# P4J

**Description**

P4J is a python package for period detection on irregularly sampled and heteroscedastic time series based on Information Theoretic objective functions. P4J was developed for astronomical light curves, irregularly sampled time series of stellar magnitude or flux. The core of this package is a class called periodogram that sweeps an array of periods/frequencies looking for the one that maximizes a given criteria. The main contribution of this work is a criterion for period detection based on the maximization of Cauchy-Schwarz Quadratic Mutual Information [1]. Information theoretic criteria incorporate information on the whole probability density function of the process and are more robust than classical second-order statistics based criteria [2, 3, 4]. For comparison P4J also incorporates other period detection methods used in astronomy such as the Phase Dispersion Minimization periodogram [5], Lafler-Kinman's string length [6] and the Orthogonal multiharmonic AoV periodogram [7]

**Contents**

- Quadratic Mutual Information periodogram for light curves
- Phase Dispersion Minimization, Analysis of Variance and String Length methods
- Basic synthetic light curve generator

**Instalation**

Dependencies

```
Numpy
GCC
Cython (optional)
```

If you have a UNIX system the GCC compiler is most likely already installed. If you have a Windows system you may want to install the Microsoft Visual C++ (MSVC) compiler. You can find relevant information at: https://wiki.python.org/moin/WindowsCompilers.

Note on Cython: If Cython is found in your system, pyx are compiled to c sources. If not the provided c sources are used.

Install from PyPI using

```
pip install P4J
```

or clone the github repository and do

```
python setup.py install --user
```

**Example**

https://github.com/phuijse/P4J/blob/master/examples/periodogram_demo.ipynb

**TODO**

- Multidimensional time series support
- More period detection criteria (Conditional Entropy)

**Authors**

- Pablo Huijse pablo.huijse@gmail.com (Millennium Institute of Astrophysics and Universidad de Chile)
- Pavlos Protopapas (Harvard Institute of Applied Computational Sciences)
- Pablo A. Estévez (Millennium Institute of Astrophysics and Universidad de Chile)
- Pablo Zegers (Universidad de los Andes, Chile)
- José C. Príncipe (University of Florida)

(P4J = Four Pablos and one Jose)

**References**

- José C. Príncipe, "Information Theoretic Learning: Renyi's Entropy and Kernel Perspectives", Springer, 2010
- Pavlos Protopapas et al., "A Novel, Fully Automated Pipeline for Period Estimation in the EROS 2 Data Set", The Astrophysical Journal Supplement, vol. 216, n. 2, 2015
- Pablo Huijse et al., "Computational Intelligence Challenges and Applications on Large-Scale Astronomical Time Series Databases", IEEE Mag. Computational Intelligence, vol. 9, n. 3, pp. 27-39, 2014
- Pablo Huijse et al., "An Information Theoretic Algorithm for Finding Periodicities in Stellar Light Curves", IEEE Trans. Signal Processing vol. 60, n. 10, pp. 5135-5145, 2012
- R. F. Stellingwerf, "Period determination using phase dispersion minimization", The Astrophysical Journal, vol. 224, pp. 953-960, 1978
- D. Clarke, "String/Rope length methods using the Lafler-Kinman statistic", Astronomy & Astrophysics, vol. 386, n. 2, pp. 763-774, 2002
- A. Schwarzenberg-Czerny "Fast and Statistically Optimal Period Search in Uneven Sampled Observations", Astrophysical Journal Letters, vol. 460, pp. 107, 1996