A mixture distribution is one that is written as a convex combination of other distributions.
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19 views
Correction term in variance of mixture [duplicate]
In the answer to this question, the correction term is claimed to be nonnegative:
$$p_A \mu_A^2 + p_B \mu_B^2 - p_A^2 \mu_A^2 - p_B^2 \mu_B^2 - 2 p_A p_B \mu_A \mu_B \geq 0$$
where $p_A + p_B = 1$, $0 ...
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0answers
29 views
How to decide whether the distribution is unimodal or bimodal in grain size distribution?
In the examples given at this link, I am not able to decide whether the distribution is unimodal or bimodal. I think it is in between unimodal and bimodal, but I do not know if this kind of class ...
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0answers
31 views
Bias in EM estimation for a mixture of normal distributions
Are the parameter estimates for a mixture of two normal distribution using EM algorithm biased or unbiased?
More specifically, if I use the EM-algorithm to obtain ML estimates of $μ_1$, $μ_2$, ...
2
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0answers
33 views
Fitting mixture distributions and computing goodness-of-fit?
This question is a follow-up from a previous question of mine here. Thanks to @Glen_b, @gung and @rbatt for teaching me so many new things yesterday. It was mentioned in passing that mixture ...
2
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0answers
21 views
Compendium or catalog of compound distributions?
Does anyone know of a good compendium or catalog of compound distributions, or finite mixture representations of those distributions?
I am trying to find out to what extent the common ...
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0answers
19 views
Constructing the generalized gamma distribution
Is there a finite mixture decomposition or representation, or a compound distribution representation, of the generalized gamma distribution? And if so, what is it? Citations welcome but not required.
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0answers
16 views
Approximating a compound distribution with a mixture: required n?
Suppose you have two continuous distributions, g(φ; θ) and f(θ; x), where φ and θ are parameter vectors of g and f, respectively, and the values of θ generated by g are all legal values for θ in f(θ; ...
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46 views
Visualizing results from multiple latent class models
I am using latent class analysis to cluster a sample of observations based on a set of binary variables. I am using R and the package poLCA. In LCA, you must specify the number of clusters you want to ...
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30 views
two component non standard mixture (normal + unknown)
I have some univariate data which might be well modeled as a two component mixture where the first component is normal with unknown mean and variance and the second is some unspecified continuous ...
3
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6answers
95 views
Time spent in an activity as an independent variable
I want to include time spent doing something (weeks breastfeeding, for example) as an independent variable in a linear model. However, some observations do not engage in the behavior at all. Coding ...
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0answers
27 views
Inference in a mix of two datasets with different features
The other day I was wondering this: Let's say that I have two experiments, a sample $N = N_1 \cup N_2$ and a set of features $F = F_1 \cup F_2$. In the experiment 1, from the $N_1$ sub sample are ...
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34 views
Does the EM algorithm for mixtures still address the missing data issue?
There is a PDF $p(D| \theta)=p(X,Z| \theta)$ with observed values $X$ but also some missing or incomplete values $Z$ (for eg. resulting from censoring).
The expectation-maximization (EM) algorithm is ...
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0answers
72 views
Ftting a mixture of two Gaussians
I want to fit a mixture of two gaussian densities to my financial data. The data can be found here: http://uploadeasy.net/upload/2a7mw.rar the variable is called dat.
The probability density of a ...
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1answer
38 views
critical value of a point mass at zero and a chi square distribution with one degree of freedom
Take a linear model with a block random effect ($b_i$)
$$
y_{ij} = \mu + b_i + \beta x_{ij} + e_{ij}
$$
with $b_i \sim N(0, \sigma_b^2)$, etc etc.
If $\sigma^2_b = 0$, then the model reduces to a ...
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0answers
24 views
If pop's have the same distribution but different parameters, can the merged population also have that distrib?
There is a substantial literature on identifying the best-fitting distribution for income and wealth data. Candidates have included the log-normal, Gamma, Singh-Maddala, Dagum type I and generalized ...
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0answers
40 views
Mixture of binomial distributions
I am experimenting with a mixture of binomial models. Consider a binary variable $y_i$. Furthermore, there are two sub-groups in the population (not known a priori and not observable): $z_i=0$ or ...
