empirical_copula is a Python library to compute and plot 2D empirical copulas of discrete
data (ordinal or categorical).
A copula is a statistical object that captures the dependencies between two random variables, independently of their marginal distribution. For instance, as in the figure below, two variables might be dependent only in their low tails (Fig. 1).
Statisticians have defined parametric families of copulas that describe different kind of dependencies between continuous variables (Fig. 2).
An empirical copula visualizes the dependency structure observed in samples from two random variables.
For discrete (or discretized) variables, the empirical copula shows on the two axes the cumulative observed frequency of the values of each variable. The values in the copulas is the empirical joint pmf in that space. Independent variables in the copula space have uniform probability, and so the values in the copula can be interpreted as deviations from independence.
An example is going to make this much clearer. Figure 3 shows the log empirical copula between a
quality score and house price from the house_price OpenML dataset.
The spacing of the lines on the axes show the frequencies of the values of the two variables. Since the area of each combination of values is the product of the frequency, independent variables would have a uniform probability of 1.0 in each of the cases, or 0.0 in logarithmic space.
The value of ~0.5 for the pair (quality=7, price=225k) means that that pair has been observed 10^0.5 ~= 3.2 times more often than one would expect in independent variables. This examples demonstrates a strong dependencies between the quality score and the hose price, especially in the high and low tails of extreme high and low quality and prices, which are observed 10x more often than in the uniform case.
We evaluate the statistical significance of the observed deviations from independence using bootstrapping: we destroy the dependencies between the variables by resampling them replacement independently from each other. Doing this many times gives us an estimate of the distribution of the frequencies we would observe with the same marginal distribution if the variables were independent. We can then look at how extreme what we observe is compared to this null-hypothesis distribution and derive a significance level.
To learn how to use emipirical_copula, it's easiest to look at the examples notebook in the
git repository: GitHub examples notebooks
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Figure 1: Cosyne08 abstract and poster: Berkes, P., Pillow, J., and Wood, F. (2008). Characterizing neural dependencies with Poisson copula models. Cosyne 2008, Salt Lake City (abstract).
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Figure 2: Berkes, P., Wood, F., and Pillow, J. (2009). Characterizing neural dependencies with copula models. Advances in Neural Information Processing Systems, 21.