Orogram

A library for working with 1D PDFs represented with piecewise linear functions. The term orogram is used to refer to any representation of a PDF with a piecewise linear function. That may be read as histogram where the histo (poles, as in telephone poles) has been replaced with oro (mountains, which are definitely triangular/piecewise-linear, at least when you're five).

This library was created for the paper "The Cross-entropy of Piecewise Linear Probability Density Functions" by Tom S. F. Haines (Published in Transactions on Machine Learning Research (04/2024)) It would be great to cite this paper if you make use of this code for a publication of your own.

Can be installed with

pip install orogram

or you can clone etc. this repo.

The most recent version of this code can be found at https://github.com/thaines/orogram.

Requirements

Needs numpy and cython (at least version 3). Whenever an array is expected it is expected to be some object compatible with a numpy array.

Contents

The (installable) code provides the following:

• RegOrogram — An orogram with regularly spaced bins, much the same as your typical histogram. Feature rich — has everything you would expect of a 1D density estimate. Designed with data capture in mind.
• Orogram — An orogram with irregularly spaced bins. Reasonable set of features, but immutable and hence not for data capture. Simpler data structures so typically faster than RegOrogram. Otherwise does its best to match the RegOrogram feature set.
• linear_crossentropy(length, p0, p1, q0, q1, fast = False) — Direct access to the internal function that implements the key result of the cross-entropy paper. Calculates cross entropy for a linear segment with the given length where the two values for each distribution are the end point.

Beyond the provided code is

• setup.py — Standard setuptools code for making a package/installing.
• test - A folder of tests and demos for the library.
• jax — A folder of demos for the paper that needed automatic differentiation. Completely ignored by the actual installable library.

RegOrogram

The regular orogram object only supports an orogram with a regular spacing, but is designed for incremental data capture, i.e. data points can be added and it will expand its range as required.

This object provides the following stable methods/capabilities (other methods may be available, but are likely to change):

• RegOrogram(spacing = 0.001, blocksize = 1024 * 16) — Constructor, with the first parameter the spacing between bin centres. The blocksize is a trade off between memory/speed and has no effect on results; it's how much extra memory to allocate each time a data point lands outside the current range.
• .copy() — Copy constructor.
• sizeof(self) — Returns how many bytes of memory the object is currently using.
• len(self) — How many bins it has, including a zero mass bin either side.
• .spacing() — Spacing between bin centers it was constructed with.
• .blocksize() — Blocksize object was constructed with.
• .add(x, weight = 1.0, smooth=False) — Adds a single value or a 1D array of data points. Can optionally provide weight as something that broadcasts to each data point. If smooth is False, the default it does the maximum likelihood solution; if True it linearly interpolates between bin centers.
• .binadd(base, density, total = None) — Lets you add to bin centers directly. Base is the index of the bin center that maps to density[0] (0 gets you the bin center at 0). total is for if you know the sum of density, to save computation.
• .bake_cdf(cdf, start, end, weight = 1, vectorised = True) — Given a function for evaluating the CDF given x adds that distribution into the model, for the range start to end. weight is the total mass that the CDF should count as while vectorised indicates if the CDF function is vectorised or not.
• .sum() — Returns the total weight that the model has, i.e. how many samples have been received in typical usage.
• .min() — Minimum x where it hits zero.
• .max() — Maximum x where it hits zero.
• .binmin() — Index of minimum bin that is zero but adjacent to a nonzero bin.
• .binmax() — Index of maximum bin that is zero but adjacent to a nonzero bin.
• .center(i) — Given the index of a bin returns the x value of its center. Vectorised.
• .weight(i) — Given the index of a bin returns how much weight has been assigned to it (fractional sample count). Vectorised.
• .prob(i) — Given the index of a bin returns the probability at its bin center. Vectorised.
• self(x) — Evaluates the probability at the given x. Vectorised.
• .modes() — Returns an array containing every mode.
• .binmodes() — Returns an array containing the bin index of every mode.
• .highest() — Returns the x with the highest probability.
• .binhighest() — Returns the index of the bin with the highest probability.
• .graph(start = None, end = None) — Convenience method that returns a tuple of (x, y) suitable for passing to a graphing tool. Can optionally provide a range.
• self += other — Adds the samples of another RegOrogram to this one. Much faster if the spacing matches but will interpolate if they do not. Matching blocksize also helps.
• self *= scalar — Lets you scale all of the weights (number of samples).
• .cdf(x) — Evaluates the CDF at any location. Vectorised.
• .bincdf(i) — Evaluates the CDF at an indexed bin's center. Vectorised.
• .cdfgraph(start = None, end = None) — Convenience method that returns a tuple of (x, y) suitable for passing to a graphing tool, this time outputting the CDF. Can optionally provide a range.
• .invcdf(uni) — Evaluates the inverse CDF for a value in [0, 1). Vectorised.
• .draw(size = None, rng = None) — Samples from the represented PDF. Can optionally provide a size (integer for 1D, or tuple for a nD array) and/or a rng, as an object that numpy.random.default_rng(rng) knows what to do with.
• .median() — Returns the median.
• .var() — Returns the variance.
• .meanvar() — Returns a tuple of (mean, variance).
• .entropy() — Calculates the entropy, in nats.
• p.crossentropy(q) — Calculates the cross entropy, H(p,q), in nats.
• p.kl(q) — Calculates the Kullback–Leibler divergence, KL(p||q), in nats.

