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GMM Gaussian mixture model
2022-01-27 05:03:07 【Vegetable sheep】
This is my participation 8 The fourth of the yuegengwen challenge 14 God , Check out the activity details ：8 Yuegengwen challenge
Gaussian mixture model GMM
Gaussian Mixture Model Gaussian mixture model , It belongs to a common generation model .
A hybrid model is one that can be used to represent the overall distribution （distribution） contains K Probability model of sub distribution , let me put it another way , The mixed model represents the probability distribution of the observed data in the population , It is a result of K A mixed distribution consisting of sub distributions . The hybrid model does not require the observed data to provide information about the sub distribution , To calculate the probability of observation data in the overall distribution .
Gaussian Mixture Model
We know that the multi-dimensional Gaussian distribution follows the following probability density function ：
Gaussian mixture model （Gaussian Mixture Model, GMM） Is an extension of a single Gaussian probability density function ,GMM It can smoothly approximate the density distribution of any shape .
Gaussian mixture model can be regarded as composed of K A model composed of a single Gaussian model , this K The sub model is the implicit variable of the mixed model .
We can think that the probability density function of Gaussian mixture model can be determined by its k Weighted by a single Gaussian probability density function .
Suppose our sample data is , share Samples , use It means the first one The weight factor of a single Gaussian model , Represents the probability density function of a single Gaussian , Yes ：
obviously ,GMM The parameters of are a set of ,
Look here and you'll find , The value of needs to be determined in advance , It's very important , similar Then you need to make sure .
In the learning of multi-dimensional Gaussian distribution , We know , It can be estimated by maximum likelihood Value , The likelihood function is
about GMM, We assume that each set of sample data is independent , Then the likelihood function is A tired ride , Considering that the probability of a single point is very small , After multiplication, the data will be smaller , It is easy to cause floating point underflow , So we can use log likelihood ：
Use EM Algorithmic solution
First, initialize a set of parameters randomly ：
E Step ：
So-called E Namely Expectation, When we know the model parameters , For implicit variables Expect , As shown below ：
That is to say, data Belong to the first Probability of sub Gaussian model .
M Step ：
Now we have , We can use the maximum likelihood to estimate the parameters of the next iteration ：
repeat E Step sum M Step until convergence
It should be noted that ,EM The algorithm has convergence , But it is not guaranteed to find the global maximum , It is possible to find the local maximum . The solution is to initialize several different parameters for iteration , The one with the best result .
- Mr. Li Hang ——《 Statistical learning method 》
author[Vegetable sheep],Please bring the original link to reprint, thank you.
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