Chi-SquaredFit v1.0.0-beta.8 Chi2fit.Fit View Source
Implements fitting a distribution function to sample data. It minimizes the liklihood function.
Asymmetric Errors
To handle asymmetric errors the module provides three ways of determining the contribution to the likelihood function:
`simple` - value difference of the observable and model divided by the averaged error lower and upper bounds;
`asimple` - value difference of the observable and model divided by the difference between upper/lower bound and the observed
value depending on whether the model is larger or smaller than the observed value;
`linear` - value difference of the observable and model divided by a linear tranformation (See below).
‘linear’: Linear transformation
Linear transformation that:
- is continuous in u=0,
- passes through the point sigma+ at u=1,
- asymptotically reaches 1-y at u->infinity
- pass through the point -sigma- at u=-1,
- asymptotically reaches -y at u->-infinity
References
[1] See https://arxiv.org/pdf/physics/0401042v1.pdf
Link to this section Summary
Types
Chi-squared statistic
Tuple holding chi-squared value, covariance matrix, parameter values, and parameter errors at the minimum chi2fit(see chi2fit/4
)
Result of chi-squared probe (see &chi2probe/4)
Tuple with chi-squared, parameter values, parameter errors, and list of intermediate fit results (see chi2probe/4
)
Tuple with chi-squared, parameter values, and parameter errors at the found minimum (see chi2probe/4
)
Covariance matrix
Cumulative distribution mapping ‘x’ and parameters to a float in the range [0,1]
Tuple describing the parameter values and the distribution function
Observation with asymmetric bounds ‘y1 < y < y2’
Observation with symmetric errors ‘dy’
List of parameter ranges
Functions
Calculates the Chi-squared function for a list of observables
Fits observables to a known model
Probes the chi-squared surface within a certain range of the parameters
Partitions the data list in segments with similar chi-squared values when fitting the model
Finds the point in the data where the chi-squared has a jump when fitting the model
Link to this section Types
Chi-squared statistic
Tuple holding chi-squared value, covariance matrix, parameter values, and parameter errors at the minimum chi2fit(see chi2fit/4
)
Result of chi-squared probe (see &chi2probe/4)
Tuple with chi-squared, parameter values, parameter errors, and list of intermediate fit results (see chi2probe/4
)
Tuple with chi-squared, parameter values, and parameter errors at the found minimum (see chi2probe/4
)
Covariance matrix
Cumulative distribution mapping ‘x’ and parameters to a float in the range [0,1].
Tuple describing the parameter values and the distribution function.
Observation with asymmetric bounds ‘y1 < y < y2’.
Observation with symmetric errors ‘dy’.
List of parameter ranges
Link to this section Functions
Calculates the Chi-squared function for a list of observables.
The observables
are given as a list. Each observation has an error associated with it. The errors can be either
symmetric or asymmetric.
A ‘penalties’-function is used to assign penalties and these contribute to the chi-squared function. It may be used to ‘forbid’ certain parameter, x combinations.
Options
`model` - Required. Determines the contribution to chi-squared taking the asymmetric errors into account.
Vaid values are `:linear`, `:simple`, and `:asimple`. See Errors below
Errors
`simple` - Use for asymmetric errors when the sigma+ and sigma- are close to each other
`asimple` - Use for asymmetric errors when y-values are not bound.
`linear` - Use this model when the y-values ar bound between 0 and 1. Linear transformation that:
- is continuous in u=0,
- passes through the point sigma+ at u=1,
- asymptotically reaches 1-y at u->infinity
- pass through the point -sigma- at u=-1,
- asymptotically reaches -y at u->-infinity
Examples
iex> fun = &(&1)
...> chi2 [{0,3,1}], fun
9.0
iex> fun = &(&1)
...> chi2 [{0,3,1},{1,7,1},{2,3,1}], fun
46.0
iex> fun = &(&1)
...> chi2 [{0,3,3},{1,7,1},{2,3,1}], fun
38.0
iex> fun = &(&1-2)
...> chi2 [{0,3,1}], fun
25.0
end
chi2fit( observables(), model(), iterations :: pos_integer(), options :: Keyword.t() ) :: chi2fit()
Fits observables to a known model.
Returns the found minimum chi-squared value, covariance matrix, gradient at the minimum, and the corresponding parameter values including error estimates. For a good fit check the following:
`chi2 per degree of freedom` - this should be about 1 or less,
`gradient` - at the minimum the gradient should be zero at all directions.
For asymmetric errors use the option model
equal to linear
.
Rough chi-squared surfaces or if numerically unstable, use the option smoothing
set to true
.
Arguments
`observables` - list of measurements including errors,
`model` - `{parameters, fun}`: set of initial parameter values and a function to fit against the measurements
Options
`onstep` - call back function; it is called with a map with keys `delta`, `chi2`, and `params`,
`smoothing` - boolean value indicating whether the chi-squared is smoothened using a Gauss distribution. This
is used in case the surface is rough because of numerical instabilities to smoothen the surface,
`model` - The same values as in [`chi2/3`](#chi2/3) and [`chi2/4`](#chi2/4)
`grid?` - Performs a grid search: per step, tries to fit only one parameter and keeps the others fixed; selects the parameter in
a round-robin fashion
`probes` -- a list of tuples containing the result of the [`chi2probe/4`](#chi2probe/4) function. Each tuple contains the chi2 value and parameter list.
Defaults to the empty list.
chi2probe(observables(), [float()], (... -> any()), Keyword.t()) :: chi2probe()
Probes the chi-squared surface within a certain range of the parameters.
It does so by randomly selecting parameter value combinations and calculate the chi-squared for the list
of observations based on the selected parameter values. This routine is used to roughly probe the chi-squared
surface and perform more detailed and expensive calculations to precisely determine the minimum by chi2fit/4
.
Returns the minimum chi-squared found, the parameter values, and all probes that resulted in chi-squared difference less than 1 with the minimum. The parameter values found in this set correspond with the errors in determining the parameters.
Options
`num` or `probes` - the number of points to calculate,
`mark` - progress indicator: a keyword list with keys `m`, `c`, `x`, and `*`; the value must be a call back
function taking zero arguments. These are called when 1000, 100, 10, probes have been done. The value of
key `*` is called when a new chi-squared minimum has been found,
`smoothing` - boolean value indicating whether the chi-squared is smoothened using a Gauss distribution. This
is used in case the surface is rough because of numerical instabilities to smoothen the surface,
`model` - See [`chi2/3`](#chi2/3) and [`chi2/4`](#chi2/4)
Partitions the data list in segments with similar chi-squared values when fitting the model
Finds the point in the data where the chi-squared has a jump when fitting the model