# FFTConvPDFV1¶

class zfit.pdf.FFTConvPDFV1(func, kernel, n=None, limits_func=None, limits_kernel=None, interpolation=None, obs=None, name='FFTConvV1')[source]

EXPERIMENTAL Numerical Convolution pdf of func convoluted with kernel using FFT

CURRENTLY ONLY 1 DIMENSIONAL!

EXPERIMENTAL: Feedback is very welcome! Performance, which parameters to tune, which fail etc.

TL;DR technical details:

• FFT-like technique: discretization of function. Number of bins splits the kernel into n bins and uses the same binwidth for the func while extending it by the kernel space. Internally, tf.nn.convolution is used.

• Then interpolation by either linear or spline function

• The kernel is assumed to be “small enough” outside of it’s space and points there won’t be evaluated.

The convolution of two (normalized) functions is defined as

$(f * g)(t) \triangleq\ \int_{-\infty}^\infty f(\tau) g(t - \tau) \, d\tau$

It defines the “smearing” of func by a kernel. This is when an element in func is randomly added to an element of kernel. While the sampling (the addition of elements) is rather simple to do computationally, the calculation of the convolutional PDF (if there is no analytic solution available) is not, as it requires:

• an integral from -inf to inf

• an integral for every point of x that is requested

This can be solved with a few tricks. Instead of integrating to infinity, it is usually sufficient to integrate from a point where the function is “small enough”.

If the functions are arbitrary and with conditional dependencies, there is no way around an integral and another PDF has to be used. If the two functions are uncorrelated, a simplified version can be done by a discretization of the space (followed by a Fast Fourier Transfrom, after which the convolution becomes a simple multiplication) and a discrete convolution can be performed.

An interpolation of the discrete convolution for the requested points x is performed afterwards.

Parameters
• func (ZfitPDF) – PDF with pdf method that takes x and returns the function value. Here x is a Data with the obs and limits of limits.

• kernel (ZfitPDF) – PDF with pdf method that takes x acting as the kernel. Here x is a Data with the obs and limits of limits.

• n (Optional[int]) – Number of points per dimension to evaluate the kernel and pdf at. The higher the number of points, the more accurate the convolution at the cost of computing time. If None, a heuristic is used (default to 100 in 1 dimension).

• limits_func (Union[Tuple[Tuple[float, …]], Tuple[float, …], bool, ForwardRef, float]) –

Specify in which limits the func should be evaluated for the convolution:

• If None, the limits from the func are used and extended by a

default value (relative 0.2).

• If float: the fraction of the limit do be extended. 0 means no extension, 1 would extend the limits to each side by the same size resulting in a tripled size (for 1 dimension). As an example, the limits (1, 5) with a limits_func of 0.5 would result in effective limits of (-1, 7), as 0.5 * (5 - 1) = 2 has been added to each side.

• If a space with limits is used, this is taken as the range.

• limits_kernel (Union[Tuple[Tuple[float, …]], Tuple[float, …], bool, ForwardRef]) – the limits of the kernel. Usually not needed to change and automatically taken from the kernel.

• interpolation (Optional[str]) –

Specify the method that is used for interpolation. Available methods are:

• ’linear’: this is the default for any convolution > 1 dimensional. It is a fast, linear interpolation between the evaluated points and approximates the function reasonably well in case of high number of points and a smooth response.

• ’spline’ or f'spline:{order}': a spline interpolation with polynomials. If the order is not specified, a default is used. To specify the order, ‘spline’ should be followed an integer, separated by a colon as e.g. in ‘spline:3’ to use a spline of order three. This method is considerably more computationally expensive as it requires to solve a system of equations. When using 1000+ points this can affect the runtime critical. However, it provides better solution, a curve that is smooth even with less points than for a linear interpolation.

• obs (Union[str, Iterable[str], ForwardRef, None]) – Observables of the class. If not specified, automatically taken from func

• name (str) – Human readable name of the PDF

Add dependencies that render the cache invalid if they change.

Parameters
• cache_deps (Union[ForwardRef, Iterable[ForwardRef]]) –

• allow_non_cachable (bool) – If True, allow cache_dependents to be non-cachables. If False, any cache_dependents that is not a ZfitCachable will raise an error.

