\sum _{i=1}^K{\tilde{\varphi }}_i\mathcal{N}({\stackrel{\mathbf{~}}{\mathit{\mu }}}_{\mathit{i}},{\stackrel{\mathbf{~}}{\mathbf{\Sigma }}}_i)}$$with new parameters${\displaystyle {\tilde{\varphi }}_i,{\stackrel{\mathbf{~}}{\mathit{\mu }}}_i}$ and${\displaystyle {\stackrel{\mathbf{~}}{\mathbf{\Sigma }}}_i}$ that are updated using theEM algorithm.^[3] Although EM-based parameter updates are well-established, providing the initial estimates for these parameters is currently an area of active research. Note that this formulation yields a closed-form solution to the complete posterior distribution. Estimations of the random variable${\displaystyle \mathit{\theta }}$ may be obtained via one of several estimators, such as the mean or maximum of the posterior distribution.

Such distributions are useful for assuming patch-wise shapes of images and clusters, for example. In the case of image representation, each Gaussian may be tilted, expanded, and warped according to the covariance matrices${\displaystyle {\mathbf{\Sigma }}_i}$. One Gaussian distribution of the set is fit to each patch (usually of size 8×8 pixels) in the image. Notably, any distribution of points around a cluster (seek-means) may be accurately given enough Gaussian components, but scarcely overK=20 components are needed to accurately model a given image distribution or cluster of data.

A typical non-Bayesian mixture model withcategorical observations looks like this:

• ${\displaystyle K,N:}$ as above
• ${\displaystyle {\varphi }_{i=1\dots K},\mathit{\varphi }:}$ as above
• ${\displaystyle z_{i=1\dots N},x_{i=1\dots N}:}$ as above
• ${\displaystyle V:}$ dimension of categorical observations, e.g., size of word vocabulary
• ${\displaystyle {\theta }_{i=1\dots K,j=1\dots V}:}$ probability for component${\displaystyle i}$ of observing item${\displaystyle j}$
• ${\displaystyle {\mathit{\theta }}_{i=1\dots K}:}$ vector of dimension${\displaystyle V,}$ composed of${\displaystyle {\theta }_{i,1\dots V};}$ must sum to 1

The random variables:

A typical Bayesian mixture model withcategorical observations looks like this:

• ${\displaystyle K,N:}$ as above
• ${\displaystyle {\varphi }_{i=1\dots K},\mathit{\varphi }:}$ as above
• ${\displaystyle z_{i=1\dots N},x_{i=1\dots N}:}$ as above
• ${\displaystyle V:}$ dimension of categorical observations, e.g., size of word vocabulary
• ${\displaystyle {\theta }_{i=1\dots K,j=1\dots V}:}$ probability for component${\displaystyle i}$ of observing item${\displaystyle j}$
• ${\displaystyle {\mathit{\theta }}_{i=1\dots K}:}$ vector of dimension${\displaystyle V,}$ composed of${\displaystyle {\theta }_{i,1\dots V};}$ must sum to 1
• ${\displaystyle \alpha :}$ shared concentration hyperparameter of${\displaystyle \mathit{\theta }}$ for each component
• ${\displaystyle \beta :}$ concentration hyperparameter of${\displaystyle \mathit{\varphi }}$

The random variables:

Financial returns often behave differently in normal situations and during crisis times. A mixture model^[4] for return data seems reasonable. Sometimes the model used is ajump-diffusion model, or as a mixture of two normal distributions. SeeFinancial economics § Challenges and criticism andFinancial risk management § Banking for further context.

Assume that we observe the prices ofN different houses. Different types of houses in different neighborhoods will have vastly different prices, but the price of a particular type of house in a particular neighborhood (e.g., three-bedroom house in moderately upscale neighborhood) will tend to cluster fairly closely around the mean. One possible model of such prices would be to assume that the prices are accurately described by a mixture model withK different components, each distributed as anormal distribution with unknown mean and variance, with each component specifying a particular combination of house type/neighborhood. Fitting this model to observed prices, e.g., using theexpectation-maximization algorithm, would tend to cluster the prices according to house type/neighborhood and reveal the spread of prices in each type/neighborhood. (Note that for values such as prices or incomes that are guaranteed to be positive and which tend to growexponentially, alog-normal distribution might actually be a better model than a normal distribution.)

