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class surpyval.univariate.parametric.distributions.gumbel.Gumbel_(name)

Bases: ParametricFitter

Hf(x, mu, sigma)

Cumulative hazard rate for the Gumbel Distribution:

\[H(x) = e^{\frac{x-\mu}{\sigma}}\]

Parameters

x (numpy array or scalar) – The values of the random variables at which the survival function will be calculated
mu (numpy array like or scalar) – The location parameter of the distribution
sigma (numpy array like or scalar) – The scale parameter of the distribution

Returns

Hf – The value(s) for the cumulative hazard rate for the Gumbel distribution.

Return type

scalar or numpy array

Examples

>>> import numpy as np
>>> from surpyval import Gumbel
>>> x = np.array([1, 2, 3, 4, 5])
>>> Gumbel.Hf(x, 3, 2)
array([0.36787944, 0.60653066, 1.        , 1.64872127, 2.71828183])

df(x, mu, sigma)

Density function (pdf) for the Gumbel Distribution:

\[f(x) = \frac{1}{\sigma}e^{\left (\frac{x - \mu}{\sigma} - e^{\frac{x-\mu}{\sigma}} \right)}\]

Parameters

x (numpy array or scalar) – The values of the random variables at which the survival function will be calculated
mu (numpy array like or scalar) – The location parameter of the distribution
sigma (numpy array like or scalar) – The scale parameter of the distribution

Returns

df – The scalar value of the density of the distribution if a scalar was passed. If an array like object was passed then a numpy array is returned with the value of the density at each corresponding value in the input array.

Return type

scalar or numpy array

Examples

>>> import numpy as np
>>> from surpyval import Gumbel
>>> x = np.array([1, 2, 3, 4, 5])
>>> Gumbel.df(x, 3, 2)
array([0.12732319, 0.16535215, 0.18393972, 0.15852096, 0.08968704])

entropy(mu, sigma)

Calculates the entropy of the Gumbel distribution.

\[S = \ln \left ( \sigma \right ) + \gamma + 1\]

Where gamma is the Euler-Mascheroni constant

Parameters

mu (numpy array or scalar) – The location parameter(s) of the distribution
sigma (numpy array or scalar) – The scale parameter(s) of the distribution

Returns

entropy – The entropy(ies) of the Gumbel distribution

Return type

scalar or numpy array

Examples

>>> from surpyval import Gumbel
>>> Gumbel.entropy(3, 2)
2.270362845461478

ff(x, mu, sigma)

CDF (or Failure) function for the Gumbel Distribution:

\[F(x) = e^{e^{-\left ( x - \mu \right )/\sigma}}\]

Parameters

x (numpy array or scalar) – The values of the random variables at which the survival function will be calculated
mu (numpy array like or scalar) – The location parameter of the distribution
sigma (numpy array like or scalar) – The scale parameter of the distribution

Returns

ff – The scalar value of the failure function of the distribution if a scalar was passed. If an array like object was passed then a numpy array is returned with the value of the failure function at each corresponding value in the input array.

Return type

scalar or numpy array

Examples

>>> import numpy as np
>>> from surpyval import Gumbel
>>> x = np.array([1, 2, 3, 4, 5])
>>> Gumbel.ff(x, 3, 2)
array([0.30779937, 0.45476079, 0.63212056, 0.80770435, 0.93401196])

fit(x: ArrayLike | None = None, c: ArrayLike | None = None, n: ArrayLike | None = None, t: ArrayLike | None = None, how: str = 'MLE', offset: bool = False, zi: bool = False, lfp: bool = False, tl: ArrayLike | numbers.Number | None = None, tr: ArrayLike | numbers.Number | None = None, xl: ArrayLike | None = None, xr: ArrayLike | None = None, fixed: dict[str, float] | None = None, heuristic: str = 'Nelson-Aalen', init: ArrayLike = [], rr: str = 'y', on_d_is_0: bool = False, turnbull_estimator: str = 'Fleming-Harrington') → Parametric

The central feature to SurPyval’s capability. This function aimed to have an API to mimic the simplicity of the scipy API. That is, to use a simple fit() call, with as many or as few parameters as is needed.