Orogram

Supports an orogram with an irregular spacing, i.e. the bin centres can be entirely arbitrary. However, the price paid for this is it's immutable, and hence can't be edited, and only does a normalised distribution, so there is no tracking of how many samples it has seen. In other words, RegOrogram is designed with data capture in mind and Orogram is designed to simple represent a PDF. It gains a speed advantage from this.

This object provides the following stable methods/capabilities (other methods may be available, but are likely to change):

• Orogram(x, y, cdf = None, norm=True, copy=True) — Lets you setup the PDF with a simple x/y pair of 1D arrays. x must be increasing. The flags are all for efficiency: cdf if you can provide a cdf array, norm can be set to False if you've provided a normalised distribution, and copy can be set to False if the object can steal the provided arrays.
• Orogram(regular) — Converts a RegOrogram into an Orogram.
• .even() — Returns True if bake_cdf() constructed this Orogram to within the given tolerance, False otherwise.
• Orogram.bake_cdf(cdf, start, end, resolution = 1024, epsilon = 1e-2, init = 8, maxiter = 128, vectorised = True) — Constructs a new Orogram (this is a static method) given a cdf function. You also provide the range to evaluate over (start and end) and how many (evenly spaced in terms of probability mass) samples to use (resolution). It uses a binary search to find the right x values — see full documentation for details.
• .copy() — Copy constructor. Of questionable value given object is immutable.
• sizeof(self) — Returns how many bytes of memory the object is currently using.
• len(self) — How many bins it has. Can be interpreted as half the parameter count.
• .min() — Minimum x.
• .max() — Maximum x.
• .center(i) — Converts the index of a bin to the location of it's bin center. Vectorised.
• .prob(i) — Given the index of a bin returns the probability at its bin center. Vectorised.
• self(x) — Evaluates the probability at the given x. Vectorised.
• .modes() — Returns an array containing every mode.
• .binmodes() — Returns an array containing the bin index of every mode.
• .highest() — Returns the x with the highest probability.
• .binhighest() — Returns the index of the bin with the highest probability.
• .graph() — Convenience method that returns a tuple of (x, y) suitable for passing to a graphing tool.
• .histogram() — An orogram constructed using the maximum likelihood technique can be interpreted as a histogram with variable bin width; this returns that histogram as the tuple (edges, weight) (edges array is one longer, to capture the last edge as well).
• Orogram.mixture(orograms, weights) — Constructs an Orogram (static method) as a mixture of Orograms (also supports RegOrograms); you provide one list of Orogram`s and another list of the respective weights, which will be normalised to sum to one. Note the return will have the union of bin centers of all inputs, which may be a lot.
• Orogram.product(orograms) — Constructs an Orogram (static method) as a product of Orograms (also supports RegOrogram`s). Note the return will have the union of bin centers of all inputs, which may be a lot and that it will raise an error if there is no probability mass.
• .clip(low, high) — Returns a new Orogram clipped to the given range. Will raise an error if there is no mass in that range to normalise.
• .binclip(low, high) — Returns a new Orogram clipped to the given range, expressed in terms of bin indices.
• .cdf(x) — Evaluates the CDF at any location. Vectorised.
• .bincdf(i) — Evaluates the CDF at an indexed bin's center. Vectorised.
• .cdfgraph(start = None, end = None) — Convenience method that returns a tuple of (x, y) suitable for passing to a graphing tool, this time outputting the CDF. Can optionally provide a range.
• .invcdf(uni) — Evaluates the inverse CDF for a value in [0, 1). Vectorised.
• .draw(size = None, rng = None) — Samples from the represented PDF. Can optionally provide a size (integer for 1D, or tuple for a nD array) and/or a rng, as an object that numpy.random.default_rng(rng) knows what to do with.
• .median() — Returns the median.
• .var() — Returns the variance.
• .meanvar() — Returns a tuple of (mean, variance).
• .credible(amount = 0.95, tolerance=1e-6) — Finds the credible interval of the distribution, as in the smallest set of ranges that cover the given amount of probability mass. Returns a list of regions, each represented as (start, end, mass). Tolerance is for the binary search it uses to find the regions.
• .entropy() — Calculates the entropy, in nats.
• p.crossentropy(q) — Calculates the cross entropy, H(p,q), in nats.
• p.kl(q) — Calculates the Kullback–Leibler divergence, KL(p||q), in nat