Assume that a document is composed ofN different words from a total vocabulary of sizeV, where each word corresponds to one ofK possible topics. The distribution of such words could be modelled as a mixture ofK differentV-dimensionalcategorical distributions. A model of this sort is commonly termed atopic model. Note thatexpectation maximization applied to such a model will typically fail to produce realistic results, due (among other things) to theexcessive number of parameters. Some sorts of additional assumptions are typically necessary to get good results. Typically two sorts of additional components are added to the model:

1. Aprior distribution is placed over the parameters describing the topic distributions, using aDirichlet distribution with aconcentration parameter that is set significantly below 1, so as to encourage sparse distributions (where only a small number of words have significantly non-zero probabilities).
2. Some sort of additional constraint is placed over the topic identities of words, to take advantage of natural clustering.
   • For example, aMarkov chain could be placed on the topic identities (i.e., the latent variables specifying the mixture component of each observation), corresponding to the fact that nearby words belong to similar topics. (This results in ahidden Markov model, specifically one where aprior distribution is placed over state transitions that favors transitions that stay in the same state.)
   • Another possibility is thelatent Dirichlet allocation model, which divides up the words intoD different documents and assumes that in each document only a small number of topics occur with any frequency.

The following example is based on an example inChristopher M. Bishop,Pattern Recognition and Machine Learning.^[5]

Imagine that we are given anN×N black-and-white image that is known to be a scan of a hand-written digit between 0 and 9, but we don't know which digit is written. We can create a mixture model with${\displaystyle K=10}$ different components, where each component is a vector of size${\displaystyle N^2}$ ofBernoulli distributions (one per pixel). Such a model can be trained with theexpectation-maximization algorithm on an unlabeled set of hand-written digits, and will effectively cluster the images according to the digit being written. The same model could then be used to recognize the digit of another image simply by holding the parameters constant, computing the probability of the new image for each possible digit (a trivial calculation), and returning the digit that generated the highest probability.

Mixture models apply in the problem of directing multiple projectiles at a target (as in air, land, or sea defense applications), where the physical and/or statistical characteristics of the projectiles differ within the multiple projectiles. An example might be shots from multiple munitions types or shots from multiple locations directed at one target. The combination of projectile types may be characterized as a Gaussian mixture model.^[6] Further, a well-known measure of accuracy for a group of projectiles is thecircular error probable (CEP), which is the numberR such that, on average, half of the group of projectiles falls within the circle of radiusR about the target point. The mixture model can be used to determine (or estimate) the valueR. The mixture model properly captures the different types of projectiles.

The financial example above is one direct application of the mixture model, a situation in which we assume an underlying mechanism so that each observation belongs to one of some number of different sources or categories. This underlying mechanism may or may not, however, be observable. In this form of mixture, each of the sources is described by a component probability density function, and its mixture weight is the probability that an observation comes from this component.

In an indirect application of the mixture model we do not assume such a mechanism. The mixture model is simply used for its mathematical flexibilities. For example, a mixture of twonormal distributions with different means may result in a density with twomodes, which is not modeled by standard parametric distributions. Another example is given by the possibility of mixture distributions to model fatter tails than the basic Gaussian ones, so as to be a candidate for modeling more extreme events.

