Inverse Exponential Distribution

This is a custom probability density function I created to solve a particular problem I had at work, however it could also be useful to others.

This distribution aims to fit exponentially ascending data on a continuous interval [a, b]. That is, it's not bound to zero like the regular exponential distribution and can be defined for any interval.

With this module, I aim to mimic the scipy API it has for its distributions (although not everything is implemented.)

Dependencies

This module uses scipy for a lot of the backend which (at the time of writing) is licensed under BSD-3.

Warnings

• When fitting your data, the lower and upper bounds are determined by the min/max of the data; this will severely impact the results if you have very low/high extremes. It's recommended to treat your data and ensure the sample you're fitting is a "natural"-looking exponentially ascending shape.

Example Data

This is an example of exponentially ascending data on the interval [300, 900]

Mathematical Details

If you're interested in the actual derivation, see the whitepaper directory in this repo (will list all revisions of the paper as a separate PDF.)

Installation

Install with pip3 install invexpo or download from the releases and install locally.

Usage

Initialize an "empty" distribution:

from invexpo.inverse_exponential import InverseExponential

invex = InverseExponential()

You can either fit the distribution to data or create a theoretical distribution.

Note that fit() only accepts python lists as of now:

# sample data that mimics an "exponentially increasing" function bounded by [600, 800]
data = [600, 625, 650, 675, 700, 710, 720, 730, 740, 750, 760, 770, 780, 790, 800]

invex = InverseExponential()

# fit to data
invex.fit(data)

# there is also a maxiter parameter if the optimizer fails to converge
# for some reason, but usually there might be a more serious problem
# if that's happening...
invex.fit(data, maxiter=12345)

# create theoretical distribution
# NOTE: the 'a' (shape) parameter is usually very small, large numbers can cause overflows.
# Larger values of 'a' will create sharper peaks. Smaller values will more smoothly
# transition over the interval.
invex.create(a = 0.007, lower_bound = 300, upper_bound = 900)

Methods

After fitting/creating a distribution you can use the following methods:

• get_parameter() -> float
  • Returns the value of a, the shape parameter
• pdf(x: float) -> float
  • Evaluates the probability density function at x (i.e., P(x))
• cdf(x: float) -> float
  • Evaluate the cumulative density function at x, (i.e., P(X <= x))
• icdf(p: float) -> float
  • Evaluate the inverse CDF to get a percentile p for 0 <= p <= 1
• ppf(p: float) -> float
  • Same as icdf just a different name (percentile point function)
• integrate(lower_bound: float, upper_bound: float) -> float
  • Integrates the pdf over the interval [lower_bound, upper_bound]
• rvs(size: int = 1) -> list[float]
  • Generate size random variables from the distribution
• moment(n: int) -> float
  • Obtain the n-th moment of the distribution
• mean() -> float
  • Obtain the mean of the distribution (equivalent to moment(1))
• median() -> float
  • Obtain the median of the distribution
• var() -> float
  • Obtain the variance of the distribution (equivalent to moment(2) - moment(1)**2)
• std() -> float
  • Obtain the standard deviation of the distribution (equivalent to np.sqrt(var()))