Parameters

x (array like, optional) – Array of observations of the random variables. If x is None, xl and xr must be provided.
c (array like, optional) – Array of censoring flag. -1 is left censored, 0 is observed, 1 is right censored, and 2 is intervally censored. If not provided will assume all values are observed.
n (array like, optional) – Array of counts for each x. If data is provided as counts, then this can be provided. If None will assume each observation is 1.
t (2D-array like, optional) – 2D array like of the left and right values at which the respective observation was truncated. If not provided it assumes that no truncation occurs.
how ({'MLE', 'MPP', 'MOM', 'MSE', 'MPS'}, optional) –
Method to estimate parameters, these are:
- MLE, Maximum Likelihood Estimation
- MPP, Method of Probability Plotting
- MOM, Method of Moments
- MSE, Mean Square Error
- MPS, Maximum Product Spacing
offset (boolean, optional) – If True finds the shifted distribution. If not provided assumes not a shifted distribution. Only works with distributions that are supported on the half-real line.
tl (array like or scalar, optional) – Values of left truncation for observations. If it is a scalar value assumes each observation is left truncated at the value. If an array, it is the respective ‘late entry’ of the observation
tr (array like or scalar, optional) – Values of right truncation for observations. If it is a scalar value assumes each observation is right truncated at the value. If an array, it is the respective right truncation value for each observation
xl (array like, optional) – Array like of the left array for 2-dimensional input of x. This is useful for data that is all intervally censored. Must be used with the xr input.
xr (array like, optional) – Array like of the right array for 2-dimensional input of x. This is useful for data that is all intervally censored. Must be used with the xl input.
fixed (dict, optional) – Dictionary of parameters and their values to fix. Fixes parameter by name.
heuristic ({"Blom", "Median", "ECDF", "Modal", "Midpoint", "Mean", "Weibull", "Benard", "Beard", "Hazen", "Gringorten", "None", "Tukey", "DPW", "Fleming-Harrington", "Kaplan-Meier", "Nelson-Aalen", "Filliben", "Larsen", "Turnbull"}, str, optional.) – Plotting method to use, if using the probability plotting, MPP, method.
init (array like, optional) – initial guess of parameters. Instead of finding an initial guess for the optimization you can provide one. Can be useful to see if optimization is failing due to poor initial guess.
rr ({'y', 'x'}, str, optional) – The dimension on which to minimise the spacing between the line and the observation. If ‘y’ the mean square error between the line and vertical distance to each point is minimised. If ‘x’ the mean square error between the line and horizontal distance to each point is minimised.
on_d_is_0 (boolean, optional) – For the case when using MPP and the highest value is right censored, you can choose to include this value into the regression analysis or not. That is, if False, all values where there are 0 deaths are excluded from the regression. If True all values regardless of whether there is a death or not are included in the regression.
turnbull_estimator ({'Nelson-Aalen', 'Kaplan-Meier', or 'Fleming-Harrington'), str, optional) – If using the Turnbull heuristic, you can elect to use either the KM, NA, or FH estimator with the Turnbull estimates of r, and d. Defaults to FH.

Returns

A parametric model with the fitted parameters and methods for all functions of the distribution using the fitted parameters.

Return type

Parametric

Examples

>>> from surpyval import Weibull
>>> import numpy as np
>>> x = Weibull.random(100, 10, 4)
>>> model = Weibull.fit(x)
>>> print(model)
Parametric SurPyval Model
=========================
Distribution        : Weibull
Fitted by           : MLE
Parameters          :
     alpha: 10.551521182640098
      beta: 3.792549834495306
>>> Weibull.fit(x, how='MPS', fixed={'alpha' : 10})
Parametric SurPyval Model
=========================
Distribution        : Weibull
Fitted by           : MPS
Parameters          :
     alpha: 10.0
      beta: 3.4314657446866836
>>> Weibull.fit(xl=x-1, xr=x+1, how='MPP')
Parametric SurPyval Model
=========================
Distribution        : Weibull
Fitted by           : MPP
Parameters          :
     alpha: 9.943092756713078
      beta: 8.613016934518258
>>> c = np.zeros_like(x)
>>> c[x > 13] = 1
>>> x[x > 13] = 13
>>> c = c[x > 6]
>>> x = x[x > 6]
>>> Weibull.fit(x=x, c=c, tl=6)
Parametric SurPyval Model
=========================
Distribution        : Weibull
Fitted by           : MLE
Parameters          :
     alpha: 10.363725328793413
      beta: 4.9886821457305865

The central feature to SurPyval’s capability. This function aimed to have an API to mimic the simplicity of the scipy API. That is, to use a simple fit() call, with as many or as few parameters as is needed.