The mixture model-based clustering is also predominantly used in identifying the state of the machine inpredictive maintenance. Density plots are used to analyze the density of high dimensional features. If multi-model densities are observed, then it is assumed that a finite set of densities are formed by a finite set of normal mixtures. A multivariate Gaussian mixture model is used to cluster the feature data into k number of groups where k represents each state of the machine. The machine state can be a normal state, power off state, or faulty state.^[7] Each formed cluster can be diagnosed using techniques such as spectral analysis. In the recent years, this has also been widely used in other areas such as early fault detection.^[8]

In image processing and computer vision, traditionalimage segmentation models often assign to onepixel only one exclusive pattern. In fuzzy or soft segmentation, any pattern can have certain "ownership" over any single pixel. If the patterns are Gaussian, fuzzy segmentation naturally results in Gaussian mixtures. Combined with other analytic or geometric tools (e.g., phase transitions over diffusive boundaries), such spatially regularized mixture models could lead to more realistic and computationally efficient segmentation methods.^[9]

Probabilistic mixture models such asGaussian mixture models (GMM) are used to resolvepoint set registration problems in image processing and computer vision fields. For pair-wisepoint set registration, one point set is regarded as the centroids of mixture models, and the other point set is regarded as data points (observations). State-of-the-art methods are e.g.coherent point drift (CPD)^[10] andStudent's t-distribution mixture models (TMM).^[11] The result of recent research demonstrate the superiority of hybrid mixture models^[12] (e.g. combining Student's t-distribution and Watson distribution/Bingham distribution to model spatial positions and axes orientations separately) compare to CPD and TMM, in terms of inherent robustness, accuracy and discriminative capacity.

Identifiability refers to the existence of a unique characterization for any one of the models in the class (family) being considered. Estimation procedures may not be well-defined and asymptotic theory may not hold if a model is not identifiable.

LetJ be the class of all binomial distributions withn = 2. Then a mixture of two members ofJ would have

andp₂ = 1 −p₀ −p₁. Clearly, givenp₀ andp₁, it is not possible to determine the above mixture model uniquely, as there are three parameters(π,θ₁,θ₂) to be determined.

Consider a mixture of parametric distributions of the same class. Let

$${\displaystyle J=\{f(\cdot ;\theta ):\theta \in \mathrm{\Omega }\}}$$

be the class of all component distributions. Then theconvex hullK ofJ defines the class of all finite mixture of distributions inJ:

$${\displaystyle K=\left\{p(\cdot ):p(\cdot )=\sum _{i=1}^n{a}_i{f}_i(\cdot ;{\theta }_i),a_i>0,\sum _{i=1}^n{a}_i=1,f_i(\cdot ;{\theta }_i)\in J\mathrm{\forall }i,n\right\}}$$

K is said to be identifiable if all its members are unique, that is, given two membersp andp′ inK, being mixtures ofk distributions andk′ distributions respectively inJ, we havep =p′ if and only if, first of all,k =k′ and secondly we can reorder the summations such thata_i =a_i′ andf_i =f_i′ for alli.

Parametric mixture models are often used when we know the distributionY and we can sample fromX, but we would like to determine thea_i andθ_i values. Such situations can arise in studies in which we sample from a population that is composed of several distinct subpopulations.

It is common to think of probability mixture modeling as a missing data problem. One way to understand this is to assume that the data points under consideration have "membership" in one of the distributions we are using to model the data. When we start, this membership is unknown, or missing. The job of estimation is to devise appropriate parameters for the model functions we choose, with the connection to the data points being represented as their membership in the individual model distributions.

A variety of approaches to the problem of mixture decomposition have been proposed, many of which focus on maximum likelihood methods such asexpectation maximization (EM) or maximuma posteriori estimation (MAP). Generally these methods consider separately the questions of system identification and parameter estimation; methods to determine the number and functional form of components within a mixture are distinguished from methods to estimate the corresponding parameter values. Some notable departures are the graphical methods as outlined in Tarter and Lock^[13] and more recentlyminimum message length (MML) techniques such as Figueiredo and Jain^[14] and to some extent the moment matching pattern analysis routines suggested by McWilliam and Loh (2009).^[15]

Expectation maximization (EM) is seemingly the most popular technique used to determine the parameters of a mixture with ana priori given number of components. This is a particular way of implementingmaximum likelihood estimation for this problem. EM is of particular appeal for finite normal mixtures where closed-form expressions are possible such as in the following iterative algorithm by Dempsteret al. (1977)^[16]