2
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0answers
66 views
Orthogonality of Hermite Polynomials
As we know probabilistic Hermite polynomials are orthogonal with respect to the weight function $\frac{1}{\sqrt{2 \pi}} e^{-x^2/2}$ (density of standard normal).
I have a distribution which is a ...
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30 views
Anti-concentration property
we say that a distribution has the anti-concentration property if it is not too concentrated around any specific value.
Let $f(x_1,x_2,. . .x_n) $ be a polynomial of degree $d$ defined over the ...
2
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1answer
81 views
Difference between the two normal distributions
I have two random variables $X$ and $Y$ which follows Normal distribution , whose pdf's are given by
$f(x)= \frac{1}{2 \sqrt{2 \pi} \sigma}[e^{\frac{-(x-1)^2}{2 \sigma^2}}+e^{\frac{-(x+1)^2}{2 ...
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1answer
75 views
Getting started with mixture models
Can you suggest a source (book, lecture notes, etc) that provides a good introduction to mixture modeling? I would like something that discusses (mixes?) theory and application at the graduate level. ...
2
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1answer
54 views
Probability distribution for varying probabilities in R
I'd like to plot the probability distribution for a set of binomial trials in R, the catch is that each trial has an independent probability of success (which I have in vector form).
So, in R if I ...
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0answers
58 views
Constraints on ML for mixture of Gaussians
I have some data sampled from a mixture of two Gaussians where one of them is known, and the density function is as follows:
$f(x, \mu, \sigma) = \frac{1}{2}\frac{1}{\sqrt{2\pi}\sigma} ...
3
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2answers
71 views
Estimating covariance of the difference of directional distributions derived from Gaussian mixtures
Given Gaussian mixtures $X_1, X_2 \in \mathbb{R}^p$ defined as $$P(X_i = x) = \sum_s \omega^{(s)}_i \mathcal{N}(x; \mu^{(s)}_i, \Sigma_i)$$ where the superscript $(s)$ indexes the $s$-th component of ...
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85 views
Merge covariance matrices
I have two 2x2 covariance matrices, stemming from bivariate datasets that are approximately normally distributed. I want to create a mixture distribution and for that I need to merge the covariance ...
4
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1answer
125 views
Latent class model with both continuous and categorical indicators in R
I have question regarding which R-package to use to create a latent class/mixture model
with both categorical and continuous indicator variables. I have not yet found a good example of this using R, ...
2
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1answer
113 views
Chi-square test in finite mixture models
I was trying to figure out whether or not the distribution of a biomarker came from heterogeneous populations. Analyzing the data with normalmixEM in the ...
3
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2answers
139 views
Dependent Bernoulli trials
The probability of a sequence of n independent Bernoulli trials can be easily expressed as
$$p(x_1,...,x_n|p_1,...,p_n)=\prod_{i=1}^np_i^{x_i}(1-p_i)^{1-x_i}$$
but what if the trials are not ...
4
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1answer
160 views
Distribution of arrival times to server for an M/M/1 queue (what the server experiences)
In an M/M/1 queue, we know that inter-arrival times are exponentially distributed, and that service times are the same. What is the distribution of to-server inter-arrival times (aka service start ...
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1answer
142 views
How to use GMM for clustering new data
I applied Gaussian Mixture Model on my data and train the model in matlab. How I can test my model or use it to cluster new data?
Thanks for any answer or comment.
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2answers
152 views
Dimension reduction for sparse matrix for clustering
I'm looking for a Sparse matrix dimension reduction. I already used some feature selection methods like PCA but it doesn't give me good results. I want to apply mixture models for clustering my data.
...
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1answer
29 views
Two-Component distribution modelling
I am trying to plot a two-component mixture distribution $F_X(x):.7N(0,1)+.3N(3,1)$ with density
$$f_X(x) = {0.7 \over {\sqrt{2\pi}}} e^{−x^2/2} + {0.3 \over {\sqrt{2\pi}}}e^{−(x−3)^2/2} $$.
I have ...
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1answer
90 views
Long-tailed distribution of time events
Suppose you have the logs of a web server. In these logs you have tuples of this kind:
...
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0answers
131 views
How can I plot the components of a mixture model obtained from pymix?
I'm trying to wrap my head around mixture modelling, and I've come across a small matlab script that seems relevant. In order to familiarize myself with pymix, I've decided to try rewriting the ...
