Similarity and Distance Quantification between Probability Functions

Describe and understand the world through data.

Data collection and data comparison are the foundations of scientific research. Mathematics provides the abstract framework to describe patterns we observe in nature and Statistics provides the framework to quantify the uncertainty of these patterns. In statistics, natural patterns are described in form of probability distributions which either follow a fixed pattern (parametric distributions) or more dynamic patterns (non-parametric distributions).

The philentropy package implements fundamental distance and similarity measures to quantify distances between probability density functions as well as traditional information theory measures. In this regard, it aims to provide a framework for comparing natural patterns in a statistical notation.

This project is born out of my passion for statistics and I hope that it will be useful to the people who share it with me.


# install philentropy version 0.4.0 from CRAN


I am developing philentropy in my spare time and would be very grateful if you would consider citing the following paper in case philentropy was useful for your own research. I plan on maintaining and extending the philentropy functionality and usability in the next years and require citations to back up these efforts. Many thanks in advance :)

HG Drost, (2018). Philentropy: Information Theory and Distance Quantification with R. Journal of Open Source Software, 3(26), 765.


# retrieve available distance metrics
 [1] "euclidean"         "manhattan"         "minkowski"        
 [4] "chebyshev"         "sorensen"          "gower"            
 [7] "soergel"           "kulczynski_d"      "canberra"         
[10] "lorentzian"        "intersection"      "non-intersection" 
[13] "wavehedges"        "czekanowski"       "motyka"           
[16] "kulczynski_s"      "tanimoto"          "ruzicka"          
[19] "inner_product"     "harmonic_mean"     "cosine"           
[22] "hassebrook"        "jaccard"           "dice"             
[25] "fidelity"          "bhattacharyya"     "hellinger"        
[28] "matusita"          "squared_chord"     "squared_euclidean"
[31] "pearson"           "neyman"            "squared_chi"      
[34] "prob_symm"         "divergence"        "clark"            
[37] "additive_symm"     "kullback-leibler"  "jeffreys"         
[40] "k_divergence"      "topsoe"            "jensen-shannon"   
[43] "jensen_difference" "taneja"            "kumar-johnson"    
[46] "avg"
# define a probability density function P
P <- 1:10/sum(1:10)
# define a probability density function Q
Q <- 20:29/sum(20:29)

# combine P and Q as matrix object
x <- rbind(P,Q)

# compute the jensen-shannon distance between
# probability density functions P and Q
distance(x, method = "jensen-shannon")
jensen-shannon using unit 'log'.

Alternatively, users can also retrieve values from all available distance/similarity metrics using dist.diversity():

dist.diversity(x, p = 2, unit = "log2")
        euclidean         manhattan 
       0.12807130        0.35250464 
        minkowski         chebyshev 
       0.12807130        0.06345083 
         sorensen             gower 
       0.17625232        0.03525046 
          soergel      kulczynski_d 
       0.29968454        0.42792793 
         canberra        lorentzian 
       2.09927095        0.49712136 
     intersection  non-intersection 
       0.82374768        0.17625232 
       wavehedges       czekanowski 
       3.16657887        0.17625232 
           motyka      kulczynski_s 
       0.58812616        2.33684211 
         tanimoto           ruzicka 
       0.29968454        0.70031546 
    inner_product     harmonic_mean 
       0.10612245        0.94948528 
           cosine        hassebrook 
       0.93427641        0.86613103 
          jaccard              dice 
       0.13386897        0.07173611 
         fidelity     bhattacharyya 
       0.97312397        0.03930448 
        hellinger          matusita 
       0.32787819        0.23184489 
    squared_chord squared_euclidean 
       0.05375205        0.01640226 
          pearson            neyman 
       0.16814418        0.36742465 
      squared_chi         prob_symm 
       0.10102943        0.20205886 
       divergence             clark 
       1.49843905        0.86557468 
    additive_symm  kullback-leibler 
       0.53556883        0.13926288 
         jeffreys      k_divergence 
       0.31761069        0.04216273 
           topsoe    jensen-shannon 
       0.07585498        0.03792749 
jensen_difference            taneja 
       0.03792749        0.04147518 
    kumar-johnson               avg 
       0.62779644        0.20797774

Install Developer Version

# install.packages("devtools")
# install the current version of philentropy on your system
install_github("HajkD/philentropy", build_vignettes = TRUE, dependencies = TRUE)


The current status of the package as well as a detailed history of the functionality of each version of philentropy can be found in the NEWS section.

Important Functions

Distance Measures

Information Theory

  • H() : Shannon’s Entropy H(X)
  • JE() : Joint-Entropy H(X,Y)
  • CE() : Conditional-Entropy H(X | Y)
  • MI() : Shannon’s Mutual Information I(X,Y)
  • KL() : Kullback–Leibler Divergence
  • JSD() : Jensen-Shannon Divergence
  • gJSD() : Generalized Jensen-Shannon Divergence

Studies that successfully applied the philentropy package

  • Single cell census of human kidney organoids shows reproducibility and diminished off-target cells after transplantation A Subramanian et al. - Nature Communications, 2019

  • Different languages, similar encoding efficiency: Comparable information rates across the human communicative niche C Coupé, YM Oh, D Dediu, F Pellegrino - Science Advances, 2019

  • Loss of adaptive capacity in asthmatic patients revealed by biomarker fluctuation dynamics after rhinovirus challenge A Sinha et al. - eLife, 2019

