amici.import_utils¶
Miscellaneous functions related to model import, independent of any specific model format
Functions Summary
Parse noise distribution string to a cost function definition amici can work with. |
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Optimized substitution that checks whether anything needs to be done first |
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Subsitutes expressions completely flattening them out. |
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Topologically sort symbol definitions according to their interdependency |
Functions
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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 (
str
) –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.
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amici.import_utils.
smart_subs
(element, old, new)[source]¶ Optimized substitution that checks whether anything needs to be done first
- Parameters
element (
sympy.core.expr.Expr
) – substitution targetold (
sympy.core.symbol.Symbol
) – to be substitutednew (
sympy.core.expr.Expr
) – subsitution value
- Return type
- Returns
substituted expression
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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
sym (
sympy.core.expr.Expr
) – Symbolic expression in which expressions will be substitutedsubs (
typing.Dict
[sympy.core.symbol.Symbol
,typing.Union
[typing.Dict
[str
,sympy.core.expr.Expr
],sympy.core.expr.Expr
]]) – Substitutionsfield (
typing.Optional
[str
]) – Field of substitution expressions in subs.values(), if applicablereverse (
bool
) – Whether ordering in subs should be reversed. Note that substitution requires the reverse order of what is required for evaluation.
- Return type
- Returns
Substituted symbolic expression
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amici.import_utils.
toposort_symbols
(symbols, field=None)[source]¶ Topologically sort symbol definitions according to their interdependency
- Parameters
symbols (
typing.Dict
[sympy.core.symbol.Symbol
,typing.Union
[typing.Dict
[str
,sympy.core.expr.Expr
],sympy.core.expr.Expr
]]) – symbol definitionsfield (
typing.Optional
[str
]) – field of definition.values() that is used to compute interdependency
- 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