package scipy

Get an attribute of this module as a Py.Object.t. This is useful to pass a Python function to another function.

A beta continuous random variable.

As an instance of the `rv_continuous` class, `beta` object inherits from it a collection of generic methods (see below for the full list), and completes them with details specific for this particular distribution.

Methods ------- rvs(a, b, loc=0, scale=1, size=1, random_state=None) Random variates. pdf(x, a, b, loc=0, scale=1) Probability density function. logpdf(x, a, b, loc=0, scale=1) Log of the probability density function. cdf(x, a, b, loc=0, scale=1) Cumulative distribution function. logcdf(x, a, b, loc=0, scale=1) Log of the cumulative distribution function. sf(x, a, b, loc=0, scale=1) Survival function (also defined as ``1 - cdf``, but `sf` is sometimes more accurate). logsf(x, a, b, loc=0, scale=1) Log of the survival function. ppf(q, a, b, loc=0, scale=1) Percent point function (inverse of ``cdf`` --- percentiles). isf(q, a, b, loc=0, scale=1) Inverse survival function (inverse of ``sf``). moment(n, a, b, loc=0, scale=1) Non-central moment of order n stats(a, b, loc=0, scale=1, moments='mv') Mean('m'), variance('v'), skew('s'), and/or kurtosis('k'). entropy(a, b, loc=0, scale=1) (Differential) entropy of the RV. fit(data) Parameter estimates for generic data. See `scipy.stats.rv_continuous.fit <https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.rv_continuous.fit.html#scipy.stats.rv_continuous.fit>`__ for detailed documentation of the keyword arguments. expect(func, args=(a, b), loc=0, scale=1, lb=None, ub=None, conditional=False, **kwds) Expected value of a function (of one argument) with respect to the distribution. median(a, b, loc=0, scale=1) Median of the distribution. mean(a, b, loc=0, scale=1) Mean of the distribution. var(a, b, loc=0, scale=1) Variance of the distribution. std(a, b, loc=0, scale=1) Standard deviation of the distribution. interval(alpha, a, b, loc=0, scale=1) Endpoints of the range that contains alpha percent of the distribution

Notes ----- The probability density function for `beta` is:

.. math::

f(x, a, b) = \frac\Gamma(a+b) x^{a-1 (1-x)^-1

}

\Gamma(a) \Gamma(b)

for :math:`0 <= x <= 1`, :math:`a > 0`, :math:`b > 0`, where :math:`\Gamma` is the gamma function (`scipy.special.gamma`).

`beta` takes :math:`a` and :math:`b` as shape parameters.

The probability density above is defined in the 'standardized' form. To shift and/or scale the distribution use the ``loc`` and ``scale`` parameters. Specifically, ``beta.pdf(x, a, b, loc, scale)`` is identically equivalent to ``beta.pdf(y, a, b) / scale`` with ``y = (x - loc) / scale``.

Examples -------- >>> from scipy.stats import beta >>> import matplotlib.pyplot as plt >>> fig, ax = plt.subplots(1, 1)

Calculate a few first moments:

>>> a, b = 2.31, 0.627 >>> mean, var, skew, kurt = beta.stats(a, b, moments='mvsk')

Display the probability density function (``pdf``):

>>> x = np.linspace(beta.ppf(0.01, a, b), ... beta.ppf(0.99, a, b), 100) >>> ax.plot(x, beta.pdf(x, a, b), ... 'r-', lw=5, alpha=0.6, label='beta pdf')

Alternatively, the distribution object can be called (as a function) to fix the shape, location and scale parameters. This returns a 'frozen' RV object holding the given parameters fixed.

Freeze the distribution and display the frozen ``pdf``:

>>> rv = beta(a, b) >>> ax.plot(x, rv.pdf(x), 'k-', lw=2, label='frozen pdf')

Check accuracy of ``cdf`` and ``ppf``:

>>> vals = beta.ppf(0.001, 0.5, 0.999, a, b) >>> np.allclose(0.001, 0.5, 0.999, beta.cdf(vals, a, b)) True

Generate random numbers:

>>> r = beta.rvs(a, b, size=1000)

And compare the histogram:

>>> ax.hist(r, density=True, histtype='stepfilled', alpha=0.2) >>> ax.legend(loc='best', frameon=False) >>> plt.show()

A binomial discrete random variable.

