Abstract
We use the definition of statistical symmetry as the invariance of a probability distribution under a given transformation and apply the concept to the underlying probability distribution of stochastic processes. To measure the degree of statistical asymmetry, we take the Kullback–Leibler divergence of a given probability distribution with respect to the corresponding transformed one and study it for the Gaussian autoregressive process using transformations on the temporal correlations’ structure. We then illustrate the employment of this notion as a time series analysis tool by measuring local statistical asymmetries of foreign exchange market price data for three transformations that capture distinct autocorrelation behaviors of the series—independence, non-negative correlations and Markovianity—obtaining a characterization of price movements in terms of each statistical symmetry.
Highlights
A basic definition of symmetry is invariance under transformation
We perform the same local statistical asymmetry analysis described in the previous section with path length m + 1 = 3 and window size w = 1000 on the market mid-quote changes time series, which are expressed in tick time, i.e., the time steps with no price change are removed
For the total independence statistical symmetry, at the beginning of the day, the price change time series presents important deviations from the symmetric process and cannot be characterized as independent, but shows statistically independent intervals after this initial period (Figure 5c); we highlight that 5 December 2011 is a Monday, when the market is reopened after the weekend, possibly explaining the initial relatively large asymmetric behavior of price changes
Summary
A basic definition of symmetry is invariance under transformation. The concept of symmetry is fundamental in mathematics, and it is readily evoked in Euclidean geometry, where a shape is symmetric if it remains unchanged after applying an operation such as a rotation about a point or a reflection with respect to a line [1]. We begin by specifying the statistical asymmetry measure as the Kullback–Leibler divergence of the probability distribution with respect to the transformed one, followed by the characterization of the Gaussian autoregressive process. Such a stochastic process is governed by a multivariate Gaussian distribution only specified by a covariance matrix and has a simple expression for the statistical asymmetry measure. Using this model as a linear approximation for the foreign exchange market price data, we exemplify the use of statistical symmetries to analyze real time series
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