Parameters

df (DataFrame) – DataFrame of data to be used to create surpyval model
x (string, optional) – column name for the column in df containing the variable data. If not provided must provide both xl and xr.
c (string, optional) – column name for the column in df containing the censor flag of x. If not provided assumes all values of x are observed.
n (string, optional) – column name in for the column in df containing the counts of x. If not provided assumes each x is one observation.
tl (string or scalar, optional) – If string, column name in for the column in df containing the left truncation data. If scalar assumes each x is left truncated by that value. If not provided assumes x is not left truncated.
tr (string or scalar, optional) – If string, column name in for the column in df containing the right truncation data. If scalar assumes each x is right truncated by that value. If not provided assumes x is not right truncated.
xl (string, optional) – column name for the column in df containing the left interval for interval censored data. If left interval is -Inf, assumes left censored. If xl[i] == xr[i] assumes observed. Cannot be provided with x, must be provided with xr.
xr (string, optional) – column name for the column in df containing the right interval for interval censored data. If right interval is Inf, assumes right censored. If xl[i] == xr[i] assumes observed. Cannot be provided with x, must be provided with xl.
fit_options (dict, optional) – dictionary of fit options that will be passed to the fit method, see that method for options.

Returns

A parametric model with the fitted parameters and methods for all functions of the distribution using the fitted parameters.

Return type

Parametric

Examples

>>> import surpyval as surv
>>> from surpyval.datasets import load_bofors_steel
>>> df = load_bofors_steel()
>>> model = surv.Weibull.fit_from_df(df, x='x', n='n', offset=True)
>>> print(model)
Parametric SurPyval Model
=========================
Distribution        : Weibull
Fitted by           : MLE
Offset (gamma)      : 39.76562962867477
Parameters          :
     alpha: 7.141925216146524
      beta: 2.6204524040137844

fit_from_surpyval_data(surv_data: SurpyvalData, how: str = 'MLE', offset: bool = False, zi: bool = False, lfp: bool = False, fixed: dict[str, float] | None = None, heuristic: str = 'Nelson-Aalen', init: ArrayLike = [], rr: str = 'y', on_d_is_0: bool = False, turnbull_estimator: str = 'Fleming-Harrington') → Parametric

The central feature to SurPyval’s capability. This function aimed to have an API to mimic the simplicity of the scipy API. That is, to use a simple fit() call, with as many or as few parameters as is needed.

Parameters: surv_data (SurpyvalData) – Survival data in the SurpyvalData class.

For other input options see fit method.

Returns: A parametric model with the fitted parameters and methods for all functions of the distribution using the fitted parameters.
Return type: Parametric

from_params(params, gamma=None, p=None, f0=None)

Creating a SurPyval Parametric class with provided parameters.

Parameters

params (array like) – array of the parameters of the distribution.
gamma (scalar, optional) – offset value for the distribution. If not provided will fit a regular, unshifted/not offset, distribution.
p (scalar, optional) – The proportion of the population that will never die or fail. If used it must be a value between 0 and 1. If None will assume 1, i.e. no proportion of the population will never die or fail.
f0 (scalar, optional) – The proportion of the population that will die or fail at time 0. If used it must be a value between 0 and 1. If None will assume 0, i.e. no proportion of the population will die or fail at time 0.

Returns

A parametric model with the parameters provided.