As an instance of the `rv_discrete` class, `binom` object inherits from it a collection of generic methods (see below for the full list), and completes them with details specific for this particular distribution.

Methods ------- rvs(n, p, loc=0, size=1, random_state=None) Random variates. pmf(k, n, p, loc=0) Probability mass function. logpmf(k, n, p, loc=0) Log of the probability mass function. cdf(k, n, p, loc=0) Cumulative distribution function. logcdf(k, n, p, loc=0) Log of the cumulative distribution function. sf(k, n, p, loc=0) Survival function (also defined as ``1 - cdf``, but `sf` is sometimes more accurate). logsf(k, n, p, loc=0) Log of the survival function. ppf(q, n, p, loc=0) Percent point function (inverse of ``cdf`` --- percentiles). isf(q, n, p, loc=0) Inverse survival function (inverse of ``sf``). stats(n, p, loc=0, moments='mv') Mean('m'), variance('v'), skew('s'), and/or kurtosis('k'). entropy(n, p, loc=0) (Differential) entropy of the RV. expect(func, args=(n, p), loc=0, lb=None, ub=None, conditional=False) Expected value of a function (of one argument) with respect to the distribution. median(n, p, loc=0) Median of the distribution. mean(n, p, loc=0) Mean of the distribution. var(n, p, loc=0) Variance of the distribution. std(n, p, loc=0) Standard deviation of the distribution. interval(alpha, n, p, loc=0) Endpoints of the range that contains alpha percent of the distribution

Notes ----- The probability mass function for `binom` is:

.. math::

f(k) = \binomnk p^k (1-p)^n-k

for ``k`` in ``

, 1,..., n

``.

`binom` takes ``n`` and ``p`` as shape parameters.

The probability mass function above is defined in the 'standardized' form. To shift distribution use the ``loc`` parameter. Specifically, ``binom.pmf(k, n, p, loc)`` is identically equivalent to ``binom.pmf(k - loc, n, p)``.

Examples -------- >>> from scipy.stats import binom >>> import matplotlib.pyplot as plt >>> fig, ax = plt.subplots(1, 1)

Calculate a few first moments:

>>> n, p = 5, 0.4 >>> mean, var, skew, kurt = binom.stats(n, p, moments='mvsk')

Display the probability mass function (``pmf``):

>>> x = np.arange(binom.ppf(0.01, n, p), ... binom.ppf(0.99, n, p)) >>> ax.plot(x, binom.pmf(x, n, p), 'bo', ms=8, label='binom pmf') >>> ax.vlines(x, 0, binom.pmf(x, n, p), colors='b', lw=5, alpha=0.5)

Alternatively, the distribution object can be called (as a function) to fix the shape and location. This returns a 'frozen' RV object holding the given parameters fixed.

Freeze the distribution and display the frozen ``pmf``:

>>> rv = binom(n, p) >>> ax.vlines(x, 0, rv.pmf(x), colors='k', linestyles='-', lw=1, ... label='frozen pmf') >>> ax.legend(loc='best', frameon=False) >>> plt.show()

Check accuracy of ``cdf`` and ``ppf``:

>>> prob = binom.cdf(x, n, p) >>> np.allclose(x, binom.ppf(prob, n, p)) True

Generate random numbers:

>>> r = binom.rvs(n, p, size=1000)

Compares the medians from two independent groups along the given axis.

The comparison is performed using the McKean-Schrader estimate of the standard error of the medians.

Parameters ---------- group_1 : array_like First dataset. Has to be of size >=7. group_2 : array_like Second dataset. Has to be of size >=7. axis : int, optional Axis along which the medians are estimated. If None, the arrays are flattened. If `axis` is not None, then `group_1` and `group_2` should have the same shape.

Returns ------- compare_medians_ms : float, ndarray If `axis` is None, then returns a float, otherwise returns a 1-D ndarray of floats with a length equal to the length of `group_1` along `axis`.

