nonlinear manifold learning for financial markets...

Nonlinear Manifold Learning for Financial Markets Integration

George Tzagkarakis1 & Thomas Dionysopoulos1,2

1 EONOS Investment Technologies, Paris (FR)2 Dalton Strategic Partnership, London (UK)

DLM International Workshop

Nice, 4 – 6 Sept 2017

Motivation


Nice, 4 – 6 Sep 2017

Extract meaningful information

from financial data

Build smart trading

strategies

Quants Traders

Investment process

Risk premia vs Risk factors

Manifold learning for financial data

Detection of critical transitions in financial markets

Conclusions

Overview


Nice, 4 – 6 Sep 2017

Investment process


Nice, 4 – 6 Sep 2017

Views(How do I produce my market signals)

Portfolio Construction(How do I aggregate my views together to

positions)

Dynamic Budgeting(How does my portfolio evolve in time)

• MVP [MinVar Portfolio]: lowest possible risk (volatility); concentration in low volatility asset classes

• MCP [MinCorr Portfolio]: lowest volatility weighted average corrcoeff between asset classes; asset classes with low corr and volatility relative to other asset classes within the portfolio receive higher weight

• RPP [Risk Parity Portfolio]: asset classes contribute the same amount of risk (volatility) to the overall portfolio; assets with lower risk (e.g. bonds) get a larger part of the portfolio than risky ones

• Driver-based models: identify mathematical relationship between business drivers and study their financial outcomes under certain operational decisions

• Rolling forecasting: continuous look forward on N-month basis updated monthly/quarterly to adjust positions in order to reach the financial goals

Repeated pattern

Statistics vs ThresholdMovAvg, MovVol, …

{-1, 1}

Investments for the long haul

Buy-and-Hold mentality

Resist the temptation to react or predict the stock market’s every next move

Successful passive investors keep their eye on the prize (returns) and ignore short-term setbacks, even sharp downturns

Passive vs Active investing


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Beat the stock market’s average returns and take full advantage of short-term price fluctuations

Involves a much deeper analysis and expertise to decide when to pivot into or out of a particular asset

Accurate determination of when and where prices change will be critical

Passive Active

Understand the true drivers of returns (From assets to small number of common factors)

Importance of true diversification/stability as opposed to decorrelation

Time-varying risk budgeting

Challenges of active investing


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Investment process




Conclusions

Overview


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Commodities

Equities

Forex

Bonds

Interest rates

Assets Systematic Exposure

From assets to risk premia


Nice, 4 – 6 Sep 2017

…

Momentum

Mean Reversion

Value

Volatility

Low Risk

Liquidity

…

RISK PREMIA

[A market segmentation]

Investors treat markets as a purely economic system

Investment decisions are made empirically based on economic interpretations of the markets

Premia vs Factors


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Segment market without relying on economic interpretations Exploit the power of mathematical and signal processing tools

Concept: Focus on achieving orthogonality

Total risk = Sum of marginal risks Risk Decomposition

Segmentation of market lacks economic interpretation BUT we are able to extract hidden information

Green World

(e.g. equities)Investors

Blue

Yellow

Quants

Commodities

Equities

Forex

Bonds

Interest rates

Assets Dimensionality Reduction

From assets to risk factors


Nice, 4 – 6 Sep 2017

…

PCA, PPCA

Fourier Transform

Wavelet Transform

Machine Learning

Pattern Analysis

Manifold Learning

…

RISK FACTORS

[Market portfolios]

We treat markets as a mathematical system

Investment decisions are made based on market information extracted via DSP/ML/… methods

Chasing the decorrelation


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[Invariance]

Time Seriesto

Returns

From ReturnsTo

Smooth Manifolds

The desired portfolio is NOT the solution

to some optimization problem but an

invariant of a dynamical system

Investment process




Conclusions

Overview


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Optimal portfolio as an invariant of a dynamical system

Optimal portfolio as an invariant


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Traditional view(optimization-based)

Utility function: max𝐄 𝒓𝒆𝒕𝒖𝒓𝒏𝒔 +min 𝑹𝒊𝒔𝒌Solution: straight line

Modern view(invariant-based)

