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A tool to conduct recurrence quantification analysis and to create recurrence plots in a massively parallel manner using the OpenCL framework.

Project description

General Information

PyRQA is a tool to conduct recurrence quantification analysis (RQA) and to create recurrence plots in a massively parallel manner using the OpenCL framework. It is designed to efficiently process time series consisting of hundreds of thousands of data points.

PyRQA supports the computation of the following RQA measures:

  • Recurrence rate (RR)
  • Determinism (DET)
  • Average diagonal line length (L)
  • Longest diagonal line length (L_max)
  • Divergence (DIV)
  • Entropy diagonal lines (L_entr)
  • Laminarity (LAM)
  • Trapping time (TT)
  • Longest vertical line length (V_max)
  • Entropy vertical lines (V_entr)
  • Average white vertical line length (W)
  • Longest white vertical line length (W_max)
  • Longest white vertical line length divergence (W_div)
  • Entropy white vertical lines (W_entr)

In addition, PyRQA allows to compute the corresponding recurrence plot and to export it as an image.


PyRQA can be installed via the following command.

pip install PyRQA


Basic Computations

RQA computations are conducted as follows.

from pyrqa.time_series import SingleTimeSeries
from pyrqa.settings import Settings
from pyrqa.neighbourhood import FixedRadius
from pyrqa.metric import EuclideanMetric
from pyrqa.computation import RQAComputation
data_points = [0.1, 0.5, 1.3, 0.7, 0.8, 1.4, 1.6, 1.2, 0.4, 1.1, 0.8, 0.2, 1.3]
time_series = SingleTimeSeries(data_points,
settings = Settings(time_series,
computation = RQAComputation.create(settings,
result =
result.min_diagonal_line_length = 2
result.min_vertical_line_length = 2
result.min_white_vertical_line_lelngth = 2

The following output is expected.

RQA Result:
Minimum diagonal line length (L_min): 2
Minimum vertical line length (V_min): 2
Minimum white vertical line length (W_min): 2

Recurrence rate (RR): 0.371901
Determinism (DET): 0.411765
Average diagonal line length (L): 2.333333
Longest diagonal line length (L_max): 3
Divergence (DIV): 0.333333
Entropy diagonal lines (L_entr): 0.636514
Laminarity (LAM): 0.400000
Trapping time (TT): 2.571429
Longest vertical line length (V_max): 4
Entropy vertical lines (V_entr): 0.955700
Average white vertical line length (W): 2.538462
Longest white vertical line length (W_max): 6
Longest white vertical line length inverse (W_div): 0.166667
Entropy white vertical lines (W_entr): 0.839796

Ratio determinism / recurrence rate (DET/RR): 1.107190
Ratio laminarity / determinism (LAM/DET): 0.971429

Recurrence plot computations can be conducted likewise.

from pyrqa.computation import RecurrencePlotComputation
from pyrqa.image_generator import ImageGenerator
computation = RecurrencePlotComputation.create(settings)
result =

Custom OpenCL Environment

The previous examples use the default OpenCL environment. A custom environment using command line input can also be created.

from pyrqa.opencl import OpenCL
opencl = OpenCL(command_line=True)

The OpenCL platform as well as the computing devices can also be selected using their IDs.

opencl = OpenCL(platform_id=0,
computation = RQAComputation.create(settings,
result =

OpenCL Compiler Optimisations Enablement

OpenCL compiler optimisations are disabled by default to ensure the comparability of computing results. They can be enabled to leverage additional performance improvements.

computation = RQAComputation.create(settings,
                                    variants_kwargs={'optimisations_enabled': True})
result =

Adaptive Implementation Selection

Adaptive implementation selection allows to automatically select well performing implementations regarding RQA and recurrence plot computations. It dynamically adapts to the current computational scenario as well as OpenCL devices employed. The selection is performed using one of multiple selection strategies, each referred to as selector. They rely on a set of customized implementation variants, which may be parameterized using a set of keyword arguments called variants_kwargs.

