amici.import_utils

Miscellaneous functions related to model import, independent of any specific model format

Classes

ObservableTransformation(value)

Different modes of observable transformation.

Functions Summary

noise_distribution_to_cost_function(…)

Parse noise distribution string to a cost function definition amici can work with.

noise_distribution_to_observable_transformation(…)

Parse noise distribution string and extract observable transformation

smart_subs(element, old, new)

Optimized substitution that checks whether anything needs to be done first

smart_subs_dict(sym, subs[, field, reverse])

Subsitutes expressions completely flattening them out.

toposort_symbols(symbols[, field])

Topologically sort symbol definitions according to their interdependency

Functions

amici.import_utils.noise_distribution_to_cost_function(noise_distribution)[source]

Parse noise distribution string to a cost function definition amici can work with.

The noise distributions listed in the following are supported. \(m\) denotes the measurement, \(y\) the simulation, and \(\sigma\) a distribution scale parameter (currently, AMICI only supports a single distribution parameter).

  • ‘normal’, ‘lin-normal’: A normal distribution:

    \[\pi(m|y,\sigma) = \frac{1}{\sqrt{2\pi}\sigma}\ exp\left(-\frac{(m-y)^2}{2\sigma^2}\right)\]
  • ‘log-normal’: A log-normal distribution (i.e. log(m) is normally distributed):

    \[\pi(m|y,\sigma) = \frac{1}{\sqrt{2\pi}\sigma m}\ exp\left(-\frac{(\log m - \log y)^2}{2\sigma^2}\right)\]
  • ‘log10-normal’: A log10-normal distribution (i.e. log10(m) is normally distributed):

    \[\pi(m|y,\sigma) = \frac{1}{\sqrt{2\pi}\sigma m \log(10)}\ exp\left(-\frac{(\log_{10} m - \log_{10} y)^2}{2\sigma^2}\right)\]
  • ‘laplace’, ‘lin-laplace’: A laplace distribution:

    \[\pi(m|y,\sigma) = \frac{1}{2\sigma} \exp\left(-\frac{|m-y|}{\sigma}\right)\]
  • ‘log-laplace’: A log-Laplace distribution (i.e. log(m) is Laplace distributed):

    \[\pi(m|y,\sigma) = \frac{1}{2\sigma m} \exp\left(-\frac{|\log m - \log y|}{\sigma}\right)\]
  • ‘log10-laplace’: A log10-Laplace distribution (i.e. log10(m) is Laplace distributed):

    \[\pi(m|y,\sigma) = \frac{1}{2\sigma m \log(10)} \exp\left(-\frac{|\log_{10} m - \log_{10} y|}{\sigma}\right)\]
  • ‘binomial’, ‘lin-binomial’: A (continuation of a discrete) binomial distribution, parameterized via the success probability \(p=\sigma\):

    \[\pi(m|y,\sigma) = \operatorname{Heaviside}(y-m) \cdot \frac{\Gamma(y+1)}{\Gamma(m+1) \Gamma(y-m+1)} \sigma^m (1-\sigma)^{(y-m)}\]
  • ‘negative-binomial’, ‘lin-negative-binomial’: A (continuation of a discrete) negative binomial distribution, with with mean = y, parameterized via success probability p:

    \[\pi(m|y,\sigma) = \frac{\Gamma(m+r)}{\Gamma(m+1) \Gamma(r)} (1-\sigma)^m \sigma^r\]

    where

    \[r = \frac{1-\sigma}{\sigma} y\]

The distributions above are for a single data point. For a collection \(D=\{m_i\}_i\) of data points and corresponding simulations \(Y=\{y_i\}_i\) and noise parameters \(\Sigma=\{\sigma_i\}_i\), AMICI assumes independence, i.e. the full distributions is

\[\pi(D|Y,\Sigma) = \prod_i\pi(m_i|y_i,\sigma_i)\]

AMICI uses the logarithm \(\log(\pi(m|y,\sigma)\).

In addition to the above mentioned distributions, it is also possible to pass a function taking a symbol string and returning a log-distribution string with variables ‘{str_symbol}’, ‘m{str_symbol}’, ‘sigma{str_symbol}’ for y, m, sigma, respectively.

Parameters

noise_distribution (typing.Union[str, typing.Callable]) –

An identifier specifying a noise model. Possible values are

{‘normal’, ‘lin-normal’, ‘log-normal’, ‘log10-normal’, ‘laplace’, ‘lin-laplace’, ‘log-laplace’, ‘log10-laplace’, ‘binomial’, ‘lin-binomial’, ‘negative-binomial’, ‘lin-negative-binomial’, <Callable>}

For the meaning of the values see above.

Return type

typing.Callable[[str], str]

Returns

A function that takes a strSymbol and then creates a cost function string (negative log-likelihood) from it, which can be sympified.

amici.import_utils.noise_distribution_to_observable_transformation(noise_distribution)[source]

Parse noise distribution string and extract observable transformation

Parameters

noise_distribution (typing.Union[str, typing.Callable]) – see noise_distribution_to_cost_function()

Return type

amici.import_utils.ObservableTransformation

Returns

observable transformation

amici.import_utils.smart_subs(element, old, new)[source]

Optimized substitution that checks whether anything needs to be done first

Parameters
Return type

sympy.core.expr.Expr

Returns

substituted expression

amici.import_utils.smart_subs_dict(sym, subs, field=None, reverse=True)[source]

Subsitutes expressions completely flattening them out. Requires sorting of expressions with toposort.

Parameters
Return type

sympy.core.expr.Expr

Returns

Substituted symbolic expression

amici.import_utils.toposort_symbols(symbols, field=None)[source]

Topologically sort symbol definitions according to their interdependency

Parameters
Return type

typing.Dict[sympy.core.symbol.Symbol, typing.Union[typing.Dict[str, sympy.core.expr.Expr], sympy.core.expr.Expr]]

Returns

ordered symbol definitions