Signatures of Quantum-like Chaos in Dow Jones Index

Scientific Paper
Title	Signatures of Quantum-like Chaos in Dow Jones Index
Read in full	Link to paper
Author(s)	A Mary Selvam
Keywords	inverse power law spectra, Dow Jones index, chaos, fractals, nonlinear dynamics, 1/f noise, self-organized criticality
Published	2003
Journal	Apeiron
Volume	10
Number	4
No. of pages	28

Read the full paper here

Abstract

Dow Jones Index time series exhibit irregular or fractal fluctuations on all time scales from days, months to years. The apparently irregular (nonlinear) fluctuations are selfsimilar as exhibited in inverse power law form for power spectra of temporal fluctuations. Inverse power law form for power spectra of fractal fluctuations in space or time is generic to all dynamical systems in nature and is identified as self-organized criticality. Selfsimilarity implies long-range space-time correlations or non-local connections. It is important to quantify the total pattern of fractal fluctuations for predictability studies, e.g., weather and climate prediction, stock market trends, etc. The author has developed a general systems theory for universal quantification of the observed inverse power law spectra in dynamical systems. The model predictions are as follows.

The power spectra of fractal fluctuations follow the universal and unique inverse power law form of the statistical normal distribution.
The nonlocal connections or long-range correlations in space or time exhibited by the fractal fluctuations are signatures of quantum-like chaos in dynamical systems.
The apparently irregular geometry of the fractal fluctuations forms the component parts of a unified whole precise geometrical pattern of the logarithmic spiral with quasiperiodic Penrose tiling pattern for the internal structure. Conventional power spectral analyses will resolve the logarithmic spiral pattern as an eddy continuum with progressive increase in eddy phase angle.
Continuous periodogram power spectral analyses of normalised daily, monthly and annual Dow Jones Index for the past 100-years show that the power spectra follow the universal inverse power law form of the statistical normal distribution in agreement with model prediction.

The fractal fluctuations of the non-stationary Dow Jones Index time series therefore exhibit signature of quantum-like chaos on all time scales from days to years.

Overview

A. M. Selvam, a retired atmospheric physicist from the Indian Institute of Tropical Meteorology, applies to a century of Dow Jones data the same "cell dynamical system" model she had earlier applied to weather records, to the spacing of adjacent prime numbers, and to the spacing of non-trivial zeros of the Riemann zeta function. The claim is not that markets obey economic laws of a novel kind but that they obey a universal law of fluctuation shared by every open, dissipative, extended system — fluid turbulence, heartbeats, atmospheric weather and stock prices alike. That law, in her formulation, is that the power spectrum of the fluctuations, when replotted in the right variables, is not merely some inverse power law f^−α with a fitted exponent but specifically the cumulative statistical normal distribution.

The move that gives the paper its title is the identification of these long-range correlations with quantum behaviour. Because the model's eddy amplitudes add while their squares give probability densities, and because the model's spiral geometry ties an increment in wavelength to an increment in phase angle, Selvam argues that macroscale turbulence and macroscale markets exhibit "quantum-like chaos" — not quantum mechanics literally, but the same amplitude-squared statistics and the same wavelength–phase relation known in quantum systems as Berry's phase. This is a far stronger claim than the now-standard econophysics observation that market returns have fat tails, and it is what separates the paper from the mainstream self-organized-criticality literature it otherwise cites at length.

The model and the analysis

Fractals, power laws and self-organized criticality

The introduction sets the context carefully and generously. Irregular fluctuations on all scales are generic; Mandelbrot's fractal dimension, the f^−α spectra of Bak, Tang and Wiesenfeld's self-organized criticality, and the renormalization-group self-similarity of continuous phase transitions are all cited as descriptions of the same fact. Selvam notes that the power law "is a distinctive experimental signature seen in a wide variety of complex systems": fat tails in economics, critical fluctuations in physics, the edge of chaos in biology, Zipf's law in demographics — and that Pareto's power-law distribution of wealth "predates any power laws in physics." She also notes Sornette's suggestion that the observed power law reflects structures similar to the Elliott waves of technical analysis, whose ratios are Fibonacci ratios, and Chen's identification of three-to-four-year persistent cycles in Standard and Poor's indices.

Two open problems are flagged: extracting a clean fractal dimension from a finite non-stationary series is unsolved, and real systems are multifractal — the exponent α drifts with time scale, approaching 1 at long periods — so a single fitted exponent does not characterize the data. Selvam's model is offered as the fix: a quantification that requires no fitted exponent at all.

The cell dynamical system

The model treats a fluctuating system as a hierarchy of eddies, each large eddy forming as an envelope of smaller ones. Its founding relation is

W² = (2/π)(r/R) w*²

where W and R are the root-mean-square circulation speed and radius of the large eddy and w* and r those of the enclosed small eddy. Large eddies grow in unit length steps at unit time intervals, the units being the small-eddy length r and circulation time.

From this Selvam draws her predictions. First, because a large eddy is the average of the enclosed small ones, the central limit theorem makes the eddy energy spectrum normal, so that "the variance represents the probability densities" — the additive amplitudes, squared, give probabilities, which she takes as the quantum-like feature. Second, the dominant peak periods in the broadband spectrum are E_n = T_s(2 + τ)τⁿ, with τ the golden mean (1 + √5)/2 ≈ 1.618 and T_s the primary perturbation scale; for n from −1 to 11 this gives 2.2, 3.6, 5.8, 9.5, 15.3, 24.8, 40.1, 64.9, 105, 170, 275, 445 and 720 units. Third, the ratio r/R also equals the increment dθ in phase angle, so wavelength and phase advance together — the Berry's-phase analogue. Fourth, the overall flow is a logarithmic spiral, W = (w*/k) log z, with k = 1/τ² ≈ 0.382 identified as "the universal constant for deterministic chaos in fluid flows" and 1/k ≈ 2.62 as the steady-state emergence of fractal structure.