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0answers
109 views
Problems with basic R functions for a two-normal mixture (EM approach of parameter estimation)
I was trying to learn EM algorithm for simple two-normal mixture models, specially writing basic R function for parameter estimation. But my codes give me the same values that I started with! I can ...
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0answers
77 views
Mixture of multiple probability distribution
This problem is related to data clustering. More specifically, related to mixture model.
I have bivariate data samples
x_i = (x_i1, x_i2)'
where the variables/features are uncorrelated and different ...
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1answer
207 views
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1answer
122 views
Computing confidence region for Gaussian mixture model
I have a 2-d Gaussian mixture model and would like to compute a confidence region for it. Our application is that the two dimensions are latitude and longitude; that is, we want to say something like ...
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0answers
43 views
Blind deconvolution of histogram
I am not sure that this is good forum to ask, but dunno where to ask.
I have a file with histogram data collected from cytometer. The problem is that these data contain several signals and I want to ...
3
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2answers
203 views
How to model data in jags/bugs generated by taking the minimum of two random variables?
I have been thinking of modeling human timing data using jags where the data comes from an experiment where participants tap in time with a very slow metronome. The data is then a number of ...
4
votes
1answer
295 views
What is the distance between a finite Gaussian mixture and a Gaussian?
Suppose I have a mixture of finitely many Gaussians with known weights, means, and standard deviations. The means are not equal. The mean and standard deviation of the mixture can be calculated, of ...
3
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0answers
113 views
Calculating the Fisher Information of bivariate normal
I'm lost. If I estimated the a Gaussian mixture model, with a shared diagonal covariance, will the Fisher information of the means be $\Sigma^{-1}$ ?
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0answers
158 views
Separation of multiple overlapping normal distributions
The problem is separation of multiple normal distributions of fish length, each normal distribution representing a distinct year class contributing to the overall population in a long-lived species. ...
1
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1answer
591 views
How do you calculate the expected value of mixed lognormal distribution?
Suppose $X=\log(Y)$ can be modeled by a mixture of two normal distributions with proportion $p$ of $X_1$ and proportion $1-p$ of $X_2$, where $X_1\sim\mathcal N(U_1, \sigma^2_1)$ and $X_2\sim\mathcal ...
3
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2answers
838 views
Connection between sum of normally distributed random variables and mixture of normal distributions
If you have two independent random variables that are normally distributed (not necessarily jointly so), then their sum is also normally distributed, which e.g. means that its excess kurtosis is $0$.
...
3
votes
1answer
191 views
Central moments of a gaussian mixture density?
Given the pdf $f(x) = \sum_i \omega_i \mathcal{N}(x; \mu_i, C_i )$ of a gaussian mixture density, where the $i$-th component has mean $\mu_i$ and covariance matrix $C_i$ and the weights $\omega_i$ sum ...
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0answers
101 views
Gaussian Mixture - Optimal number of components
So, getting an "idea" of the optimal number of clusters in k-means is well documented. I found an article on doing this in gaussian mixtures, but not sure I am convinced by it, don't understand it ...
10
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1answer
202 views
Why is clutter problem intractable for large sample sizes?
Suppose we have a set of points $\mathbf{y} = \{y_1, y_2, \ldots, y_N \}$.
Each point $y_i$ is generated using distribution
$$
p(y_i| x) = \frac12 \mathcal{N}(x, 1) + \frac12 \mathcal{N}(0, 10).
$$
...
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1answer
110 views
Derivation of a the log-likelihood for a regression model where the outcome is a mixture between Poisson and a point mass at zero
Suppose $ \textbf{Y} = (Y_1, \dots, Y_n)'$ are independent and
$$\eqalign{
Y_i = 0 & \text{with probability} \ p_i+(1-p_i)e^{-\lambda_i}\\
Y_i = k & \text{with probability} \ ...
4
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1answer
129 views
Mixture distributions moments if one distribution has undefined/infinite moments
Consider probability density functions $f_{1}\left(x\right)$ and $f_{2}\left(x\right)$ and the mixture distribution
$$f_{3}\left(x\right)\equiv ...
7
votes
3answers
479 views
Mixture Models and Dirichlet Process Mixtures (beginner lectures or papers)
In the context of online clustering, I often find many papers talking about: "dirichlet process" and "finite/infinite mixture models".
Given that I've never used or read about dirichlet process or ...