${\displaystyle w_s^{(j+1)}=\frac{1}{N}\sum _{t=1}^N{h}_s^{(j)}(t)}$

${\displaystyle {\mu }_s^{(j+1)}=\frac{\sum _{t=1}^N{h}_s^{(j)}(t)x^{(t)}}{\sum _{t=1}^N{h}_s^{(j)}(t)}}$

${\displaystyle {\mathrm{\Sigma }}_s^{(j+1)}=\frac{\sum _{t=1}^N{h}_s^{(j)}(t)[x^{(t)}-{\mu }_s^{(j+1)}][x^{(t)}-{\mu }_s^{(j+1)}{]}^{\mathrm{\top }}}{\sum _{t=1}^N{h}_s^{(j)}(t)}}$

with the posterior probabilities

${\displaystyle h_s^{(j)}(t)=\frac{w_s^{(j)}{p}_s(x^{(t)};{\mu }_s^{(j)},{\mathrm{\Sigma }}_s^{(j)})}{\sum _{i=1}^n{w}_i^{(j)}{p}_i(x^{(t)};{\mu }_i^{(j)},{\mathrm{\Sigma }}_i^{(j)})}.}$

Thus on the basis of the current estimate for the parameters, theconditional probability for a given observationx^(t) being generated from states is determined for eacht = 1, …,N ;N being the sample size. The parameters are then updated such that the new component weights correspond to the average conditional probability and each component mean and covariance is the component specific weighted average of the mean and covariance of the entire sample.

Dempster^[16] also showed that each successive EM iteration will not decrease the likelihood, a property not shared by other gradient based maximization techniques. Moreover, EM naturally embeds within it constraints on the probability vector, and for sufficiently large sample sizes positive definiteness of the covariance iterates. This is a key advantage since explicitly constrained methods incur extra computational costs to check and maintain appropriate values. Theoretically EM is a first-order algorithm and as such converges slowly to a fixed-point solution. Redner and Walker (1984)^{[full citation needed]} make this point arguing in favour of superlinear and second order Newton and quasi-Newton methods and reporting slow convergence in EM on the basis of their empirical tests. They do concede that convergence in likelihood was rapid even if convergence in the parameter values themselves was not. The relative merits of EM and other algorithms vis-à-vis convergence have been discussed in other literature.^[17]

Other common objections to the use of EM are that it has a propensity to spuriously identify local maxima, as well as displaying sensitivity to initial values.^[18][19] One may address these problems by evaluating EM at several initial points in the parameter space but this is computationally costly and other approaches, such as the annealing EM method of Udea and Nakano (1998) (in which the initial components are essentially forced to overlap, providing a less heterogeneous basis for initial guesses), may be preferable.

Figueiredo and Jain^[14] note that convergence to 'meaningless' parameter values obtained at the boundary (where regularity conditions breakdown, e.g., Ghosh and Sen (1985)) is frequently observed when the number of model components exceeds the optimal/true one. On this basis they suggest a unified approach to estimation and identification in which the initialn is chosen to greatly exceed the expected optimal value. Their optimization routine is constructed via a minimum message length (MML) criterion that effectively eliminates a candidate component if there is insufficient information to support it. In this way it is possible to systematize reductions inn and consider estimation and identification jointly.

With initial guesses for the parameters of our mixture model, "partial membership" of each data point in each constituent distribution is computed by calculatingexpectation values for the membership variables of each data point. That is, for each data pointx_j and distributionY_i, the membership valuey_i,j is:

${\displaystyle y_{i,j}=\frac{a_i{f}_Y(x_j;{\theta }_i)}{f_X(x_j)}.}$

With expectation values in hand for group membership,plug-in estimates are recomputed for the distribution parameters.

The mixing coefficientsa_i are themeans of the membership values over theN data points.

${\displaystyle a_i=\frac{1}{N}\sum _{j=1}^N{y}_{i,j}}$

The component model parametersθ_i are also calculated by expectation maximization using data pointsx_j that have been weighted using the membership values. For example, ifθ is a meanμ

${\displaystyle {\mu }_i=\frac{\sum _j{y}_{i,j}{x}_j}{\sum _j{y}_{i,j}}.}$

With new estimates fora_i and theθ_i's, the expectation step is repeated to recompute new membership values. The entire procedure is repeated until model parameters converge.

