Chi2fit.Statistics (Chi-SquaredFit v2.0.0) View Source

Link to this section Summary

Types

Algorithm used to assign errors to frequencey data: Wald score and Wilson score.

Cumulative Distribution Function

Binned data with error bounds specified through low and high values

Functions

Calculates the autocorrelation coefficient of a list of observations.

Calculates the systematic errors for bins due to uncertainty in assigning data to bins.

Implements bootstrapping procedure as resampling with replacement.

Converts a CDF function to a list of data points.

Generates a Cullen & Frey plot for the sample data.

Extracts data point with standard deviation from Cullen & Frey plot data.

Generates an empirical Cumulative Distribution Function from sample data.

Calculates and returns the error associated with a list of observables.

Converts a list of numbers to frequency data.

Calculates the nth moment of the sample.

Calculates the nth centralized moment of the sample.

Calculates the nth centralized moment of the sample.

Calculates the nth normalized moment of the sample.

Calculates the nth normalized moment of the sample.

Calculates the nth normalized moment of the sample.

Converts the input so that the result is a Puiseaux diagram, that is a strict convex shape.

Resamples the subsequences of numbers contained in the list as determined by analyze/2

Calculates the test statistic for subexponentiality of a sample.

Converts raw data to binned data with (asymmetrical) errors.

Link to this section Types

Specs

algorithm() :: :wilson | :wald

Algorithm used to assign errors to frequencey data: Wald score and Wilson score.

Specs

cdf() :: (number() -> {number(), number(), number()})

Cumulative Distribution Function

Specs

cullenfrey() :: [{squared_skewness :: float(), kurtosis :: float()} | nil]

Specs

ecdf() :: [{float(), float(), float(), float()}]

Binned data with error bounds specified through low and high values

Specs

range() :: {float(), float()} | [float(), ...]

Link to this section Functions

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auto(list, opts \\ [nproc: 1])

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Specs

auto([number()], Keyword.t()) :: [number()]

Calculates the autocorrelation coefficient of a list of observations.

The implementation uses the discrete Fast Fourier Transform to calculate the autocorrelation. For available options see Chi2fit.FFT.fft/2. Returns a list of the autocorrelation coefficients.

Example

iex> auto [1,2,3]
[14.0, 7.999999999999999, 2.999999999999997]
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binerror(data, noise_fun, options \\ [])

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Specs

binerror(
  data :: [number()],
  noise_fun :: (Enumerable.t() -> Enumerable.t()),
  options :: Keyword.t()
) :: [{bin :: number(), avg :: number(), error :: number()}]

Calculates the systematic errors for bins due to uncertainty in assigning data to bins.

Options

`bin` - the size of bins to use (defaults to 1)
`iterations` - the number of iterations to use to estimate the error due to noise (defatuls to 100)
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bootstrap(total, data, fun, options \\ [])

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Specs

bootstrap(
  total :: integer(),
  data :: [number()],
  fun :: ([number()], integer() -> number()),
  options :: Keyword.t()
) :: [any()]

Implements bootstrapping procedure as resampling with replacement.

It supports saving intermediate results to a file using :dets. Use the options :safe and :filename (see below)

Arguments:

`total` - Total number resamplings to perform
`data` - The sample data
`fun` - The function to evaluate
`options` - A keyword list of options, see below.

Options

`:safe` - Whether to safe intermediate results to a file, so as to support continuation when it is interrupted.
      Valid values are `:safe` and `:cont`.
`:filename` - The filename to use for storing intermediate results

Specs

convert_cdf({cdf(), range()}) :: [{float(), float(), float(), float()}]

Converts a CDF function to a list of data points.

Example

iex> convert_cdf {fn x->{:math.exp(-x),:math.exp(-x)/16,:math.exp(-x)/4} end, {1,4}}
[{1, 0.36787944117144233, 0.022992465073215146, 0.09196986029286058},
 {2, 0.1353352832366127, 0.008458455202288294, 0.033833820809153176},
 {3, 0.049787068367863944, 0.0031116917729914965, 0.012446767091965986},
 {4, 0.01831563888873418, 0.0011447274305458862, 0.004578909722183545}]
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cullen_frey(sample, n \\ 100)

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Specs

cullen_frey(sample :: [number()], n :: integer()) :: cullenfrey()

Generates a Cullen & Frey plot for the sample data.