Raises

TypeError – if one of the cache_dependents is not a ZfitCachable _and_ allow_non_cachable if False.

analytic_integrate(limits, norm=None, *, norm_range=None)

Analytical integration over function and raise Error if not possible.

Parameters
• limits (Union[Tuple[Tuple[float, …]], Tuple[float, …], bool, ForwardRef]) – the limits to integrate over

• norm (Union[Tuple[Tuple[float, …]], Tuple[float, …], bool, ForwardRef]) – the limits to normalize over

Return type

Union[float, Tensor]

Returns

The integral value

Raises
• AnalyticIntegralNotImplementedError – If no analytical integral is available (for this limits).

• NormRangeNotImplementedError – if the norm argument is not supported. This means that no analytical normalization is available, explicitly: the analytical integral over the limits = norm is not available.

apply_yield(value, norm=False, log=False)

If a norm_range is given, the value will be multiplied by the yield.

Parameters
• value (Union[float, Tensor]) –

• norm (Union[ZfitLimit, Tensor, ndarray, Iterable[float], float, Tuple[float], List[float], bool, None]) –

• log (bool) –

Return type

Union[float, Tensor]

Returns

Numerical

as_func(norm=False, *, norm_range=None)

Return a Function with the function model(x, norm=norm).

Parameters

norm (Union[Tuple[Tuple[float, …]], Tuple[float, …], bool, ForwardRef]) –

copy(**override_parameters)

Creates a copy of the model.

Note: the copy model may continue to depend on the original initialization arguments.

Parameters

**override_parameters – String/value dictionary of initialization arguments to override with new value.

Return type

BasePDF

Returns

A new instance of type(self) initialized from the union

of self.parameters and override_parameters, i.e., dict(self.parameters, **override_parameters).

Return an extended version of this pdf with yield yield_. The parameters are shared.

Parameters
• yield

Return type

ZfitPDF

Returns

ZfitPDF

create_projection_pdf(limits, *, options=None, limits_to_integrate=None)

Create a PDF projection by integrating out some of the dimensions. (deprecated arguments)

Warning: SOME ARGUMENTS ARE DEPRECATED: (limits_to_integrate). They will be removed in a future version. Instructions for updating: Use limits instead.

The new projection pdf is still fully dependent on the pdf it was created with.

Parameters
• () (options) –

• ()

• limits (Union[ZfitLimit, Tensor, ndarray, Iterable[float], float, Tuple[float], List[float], bool, None]) –

Return type

ZfitPDF

Returns

A pdf without the dimensions from limits_to_integrate.

create_sampler(n=None, limits=None, fixed_params=True)

Create a Sampler that acts as Data but can be resampled, also with changed parameters and n.

If limits is not specified, space is used (if the space contains limits). If n is None and the model is an extended pdf, ‘extended’ is used by default.

Parameters
• n (Union[int, Tensor, str]) –

The number of samples to be generated. Can be a Tensor that will be or a valid string. Currently implemented:

• ’extended’: samples poisson(yield) from each pdf that is extended.

• limits (Union[Tuple[Tuple[float, …]], Tuple[float, …], bool, ForwardRef]) – From which space to sample.

• fixed_params (Union[bool, List[ZfitParameter], Tuple[ZfitParameter]]) – A list of Parameters that will be fixed during several resample calls. If True, all are fixed, if False, all are floating. If a Parameter is not fixed and its value gets updated (e.g. by a Parameter.set_value() call), this will be reflected in resample. If fixed, the Parameter will still have the same value as the Sampler has been created with when it resamples.

Return type

Sampler

Returns

Sampler

Raises
• NotExtendedPDFError – if ‘extended’ is chosen (implicitly by default or explicitly) as an option for n but the pdf itself is not extended.

• ValueError – if n is an invalid string option.

• InvalidArgumentError – if n is not specified and pdf is not extended.

property dtype: tensorflow.python.framework.dtypes.DType

The dtype of the object.

Return type

DType

get_cache_deps(only_floating=True)

Return a set of all independent Parameter that this object depends on.

