A generic continuous random variable class meant for subclassing.
`rv_continuous` is a base class to construct specific distribution classes and instances for continuous random variables. It cannot be used directly as a distribution.
Parameters ---------- momtype : int, optional The type of generic moment calculation to use: 0 for pdf, 1 (default) for ppf. a : float, optional Lower bound of the support of the distribution, default is minus infinity. b : float, optional Upper bound of the support of the distribution, default is plus infinity. xtol : float, optional The tolerance for fixed point calculation for generic ppf. badvalue : float, optional The value in a result arrays that indicates a value that for which some argument restriction is violated, default is np.nan. name : str, optional The name of the instance. This string is used to construct the default example for distributions. longname : str, optional This string is used as part of the first line of the docstring returned when a subclass has no docstring of its own. Note: `longname` exists for backwards compatibility, do not use for new subclasses. shapes : str, optional The shape of the distribution. For example ``'m, n'`` for a distribution that takes two integers as the two shape arguments for all its methods. If not provided, shape parameters will be inferred from the signature of the private methods, ``_pdf`` and ``_cdf`` of the instance. extradoc : str, optional, deprecated This string is used as the last part of the docstring returned when a subclass has no docstring of its own. Note: `extradoc` exists for backwards compatibility, do not use for new subclasses. seed : None or int or ``numpy.random.RandomState`` instance, optional This parameter defines the RandomState object to use for drawing random variates. If None (or np.random), the global np.random state is used. If integer, it is used to seed the local RandomState instance. Default is None.
Methods ------- rvs pdf logpdf cdf logcdf sf logsf ppf isf moment stats entropy expect median mean std var interval __call__ fit fit_loc_scale nnlf support
Notes ----- Public methods of an instance of a distribution class (e.g., ``pdf``, ``cdf``) check their arguments and pass valid arguments to private, computational methods (``_pdf``, ``_cdf``). For ``pdf(x)``, ``x`` is valid if it is within the support of the distribution. Whether a shape parameter is valid is decided by an ``_argcheck`` method (which defaults to checking that its arguments are strictly positive.)
**Subclassing**
New random variables can be defined by subclassing the `rv_continuous` class and re-defining at least the ``_pdf`` or the ``_cdf`` method (normalized to location 0 and scale 1).
If positive argument checking is not correct for your RV then you will also need to re-define the ``_argcheck`` method.
For most of the scipy.stats distributions, the support interval doesn't depend on the shape parameters. ``x`` being in the support interval is equivalent to ``self.a <= x <= self.b``. If either of the endpoints of the support do depend on the shape parameters, then i) the distribution must implement the ``_get_support`` method; and ii) those dependent endpoints must be omitted from the distribution's call to the ``rv_continuous`` initializer.
Correct, but potentially slow defaults exist for the remaining methods but for speed and/or accuracy you can over-ride::
_logpdf, _cdf, _logcdf, _ppf, _rvs, _isf, _sf, _logsf
The default method ``_rvs`` relies on the inverse of the cdf, ``_ppf``, applied to a uniform random variate. In order to generate random variates efficiently, either the default ``_ppf`` needs to be overwritten (e.g. if the inverse cdf can expressed in an explicit form) or a sampling method needs to be implemented in a custom ``_rvs`` method.
If possible, you should override ``_isf``, ``_sf`` or ``_logsf``. The main reason would be to improve numerical accuracy: for example, the survival function ``_sf`` is computed as ``1 - _cdf`` which can result in loss of precision if ``_cdf(x)`` is close to one.
**Methods that can be overwritten by subclasses** ::
_rvs _pdf _cdf _sf _ppf _isf _stats _munp _entropy _argcheck _get_support
There are additional (internal and private) generic methods that can be useful for cross-checking and for debugging, but might work in all cases when directly called.
A note on ``shapes``: subclasses need not specify them explicitly. In this case, `shapes` will be automatically deduced from the signatures of the overridden methods (`pdf`, `cdf` etc). If, for some reason, you prefer to avoid relying on introspection, you can specify ``shapes`` explicitly as an argument to the instance constructor.
**Frozen Distributions**
Normally, you must provide shape parameters (and, optionally, location and scale parameters to each call of a method of a distribution.
Alternatively, the object may be called (as a function) to fix the shape, location, and scale parameters returning a 'frozen' continuous RV object:
rv = generic(<shape(s)>, loc=0, scale=1) `rv_frozen` object with the same methods but holding the given shape, location, and scale fixed
**Statistics**
Statistics are computed using numerical integration by default. For speed you can redefine this using ``_stats``:
- take shape parameters and return mu, mu2, g1, g2
- If you can't compute one of these, return it as None
- Can also be defined with a keyword argument ``moments``, which is a string composed of 'm', 'v', 's', and/or 'k'. Only the components appearing in string should be computed and returned in the order 'm', 'v', 's', or 'k' with missing values returned as None.
Alternatively, you can override ``_munp``, which takes ``n`` and shape parameters and returns the n-th non-central moment of the distribution.
Examples -------- To create a new Gaussian distribution, we would do the following:
>>> from scipy.stats import rv_continuous >>> class gaussian_gen(rv_continuous): ... 'Gaussian distribution' ... def _pdf(self, x): ... return np.exp(-x**2 / 2.) / np.sqrt(2.0 * np.pi) >>> gaussian = gaussian_gen(name='gaussian')
``scipy.stats`` distributions are *instances*, so here we subclass `rv_continuous` and create an instance. With this, we now have a fully functional distribution with all relevant methods automagically generated by the framework.
Note that above we defined a standard normal distribution, with zero mean and unit variance. Shifting and scaling of the distribution can be done by using ``loc`` and ``scale`` parameters: ``gaussian.pdf(x, loc, scale)`` essentially computes ``y = (x - loc) / scale`` and ``gaussian._pdf(y) / scale``.