Returns the Harrell-Davis estimate of the median along the given axis.

Parameters ---------- data : ndarray Data array. axis : int, optional Axis along which to compute the quantiles. If None, use a flattened array. var : bool, optional Whether to return the variance of the estimate.

Returns ------- hdmedian : MaskedArray The median values. If ``var=True``, the variance is returned inside the masked array. E.g. for a 1-D array the shape change from (1,) to (2,).

Computes quantile estimates with the Harrell-Davis method.

The quantile estimates are calculated as a weighted linear combination of order statistics.

Parameters ---------- data : array_like Data array. prob : sequence, optional Sequence of quantiles to compute. axis : int or None, optional Axis along which to compute the quantiles. If None, use a flattened array. var : bool, optional Whether to return the variance of the estimate.

Returns ------- hdquantiles : MaskedArray A (p,) array of quantiles (if `var` is False), or a (2,p) array of quantiles and variances (if `var` is True), where ``p`` is the number of quantiles.

See Also -------- hdquantiles_sd

The standard error of the Harrell-Davis quantile estimates by jackknife.

Parameters ---------- data : array_like Data array. prob : sequence, optional Sequence of quantiles to compute. axis : int, optional Axis along which to compute the quantiles. If None, use a flattened array.

Returns ------- hdquantiles_sd : MaskedArray Standard error of the Harrell-Davis quantile estimates.

See Also -------- hdquantiles

Returns an estimate of the lower and upper quartiles.

Uses the ideal fourths algorithm.

Parameters ---------- data : array_like Input array. axis : int, optional Axis along which the quartiles are estimated. If None, the arrays are flattened.

Returns ------- idealfourths :

st of floats, masked array

}

Returns the two internal values that divide `data` into four parts using the ideal fourths algorithm either along the flattened array (if `axis` is None) or along `axis` of `data`.

Computes the alpha-level confidence interval for the median of the data.

Uses the Hettmasperger-Sheather method.

Parameters ---------- data : array_like Input data. Masked values are discarded. The input should be 1D only, or `axis` should be set to None. alpha : float, optional Confidence level of the intervals. axis : int or None, optional Axis along which to compute the quantiles. If None, use a flattened array.

Returns ------- median_cihs Alpha level confidence interval.

Returns the Maritz-Jarrett estimators of the standard error of selected experimental quantiles of the data.

Parameters ---------- data : ndarray Data array. prob : sequence, optional Sequence of quantiles to compute. axis : int or None, optional Axis along which to compute the quantiles. If None, use a flattened array.

Computes the alpha confidence interval for the selected quantiles of the data, with Maritz-Jarrett estimators.

Parameters ---------- data : ndarray Data array. prob : sequence, optional Sequence of quantiles to compute. alpha : float, optional Confidence level of the intervals. axis : int or None, optional Axis along which to compute the quantiles. If None, use a flattened array.

Returns ------- ci_lower : ndarray The lower boundaries of the confidence interval. Of the same length as `prob`. ci_upper : ndarray The upper boundaries of the confidence interval. Of the same length as `prob`.

A normal continuous random variable.

The location (``loc``) keyword specifies the mean. The scale (``scale``) keyword specifies the standard deviation.

As an instance of the `rv_continuous` class, `norm` object inherits from it a collection of generic methods (see below for the full list), and completes them with details specific for this particular distribution.