Parameters

only_floating (bool) – If True, only return floating Parameter

Return type

OrderedSet

get_dependencies(only_floating=True)

DEPRECATED FUNCTION

Warning: THIS FUNCTION IS DEPRECATED. It will be removed in a future version. Instructions for updating: Use get_params instead if you want to retrieve the independent parameters or get_cache_deps in case you need the numerical cache dependents (advanced).

get_params(floating=True, is_yield=None, extract_independent=True, only_floating=<class 'zfit.util.checks.NotSpecified'>)

Recursively collect parameters that this object depends on according to the filter criteria.

Which parameters should be included can be steered using the arguments as a filter.
• None: do not filter on this. E.g. floating=None will return parameters that are floating as well as

parameters that are fixed.

• True: only return parameters that fulfil this criterion

• False: only return parameters that do not fulfil this criterion. E.g. floating=False will return

only parameters that are not floating.

Parameters
• floating (Optional[bool]) – if a parameter is floating, e.g. if floating() returns True

• is_yield (Optional[bool]) – if a parameter is a yield of the _current_ model. This won’t be applied recursively, but may include yields if they do also represent a parameter parametrizing the shape. So if the yield of the current model depends on other yields (or also non-yields), this will be included. If, however, just submodels depend on a yield (as their yield) and it is not correlated to the output of our model, they won’t be included.

• extract_independent (Optional[bool]) – If the parameter is an independent parameter, i.e. if it is a ZfitIndependentParameter.

Return type

Set[ZfitParameter]

get_yield()

Return the yield (only for extended models).

Return type

Optional[Parameter]

Returns

The yield of the current model or None

property is_extended: bool

Flag to tell whether the model is extended or not.

Return type

bool

Returns

A boolean.

log_pdf(x, norm=None, *, norm_range=None)

Log probability density function normalized over norm_range.

Parameters
• x (Union[float, Tensor]) – float or double Tensor.

• norm (Union[Tuple[Tuple[float, …]], Tuple[float, …], bool, ForwardRef]) – Space to normalize over

Return type

Union[float, Tensor]

Returns

A Tensor of type self.dtype.

property models: List[zfit.core.interfaces.ZfitModel]

Return the models of this Functor.

Can be pdfs or funcs.

Return type

List[ZfitModel]

property name: str

The name of the object.

Return type

str

property norm: Union[zfit.core.space.Space, None, bool]

Return the current normalization range. If None and the obs have limits, they are returned.

Return type

Union[Space, None, bool]

Returns

The current normalization range.

property norm_range: Union[zfit.core.space.Space, None, bool]

Return the current normalization range. If None and the obs have limits, they are returned. (deprecated)

Warning: THIS FUNCTION IS DEPRECATED. It will be removed in a future version. Instructions for updating: Use the norm attribute instead.

Return type

Union[Space, None, bool]

Returns

The current normalization range.

normalization(limits, *, options=None)

Return the normalization of the function (usually the integral over limits).

Parameters
• () (options) –

• ()

• limits (Union[Tuple[Tuple[float, …]], Tuple[float, …], bool, ForwardRef]) – The limits on where to normalize over

Return type

Union[float, Tensor]

Returns

The normalization value

numeric_integrate(limits, norm=None, *, options=None, norm_range=None)

Numerical integration over the model.

Parameters
• limits (Union[Tuple[Tuple[float, …]], Tuple[float, …], bool, ForwardRef]) – the limits to integrate over

• norm (Union[Tuple[Tuple[float, …]], Tuple[float, …], bool, ForwardRef]) – the limits to normalize over

Return type

Union[float, Tensor]

Returns

The integral value

Parameters
• an (any keyword argument. The value has to be gettable from the instance (has to be) –

• self. (attribute or callable method of) –

classmethod register_analytic_integral(cls, func, limits=None, priority=50, *, supports_norm=None, supports_norm_range=None, supports_multiple_limits=None)

Register an analytic integral with the class. (deprecated arguments)

Warning: SOME ARGUMENTS ARE DEPRECATED: (supports_norm_range). They will be removed in a future version. Instructions for updating: Use supports_norm instead.

