Entropy Test: Identifying the Dimension Where Deterministic Chaos is alive in Financial Markets

Sergey Okun This article “Entropy test: identify the dimenstion where deterministic chaos is alive in financial markets” was written by Sergey Okun – Senior Financial Analyst at I Know First, Ph.D. in Economics.

(Source: flickr.com)

Highlights:

  • The approximate entropy test enables us to identify when deterministic chaotic patterns start emerging in financial assets.
  • There are non-linear dependence patterns in the 3-day interval for the S&P500, precious metal ETFs, volatility ETFs, debt market ETFs, real-estate ETF, US dollar ETF, and cryptocurrencies.
  • We could not reject the hypothesis of linear dependence for the platinum ETF.
  • The I Know First AI algorithm allows for identifying non-linear dependencies in financial assets to find the most promising investment opportunities.

In general, randomness is defined as lack of patterns. We would say that a market is somehow predictable if it always follows the same price patterns, and totally random if there is no repetition of the patterns and the participants buy or sell without any way to determine their next move. While entropy is a measurement of disorganization, the information determines a measure of certainty. Approximate entropy allows us to detect dependencies and deterministic chaotic patterns in financial time series.

In our previous article, we tested the S&P500 and its ETF SPDRs, and we found that all of them featured non-linear dependence patterns in the 4-day interval. In this article, we aim to identify the existence of deterministic chaotic patterns in a broad range of financial assets.

AssetDescriptionData
S&P500Tracking the stock performance of 500 large companies listed on stock exchanges in the United States.Oct 18th, 2022 – Mar 14th, 2023
ETF GLDGLD tracks the gold spot price, less expenses and liabilities, using gold bars held in London vaults.Oct 18th, 2022 – Mar 14th, 2023
ETF SLVSLV tracks the silver spot price, less expenses and liabilities, using silver bullion held in London.Oct 18th, 2022 – Mar 14th, 2023
ETF PPLTPPLT tracks the platinum spot price, less trust expenses, using platinum bullion.Oct 18th, 2022 – Mar 14th, 2023
BTCBitcoinDec 4th, 2022- Mar 14th, 2023
ETHEthereumDec 4th, 2022- Mar 14th, 2023
ETF VNQ VNQ tracks a market-cap-weighted index of companies involved in the ownership and operation of real estate in the United States.Oct 18th, 2022 – Mar 14th, 2023
ETF VIXYVIXY tracks an index with exposure to futures contracts on the CBOE Volatility Index with average one-month maturity. Exposure resets daily.Oct 18th, 2022 – Mar 14th, 2023
ETF VIXMVIXM tracks an index of futures contracts on the CBOE Volatility Index with an average of five months until maturity. Exposure resets daily.Oct 18th, 2022 – Mar 14th, 2023
ETF BILBIL tracks a market-weighted index of all publicly issued zero-coupon US Treasury bills with a maturity of at least 1 month, but less than 3 months.Oct 18th, 2022 – Mar 14th, 2023
ETF IEFIEF tracks a market-value-weighted index of debt issued by the US Treasury with 7-10 years to maturity remaining. Treasury STRIPS are excluded.Oct 18th, 2022 – Mar 14th, 2023
ETF IGIBIGIB tracks a market-value-weighted index of USD-denominated, investment-grade corporate debt with maturities between 5-10 years.Oct 18th, 2022 – Mar 14th, 2023
ETF UUPUUP tracks the changes in value of the US dollar relative to a basket of world currencies via USDX future contracts.Oct 18th, 2022 – Mar 14th, 2023
*Source: etf.com and Yahoo Finance
(Table 1: Assets Description)

The advantage of the approximate entropy test is that it can be successfully implemented in small samples. This test enables us to identify regularity in binary signals (price goes up or goes down) and determine if financial asset returns exhibit some non-linear dependence. The entropy test is based on the information theory and the test measures the amount of information that each new observation releases to the observer. We can compare the amount of information, or at least approximate, that one additional observation of our series releases to us, and compare it to the amount of information that we would obtain from a purely independent or random series.

For instance, the entropy of a fair coin toss is one bit, or in logarithmic entropy that would be log 2 because we have no pressure position what is more likely to happen before the toss – it is a head or it is a tail. Similarly, we can apply the same logic to stock returns and code them zero and ones regarding their particular patterns.

