using StatsOrdinalPatterns, ComplexityMeasures, Random
Random.seed!(1)
x = randn(500)ComplexityMeasures.jl API
StatsOrdinalPatterns.jl builds on ComplexityMeasures.jl. The entropy statistics are the types Shannon and ShannonExtropy of ComplexityMeasures.jl, and the ordinal pattern probabilities of both packages coincide: both number the \(m!\) patterns in the same lexicographic order. ComplexityMeasures.jl estimates probabilities and complexity measures. This package adds the statistical inference on top of them: asymptotic and bootstrap tests, critical values, control charts and their run lengths.
The statistics of this package can therefore also be computed through the functions complexity and information of ComplexityMeasures.jl. These methods are an additional interface. stat_op and stat_sop remain the main interface of the package, since they also return the pattern probabilities and the EWMA statistics that the tests and control charts need.
Information measures and complexity measures
ComplexityMeasures.jl distinguishes two kinds of measures. An information measure is a functional of a probability mass function: it must be computable from any probability vector, by information(measure, probabilities). Entropies and extropies such as Shannon and ShannonExtropy are of this kind. Everything else is a complexity measure, a subtype of ComplexityEstimator, computed by complexity(measure, data).
All chart types of this package are complexity measures. The statistics of Bandt (2019) and the SOP statistics need the probabilities of particular patterns, and the distance to white noise \(\widehat{\Delta} = \sum_i (\hat p_i - 1/m!)^2\) needs the total number \(m!\) of patterns, including those that do not occur. \(\widehat{\Delta}\) is a disequilibrium in the sense of StatisticalComplexity of ComplexityMeasures.jl, which is a complexity measure for the same reason. The entropy statistics of this package use the information measures Shannon and ShannonExtropy of ComplexityMeasures.jl.
Correspondence of the calls
| Statistic | This package | ComplexityMeasures.jl |
|---|---|---|
| \(\hat{\beta}\), \(\hat{\tau}\), \(\hat{\gamma}\), \(\hat{\delta}\) | stat_op(x; chart_choice=Persistence(), d=d)[1] |
complexity(Persistence(), probabilities(OrdinalPatterns{3}(d), x)) |
| same, delay 1 | stat_op(x; chart_choice=Persistence())[1] |
complexity(Persistence(), x) |
| \(\widehat{H}\) | stat_op(x; chart_choice=Shannon(base=exp(1)), m=m, d=d)[1] |
information(Shannon(base=exp(1)), OrdinalPatterns{m}(d), x) |
| \(\widehat{H}_{\text{ex}}\) | stat_op(x; chart_choice=ShannonExtropy(base=exp(1)), m=m, d=d)[1] |
information(ShannonExtropy(base=exp(1)), OrdinalPatterns{m}(d), x) |
| \(\widehat{\Delta}\) | stat_op(x; chart_choice=DistanceToWhiteNoise(), m=m, d=d)[1] |
complexity(DistanceToWhiteNoise(), probabilities(OrdinalPatterns{m}(d), x)) |
| \(\hat{\tau}\), \(\hat{\kappa}\), \(\tilde{\tau}\), \(\tilde{\kappa}\) (SOP) | stat_sop(X, 1, 1; chart_choice=TauTilde())[1] |
complexity(TauTilde(), X) |
Persistence() in the first two rows stands for any of the four statistics of Bandt (2019): UpDownBalance, Persistence, RotationalAsymmetry and UpDownScaling. They require patterns of length 3, and UpDownBalance also accepts length 2. TauTilde() stands for any of the SOP statistics TauHat, KappaHat, TauTilde and KappaTilde.
The statistics of Bandt (2019)
complexity with a time series uses patterns of length 3 and delay 1. Other delays go through the ordinal pattern probabilities of ComplexityMeasures.jl:
p = probabilities(OrdinalPatterns{3}(2), x) # patterns of length 3, delay 2
(complexity_route=complexity(Persistence(), p),
stat_op_route=stat_op(x; chart_choice=Persistence(), m=3, d=2)[1])(complexity_route = -0.018817204301075252, stat_op_route = -0.018817204301075252)Entropy statistics
Shannon and ShannonExtropy are information measures of ComplexityMeasures.jl, so information computes them directly:
o = OrdinalPatterns{3}(1)
[(statistic=name,
information_route=information(e, o, x),
stat_op_route=stat_op(x; chart_choice=e, m=3, d=1)[1])
for (name, e) in (("H", Shannon(base=exp(1))), ("H_ex", ShannonExtropy(base=exp(1))))]2-element Vector{@NamedTuple{statistic::String, information_route::Float64, stat_op_route::Float64}}:
(statistic = "H", information_route = 1.785891169486249, stat_op_route = 1.785891169486249)
(statistic = "H_ex", information_route = 0.9104014512723928, stat_op_route = 0.9104014512723928)Note that Shannon() and ShannonExtropy() use base 2 by default. The papers and the asymptotic tests use the natural logarithm; see Logarithm base.
The distance to white noise
DistanceToWhiteNoise is a complexity measure and accepts any pattern length:
p = probabilities(OrdinalPatterns{4}(1), x) # patterns of length 4, delay 1
(complexity_route=complexity(DistanceToWhiteNoise(), p),
stat_op_route=stat_op(x; chart_choice=DistanceToWhiteNoise(), m=4, d=1)[1])(complexity_route = 0.001833772588583142, stat_op_route = 0.001833772588583142)Spatial ordinal patterns
For a matrix, complexity computes the SOP statistics of Weiß and Kim (2024) with the classical SOP classification at delays \((1, 1)\):
X = rand(20, 20)
(complexity_route=complexity(TauTilde(), X),
stat_sop_route=stat_sop(X, 1, 1; chart_choice=TauTilde())[1])(complexity_route = 0.0018467220683287544, stat_sop_route = 0.0018467220683287544)Other delays, the refined classifications and the entropy statistics of spatial ordinal patterns are available through stat_sop. The spatial outcome space SpatialOrdinalPatterns of ComplexityMeasures.jl classifies patterns differently, so its probabilities cannot be used for the SOP statistics.
Agreement of the two packages
Both routes compute the same numbers, not merely similar ones. The test suite of the package checks this in test/test_complexitymeasures_api.jl: for time series with different dependence structures, pattern lengths 2 to 4 and delays 1 to 4, it compares
- the ordinal pattern probabilities of both packages,
complexitywithstat_opfor the four statistics of Bandt (2019) and forDistanceToWhiteNoise,informationwithstat_opforShannonandShannonExtropy, in several logarithm bases,- that every chart type of the package is a complexity measure and none an information measure,
complexitywithstat_sopfor the four SOP statistics.
The probabilities agree for data without ties. When values are tied, ComplexityMeasures.jl orders them at random, while stat_op orders them by position. For discrete data, stat_op offers add_noise=true to break ties.
References
Bandt, Christoph. 2019. “Small Order Patterns in Big Time Series: A Practical Guide.” Entropy 21 (6): 613. https://doi.org/10.3390/e21060613.
Weiß, Christian H, and Hee-Young Kim. 2024. “Using Spatial Ordinal Patterns for Non-Parametric Testing of Spatial Dependence.” Spatial Statistics 59: 100800.