from pyrqa.variants.rqa.fixed_radius.column_materialisation_bit_no_recycling import ColumnMaterialisationBitNoRecycling
from pyrqa.variants.rqa.fixed_radius.column_materialisation_bit_recycling import ColumnMaterialisationBitRecycling
from pyrqa.variants.rqa.fixed_radius.column_materialisation_byte_no_recycling import ColumnMaterialisationByteNoRecycling
from pyrqa.variants.rqa.fixed_radius.column_materialisation_byte_recycling import ColumnMaterialisationByteRecycling
from pyrqa.variants.rqa.fixed_radius.column_no_materialisation import ColumnNoMaterialisation
from pyrqa.selector import EpsilonGreedySelector
computation = RQAComputation.create(settings,
                                    variants_kwargs={'optimisations_enabled': True})
result =

Floating Point Precision

It is possible to specify the precision of the time series data, which determines the precision of the computations using the OpenCL devices. Currently, the following precisions are supported:

  • Half precision (16 bit)
  • Single precision (32 bit)
  • Double precision (64 bit)

Note that not all precisions may be supported by the OpenCL devices that are used to conduct the computations. Furthermore, the precision selected influences the performance of those computations. The following example depicts the usage of double precision floating point values.

import numpy as np
time_series = SingleTimeSeries(data_points,


The basic tests referring to RQA and recurrence plot computations can be executed cumulatively.

python -m pyrqa.test

The complete set of tests can be executed by adding --extended.

python -m pyrqa.test --extended


The PyRQA package was initiated by computer scientists from the Humboldt-Universität zu Berlin and the GFZ German Research Centre for Geosciences.


We would like to thank Norbert Marwan from the Potsdam Institute for Climate Impact Research for his continuous support of the project. Please visit his website for further information on recurrence analysis.


The underlying computational approach of PyRQA is described in detail within the following thesis, which is openly accessible under

Rawald, T. (2018): Scalable and Efficient Analysis of Large High-Dimensional Data Sets in the Context of Recurrence Analysis, PhD Thesis, Berlin : Humboldt-Universität zu Berlin, 299 p.

Selected aspects of the computational approach are presented within the following publications.

Rawald, T., Sips, M., Marwan, N., Dransch, D. (2014): Fast Computation of Recurrences in Long Time Series. - In: Marwan, N., Riley, M., Guiliani, A., Webber, C. (Eds.), Translational Recurrences. From Mathematical Theory to Real-World Applications, (Springer Proceedings in Mathematics and Statistics ; 103), p. 17-29.

Rawald, T., Sips, M., Marwan, N., Leser, U. (2015): Massively Parallel Analysis of Similarity Matrices on Heterogeneous Hardware. - In: Fischer, P. M., Alonso, G., Arenas, M., Geerts, F. (Eds.), Proceedings of the Workshops of the EDBT/ICDT 2015 Joint Conference (EDBT/ICDT), (CEUR Workshop Proceedings ; 1330), p. 56-62.

Release Notes



  • Updated documentation.


  • Major refactoring.
  • Removal of operator and variant implementations that do not refer to OpenCL brute force computing.
  • Time series data may be represented using half, single and double precision floating point values, which is reflected in the computations on the OpenCL devices.
  • Several changes to the public API.


  • Changes to the public API have been made, e.g., to the definition of the settings. This leads to an increase in the major version number (see
  • Time series objects either consist of one or multiple series. The former requires to specify a value for the embedding delay as well as the time delay parameter.
  • Regarding the RQA computations, minimum line lengths are now specified on the result object. This allows to compute quantitative results using different lengths without having to inspect the matrix using the same parametrisation multiple times.
  • Modules for selecting well-performing implementations based on greedy selection strategies have been added. By default, the selection pool consists of a single pre-defined implementation.
  • Operators and implementation variants based on multidimensional search trees and grid data structures have been added.
  • The diagonal line based quantitative measures are modified regarding the semantics of the Theiler corrector.
  • The creation of the OpenCL environment now supports device fission.


  • Initial release.

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