The kurtosis returned is the 'excess kurtosis'.

Specs

cullen_frey_point(data :: cullenfrey()) ::
  {{x :: float(), dx :: float()}, {y :: float(), dy :: float()}}

Extracts data point with standard deviation from Cullen & Frey plot data.

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empirical_cdf(data, bin \\ {1.0, 0.5}, algorithm \\ :wilson, correction \\ 0)

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Specs

empirical_cdf(
  [{float(), number()}],
  {number(), number()},
  algorithm(),
  integer()
) ::
  {cdf(), bins :: [float()], numbins :: pos_integer(), sum :: float()}

Generates an empirical Cumulative Distribution Function from sample data.

Three parameters determine the resulting empirical distribution:

  1. algorithm for assigning errors,

  2. the size of the bins,

  3. a correction for limiting the bounds on the 'y' values

When e.g. task effort/duration is modeled, some tasks measured have 0 time. In practice what is actually is meant, is that the task effort is between 0 and 1 hour. This is where binning of the data happens. Specify a size of the bins to control how this is done. A bin size of 1 means that 0 effort will be mapped to 1/2 effort (at the middle of the bin). This also prevents problems when the fited distribution cannot cope with an effort os zero.

Supports two ways of assigning errors: Wald score or Wilson score. See [1]. Valie values for the algorithm argument are :wald or :wilson.

In the handbook of MCMC [1] a cumulative distribution is constructed. For the largest 'x' value in the sample, the 'y' value is exactly one (1). In combination with the Wald score this gives zero errors on the value '1'. If the resulting distribution is used to fit a curve this may give an infinite contribution to the maximum likelihood function. Use the correction number to have a 'y' value of slightly less than 1 to prevent this from happening. Especially the combination of 0 correction, algorithm :wald, and 'linear' model for handling asymmetric errors gives problems.

The algorithm parameter determines how the errors onthe 'y' value are determined. Currently supported values include :wald and :wilson.

References

[1] "Handbook of Monte Carlo Methods" by Kroese, Taimre, and Botev, section 8.4
[2] See https://en.wikipedia.org/wiki/Cumulative_frequency_analysis
[3] https://arxiv.org/pdf/1112.2593v3.pdf
[4] See https://en.wikipedia.org/wiki/Student%27s_t-distribution:
    90% confidence ==> t = 1.645 for many data points (> 120)
    70% confidence ==> t = 1.000

Specs

error([{gamma :: number(), k :: pos_integer()}], :initial_sequence_method) ::
  {var :: number(), lag :: number()}

Calculates and returns the error associated with a list of observables.

Usually these are the result of a Markov Chain Monte Carlo simulation run.

The only supported method is the so-called Initial Sequence Method. See section 1.10.2 (Initial sequence method) of [1].

Input is a list of autocorrelation coefficients. This may be the output of auto/2.

References

[1] 'Handbook of Markov Chain Monte Carlo'
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get_cdf(data, binsize \\ {1.0, 0.5}, algorithm \\ :wilson, correction \\ 0)

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Specs

get_cdf([number()], number() | {number(), number()}, algorithm(), integer()) ::
  {cdf(), bins :: [float()], numbins :: pos_integer(), sum :: float()}

Calculates the empirical CDF from a sample.

Convenience function that chains make_histogram/2 and empirical_cdf/3.

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make_histogram(list, binsize \\ 1.0, offset \\ 0.0)

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Specs

make_histogram([number()], number(), number()) :: [
  {non_neg_integer(), pos_integer()}
]

Converts a list of numbers to frequency data.

The data is divided into bins of size binsize and the number of data points inside a bin are counted. A map is returned with the bin's index as a key and as value the number of data points in that bin.

