From data to probability densities without histograms

Abstract

When one deals with data drawn from continuous variables, a histogram is often inadequate to display their probability density. It deals inefficiently with statistical noise, and binsizes are free parameters. In contrast to that, the empirical cumulative distribution function (obtained after sorting the data) is parameter free. But it is a step function, so that its differentiation does not give a smooth probability density. Based on Fourier series expansion and Kolmogorov tests, we introduce a simple method, which overcomes this problem. Error bars on the estimated probability density are calculated using a jackknife method. We give several examples and provide computer code reproducing them. You may want to look at the corresponding figures 4 to 9 first. Program summary Program title: cdf_to_pd Catalogue identifier: AEBC_v1_0 Program summary URL: http://cpc.cs.qub.ac.uk/summaries/AEBC_v1_0.html Program obtainable from: CPC Program Library, Queen's University, Belfast, N. Ireland Licensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.html No. of lines in distributed program, including test data, etc.: 2758 No. of bytes in distributed program, including test data, etc.: 18 594 Distribution format: tar.gz Programming language: Fortran 77 Computer: Any capable of compiling and executing Fortran code Operating system: Any capable of compiling and executing Fortran code Classification: 4.14, 9 Nature of problem: When one deals with data drawn from continuous variables, a histogram is often inadequate to display the probability density. It deals inefficiently with statistical noise, and binsizes are free parameters. In contrast to that, the empirical cumulative distribution function (obtained after sorting the data) is parameter free. But it is a step function, so that its differentiation does not give a smooth probability density. Solution method: Based on Fourier series expansion and Kolmogorov tests, we introduce a simple method, which overcomes this problem. Error bars on the estimated probability density are calculated using a jackknife method. Several examples are included in the distribution file. Running time: The test runs provided take only a few seconds to execute.