Chi-SquaredFit v0.9.2 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.
Average and standard deviationm (error)
Cumulative Distribution Function
Binned data with error bounds specified through low and high values
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.
Displays results of the function Chi2fit.Fit.chi2probe/4
Displays results of the function Chi2fit.Fit.chi2fit/4
Generates an empirical Cumulative Distribution Function from sample data.
Calculates and returns the error associated with a list of observables.
Forecasts how many time periods are needed to complete size
items
Calculates the empirical CDF from a sample.
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.
Basic Monte Carlo simulation to repeatedly run a simulation multiple times.
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 Chi2fit.Fit.chi2/4
Reads data from a file specified by filename
and returns a stream with the data parsed as floats.
Richardson extrapolation.
Converts raw data to binned data with (asymmetrical) errors.
Unzips lists of 1-, 2-, 3-, 4-, and 5-tuples.
Link to this section Types
algorithm()
View Source
algorithm() :: :wilson | :wald
algorithm() :: :wilson | :wald
Algorithm used to assign errors to frequencey data: Wald score and Wilson score.
avgsd() View Source
Average and standard deviationm (error)
cdf() View Source
Cumulative Distribution Function
cullenfrey() View Source
ecdf() View Source
Binned data with error bounds specified through low and high values
method()
View Source
method() :: :gauss | :gauss2 | :gauss3 | :romberg | :romberg2 | :romberg3
method() :: :gauss | :gauss2 | :gauss3 | :romberg | :romberg2 | :romberg3
Supported numerical integration methods
range() View Source
Link to this section Functions
auto(list, opts \\ [nproc: 1]) View Source
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]
bootstrap(total, data, fun, options \\ []) View Source
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
convert_cdf(arg) View Source
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}]
cullen_frey(sample, n \\ 100)
View Source
cullen_frey(sample :: [number()], n :: integer()) :: cullenfrey()
cullen_frey(sample :: [number()], n :: integer()) :: cullenfrey()
Generates a Cullen & Frey plot for the sample data.
The kurtosis returned is the 'excess kurtosis'.
cullen_frey_point(data)
View Source
cullen_frey_point(data :: cullenfrey()) ::
{{x :: float(), dx :: float()}, {y :: float(), dy :: float()}}
cullen_frey_point(data :: cullenfrey()) :: {{x :: float(), dx :: float()}, {y :: float(), dy :: float()}}
Extracts data point with standard deviation from Cullen & Frey plot data.
der(parameters, fun, options \\ []) View Source
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
display(device \\ :stdio, results)
View Source
display(device :: IO.device(), Chi2fit.Fit.chi2probe() | avgsd()) :: none()
display(device :: IO.device(), Chi2fit.Fit.chi2probe() | avgsd()) :: none()
Displays results of the function Chi2fit.Fit.chi2probe/4
display(device \\ :stdio, hdata, model, arg, options)
View Source
display(
device :: IO.device(),
hdata :: ecdf(),
model :: Chi2fit.Distribution.model(),
Chi2fit.Fit.chi2fit(),
options :: Keyword.t()
) :: none()
display( device :: IO.device(), hdata :: ecdf(), model :: Chi2fit.Distribution.model(), Chi2fit.Fit.chi2fit(), options :: Keyword.t() ) :: none()
Displays results of the function Chi2fit.Fit.chi2fit/4
empirical_cdf(data, bin \\ {1.0, 0.5}, algorithm \\ :wilson, correction \\ 0) View Source
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
error(nauto, atom)
View Source
error([{gamma :: number(), k :: pos_integer()}], :initial_sequence_method) ::
{var :: number(), lag :: number()}
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'
forecast(fun, size, tries \\ 0)
View Source
forecast(
fun :: (() -> non_neg_integer()),
size :: pos_integer(),
tries :: pos_integer()
) :: pos_integer()
forecast( fun :: (() -> non_neg_integer()), size :: pos_integer(), tries :: pos_integer() ) :: pos_integer()
Forecasts how many time periods are needed to complete size
items
get_cdf(data, binsize \\ {1.0, 0.5}, algorithm \\ :wilson, correction \\ 0) View Source
Calculates the empirical CDF from a sample.
Convenience function that chains make_histogram/2
and empirical_cdf/3
.
integrate(method, func, a, b, options \\ []) View Source
Numerical integration providing Gauss and Romberg types.
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]
make_histogram(list, binsize \\ 1.0, offset \\ 0.5)
View Source
make_histogram([number()], number(), number()) :: [
{non_neg_integer(), pos_integer()}
]
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}]
mc(iterations, fun, all? \\ false)
View Source
mc(
iterations :: pos_integer(),
fun :: (pos_integer() -> float()),
all? :: boolean()
) ::
{avg :: float(), sd :: float(), tries :: [float()]}
| {avg :: float(), sd :: float()}
mc( iterations :: pos_integer(), fun :: (pos_integer() -> float()), all? :: boolean() ) :: {avg :: float(), sd :: float(), tries :: [float()]} | {avg :: float(), sd :: float()}
Basic Monte Carlo simulation to repeatedly run a simulation multiple times.
moment(sample, n)
View Source
moment(sample :: [number()], n :: pos_integer()) :: float()
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
momentc(sample, n)
View Source
momentc(sample :: [number()], n :: pos_integer()) :: float()
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
momentc(sample, n, mu)
View Source
momentc(sample :: [number()], n :: pos_integer(), mu :: float()) :: float()
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
momentn(sample, n)
View Source
momentn(sample :: [number()], n :: pos_integer()) :: float()
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
momentn(sample, n, mu)
View Source
momentn(sample :: [number()], n :: pos_integer(), mu :: float()) :: float()
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
momentn(sample, n, mu, sigma)
View Source
momentn(
sample :: [number()],
n :: pos_integer(),
mu :: float(),
sigma :: float()
) :: float()
momentn( sample :: [number()], n :: pos_integer(), mu :: float(), sigma :: float() ) :: float()
Calculates the nth normalized moment of the sample.
newton(a, b, func, maxiter \\ 10, options) View Source
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]
puiseaux(list, result \\ [], flag \\ false) View Source
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]
puts_errors(device \\ :stdio, errors) View Source
Outputs and formats the errors that result from a call to Chi2fit.Fit.chi2/4
Errors are tuples of length 2 and larger: {[min1,max1], [min2,max2], ...}
.
read_data(filename) View Source
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)
richardson(func, init, factor, results \\ [], options) View Source
Richardson extrapolation.
to_bins(data, binsize \\ {1.0, 0.5}) View Source
Converts raw data to binned data with (asymmetrical) errors.
unzip(list) View Source
Unzips lists of 1-, 2-, 3-, 4-, and 5-tuples.