Parameters
• func (Callable) –

A function that calculates the (partial) integral over the axes limits. The signature has to be the following:

• x (ZfitData, None): the data for the remaining axes in a partial

integral. If it is not a partial integral, this will be None.

• limits (ZfitSpace): the limits to integrate over.

• norm_range (ZfitSpace, None): Normalization range of the integral.

If not supports_supports_norm_range, this will be None.

• params (Dict[param_name, zfit.Parameters]): The parameters of the model.

• model (ZfitModel):The model that is being integrated.

• limits (Union[Tuple[Tuple[float, …]], Tuple[float, …], bool, ForwardRef]) – If a Space is given, it is used as limits. Otherwise arguments to instantiate a Range class can be given as follows.|limits_init|

• priority (Union[int, float]) – Priority of the function. If multiple functions cover the same space, the one with the highest priority will be used.

• supports_multiple_limits (bool) – If True, the limits given to the integration function can have multiple limits. If False, only simple limits will pass through and multiple limits will be auto-handled.

• supports_norm (bool) – If True, norm argument to the function may not be None. If False, norm will always be None and care is taken of the normalization automatically.

Return type

None

register_cacher(cacher)

Register a cacher that caches values produces by this instance; a dependent.

Parameters

cacher (Union[ForwardRef, Iterable[ForwardRef]]) –

classmethod register_inverse_analytic_integral(func)

Register an inverse analytical integral, the inverse (unnormalized) cdf.

Parameters

func (Callable) – A function with the signature func(x, params), where x is a Data object and params is a dict.

Return type

None

reset_cache_self()

Clear the cache of self and all dependent cachers.

sample(n=None, limits=None, x=None)

Sample n points within limits from the model.

If limits is not specified, space is used (if the space contains limits). If n is None and the model is an extended pdf, ‘extended’ is used by default.

Parameters
• n (Union[int, Tensor, str]) –

The number of samples to be generated. Can be a Tensor that will be or a valid string. Currently implemented:

• ’extended’: samples poisson(yield) from each pdf that is extended.

• limits (Union[Tuple[Tuple[float, …]], Tuple[float, …], bool, ForwardRef]) – In which region to sample in

Return type

SampleData

Returns

SampleData(n_obs, n_samples)

Raises
• NotExtendedPDFError – if ‘extended’ is (implicitly by default or explicitly) chosen as an option for n but the pdf itself is not extended.

• ValueError – if n is an invalid string option.

• InvalidArgumentError – if n is not specified and pdf is not extended.

set_norm_range(norm)

Set the normalization range (temporarily if used with contextmanager).

Parameters

norm (Union[ZfitLimit, Tensor, ndarray, Iterable[float], float, Tuple[float], List[float], bool, None]) –

set_yield(value)

Make the model extended by setting a yield. If possible, prefer to use create_extended.

This does not alter the general behavior of the PDF. The pdf and integrate and similar methods will continue to return the same - normalized to 1 - values. However, not only can this parameter be accessed via get_yield, the methods ext_pdf and ext_integral provide a version of pdf and integrate respecetively that is multiplied by the yield.

These can be useful for plotting and for binned likelihoods.

Parameters

() (value) –

unnormalized_pdf(x)

PDF “unnormalized”. Use functions for unnormalized pdfs. this is only for performance in special cases. (deprecated)

Warning: THIS FUNCTION IS DEPRECATED. It will be removed in a future version. Instructions for updating: Use pdf(norm=False) instead

Parameters

x (Union[float, Tensor]) – The value, have to be convertible to a Tensor

Return type

Union[float, Tensor]

Returns

1-dimensional tf.Tensor containing the unnormalized pdf.

update_integration_options(draws_per_dim=None, mc_sampler=None, tol=None, max_draws=None, draws_simpson=None)

Set the integration options.

Parameters
• max_draws (default ~1'000'000) – Maximum number of draws when integrating . Typically 500’000 - 5’000’000.

• tol – Tolerance on the error of the integral. typically 1e-4 to 1e-8

• draws_per_dim – The draws for MC integration to do per iteration. Can be set to 'auto’.

• draws_simpson – Number of points in one dimensional Simpson integration. Can be set to 'auto'.