As an alternative to the EM algorithm, the mixture model parameters can be deduced usingposterior sampling as indicated byBayes' theorem. This is still regarded as an incomplete data problem in which membership of data points is the missing data. A two-step iterative procedure known asGibbs sampling can be used.

The previous example of a mixture of twoGaussian distributions can demonstrate how the method works. As before, initial guesses of the parameters for the mixture model are made. Instead of computing partial memberships for each elemental distribution, a membership value for each data point is drawn from aBernoulli distribution (that is, it will be assigned to either the first or the second Gaussian). The Bernoulli parameterθ is determined for each data point on the basis of one of the constituent distributions.^[vague] Draws from the distribution generate membership associations for each data point. Plug-in estimators can then be used as in the M step of EM to generate a new set of mixture model parameters, and the binomial draw step repeated.

Themethod of moment matching is one of the oldest techniques for determining the mixture parameters dating back to Karl Pearson's seminal work of 1894.In this approach the parameters of the mixture are determined such that the composite distribution has moments matching some given value. In many instances extraction of solutions to the moment equations may present non-trivial algebraic or computational problems. Moreover, numerical analysis by Day^[20] has indicated that such methods may be inefficient compared to EM. Nonetheless, there has been renewed interest in this method, e.g., Craigmile and Titterington (1998) and Wang.^[21]

McWilliam and Loh (2009) consider the characterisation of a hyper-cuboid normal mixturecopula in large dimensional systems for which EM would be computationally prohibitive. Here a pattern analysis routine is used to generate multivariate tail-dependencies consistent with a set of univariate and (in some sense) bivariate moments. The performance of this method is then evaluated using equity log-return data withKolmogorov–Smirnov test statistics suggesting a good descriptive fit.

Some problems in mixture model estimation can be solved usingspectral methods.In particular it becomes useful if data pointsx_i are points in high-dimensionalreal space, and the hidden distributions are known to belog-concave (such asGaussian distribution orExponential distribution).

Spectral methods of learning mixture models are based on the use ofSingular Value Decomposition of a matrix which contains data points.The idea is to consider the topk singular vectors, wherek is the number of distributions to be learned. The projectionof each data point to alinear subspace spanned by those vectors groups points originating from the same distributionvery close together, while points from different distributions stay far apart.

One distinctive feature of the spectral method is that it allows us toprove that ifdistributions satisfy certain separation condition (e.g., not too close), then the estimated mixture will be very close to the true one with high probability.

Tarter and Lock^[13] describe a graphical approach to mixture identification in which a kernel function is applied to an empirical frequency plot so to reduce intra-component variance. In this way one may more readily identify components having differing means. While thisλ-method does not require prior knowledge of the number or functional form of the components its success does rely on the choice of the kernel parameters which to some extent implicitly embeds assumptions about the component structure.

Some of them can even probably learn mixtures ofheavy-tailed distributions including those withinfinitevariance (seelinks to papers below).In this setting, EM based methods would not work, since the Expectation step would diverge due to presence ofoutliers.

To simulate a sample of sizeN that is from a mixture of distributionsF_i,i=1 ton, with probabilitiesp_i (sum= p_i = 1):

1. GenerateN random numbers from acategorical distribution of sizen and probabilitiesp_i fori= 1= to n. These tell you which of theF_i each of theN values will come from. Denote bym_i the quantity of random numbers assigned to thei^th category.
2. For eachi, generatem_i random numbers from theF_i distribution.

In aBayesian setting, additional levels can be added to thegraphical model defining the mixture model. For example, in the commonlatent Dirichlet allocationtopic model, the observations are sets of words drawn fromD different documents and theK mixture components represent topics that are shared across documents. Each document has a different set of mixture weights, which specify the topics prevalent in that document. All sets of mixture weights share commonhyperparameters.