The function returns a list of 2-tuples. Each tuple contains the index of the bin and the value of the count of the number of items in the bin. The index of the bins start at 1 in the following way:

  • [0..1) has index 1 (including 0 and excludes 1),
  • [1..2) has index 2,
  • etc.

When an offset is used, the bin starting from the offset, i.e. [offset..offset+1) gets index 1. Values less than the offset are gathered in a bin with index 0.

Examples

iex> make_histogram [1,2,3]
[{2, 1}, {3, 1}, {4, 1}]

iex> make_histogram [1,2,3], 1.0, 0
[{2, 1}, {3, 1}, {4, 1}]

iex> make_histogram [1,2,3,4,5,6,5,4,3,4,5,6,7,8,9]
[{2, 1}, {3, 1}, {4, 2}, {5, 3}, {6, 3}, {7, 2}, {8, 1}, {9, 1}, {10  , 1}]

iex> make_histogram [1,2,3,4,5,6,5,4,3,4,5,6,7,8,9], 3, 1.5
[{0, 1}, {1, 6}, {2, 6}, {3, 2}]

iex> make_histogram [0,0,0,1,3,4,3,2,6,7],1
[{1,3},{2,1},{3,1},{4,2},{5,1},{7,1},{8,1}]

iex> make_histogram [0,0,0,1,3,4,3,2,6,7],1,0.5
[{0,3},{1,1},{2,1},{3,2},{4,1},{6,1},{7,1}]

Specs

moment(sample :: [number()], n :: pos_integer()) :: float()

Calculates the nth moment of the sample.

Example

iex> moment [1,2,3,4,5,6], 1
3.5

Specs

momentc(sample :: [number()], n :: pos_integer()) :: float()

Calculates the nth centralized moment of the sample.

Example

iex> momentc [1,2,3,4,5,6], 1
0.0

iex> momentc [1,2,3,4,5,6], 2
2.9166666666666665

Specs

momentc(sample :: [number()], n :: pos_integer(), mu :: float()) :: float()

Calculates the nth centralized moment of the sample.

Example

iex> momentc [1,2,3,4,5,6], 2, 3.5
2.9166666666666665

Specs

momentn(sample :: [number()], n :: pos_integer()) :: float()

Calculates the nth normalized moment of the sample.

Example

iex> momentn [1,2,3,4,5,6], 1
0.0

iex> momentn [1,2,3,4,5,6], 2
1.0

iex> momentn [1,2,3,4,5,6], 4
1.7314285714285718

Specs

momentn(sample :: [number()], n :: pos_integer(), mu :: float()) :: float()

Calculates the nth normalized moment of the sample.

Example

iex> momentn [1,2,3,4,5,6], 4, 3.5
1.7314285714285718
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momentn(sample, n, mu, sigma)

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Specs

momentn(
  sample :: [number()],
  n :: pos_integer(),
  mu :: float(),
  sigma :: float()
) :: float()

Calculates the nth normalized moment of the sample.

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puiseaux(list, result \\ [], flag \\ false)

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Specs

puiseaux([number()], [number()], boolean()) :: [number()]

Converts the input so that the result is a Puiseaux diagram, that is a strict convex shape.

Examples

iex> puiseaux [1]
[1]

iex> puiseaux [5,3,3,2]
[5, 3, 2.5, 2]

Specs

resample(data :: [number()], options :: Keyword.t()) :: [number()]

Resamples the subsequences of numbers contained in the list as determined by analyze/2

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subexponential_stat(data, test \\ :sum, n \\ 2, binsize \\ {1, 0})

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Calculates the test statistic for subexponentiality of a sample.

A value close to 0 is a strong indication that the sample shows subexponential behaviour (extremistan), i.e. is fat-tailed.

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to_bins(data, binsize \\ {1.0, 0.5})

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Specs

to_bins(data :: [number()], binsize :: {number(), number()}) :: ecdf()

Converts raw data to binned data with (asymmetrical) errors.