The crucial testable step is the change of variable. Since log(wavelength) represents the r.m.s. eddy amplitude, a normalized standard deviation is defined as

t = log L / log T₅₀ − 1

where T₅₀ is the period below which the cumulative contribution to total variance reaches 50%. Plotting cumulative percentage variance against t should then trace the statistical normal distribution — with no free parameters. And since t = 0 corresponds to eddy growth step n = 0, the model predicts T₅₀ = (2 + τ)τ⁰ ≈ 3.6 unit time intervals, whatever the unit.

Data and results

The data are Dow Jones closing prices from 3 January 1900 to 5 June 2000, 27,523 trading days, from the Carnegie Mellon statistics archive. Day-to-day changes were normalized as percentages of the previous day's value, and monthly and annual means built from those. Spectral analysis used Jenkinson's continuous periodogram, a quasi-continuous classical periodogram over 10,000 geometrically spaced periodicities, with variance and normalized phase accumulated from the high-frequency end. The series was cut into 115 daily data sets (lengths 100 to 10,000 days, started from three different origins), 11 monthly sets and 5 annual sets.

The results, by chi-square goodness-of-fit at the 5% level: the variance spectra follow the normal distribution for every daily and monthly set, and for every annual set except the first twenty years (1900–1919). Phase spectra follow it for 100% of the monthly and annual sets but only 66% of the daily sets; where the daily phase spectrum fails globally, variance and phase spectra nonetheless coincide within individual dominant wavebands, which Selvam reads as the Berry's-phase signature. T₅₀ came out close to the predicted 3.6 units for the monthly (≈4.2–4.6 months) and annual (≈3.45–3.55 years) sets and was near-constant across daily set lengths from 100 to 10,000 days — with a marked excursion in daily sets 81 to 108, which Selvam observes "correspond to the period just prior to and following the oil shock of the year 1973."

The conclusion is that the Dow's long-range temporal correlations mean persistence, or long-term memory of short-term fluctuations: "the cumulative integration of short-term fluctuations generates long-term fluctuations," so the eddy continuum "acts as a robust unified whole fuzzy logic network with global response to local perturbations."

Assessment

The paper's real strength is that it makes a parameter-free prediction and then tests it against a large, public, unambiguous data set. Most power-law claims in econophysics fit an exponent and report the fit; Selvam instead predicts a specific functional form and a specific number, T₅₀ ≈ 3.6, and the number recurs across three independent time resolutions and 131 data sets. The reported near-constancy of T₅₀ across daily windows from 100 to 10,000 days is a genuinely non-trivial regularity in a non-stationary series, and the identification of the anomalous window with the 1973 oil shock is a nice, checkable detail rather than a post hoc gloss. Methodologically the analysis is transparent: source, normalization, periodogram method and test statistic are all stated, so the work can be repeated.

The difficulties begin with what the test can distinguish. A cumulative curve plotted against a logarithmic variable and compared to a normal ogive is a forgiving comparison; the chi-square test as applied here is not accompanied by any null model, so we are never shown what a shuffled series, a random walk, or a plain fractional Gaussian noise would produce under the same procedure. Since T₅₀ is defined from the spectrum and then used to normalize the abscissa, the construction is partly self-referential, and it is not demonstrated that a series with no long-range correlation at all would fail the test. Without that control the agreement, however wide, cannot bear the weight placed on it.

The second difficulty is the leap from statistics to physics. That amplitudes add and squared amplitudes give probabilities follows here from the central limit theorem applied to an assumed eddy hierarchy; it does not require, or evidence, anything quantum. Genuine quantum mechanics adds complex amplitudes, which produces interference — negative contributions to probability — and it is that, not the mere squaring of a real amplitude, which distinguishes quantum from classical statistics. Nothing in the Dow analysis tests for interference, and no violation of a Bell-type or Leggett-Garg-type inequality is offered, so "non-local connections" here means only long-range correlation in the ordinary statistical sense. Likewise, the identification of the wavelength–phase relation with Berry's phase rests on an analogy of form, not on a demonstration that the market spectrum has a holonomy in a parameter space. The label "quantum-like" is doing a great deal of work that the analysis does not support.

Third, several central quantities are asserted rather than derived within the paper: the factor 2/π in the eddy relation, the identification of k = 1/τ² as a universal constant of fluid chaos, and the appearance of the golden mean in the peak-period spectrum are all carried over from Selvam's earlier work, and the reader cannot check them here. The golden-mean period sequence, moreover, is never actually tested against the Dow spectra; only the n = 0 member, T₅₀, is examined. A prediction of thirteen dominant periods of which one is checked is a weak test of the sequence, particularly since a geometric sequence with ratio 1.618 is dense enough on a log axis that spectral peaks will often fall near some member of it. And the fitted T₅₀ values for monthly data, 4.2 to 4.6, sit noticeably above the predicted 3.6 — a discrepancy the paper records in its figures but does not discuss.

None of this makes the empirical regularity uninteresting. Read as an econophysics result — that Dow fluctuations show scale-invariant persistence with a stable characteristic variance-partition scale of a few time units, across three resolutions and a century of data — the paper stands on its own and is compatible with the mainstream literature it cites. Read as evidence that markets are quantum-like, it is an argument from analogy that has not yet been given a discriminating test.