Session Details: This session explains Mixture Models and explores its application to predict an asset’s return distribution and identify outlier returns that are likely to mean revert. The objective of this session is to explain and illustrated the use of Mixture Models with a sample strategy in Python. Who should attend? - Traders/quants/analysts interested in algorithmic trading research - Python/software/strategy developers - Algorithmic/Systematic traders - Portfolio Managers and consultants - Students and academicians Guest Speaker Mr. Brian Christopher Quantitative researcher, Python developer, CFA charterholder, and founder of Blackarbs LLC, a quantitative research firm. Six years ago he learned to code using Python for the purpose of creating algorithmic trading strategies. Four years ago he decided to self publish his research with a focus on practical, reproducible application. Now he continues his open research initiatives for a growing community of traders, researchers, developers, engineers, architects and practitioners across various industries. He attained a BSc in Economics from Northeastern University in Boston, MA and received the Chartered Financial Analyst (CFA) designation in 2016. Access the webinar recording here: https://www.youtube.com/watch?v=o5BFAQK_Acw Know more about EPAT™ by QuantInsti™ at http://www.quantinsti.com/epat/

1. BlackArbs LLC
Brian Christopher, CFA
April 25, 2017

2. Brian Christopher
Quantitative researcher, Python developer, CFA charterholder, and founder of Blackarbs LLC, a quantitative research firm.
Six years ago he learned to code using Python for the purpose of creating
algorithmic trading strategies. Four years ago he decided to self publish his
research with a focus on practical, reproducible application.
Now he continues his open research initiatives for a growing community of
traders, researchers, developers, engineers, architects and practitioners
across various industries.
He attained a BSc in Economics from Northeastern University in Boston,
MA and received the Chartered Financial Analyst (CFA) designation in 2016.
About the Speaker
2

3.  Return prediction is difficult.
 In Time Series Analysis (TSA) we observe multiple phenomena
that violate the rules of stationarity (non-linear processes,
volatility clustering, autocorrelation, etc.)
 Traditional TSA techniques don’t generalize well for asset
return prediction but may be useful in volatility prediction
(ARIMA, ARCH, GARCH)
The Problem with Traditional Time Series Analysis
(Defining the Problem)
3

4. Traditional Issues with Asset Return Prediction
4
Blackarbs credit: SeanAbu.com

5. What Can We Do?
5
 We need a model that doesn’t require stationary data.
 We need a model that can approximate non-linear
distributions.

6. Are mixture models a solution?
6
 Mixture models build on several established concepts.
 Markov models - Sequences where future state
depends only upon current state aka memoryless
processes.
 Hidden Markov models – The true state is unobserved
(hidden) but there are observable factors that give us
information about the true state.
 Expectation-Maximization (E-M) – Class of algorithms
that iterates between computing class parameters and
evaluating the likelihood of the data given those
parameters.

7. What are mixture models and how do they work?
7
 We know that time series exhibit temporary periods where
the expected means and variances are stable through time.
These periods or regimes can be likened to hidden states.
 If we can better estimate an asset's most likely regime or
hidden state, including the associated means and variances,
then our predictive models become more adaptable and will
likely improve.
 If that's the case, then all we need are observable variables
whose behavior allows us to infer the true hidden state(s).

8. What are mixture models and how do they work?
Summary
8
 Think of an asset’s returns as a sequence of states or
“regimes”.
 Each regime has its own descriptive statistics including mean
and volatility.
 Examples include “low-vol” vs “high-vol” regimes.
 The model assumes that over time, the asset’s returns will
transition between regimes based on probability.
 Mixture models try to estimate the sequence of regimes,
each regime’s descriptive attributes, and the transition
probabilities between regimes.
 The most well known is the Gaussian Mixture Model (GMM).

9. What are mixture models and how do they work?
9
 GMM assumes each regime (aka component, class, state) is
generated by a Gaussian process with discoverable
parameters.
 GMM uses E-M under the hood alternating between
estimating the regime parameters (mean, variance) and
likelihood that those parameters could generate the data,
until convergence or another stopping criteria is met.

10. Can GMMs model non-stationary data sets?
10
 Using SPY return data from Yahoo Finance I fit the GMM with
3 components representing low-volatility, neutral, and high-
volatility regimes.

11. Can GMMs model non-stationary data sets?
11
 Notice the 2th hidden state or “regime” has a negative mean
return, and the largest variance, which represents our high-
vol regime. The 1th hidden state looks like our low-vol
regime, and the 0th hidden state is somewhere in between.
 The GMM accounts for non-stationarity by design. It chops
the data into regimes and provides us with an estimate of the
mean and variance for each one.