  • The Tug1 lncRNA locus is essential for male fertility JP Lewandowski et al. - Genome Biology, 2020

  • Sex and hatching order modulate the association between MHC‐II diversity and fitness in early‐life stages of a wild seabird M Pineaux et al - Molecular Ecology, 2020

  • How the Choice of Distance Measure Influences the Detection of Prior-Data Conflict K Lek, R Van De Schoot - Entropy, 2019

  • Differential variation analysis enables detection of tumor heterogeneity using single-cell RNA-sequencing data EF Davis-Marcisak, TD Sherman et al. - Cancer research, 2019

  • Multi-Omics Investigation of Innate Navitoclax Resistance in Triple-Negative Breast Cancer Cells M Marczyk et al. - Cancers, 2020

  • Impact of Gut Microbiome on Hypertensive Patients with Low-Salt Intake: Shika Study Results S Nagase et al. - Frontiers in Medicine, 2020

  • Combined TCR Repertoire Profiles and Blood Cell Phenotypes Predict Melanoma Patient Response to Personalized Neoantigen Therapy plus Anti-PD-1 A Poran et al. - Cell Reports Medicine, 2020

  • SEDE-GPS: socio-economic data enrichment based on GPS information T Sperlea, S Füser, J Boenigk, D Heider - BMC bioinformatics, 2018

  • Evacuees and Migrants Exhibit Different Migration Systems after the Great East Japan Earthquake and Tsunami M Hauer, S Holloway, T Oda – 2019

  • Robust comparison of similarity measures in analogy based software effort estimation P Phannachitta - 11th International Conference on Software, 2017

  • Expression variation analysis for tumor heterogeneity in single-cell RNA-sequencing data EF Davis-Marcisak, P Orugunta et al. - BioRxiv, 2018

  • Concept acquisition and improved in-database similarity analysis for medical data I Wiese, N Sarna, L Wiese, A Tashkandi, U Sax - Distributed and Parallel Databases, 2019

  • Dynamics of Vaginal and Rectal Microbiota over Several Menstrual Cycles in Female Cynomolgus Macaques MT Nugeyre, N Tchitchek, C Adapen et al. - Frontiers in Cellular and Infection Microbiology, 2019

  • Inferring the quasipotential landscape of microbial ecosystems with topological data analysis WK Chang, L Kelly - BioRxiv, 2019

  • Shifts in the nasal microbiota of swine in response to different dosing regimens of oxytetracycline administration KT Mou, HK Allen, DP Alt, J Trachsel et al. - Veterinary microbiology, 2019

  • The Patchy Distribution of Restriction–Modification System Genes and the Conservation of Orphan Methyltransferases in Halobacteria MS Fullmer, M Ouellette, AS Louyakis et al. - Genes, 2019

  • Genetic differentiation and intrinsic genomic features explain variation in recombination hotspots among cocoa tree populations EJ Schwarzkopf, JC Motamayor, OE Cornejo - BioRxiv, 2019

  • Metastable regimes and tipping points of biochemical networks with potential applications in precision medicine SS Samal, J Krishnan, AH Esfahani et al. - Reasoning for Systems Biology and Medicine, 2019

  • Genome‐wide characterization and developmental expression profiling of long non‐coding RNAs in Sogatella furcifera ZX Chang, OE Ajayi, DY Guo, QF Wu - Insect science, 2019

  • Development of a simulation system for modeling the stock market to study its characteristics P Mariya – 2018

  • The Tug1 Locus is Essential for Male Fertility JP Lewandowski, G Dumbović, AR Watson, T Hwang et al. - BioRxiv, 2019

  • Microbiotyping the sinonasal microbiome A Bassiouni, S Paramasivan, A Shiffer et al. - BioRxiv, 2019

  • Critical search: A procedure for guided reading in large-scale textual corpora J Guldi - Journal of Cultural Analytics, 2018

  • A Bibliography of Publications about the R, S, and S-Plus Statistics Programming Languages NHF Beebe – 2019

  • Improved state change estimation in dynamic functional connectivity using hidden semi-Markov models H Shappell, BS Caffo, JJ Pekar, MA Lindquist - NeuroImage, 2019

  • A Smart Recommender Based on Hybrid Learning Methods for Personal Well-Being Services RM Nouh, HH Lee, WJ Lee, JD Lee - Sensors, 2019

  • Cognitive Structural Accuracy V Frenz – 2019

  • Kidney organoid reproducibility across multiple human iPSC lines and diminished off target cells after transplantation revealed by single cell transcriptomics A Subramanian, EH Sidhom, M Emani et al. - BioRxiv, 2019

  • Multi-classifier majority voting analyses in provenance studies on iron artefacts G Żabiński et al. - Journal of Archaeological Science, 2020

  • Identifying inhibitors of epithelial–mesenchymal plasticity using a network topology-based approach K Hari et al. - NPJ systems biology and applications, 2020

  • Genetic differentiation and intrinsic genomic features explain variation in recombination hotspots among cocoa tree populations EJ Schwarzkopf et al. - BMC Genomics, 2020

Discussions and Bug Reports

I would be very happy to learn more about potential improvements of the concepts and functions provided in this package.

Furthermore, in case you find some bugs or need additional (more flexible) functionality of parts of this package, please let me know:

or find me on twitter: HajkDrost