Return type

Parametric

Examples

>>> from surpyval import Weibull
>>> model = Weibull.from_params([10, 4])
>>> print(model)
Parametric SurPyval Model
=========================
Distribution        : Weibull
Fitted by           : given parameters
Parameters          :
     alpha: 10
      beta: 4
>>> model = Weibull.from_params([10, 4], gamma=2)
>>> print(model)
Parametric SurPyval Model
=========================
Distribution        : Weibull
Fitted by           : given parameters
Offset (gamma)      : 2
Parameters          :
     alpha: 10
      beta: 4

hf(x, mu, sigma)

Instantaneous hazard rate for the Gumbel Distribution:

\[h(x) = \frac{1}{\sigma} e^{\frac{x-\mu}{\sigma}}\]

Parameters

x (numpy array or scalar) – The values of the random variables at which the survival function will be calculated
mu (numpy array like or scalar) – The location parameter of the distribution
sigma (numpy array like or scalar) – The scale parameter of the distribution

Returns

hf – The value(s) for the instantaneous hazard rate for the Gumbel distribution.

Return type

scalar or numpy array

Examples

>>> import numpy as np
>>> from surpyval import Gumbel
>>> x = np.array([1, 2, 3, 4, 5])
>>> Gumbel.hf(x, 3, 2)
array([0.18393972, 0.30326533, 0.5       , 0.82436064, 1.35914091])

mean(mu, sigma)

Calculates the mean of the Gumbel distribution with given parameters.

\[E = \mu - \sigma\gamma\]

Where gamma is the Euler-Mascheroni constant. The Gumbel distribution here is the smallest extreme value distribution, so the mean sits below the location parameter.

Parameters

mu (numpy array like or scalar) – The location parameter(s) of the distribution
sigma (numpy array like or scalar) – The scale parameter(s) of the distribution

Returns

mean – The mean(s) of the Gumbel distribution

Return type

scalar or numpy array

Examples

>>> from surpyval import Gumbel
>>> Gumbel.mean(3, 2)
1.8455686701969343

qf(p, mu, sigma)

Quantile function for the Gumbel Distribution:

\[q(p) = \mu + \sigma\ln\left ( -\ln\left ( 1 - p \right ) \right )\]

Parameters

p (numpy array or scalar) – The percentiles at which the quantile will be calculated
mu (numpy array like or scalar) – The location parameter(s) of the distribution
sigma (numpy array like or scalar) – The scale parameter(s) of the distribution

Returns

q – The quantiles for the Gumbel distribution at each value p.

Return type

scalar or numpy array

Examples

>>> import numpy as np
>>> from surpyval import Gumbel
>>> p = np.array([.1, .2, .3, .4, .5])
>>> Gumbel.qf(p, 3, 2)
array([-1.50073465e+00, 1.20026481e-04, 9.38139134e-01, 1.65654602e+00,
2.26697416e+00])

random(size, *params)

Draws random samples from the distribution in shape size, using the inverse transform method with the distribution’s quantile function.

Parameters

size (integer or tuple of positive integers) – Shape or size of the random draw
params (numpy array or scalar) – The parameters of the distribution

Returns

random – Random values drawn from the distribution in shape size

Return type

scalar or numpy array

Examples

>>> import numpy as np
>>> from surpyval import Weibull
>>> np.random.seed(1)
>>> Weibull.random(5, 3, 4)
array([2.57122697, 3.18730986, 0.31024877, 2.32381059, 1.89352939])

sf(x, mu, sigma)

Survival (or Reliability) function for the Gumbel Distribution:

\[R(x) = 1 - e^{e^{-\left ( x - \mu \right ) / \sigma}}\]

Parameters

x (numpy array or scalar) – The values of the random variables at which the survival function will be calculated
mu (numpy array like or scalar) – The location parameter of the distribution
sigma (numpy array like or scalar) – The scale parameter of the distribution

Returns

sf – The scalar value of the survival function of the distribution if a scalar was passed. If an array like object was passed then a numpy array is returned with the value of the survival function at each corresponding value in the input array.

Return type

scalar or numpy array

Examples

>>> import numpy as np
>>> from surpyval import Gumbel
>>> x = np.array([1, 2, 3, 4, 5])
>>> Gumbel.sf(x, 3, 2)
array([0.69220063, 0.54523921, 0.36787944, 0.19229565, 0.06598804])