Suppose I have a mixture of Gaussians and I know the mean and variance of each separate Gaussian. How can I tell whether or not the resulting distribution will be multimodal or, more specifically, that there will be be a number of modes equal to the number of Gaussians in the mixture? I see that there are some formulas that can answer this question in the case of a mixture of two Gaussians, but I cannot find any literature related to a more general case. I only need to deal with a one dimensional Gaussian if this is of any relevance.
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4$\begingroup$ I believe the situation is too complicated to admit a general formula that is any simpler than actually finding the desired number of modes. The only mitigating factor is that numerical searches should work well because (a) all the components are highly differentiable everywhere and (b) you can make reasonable initial estimates of modes based on the modes of the components. $\endgroup$– whuber ♦Commented Jan 2, 2015 at 22:55
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2 Answers
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I doubt you will find an explicit solution. The number of modes is the number of roots of the first derivative of the density function - which is quite messy - I am quite sure you can't find an explicit solution.
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$\begingroup$ That was my first thought on the matter. After more searching I am thinking that the problem may not have an explicit solution. $\endgroup$– mjnicholCommented Jan 2, 2015 at 23:16
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1$\begingroup$ Worked out the formula myself and am convinced that the equation is transcendental. Thanks for the help! $\endgroup$– mjnicholCommented Jan 2, 2015 at 23:36
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Miguel Carrera-Perpinan has a webpage on this topic with associated software. This does not directly solve your question, but indicates that
- unidimensional Gaussian mixtures with $k$ components have at most $k$ modes;
- unidimensional non-Gaussian mixtures with $k$ components may have more than $k$ modes;
- multidimensional mixtures with $k$ components may have more than $k$ modes.
In the unidimensional Gaussian, a component will induce a mode near its mean if the variance is small enough (relative to the other variances) and if the weight is large enough (relative to the other weights). This may suggest a specific numerical way to check for modes by changing the variance of one component until a mode appears or disappears near a component mean.
Obviously, running EM algorithms starting from each normal mean should indicate whether or not this component induces a local mode.
Else, you may always follow Larry Wasserman's advice: "mixtures, like tequila, are inherently evil and should be avoided at all costs."
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1$\begingroup$ +1 The suggestion to vary one variance at a time appears likely to be quite effective. $\endgroup$– whuber ♦Commented Jan 3, 2015 at 16:24