Invariance with respect to the 1st (returns) and 2nd

(Var) derivatives(changes of these two derivatives affect portfolio performance)

Allow more features of the portfolio to remain invariant (e.g. correlations (= angles) between assets)

Financial data as dynamical systems


Nice, 4 – 6 Sep 2017

Time-Delay Embedding

Financial Time Series

Manifold Learning

Early Warning

• Financial markets are highly complex,nonlinear dynamical systems

• Financial time series are comprehensivereflections of market condition/operationsand provide the ground for market analysis

• Dynamic nature of original system is oftencorrupted by irrelevant components thatdisturb useful intrinsic features

• Extract underlying manifold structure thatgoverns the dynamical system; embeddingin a more stable/smooth low-dimensionalspace

Taken’s theorem: complete information about the hidden state of dynamical systems can be preserved in observed time series

Phase space reconstruction


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Critical Parameters

τ Delay

m Embedding dimension

N ( = n - (m - 1)τ ) Number of states

1st min of Average Mutual Information

1st min of False Nearest Neighbors (%)

AM

I

Time lag

FN

N (

%)

Dimension

Information-based manifold learning


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Obtain the attractor manifolds by preserving geodesic distances between points in the state space

Classical Manifold Learning

Data points in state space

Data representation via probability distributions (e.g. risk quantification)

Considering only the geometric structure of a data space hides essential characteristics of the data and destroys the proximity relations (topology) of the original data space

Financial Practice

PDFs of data points in state space

Measure informationchange between data points



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Sta

te (p

has

e) s

pac

e

Kernel Density Estimator

K: Gaussian kernelh: plug-in bandwidth selection

Information SimilarityGlobal Relationship

Matrix

Extract an extended local linear structure (adjacency doesnot entirely depend on the states’ geometric relations)while retaining the global topological characteristics in theinherent low-dimensional manifold

Employ locally linear embedding (LLE)Maps its inputs into a single global coordinate system of lower dimensionality

Optimizations do not involve local minima

Recovers global nonlinear structure from locally linear fits; local geometry of locally linear patches characterized by linear coefficients that reconstruct each data point from its neighbors



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Embedding Cost FunctionTranslation-free embedding

Rotation/Scaling-free embedding

Solution: d eigenvectorscorresponding to the smallest

d eigenvalues of A

Investment process




Conclusions

Overview


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Hard to predict accurately the market shifting points

Detect gradual increase of transition points’ likelihood

HMM classifier on the learned manifold of probability distributions 3 states (classes) [High, Medium, Low risk]

Early warning for market transitions


Nice, 4 – 6 Sep 2017

Learned Manifold

Initial State Probability Distribution (GMMs)

Construct State Transition Matrix P

Compute Posterior Probability

Classification of the corresponding time series point (end of phase space

vectors)

S&P500 Index: daily closing prices in the period 2005-2016

Estimated phase space parameters: m = 8, τ = 23

Manifold learning/Early warning over sliding windows: length = 250, step = 25

3 warning states (posterior thresholds): 50%-70% Low, 70%-90% Medium, 90%-100% High

Early warning for market transitions


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Manifold of S&P500

Early warning signals

Pre-crisis period Crisis period

• High posteriors concentratedin 2007-2009

• Medium posteriors (earlysigns) in 2005-2006 (US realestate bubble)

Investment process




Conclusions

Overview


Nice, 4 – 6 Sep 2017

Take home messages

Manifold learning (ML) is an efficient framework for unveiling intrinsic dynamic structure of financial systems

Traditional geometry-based ML methods are not proper to handle investors’ probabilistic (risk-based) view

Information distance-based ML measures more complex relationships between financial data in a phase space

ML coupled with a conventional HMM enabled accurate identification of critical market transitions, providing reliable early warnings for investors

Future work: Study the effect of differential curvature of a financial system through its attractor manifold

as an indicator of market resilience against external disturbances Examine alternative manifold learning techniques and distance measures adapted to

financial data


Nice, 4 – 6 Sep 2017


Nice, 4 – 6 Sep 2017

nonlinear manifold learning for financial markets...

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