Methods ------- rvs(loc=0, scale=1, size=1, random_state=None) Random variates. pdf(x, loc=0, scale=1) Probability density function. logpdf(x, loc=0, scale=1) Log of the probability density function. cdf(x, loc=0, scale=1) Cumulative distribution function. logcdf(x, loc=0, scale=1) Log of the cumulative distribution function. sf(x, loc=0, scale=1) Survival function (also defined as ``1 - cdf``, but `sf` is sometimes more accurate). logsf(x, loc=0, scale=1) Log of the survival function. ppf(q, loc=0, scale=1) Percent point function (inverse of ``cdf`` --- percentiles). isf(q, loc=0, scale=1) Inverse survival function (inverse of ``sf``). moment(n, loc=0, scale=1) Non-central moment of order n stats(loc=0, scale=1, moments='mv') Mean('m'), variance('v'), skew('s'), and/or kurtosis('k'). entropy(loc=0, scale=1) (Differential) entropy of the RV. fit(data) Parameter estimates for generic data. See `scipy.stats.rv_continuous.fit <https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.rv_continuous.fit.html#scipy.stats.rv_continuous.fit>`__ for detailed documentation of the keyword arguments. expect(func, args=(), loc=0, scale=1, lb=None, ub=None, conditional=False, **kwds) Expected value of a function (of one argument) with respect to the distribution. median(loc=0, scale=1) Median of the distribution. mean(loc=0, scale=1) Mean of the distribution. var(loc=0, scale=1) Variance of the distribution. std(loc=0, scale=1) Standard deviation of the distribution. interval(alpha, loc=0, scale=1) Endpoints of the range that contains alpha percent of the distribution

Notes ----- The probability density function for `norm` is:

.. math::

f(x) = \frac\exp(-x^2/2)\sqrt{2\pi

}

for a real number :math:`x`.

The probability density above is defined in the 'standardized' form. To shift and/or scale the distribution use the ``loc`` and ``scale`` parameters. Specifically, ``norm.pdf(x, loc, scale)`` is identically equivalent to ``norm.pdf(y) / scale`` with ``y = (x - loc) / scale``.

Examples -------- >>> from scipy.stats import norm >>> import matplotlib.pyplot as plt >>> fig, ax = plt.subplots(1, 1)

Calculate a few first moments:

>>> mean, var, skew, kurt = norm.stats(moments='mvsk')

Display the probability density function (``pdf``):

>>> x = np.linspace(norm.ppf(0.01), ... norm.ppf(0.99), 100) >>> ax.plot(x, norm.pdf(x), ... 'r-', lw=5, alpha=0.6, label='norm pdf')

Alternatively, the distribution object can be called (as a function) to fix the shape, location and scale parameters. This returns a 'frozen' RV object holding the given parameters fixed.

Freeze the distribution and display the frozen ``pdf``:

>>> rv = norm() >>> ax.plot(x, rv.pdf(x), 'k-', lw=2, label='frozen pdf')

Check accuracy of ``cdf`` and ``ppf``:

>>> vals = norm.ppf(0.001, 0.5, 0.999) >>> np.allclose(0.001, 0.5, 0.999, norm.cdf(vals)) True

Generate random numbers:

>>> r = norm.rvs(size=1000)

And compare the histogram:

>>> ax.hist(r, density=True, histtype='stepfilled', alpha=0.2) >>> ax.legend(loc='best', frameon=False) >>> plt.show()

Evaluates Rosenblatt's shifted histogram estimators for each data point.

Rosenblatt's estimator is a centered finite-difference approximation to the derivative of the empirical cumulative distribution function.

Parameters ---------- data : sequence Input data, should be 1-D. Masked values are ignored. points : sequence or None, optional Sequence of points where to evaluate Rosenblatt shifted histogram. If None, use the data.

A Student's t continuous random variable.

As an instance of the `rv_continuous` class, `t` object inherits from it a collection of generic methods (see below for the full list), and completes them with details specific for this particular distribution.

