Chi-SquaredFit v0.9.1 Chi2fit.Utilities View Source

Provides various utilities:

  • Bootstrapping
  • Derivatives
  • Creating Cumulative Distribution Functions / Histograms from sample data
  • Solving linear, quadratic, and cubic equations
  • Autocorrelation coefficients

Link to this section Summary

Types

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

Cumulative Distribution Function

Supported numerical integration methods

Functions

Calculates the autocorrelation coefficient of a list of observations.

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.

Calculates the partial derivative of a function and returns the value.

Generates an empirical Cumulative Distribution Function from sample data.

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

Numerical integration providing Gauss and Romberg types.

Calculates the jacobian of the function at the point x.

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.

Newton-Fourier method for locating roots and returning the interval where the root is located.

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

Outputs and formats the errors that result from a call to chi2/4

Reads data from a file specified by filename and returns a stream with the data parsed as floats.

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

Unzips lists of 1-, 2-, 3-, 4-, and 5-tuples.

Link to this section Types

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algorithm() View Source
algorithm() :: :wilson | :wald

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

Cumulative Distribution Function

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

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method() View Source
method() :: :gauss | :gauss2 | :gauss3 | :romberg | :romberg2 | :romberg3

Supported numerical integration methods

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

Link to this section Functions

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auto(list, opts \\ [nproc: 1]) View Source
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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bootstrap(total, data, fun, options \\ []) View Source
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 resmaplings 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
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convert_cdf(arg) View Source
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) View Source
cullen_frey(sample :: [number()], n :: integer()) :: cullenfrey()

Generates a Cullen & Frey plot for the sample data.

The kurtosis returned is the 'excess kurtosis'.

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cullen_frey_point(data) View Source
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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der(parameters, fun, options \\ []) View Source
der([float() | {float(), integer()}], ([float()] -> float()), Keyword.t()) ::
  float()

Calculates the partial derivative of a function and returns the value.

Examples

The function value at a point:
iex> der([3.0], fn [x]-> x*x end) |> Float.round(3)
9.0

The first derivative of a function at a point:
iex> der([{3.0,1}], fn [x]-> x*x end) |> Float.round(3)
6.0

The second derivative of a function at a point:
iex> der([{3.0,2}], fn [x]-> x*x end) |> Float.round(3)
2.0

Partial derivatives with respect to two variables:
iex> der([{2.0,1},{3.0,1}], fn [x,y] -> 3*x*x*y end) |> Float.round(3)
12.0
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empirical_cdf(data, bin \\ {1.0, 0.5}, algorithm \\ :wilson, correction \\ 0) View Source
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
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error(nauto, atom) View Source
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) View Source
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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integrate(method, func, a, b, options \\ []) View Source
integrate(
  method(),
  (float() -> float()),
  a :: float(),
  b :: float(),
  options :: Keyword.t()
) :: float()

Numerical integration providing Gauss and Romberg types.

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jacobian(x, fun, options \\ []) View Source

Calculates the jacobian of the function at the point x.

Examples

iex> jacobian([2.0,3.0], fn [x,y] -> x*y end) |> Enum.map(&Float.round(&1))
[3.0, 2.0]
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make_histogram(list, binsize \\ 1.0, offset \\ 0.5) View Source
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.

Examples

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

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

iex> make_histogram [1,2,3,4,5,6,5,4,3,4,5,6,7,8,9]
[{1, 1}, {2, 1}, {3, 2}, {4, 3}, {5, 3}, {6, 2}, {7, 1}, {8, 1}, {9, 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}]
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moment(sample, n) View Source
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
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momentc(sample, n) View Source
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
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momentc(sample, n, mu) View Source
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
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momentn(sample, n) View Source
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
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momentn(sample, n, mu) View Source
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) View Source
momentn(
  sample :: [number()],
  n :: pos_integer(),
  mu :: float(),
  sigma :: float()
) :: float()

Calculates the nth normalized moment of the sample.

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newton(a, b, func, maxiter \\ 10, options) View Source
newton(
  a :: float(),
  b :: float(),
  func :: (x :: float() -> float()),
  maxiter :: non_neg_integer(),
  options :: Keyword.t()
) :: {float(), {float(), float()}, {float(), float()}}

Newton-Fourier method for locating roots and returning the interval where the root is located.

See [https://en.wikipedia.org/wiki/Newton%27s_method#Newton.E2.80.93Fourier_method]

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puiseaux(list, result \\ [], flag \\ false) View Source
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]
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puts_errors(device \\ :stdio, errors) View Source
puts_errors(device :: IO.device(), errors :: tuple()) :: none()

Outputs and formats the errors that result from a call to chi2/4

Errors are tuples of length 2 and larger: {[min1,max1], [min2,max2], ...}.

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read_data(filename) View Source
read_data(filename :: String.t()) :: Stream.t()

Reads data from a file specified by filename and returns a stream with the data parsed as floats.

It expects a single data point on a separate line and removes entries that:

  • are not floats, and
  • smaller than zero (0)
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richardson(func, init, factor, results \\ [], options) View Source
richardson(
  func :: (term() -> {float(), term()}),
  init :: term(),
  factor :: float(),
  results :: [float()],
  options :: Keyword.t()
) :: float()

Richardson extrapolation.

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to_bins(data, binsize \\ {1.0, 0.5}) View Source
to_bins(data :: [number()], binsize :: {number(), number()}) :: [
  {float(), float(), float(), float()}
]

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

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unzip(list) View Source
unzip(list :: [tuple()]) :: tuple()

Unzips lists of 1-, 2-, 3-, 4-, and 5-tuples.