A very common extension is to connect thelatent variables defining the mixture component identities into aMarkov chain, instead of assuming that they areindependent identically distributed random variables. The resulting model is termed ahidden Markov model and is one of the most common sequential hierarchical models. Numerous extensions of hidden Markov models have been developed; see the resulting article for more information.

Mixture distributions and the problem of mixture decomposition, that is the identification of its constituent components and the parameters thereof, has been cited in the literature as far back as 1846 (Quetelet in McLachlan,^[18] 2000) although common reference is made to the work ofKarl Pearson (1894)^[22] as the first author to explicitly address the decomposition problem in characterising non-normal attributes of forehead to body length ratios in female shore crab populations. The motivation for this work was provided by the zoologistWalter Frank Raphael Weldon who had speculated in 1893 (in Tarter and Lock^[13]) that asymmetry in the histogram of these ratios could signal evolutionary divergence. Pearson's approach was to fit a univariate mixture of two normals to the data by choosing the five parameters of the mixture such that the empirical moments matched that of the model.

While his work was successful in identifying two potentially distinct sub-populations and in demonstrating the flexibility of mixtures as a moment matching tool, the formulation required the solution of a 9th degree (nonic) polynomial which at the time posed a significant computational challenge.

Subsequent works focused on addressing these problems, but it was not until the advent of the modern computer and the popularisation ofMaximum Likelihood (MLE) parameterisation techniques that research really took off.^[23] Since that time there has been a vast body of research on the subject spanning areas such asfisheries research,agriculture,botany,economics,medicine,genetics,psychology,palaeontology,electrophoresis,finance,geology andzoology.^[24]

• Mixture density
• Mixture (probability)
• Flexible Mixture Model (FMM)
• Subspace Gaussian mixture model
• Giry monad
• HELIOS Hybrid Evaluation of Lifecycle and Impact of Outstanding Science

• Graphical model
• Hierarchical Bayes model

• RANSAC

• Everitt, B.S.; Hand, D.J. (1981).Finite mixture distributions. Chapman & Hall.ISBN 978-0-412-22420-1.
• Lindsay, B. G. (1995).Mixture Models: Theory, Geometry, and Applications. NSF-CBMS Regional Conference Series in Probability and Statistics. Vol. 5. Hayward: Institute of Mathematical Statistics.
• Marin, J.M.;Mengersen, K.;Robert, C. P. (2011)."Bayesian modelling and inference on mixtures of distributions"(PDF). In Dey, D.; Rao, C.R. (eds.).Essential Bayesian models. Handbook of statistics: Bayesian thinking - modeling and computation. Vol. 25. Elsevier.ISBN 9780444537324.
• McLachlan, G.J.; Peel, D. (2000).Finite Mixture Models. Wiley.ISBN 978-0-471-00626-8.
• Press, WH; Teukolsky, SA; Vetterling, WT; Flannery, BP (2007)."Section 16.1. Gaussian Mixture Models and k-Means Clustering".Numerical Recipes: The Art of Scientific Computing (3rd ed.). New York: Cambridge University Press.ISBN 978-0-521-88068-8.
• Titterington, D.; Smith, A.; Makov, U. (1985).Statistical Analysis of Finite Mixture Distributions. Wiley.ISBN 978-0-471-90763-3.
• Yao, W.; Xiang, S. (2024).Mixture Models: Parametric, Semiparametric, and New Directions. Chapman & Hall/CRC Press.ISBN 978-0367481827.

