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Bootstrap method for standard errors, confidence intervals, and more!

Project description

bootstrap-stat

Methods relating to the Bootstrap.

Estimates of standard errors, bias, confidence intervals, prediction errors, and more!

Getting Started

Bootstrap-Stat is hosted on PyPI. Install as you would any other library, e.g.: poetry add bootstrap-stat.

Documentation is available at Convex Analytics but I also recommend reviewing the rest of this README.

Brief Overview

Quoting [ET93], "The bootstrap is a data-based simulation method for statistical inference...The use of the term bootstrap derives from the phrase to pull oneself up by one's bootstrap..."

The bootstrap is a collection of methodologies for estimating errors and uncertainties associated with estimates and predictions. For example, it can be used to compute the bias and standard error of a particular estimator. It can be used to estimate the prediction error of a particular ML model (as a competitor to Cross Validation). And it can be used to compute confidence intervals. Most of the techniques described in [ET93] have now been implemented. (Notable exceptions include importance resampling, likelihood-based methods, and better support for the parametric bootstrap.)

Basic Terminology

To apply bootstrap techniques, it is important to understand the following terminology.

  • Distribution: an entity assigning probabilities to observations or sets of observations.
  • Population: the complete set of data, typically only partially observed.
  • Sample: the observed data, assumed to be drawn with replacement from the population. In real life, samples are usually drawn without replacement from the population, but provided the sample is a small fraction of the population, this is a negligible concern.
  • Empirical Distribution: the distribution assigning probability 1/n to each observation in a sample of size n.
  • Parameter: some function of a distribution. See statistic.
  • Statistic: some function of a collection of observations. We will assume all statistics are real-valued scalars, such as mean, median, or variance. Parameters and statistics are similar: it makes sense to talk about the mean of a distribution, which is a parameter of that distribution, and it also makes sense to talk about the mean of a collection of numbers, which is a statistic. For this reason, it is important to keep it straight whether we are talking about a parameter or a statistic! For example, it makes sense to talk about the bias of a statistic, but it does not make sense to talk about the bias of a parameter.
  • Plug-in estimate of a parameter: an estimate of a parameter calculated by "plugging-in" the Empirical Distribution. For example, to estimate the mean of an unobserved distribution, simply calculate the mean of the (observed) Empirical Distribution. The plug-in estimate is a statistic.
  • Bootstrap sample: a sample drawn with replacement from the Empirical Distribution, having size equal to the size of the original dataset.
  • Standard Error: the square root of the variance of a statistic, typically used to quantify accuracy.
  • Bias: the difference between the expected value of a statistic and the parameter it purports to estimate.
  • Confidence Interval: a range of plausible values of a parameter consistent with the data.

Examples

This library includes some datasets that can be used for trying out methods. The test cases themselves (in tests/) contain many practical examples.

>>> import numpy as np
>>> from bootstrap_stat import bootstrap_stat as bp
>>> from bootstrap_stat import datasets as d
>>>
>>> df = d.law_data()
>>> print(df)
    LSAT   GPA
0    576  3.39
1    635  3.30
2    558  2.81
3    578  3.03
4    666  3.44
5    580  3.07
6    555  3.00
7    661  3.43
8    651  3.36
9    605  3.13
10   653  3.12
11   575  2.74
12   545  2.76
13   572  2.88
14   594  2.96

The law data are a collection of N = 82 American law schools participating in a large study of admissions practices. Two measurements were made on the entering classes of each school in 1973: LSAT, the average score for the class on a national law test, and GPA, the average undergraduate grade-point average for the class. Both the full data set, and a sample are available. The above is a sample of 15 schools. The law data are taken from [EF93].

Suppose we are interested in the correlation between LSAT and GPA. Numpy can be used to compute the observed correlation for the sample (a statistic), but we hope to draw inferences about the population (all 82 schools) correlation coefficient (a parameter) based just on the sample. In this case, the entire population is available, and we could just compute the parameter directly. In most cases, the entire population is not available.

