Sisi – Page 3 – Sisi Tang

Skewness describes how symmetric distribution is. It is defined as the third moment of the distribution after normalization:

$\mbox{Skew}[X]:= \mathbb{E}[(\frac{X-\mu}{\sigma})^3]$

Kurtosis is a measure of tailedness. The heavier the tail is, the larger its Kurtosis is. Mathematically, it is defined as

$\mbox{Kurt}[X]:= \mathbb{E}[(\frac{X-\mu}{\sigma})^4]=\frac{\mu_4}{\sigma^4}=\frac{\mu_4}{\mu_2^2},$

where $\mu$ is the mean, and $\sigma$ is the standard deviation of the distribution of $X$ ; and $\mu_k$ is $k$ -th central moment.

It can be shown that the Kurtosis of Gaussian distribution is $\mbox{Kurt}[N(0,1)] = 3$ . People usually use excess Kurtosis as the extra Kurtosis of a distribution compared with standard Gaussian distribution. Namely,

$\mbox{Excess Kurt}[X] := \mbox{Kurt}[X] - 3.$

We can create a plot with the square of Skewness as its x-axis and Kurtosis as its y-axis. This plot is called Cullen and Frey graph.

This graph helps us to determine which distribution our data is closest to.

Gaussian mixture model (GMM) is a probability model for a mixture of several Gaussian distributions with possibly different mean and variance.

For example, we can model the 100m race time of all grade 12 students in a high school as two normal distributions: one for female students and one for male students. It is reasonable to expect two groups have different mean and may different variance.

When to use Gaussian mixture model?

1. Data has more than one clusters.
In the following picture, the left one models the data with one normal distribution; the right one models the data by two normal distribution, Gaussian mixture model. Obviously, the right one better describes the data.

Pictures are from https://brilliant.org/wiki/gaussian-mixture-model/

2. Each cluster is theoretically normally distributed.

Theory of Gaussian Mixture Model

1. Gaussian distribution in 1 dimension
Since there are several Gaussian distributions in the GMM. We assign an index to each Gaussian distribution: $k$ for $k=1,2,..., K$ where K is the number of clusters. For a given mean $\mu_k$ and variance $\sigma_k$ , the probability density function is

$p_k(x|\mu_k,\sigma_k) = \frac{1}{\sigma_k\sqrt{2\pi}}exp\left(-\frac{(x-\mu_k)^2}{2\sigma_k^2}\right)$

Above $|$ is not the mathematical conditional expectation, but a statistical way of saying we know true parameters $\mu_k,\sigma_k$ in advance.

2. Gaussian mixture model in 1 dimension
The probability density function of GMM is the weighted average of several Gaussian densities:

$p(x|\mu_k,\sigma_k, k=1,2,...,K) = \sum_{k=1}^K w_k \cdot p_k(x|\mu_k,\sigma_k ),$

where $w_k \in [0,1]$ satisfies

$\sum_{k=1}^K w_k = 1$

Plug in the Gaussian density,

$p(x) = \sum_{k=1}^K w_k \frac{1}{\sigma_k\sqrt{2\pi}}exp\left(-\frac{(x-\mu_k)^2}{2\sigma_k^2}\right)$

Note that this is a density function because its integral on $\mathbb{R}$ is 1.

3. Gaussian mixture model in n-dimension
Let $x \in \mathbb{R}^n$ be an n-dimension multivariate Gaussian random variable with mean vector
$\mathbf{\mu_k}$ and covariance matrix $\Sigma_k$ . Then the probability density function is

$p_k(x|\mathbf{\mu_k} , \Sigma_k ) = \frac{1}{\sqrt{(2\pi)^K|\Sigma_k|}}exp\left(-\frac{1}{2}(x-\mathbf{\mu}_k)^T\Sigma_k^{-1}(x-\mathbf{\mu}_k )\right)$

Then, the probability density function of GMM, which is the weighted average of serveral multivariate Gaussian density, is

$p(x) = \sum_{k=1}^K w_k \frac{1}{\sqrt{(2\pi)^K|\Sigma_k|}}exp\left(-\frac{1}{2}(x-\mathbf{\mu}_k)^T\Sigma_k^{-1}(x-\mathbf{\mu}_k )\right)$

with

$\sum_{k=1}^K w_k = 1$

Training the Model

Suppose that we know the number of clusters $K$ a priori. (The choice of $K$ relies on statistician’s experience.) Then, we can use Expectation Maximization (EM) algorithm to find the parameters $\hat{w_k}, \hat{\mu_k}$ and $\hat{\sigma_k}$ or $\hat{\Sigma_k}$ for multi-dimensional model. Let $K$ be the number of clusters, and $N$ be the number of samples.

Step 1: Initialize

Randomly choose $K$ samples and set them to be the group mean. For example, in the case of $K=2, N=300$ , $\hat{\mu_1} = x_{57}$ , $\hat{\mu_2} = x_{193}$ . (note that this is also valid for multi-dimensional case)
Set all variances (resp. covariance matrices) to be the same value: sample variance (resp. sample covariance matrix). Namely,
$\hat{\sigma_k}^2 = \frac{1}{N}\sum_i^N(x_i-\bar{x})^2,$
where $\bar{x}=\sum_1^Nx_i/N$ .
Set all weights equal to $1/K$ , i.e.,
$\hat{w_k} = \frac{1}{K}.$

Step 2: Expectation

We compute the probability that a sample $x_n$ belongs to cluster $C_k$ .

$\begin{aligned} \gamma_{nk} = &~ \mathbf{P}(x_n \in C_k|x_n, \hat{w_k}, \hat{\mu_k},\hat{\sigma_k} ) \\ = &~ \frac{\hat{w_k}p_k(x_n|\hat{\mu_k},\hat{\sigma_k})}{\sum_{j=1}^K \hat{w_j}p_j(x_n|\hat{\mu_j},\hat{\sigma_j}) } \end{aligned}$

Step 3: Maximization

Update parameters then go back to step 2 until converge

$\hat{w_k} = \frac{\sum_{n=1}^N\gamma_{nk}}{N}$
$\hat{\mu_k} = \frac{\sum_{n=1}^N\gamma_{nk}x_i}{ \sum_{n=1}^N\gamma_{nk} }$
$\hat{\sigma_k} = \frac{\sum_{n=1}^N\gamma_{nk}(x_n-\hat{\mu_k} )^2}{ \sum_{n=1}^N\gamma_{nk} }$
resp.
$\hat{\Sigma_k} = \frac{\sum_{n=1}^N\gamma_{nk}(x_n-\hat{\mu_k} )(x_n-\hat{\mu_k} )^T }{ \sum_{n=1}^N\gamma_{nk} }$

Author: Sisi

Skewness and Tailedness

SIMM

Gaussian Mixture Model