Entropy Test: Indicator Function for Dimansion
m – indicator Function for Dimension m; m – Dimension; N – Number of Observations; r – Median of Distance Returns; X – vector of distance returns
(Equation 1: The Indicator Function)
Entropy test: approximate entropy for different dimansions
(Equation 2: Approximate Entropy for Different Dimensions)
Entropy test: Chi-square
s – Number of Stages; X2 – Chi-Square
(Equation 3: The Entropy Test)

We try to mimic the logic of theoretical entropy by calculating the indicator function in Equation 1 which is equal to 1 if observations are close apart and zero if they are far apart from the median return distance. Equation 2 is the difference between values of the aggregated correlation integrals between different embedding dimensions. It allows us to proxy the approximate entropy. Equation 3 allows us to calculate the difference between the entropy which we observe in our data and the entropy that we would observe if the data were perfectly random that is the entropy of a coin toss. We assume that our entropy would be equal to or less than the entropy of a coin toss or in our case the median return distance.

We implement the approximate entropy test by using 100 observations of series returns. The number of stages is equal to 2, if an observation has a distance return close to the median distance return, it gets to stage 1, and 0 otherwise. Below, we can notice entropy evaluations for different lags, where m=2 is the 1-day lag, m=3 is the 2-day lag and etc.

Entropy test of financial assets
(Table 2: Approximate Entropy Test for Financial Assets)

According to Table 1, we can notice that BTC has non-linear dependence patterns for the dimension of 2 (1 day) with a confidence level of 95%. Also, we cannot estimate the p-value for the dimension of 2 (1 day) for the S&P500 index, SLV, PPLT, ETH, VNQ, VIXY, VIXM, BIL, IGIB, and UUP, because Equation 3 does not allow us to correctly approximate the Chi-square, which gets negative numbers. Therefore, we cannot say that there are non-linear dependences or deterministic chaos in the dimension of 2 for these assets. If we change to the dimension of 3 (2 days), we can conclude that there are non-linear dependency patterns in GLD, SLV, BTC, VNQ, VIXY, VIXM, IEF, and UUP with a confidence of 95%, and in the S&P500, BIL, and IGIB with a confidence of 90%. All assets have non-linear dependence patterns for the dimension of 4 or 3 days with a confidence of 95%.

Ordering Chaos with I Know First

The I Know First AI algorithm allows for identifying non-linear dependencies in financial assets to find the most promising investment opportunities. Having non-linear dependencies in financial data series tells us that implying sophisticated strategies enables an investor to systematically beat the market and extract additional profit from its investment positions. Previously in our articles, we discussed market chaos, the conceptual framework of applying ML and AI models to analyze and forecast financial assets, and we examined the presence of long-term memory in world stock market indices and the chaotic behavior of the S&P500 index in the pandemic time.

The I Know First predictive algorithm is a successful attempt to discover the rules of the market that enable us to make accurate stock market forecasts. Taking advantage of artificial intelligence and machine learning and using insights into chaos theory and self-similarity (the fractals), the algorithmic system is able to predict the behavior of over 13,500 markets. The key principle of the algorithm lies in the fact that a stock’s price is a function of many factors interacting non-linearly. Therefore, it is advantageous to use elements of artificial neural networks and genetic algorithms. How does it work? At first, an analysis of inputs is performed, ranking them according to their significance in predicting the target stock price. Then multiple models are created and tested utilizing 15 years of historical data. Only the best-performing models are kept while the rest are rejected. Models are refined every day, as new data becomes available. As the algorithm is purely empirical and self-learning, there is no human bias in the models and the market forecast system adapts to the new reality every day while still following general historical rules.

Basic Principle of the "I Know First" Predictive Algorithm

I Know First has used algorithmic outputs to provide an investment strategy for institutional investors. Below you can see the investment result of our S&P 500 Stocks package which was recommended to our clients for the period from January 1st, 2020 to March 8th, 2023 (you can access our forecast packages here).

The Investment Result for the period from January 1st, 2020 to March 8th, 2023

The investment strategy that was recommended by I Know First accumulated a return of 78.06%, which exceeded the S&P 500 return by 54.67%.

Conclusion

The appropriate entropy test enables us to identify the existence of non-linear dependencies in different dimensions of financial data series. We have estimated approximate entropy for different kinds of financial assets and we have noticed that the hypothesis of linear dependence could be rejected for all analyzed assets with a confidence of 95%. Consequently, implementing sophisticated portfolio strategies by investors allows them to beat the market and extract additional profit.

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