1. Reynolds, D.A.; Rose, R.C. (January 1995). "Robust text-independent speaker identification using Gaussian mixture speaker models".IEEE Transactions on Speech and Audio Processing.3 (1):72–83.Bibcode:1995ITSAP...3...72R.doi:10.1109/89.365379.S2CID 7319345.
2. Permuter, H.; Francos, J.; Jermyn, I.H. (2003).Gaussian mixture models of texture and colour for image database retrieval. IEEEInternational Conference on Acoustics, Speech, and Signal Processing, 2003. Proceedings (ICASSP '03).doi:10.1109/ICASSP.2003.1199538.
   • Permuter, Haim; Francos, Joseph; Jermyn, Ian (2006)."A study of Gaussian mixture models of color and texture features for image classification and segmentation"(PDF).Pattern Recognition.39 (4):695–706.Bibcode:2006PatRe..39..695P.doi:10.1016/j.patcog.2005.10.028.S2CID 8530776.
3. Lemke, Wolfgang (2005).Term Structure Modeling and Estimation in a State Space Framework. Springer Verlag.ISBN 978-3-540-28342-3.
4. Brigo, Damiano;Mercurio, Fabio (2001).Displaced and Mixture Diffusions for Analytically-Tractable Smile Models. Mathematical Finance – Bachelier Congress 2000. Proceedings. Springer Verlag.
5. Brigo, Damiano; Mercurio, Fabio (June 2002). "Lognormal-mixture dynamics and calibration to market volatility smiles".International Journal of Theoretical and Applied Finance.5 (4): 427.CiteSeerX 10.1.1.210.4165.doi:10.1142/S0219024902001511.
6. Spall, J. C.; Maryak, J. L. (1992). "A feasible Bayesian estimator of quantiles for projectile accuracy from non-i.i.d. data".Journal of the American Statistical Association.87 (419):676–681.doi:10.1080/01621459.1992.10475269.JSTOR 2290205.
7. Alexander, Carol (December 2004)."Normal mixture diffusion with uncertain volatility: Modelling short- and long-term smile effects"(PDF).Journal of Banking & Finance.28 (12):2957–80.doi:10.1016/j.jbankfin.2003.10.017.
8. Stylianou, Yannis; Pantazis, Yannis; Calderero, Felipe; Larroy, Pedro; Severin, Francois; Schimke, Sascha; Bonal, Rolando; Matta, Federico; Valsamakis, Athanasios (2005).GMM-Based Multimodal Biometric Verification(PDF).
9. Chen, J.; Adebomi, 0.E.; Olusayo, O.S.; Kulesza, W. (2010).The Evaluation of the Gaussian Mixture Probability Hypothesis Density approach for multi-target tracking. IEEEInternational Conference on Imaging Systems and Techniques, 2010.doi:10.1109/IST.2010.5548541.{{cite conference}}: CS1 maint: numeric names: authors list (link)

• Nielsen, Frank (23 March 2012). "K-MLE: A fast algorithm for learning statistical mixture models".2012 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP). pp. 869–872.arXiv:1203.5181.Bibcode:2012arXiv1203.5181N.doi:10.1109/ICASSP.2012.6288022.ISBN 978-1-4673-0046-9.S2CID 935615.
• TheSOCR demonstrations of EM and Mixture Modeling
• Mixture modelling page (and theSnob program forMinimum Message Length (MML) applied to finite mixture models), maintained by D.L. Dowe.
• PyMix – Python Mixture Package, algorithms and data structures for a broad variety of mixture model based data mining applications in Python
• sklearn.mixture – A module from thescikit-learn Python library for learning Gaussian Mixture Models (and sampling from them), previously packaged withSciPy and now packaged as aSciKit
• GMM.m Matlab code for GMM Implementation
• GPUmix C++ implementation of Bayesian Mixture Models using EM and MCMC with 100x speed acceleration using GPGPU.
• [2] Matlab code for GMM Implementation using EM algorithm
• [3] jMEF: A Java open source library for learning and processing mixtures of exponential families (using duality with Bregman divergences). Includes a Matlab wrapper.
• Very Fast and clean C implementation of theExpectation Maximization (EM) algorithm for estimatingGaussian Mixture Models (GMMs).
• mclust is an R package for mixture modeling.
• dpgmm Pure Python Dirichlet process Gaussian mixture model implementation (variational).
• Gaussian Mixture Models Blog post on Gaussian Mixture Models trained via Expectation Maximization, with an implementation in Python.

• Cluster analysis
• Latent variable models
• Probabilistic models

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