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| [/ | ||
| Copyright (c) 2019 Nick Thompson | ||
| Use, modification and distribution are subject to the | ||
| Boost Software License, Version 1.0. (See accompanying file | ||
| LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) | ||
| ] | ||
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| [section:empirical_cdf Empirical Cumulative Distribution Function] | ||
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| [heading Synopsis] | ||
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| ``` | ||
| #include <boost/math/distributions/empirical_cumulative_distribution_function.hpp> | ||
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| namespace boost{ namespace math{ | ||
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| template <class RandomAccessContainer> | ||
| class empirical_cumulative_distribution_function | ||
| { | ||
| public: | ||
| using Real = typename RandomAccessContainer::value_type; | ||
| empirical_cumulative_distribution_function(RandomAccessContainer && v, bool sorted = false); | ||
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| auto operator()(Real t) const; | ||
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| RandomAccessContainer&& return_data(); | ||
| }; | ||
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| }} | ||
| ``` | ||
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| [heading Empirical Cumulative Distribution Function] | ||
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| The empirical cumulative distribution function is a step function constructed from observed data which converges to the true cumulative distribution function in the limit of infinite data. | ||
| This function is a basic building block of hypothesis testing workflows that attempt to answer the question "does my data come from a given distribution?" | ||
| These tests require computing quadratures over some function of the empirical CDF and the supposed CDF to create a distance measurement, and hence it is occasionally useful to construct a continuous callable from the data. | ||
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| An example usage is demonstrated below: | ||
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| ``` | ||
| #include <vector> | ||
| #include <random> | ||
| #include <boost/math/distributions/empirical_cumulative_distribution_function.hpp> | ||
| using boost::math::empirical_cumulative_distribution_function; | ||
| std::random_device rd; | ||
| std::mt19937 gen{rd()}; | ||
| std::normal_distribution<double> dis(0, 1); | ||
| size_t n = 128; | ||
| std::vector<double> v(n); | ||
| for (size_t i = 0; i < n; ++i) { | ||
| v[i] = dis(gen); | ||
| } | ||
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| auto ecdf = empirical_cumulative_distribution_function(std::move(v)); | ||
| std::cout << "ecdf(0.0) = " << ecdf(0.0) << "\n"; | ||
| // should print approximately 0.5 . . . | ||
| ``` | ||
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| The empirical distribution function requires sorted data. | ||
| By default, the constructor sorts it for you at O(Nlog(N)) cost. | ||
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| If your data is already sorted, you can specify this and the constructor simply moves your data into the class: | ||
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| ``` | ||
| std::sort(v.begin(), v.end()); | ||
| auto ecdf = empirical_cumulative_distribution_function(std::move(v), /* already sorted = */ true); | ||
| ``` | ||
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| If you want your data back after being done with the object, use | ||
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| ``` | ||
| v = ecdf.return_data(); | ||
| ``` | ||
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| This operation invalidates `ecdf`; it can no longer be used. | ||
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| The call operator complexity is O(log(N)), as it requires a call to `std::upper_bound`. | ||
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| Works with both integer and floating point types. | ||
| If the input data consists of integers, the output of the call operator is a double. Requires C++17. | ||
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| [$../graphs/empiricial_cumulative_distribution_gauss.svg] | ||
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| [$../graphs/empiricial_cumulative_distribution_uniform.svg] | ||
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| [heading Performance] | ||
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| ``` | ||
| ------------------------------------------------------ | ||
| Benchmark Time | ||
| ------------------------------------------------------ | ||
| ECDFConstructorSorted<double>/8 4.52 ns | ||
| ECDFConstructorSorted<double>/16 5.20 ns | ||
| ECDFConstructorSorted<double>/32 5.22 ns | ||
| ECDFConstructorSorted<double>/64 7.37 ns | ||
| ECDFConstructorSorted<double>/128 7.16 ns | ||
| ECDFConstructorSorted<double>/256 8.97 ns | ||
| ECDFConstructorSorted<double>/512 8.44 ns | ||
| ECDFConstructorSorted<double>/1024 9.07 ns | ||
| ECDFConstructorSorted<double>/2048 11.4 ns | ||
| ECDFConstructorSorted<double>/4096 12.6 ns | ||
| ECDFConstructorSorted<double>/8192 11.4 ns | ||
| ECDFConstructorSorted<double>/16384 16.0 ns | ||
| ECDFConstructorSorted<double>/32768 17.0 ns | ||
| ECDFConstructorSorted<double>/65536 19.5 ns | ||
| ECDFConstructorSorted<double>/131072 15.8 ns | ||
| ECDFConstructorSorted<double>/262144 17.9 ns | ||
| ECDFConstructorSorted<double>/524288 26.7 ns | ||
| ECDFConstructorSorted<double>/1048576 29.5 ns | ||
| ECDFConstructorSorted<double>/2097152 31.8 ns | ||
| ECDFConstructorSorted<double>/4194304 32.8 ns | ||
| ECDFConstructorSorted<double>/8388608 35.4 ns | ||
| ECDFConstructorSorted<double>/16777216 30.4 ns | ||
| ECDFConstructorSorted<double>_BigO 1.27 lgN | ||
| ECDFConstructorSorted<double>_RMS 20 % | ||
| ECDFConstructorUnsorted<double>/8 155 ns | ||
| ECDFConstructorUnsorted<double>/64 2095 ns | ||
| ECDFConstructorUnsorted<double>/512 22212 ns | ||
| ECDFConstructorUnsorted<double>/4096 220821 ns | ||
| ECDFConstructorUnsorted<double>/32768 1996380 ns | ||
| ECDFConstructorUnsorted<double>/262144 18916039 ns | ||
| ECDFConstructorUnsorted<double>/2097152 194250013 ns | ||
| ECDFConstructorUnsorted<double>/16777216 2281469424 ns | ||
| ECDFConstructorUnsorted<double>_BigO 5.65 NlgN | ||
| ECDFConstructorUnsorted<double>_RMS 6 % | ||
| Shuffle<double>/8 82.4 ns | ||
| Shuffle<double>/64 731 ns | ||
| Shuffle<double>/512 5876 ns | ||
| Shuffle<double>/4096 46864 ns | ||
| Shuffle<double>/32768 385265 ns | ||
| Shuffle<double>/262144 4663866 ns | ||
| Shuffle<double>/2097152 54686332 ns | ||
| Shuffle<double>/16777216 875309099 ns | ||
| Shuffle<double>_BigO 2.16 NlgN | ||
| Shuffle<double>_RMS 12 % | ||
| ECDFEvaluation<double>/8 48.6 ns | ||
| ECDFEvaluation<double>/64 61.3 ns | ||
| ECDFEvaluation<double>/512 85.1 ns | ||
| ECDFEvaluation<double>/4096 105 ns | ||
| ECDFEvaluation<double>/32768 131 ns | ||
| ECDFEvaluation<double>/262144 196 ns | ||
| ECDFEvaluation<double>/2097152 391 ns | ||
| ECDFEvaluation<double>/16777216 715 ns | ||
| ECDFEvaluation<double>_BigO 18.19 lgN | ||
| ECDFEvaluation<double>_RMS 60 % | ||
| ``` | ||
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| The call to the unsorted constructor is in fact a little faster than indicated, as the data must be shuffled after being sorted in the benchmark. | ||
| This is itself a fairly expensive operation. | ||
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| [endsect] | ||
| [/section:empirical_cdf] |
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