Methods ------- rvs(df, loc=0, scale=1, size=1, random_state=None) Random variates. pdf(x, df, loc=0, scale=1) Probability density function. logpdf(x, df, loc=0, scale=1) Log of the probability density function. cdf(x, df, loc=0, scale=1) Cumulative distribution function. logcdf(x, df, loc=0, scale=1) Log of the cumulative distribution function. sf(x, df, loc=0, scale=1) Survival function (also defined as ``1 - cdf``, but `sf` is sometimes more accurate). logsf(x, df, loc=0, scale=1) Log of the survival function. ppf(q, df, loc=0, scale=1) Percent point function (inverse of ``cdf`` --- percentiles). isf(q, df, loc=0, scale=1) Inverse survival function (inverse of ``sf``). moment(n, df, loc=0, scale=1) Non-central moment of order n stats(df, loc=0, scale=1, moments='mv') Mean('m'), variance('v'), skew('s'), and/or kurtosis('k'). entropy(df, loc=0, scale=1) (Differential) entropy of the RV. fit(data) Parameter estimates for generic data. See `scipy.stats.rv_continuous.fit <https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.rv_continuous.fit.html#scipy.stats.rv_continuous.fit>`__ for detailed documentation of the keyword arguments. expect(func, args=(df,), loc=0, scale=1, lb=None, ub=None, conditional=False, **kwds) Expected value of a function (of one argument) with respect to the distribution. median(df, loc=0, scale=1) Median of the distribution. mean(df, loc=0, scale=1) Mean of the distribution. var(df, loc=0, scale=1) Variance of the distribution. std(df, loc=0, scale=1) Standard deviation of the distribution. interval(alpha, df, loc=0, scale=1) Endpoints of the range that contains alpha percent of the distribution

Notes ----- The probability density function for `t` is:

.. math::

f(x, \nu) = \frac\Gamma((\nu+1)/2) \sqrt{\pi \nu \Gamma(\nu/2)

}

(1+x^2/\nu)^

(\nu+1)/2

}

where :math:`x` is a real number and the degrees of freedom parameter :math:`\nu` (denoted ``df`` in the implementation) satisfies :math:`\nu > 0`. :math:`\Gamma` is the gamma function (`scipy.special.gamma`).

The probability density above is defined in the 'standardized' form. To shift and/or scale the distribution use the ``loc`` and ``scale`` parameters. Specifically, ``t.pdf(x, df, loc, scale)`` is identically equivalent to ``t.pdf(y, df) / scale`` with ``y = (x - loc) / scale``.

Examples -------- >>> from scipy.stats import t >>> import matplotlib.pyplot as plt >>> fig, ax = plt.subplots(1, 1)

Calculate a few first moments:

>>> df = 2.74 >>> mean, var, skew, kurt = t.stats(df, moments='mvsk')

Display the probability density function (``pdf``):

>>> x = np.linspace(t.ppf(0.01, df), ... t.ppf(0.99, df), 100) >>> ax.plot(x, t.pdf(x, df), ... 'r-', lw=5, alpha=0.6, label='t pdf')

Alternatively, the distribution object can be called (as a function) to fix the shape, location and scale parameters. This returns a 'frozen' RV object holding the given parameters fixed.

Freeze the distribution and display the frozen ``pdf``:

>>> rv = t(df) >>> ax.plot(x, rv.pdf(x), 'k-', lw=2, label='frozen pdf')

Check accuracy of ``cdf`` and ``ppf``:

>>> vals = t.ppf(0.001, 0.5, 0.999, df) >>> np.allclose(0.001, 0.5, 0.999, t.cdf(vals, df)) True

Generate random numbers:

>>> r = t.rvs(df, size=1000)

And compare the histogram:

>>> ax.hist(r, density=True, histtype='stepfilled', alpha=0.2) >>> ax.legend(loc='best', frameon=False) >>> plt.show()

Selected confidence interval of the trimmed mean along the given axis.

Parameters ---------- data : array_like Input data. limits : None, tuple, optional None or a two item tuple. Tuple of the percentages to cut on each side of the array, with respect to the number of unmasked data, as floats between 0. and 1. If ``n`` is the number of unmasked data before trimming, then (``n * limits0``)th smallest data and (``n * limits1``)th largest data are masked. The total number of unmasked data after trimming is ``n * (1. - sum(limits))``. The value of one limit can be set to None to indicate an open interval.

Defaults to (0.2, 0.2). inclusive : (2,) tuple of boolean, optional If relative==False, tuple indicating whether values exactly equal to the absolute limits are allowed. If relative==True, tuple indicating whether the number of data being masked on each side should be rounded (True) or truncated (False).

Defaults to (True, True). alpha : float, optional Confidence level of the intervals.

Defaults to 0.05. axis : int, optional Axis along which to cut. If None, uses a flattened version of `data`.

Defaults to None.

Returns ------- trimmed_mean_ci : (2,) ndarray The lower and upper confidence intervals of the trimmed data.