What is a Student-T Mixture Model?
A probabilistic model with heavier tails
Why are there often black swan events in the stock market?
How come 100-year events seem to happen more frequently than once ever hundred years?
The unexpected is more common than we think, we need tools to handle this in time series data.
The Problem with Normal Distributions
Most forecasting models assume data follows a normal (Gaussian) distribution - that bell curve we all learned about in high school. But real-world time series data rarely fits.
Back when I worked as a day trader, I saw firsthand how markets can make "impossible" moves (these were often the best days for our style of discretionary trading).
A stock that should move 2% in a day based on its historical volatility suddenly drops 10% (a 5σ event). According to normal distribution models, such events should happen once every few centuries - yet I've seen them multiple times in my 7 year trading career.
This means we can't always assume normal distributions in certain time series.
Enter the Student-T Distribution
The Student-T distribution is like the normal distribution's more realistic cousin.
It has:
A similar bell-shape in the middle
Much "fatter tails" - meaning it assigns higher probability to extreme events
A parameter called "degrees of freedom" that controls exactly how fat those tails are
This makes it better suited for modeling data where outliers and extreme values happen more often than a normal distribution would predict.
What Makes it a "Mixture" Model?
A mixture model combines multiple probability distributions, allowing us to represent complex patterns that a single distribution can't capture.
Think of it like this: Instead of forcing all your data to fit into one pattern, you're saying "some of my data points come from this distribution, and others come from that distribution."
The model learns:
The parameters of each distribution
The probability of a data point belonging to each distribution
This makes for a model that can be more robust and generalize to different types of time series.
Putting It All Together: The Student-T Mixture Model
A Student-T Mixture Model (SMM) combines multiple Student-T distributions to create a flexible, robust forecasting model that can:
Handle multiple patterns in your data
Better represent the uncertainty in predictions
Account for outliers and extreme events naturally
Adapt to changing volatility over time
For time series forecasting, this means more realistic prediction intervals - instead of being overly confident about the future, the model acknowledges the true range of possibilities, including the occasional extreme event.
View all related code here on my GitHub.