12. Can GMMs approximate nonlinear data
distributions?
12
 Short answer: Yes
 The top image shows the
GMM trying to fit 2
Gaussians to the data with
predictably bad results.
 The bottom image shows
the GMM trying to fit 16
Gaussians to the data with
more success.
 You can approx. many
complex shapes with
enough Gaussian
components but you risk
overfitting.

13. Strategy Design
13
Now that we have determined that GMMs can overcome the
weaknesses of the TSA models, we can design a trading
strategy to test its application:
 The strategy uses a total of 4 factors to estimate the
sequence of regimes and their parameters.
• The asset’s returns (Yahoo Finance)
• TEDRATE (FRED)
• T10Y2Y (FRED)
• T10Y3M (FRED)

14. Strategy Design
14
 The strategy works like so.
 Feed market data to the GMM.
 Call predict method to extract model estimate of last regime
“label”.
 Once we have the “label” we can get the model estimate of
the mean and variance for that regime.
 We feed the parameter estimates to the JohnsonSU
distribution and draw N samples to construct our 99%
confidence intervals (CIs).
 Any returns that fall outside the estimated CIs are defined as
outliers and assumed to mean revert. This strategy assumes
a “buy” of the etf for M number of steps (days) after the
outlier event of a return falling below the lower CI.

15. Evaluate the Strategy
15
 For a single ETF with a set of parameters, for the walk-
forward period, we can plot the model accuracy as well as an
“equity” plot.
 Below is an example output for SPY:

16. Evaluate the Strategy
16
To really dive in we have to analyze the aggregated results. By
organizing the data properly...
 We can evaluate each ETF individually across lookbacks, steps
or ks
 We can evaluate the ETFs collectively across lookbacks, steps
or ks
 We’ll look at a few examples in the following slides...

17. Evaluate the Strategy
17
We can see that across lookbacks SPY, QQQ, TLT have positive
median returns per event. EFA and EEM have more symmetric
distributions.

18. Evaluate the Strategy
18
We can see a similar distribution pattern here using the
sum_ratio statistic. The sum_ratio statistic is the sum of returns
> 0 divided by the sum of returns < 0. Values greater than 1 are
considered a “success”.

19. Evaluate the Strategy
19
In this example we evaluate the median return for each ETF by
lookback and across steps. Again we see SPY, QQQ, and TLT
dominating. EFA is also mostly > 0 across steps.

20. Evaluate the Strategy
20
In this plot we see the sum_ratio metric is > 1 across multiple
dimensions for SPY, QQQ, TLT, and EEM. GLD’s sum_ratio
depends on lookback. EFA is weak, and EEM suffers from
exposure at 21 steps.

21. Evaluate the Strategy
21
Here we can evaluate the median returns across k (# of
components). Again we see broad success across k for SPY,
QQQ, TLT, and surprisingly EFA. GLD and EEM are more varied
and dependent on lookback.

22. Evaluate the Strategy
22
Here we can evaluate the median returns across k (# of
components). Again we see broad success across k for SPY,
QQQ, TLT, and surprisingly EFA. GLD and EEM are more varied
and dependent on lookback.

23. Evaluate the Strategy
23
Here we see a similar pattern among the ETFs with the
sum_ratio for SPY, QQQ, TLT are > 1 across multiple
dimensions. EEM and GLD show similar variability based on
lookback but with passable performance overall.

24. Conclusions
24
 The secret sauce for GMM accuracy when predicting the
return distribution is choosing the proper sampling
distribution (JohnsonSU vs. Normal)
 US based ETFs showed consistently positive results across
many parameter combinations.
 There was more variability with GLD, EFA, and EEM depending
on metric evaluated, steps (exposure), and k which warrants
an expanded study.
 Must note that there is a risk that these results are simply the
result of being long the US market post 2009!

25. Further Areas to Explore
25
 GMM with large cross section of single-issuer stocks.
 GMM testing with multiple time frames (intraday, weekly,
monthly).
 GMM as basis for volatility strategy (testing volatility trends
post outlier event.
 Are the outlier events themselves randomly distributed? Or are
they clustered?
 GMM as a basis for multi-asset class strategy (combinations of
equity, options, futures)

26. Resources
26
* Introduction to Hidden Markov Models with Python, Networkx
and Sklearn - http://blkarb.se/2l6Lndw
* Intro to Expectation-Maximization, K-Means, Gaussian Mixture
Models with Python, Sklearn - http://blkarb.se/2n1A7kn
* Python Data Science Handbook by Jake Vanderplas -
https://github.com/jakevdp/PythonDataScienceHandbook
* Blackarbs blog – Blackarbs.com/blog
* Blackarbs Github -
https://github.com/BlackArbsCEO/Mixture_Models

27. Contact
27
* Website: Blackarbs.com
* Twitter: @blackarbsCEO
* Linkedin: www.linkedin.com/in/bcrblackarbs/
* Email: bcr@blackarbs.com

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