To use the bootrap method, we need to specify the statistic as well as the dataset. Specifically, we need to be able to sample with replacement from the Empirical Distribution. bootstrap_stat has a class facilitating just that.

>>> dist = bp.EmpiricalDistribution(df)
>>> dist.sample(reset_index=False)
    LSAT   GPA
14   594  2.96
3    578  3.03
0    576  3.39
6    555  3.00
10   653  3.12
12   545  2.76
6    555  3.00
0    576  3.39
10   653  3.12
8    651  3.36
3    578  3.03
4    666  3.44
13   572  2.88
5    580  3.07
8    651  3.36

Generating the Empirical Distribution is as simple as feeding either an array, pandas Series, or pandas DataFrame into the constructor. Under the hood, the bootstrap methods make frequent use of the sample method, which samples with replacement from the original dataset. Such samples are called bootstrap samples. Notice in the example above how school 0 makes multiple appearances in the bootstrap sample. Since the sampling is random, if you run the above you will likely get different results than in this example. (In some of the more exotic use cases, we need to reset the index for technical reasons relating to pandas indexing, so the default behavior is to reset, hence the reset_index=False in this example.)

Next we need to implement the statistic, which will be applied to bootstrap samples.

>>> def statistic(df):
...     return np.corrcoef(df["LSAT"], df["GPA"])[0, 1]
...
>>> obs_correlation = statistic(df)  # Observed correlation coefficient
>>> print(obs_correlation)
0.776374491289407

Notice how we can apply the statistic to the original dataset to calculate the observed value. The statistic should take as input either an array or a pandas Series or DataFrame, whatever was used to generate the Empirical Distribution. It should output a single number. Other than that, it can be anything: a simple calculation like a mean, a parameter from a linear regression model, or even a prediction from a neural network.

Now we can compute the standard error, which is a way of quantifying the variability of a statistic:

>>> se = bp.standard_error(dist, statistic)
>>> print(se)
0.13826565276176475

Since the bootstrap involves random sampling, you will likely get a slightly different answer than above, but it should be within 1% or so.

Or we can compute a confidence interval, a range of plausible values for the parameter consistent with the data.

>>> ci_low, ci_high = bp.bcanon_interval(dist, statistic, df)
>>> print(ci_low, ci_high)
0.44968698948896413 0.9230026418265834

These represent lower and upper bounds on a 90% confidence interval, the default behavior of bcanon_interval. We can do a 95% confidence interval by specifying alpha:

>>> ci_low, ci_high = bp.bcanon_interval(dist, statistic, df, alpha=0.025)
>>> print(ci_low, ci_high)
0.3120414479586675 0.9425059323691073

In general, bcanon_interval returns a 100(1-2alpha)% confidence interval. The bcanon terminology is a nod to the S implementation discussed in [ETF93]. (BCa is an algorithm for Bias-Corrected and Accelerated confidence intervals, and the function is NONparametric.)

Basic multicore functionality is implemented, allowing parallel calculation of bootstrap samples. Simply specify the num_threads argument in applicable functions. See the function documentation for details.

Running the test cases

$ poetry shell
$ python -m pytest

Documentation

Documentation is built using Sphinx and is hosted at Convex Analytics.

To update the docs (e.g. after updating the code), just change directory to docs and type make html. You'll need to be in a poetry shell.

Architecture

We use Poetry to manage dependencies, pytest as our test runner, black for code formatting, and sphinx for generating documentation.

Basic multicore functionality is implemented, using the pathos version of multiprocessing. We chose this version over the official python multiprocessing library since pathos uses dill instead of pickle to manage shared memory, and pickle cannot be used with locally defined functions. For users of this library, hopefully that implementation detail is irrelevant.

Licensing

Bootstrap-Stat is licensed under the Apache License, Version 2.0. See LICENSE.txt for the full license text.

References

[ET93] Bradley Efron and Robert J. Tibshirani, "An Introduction to the Bootstrap". Chapman & Hall, 1993.

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