Dist_reject

class stats.Dist_reject(dist_ran, f_fx, f_tot=None)

Bases: stats.Dist_qilum

Distribution with rejection method. Give a distribution with a known method for both random number generation and function value, and apply rejection method to keep values

Parameters

• dist_ran (distribution scipy or qilum.Dist) – we can call a random number generator rvs()

• f_fx (function) – objective

• f_tot (double) – Cumulative of the function f_fx() on the whole valid range x, Only used if we want to use this distribution as part of Dist_sum

Examples

>>> import qilum.stats as qs
>>> import scipy.stats
>>> # gaussian function
>>> def f_f(xs):
...     return 3.*np.exp(-np.square(xs)/10.)
>>>
>>> # 1.create a function f_sum(x) above the function f_f(x).
>>> #   f_sum(x) = y0*exp(+a*x)  if x_min < x
>>> #   f_sum(x) = step function if x_min < x <= x_max
>>> #   f_sum(x) = y0*exp(-a*x)  if x_max < x
>>> xs = np.linspace(-5,5, 1001)
>>> ys = f_f(xs)
>>> xs_inter, ys_inter = qs.f_max(xs, ys, 6)
>>> ys_inter *= 1.2 # just to be sure that our step function >= f_f()
>>> # histogram distribution with the step function
>>> hist_dist = scipy.stats.rv_histogram((ys_inter, xs_inter))
>>> # scale this diribution
>>> cumulative = ((np.roll(xs_inter,-1)-xs_inter)[:-1]*ys_inter).sum()
>>> step = qs.Dist_scale(hist_dist, scale_y = cumulative)
>>>
>>> # exponential distribution at the left and right of the step distribution
>>> exp_left  = qs.Dist_scale(scipy.stats.expon(),loc_x=xs_inter[-1], scale_x=1, scale_y=ys_inter[-1], name='Exp+')
>>> exp_right = qs.Dist_scale(scipy.stats.expon(),loc_x=xs_inter[0]-1e-10, scale_x=-1,scale_y=ys_inter[0], name='Exp-')
>>>
>>> # 2. create a sum of the known distributions for each part of f2 function
>>> dist_sum    = qs.Dist_sum([exp_left, step, exp_right])
>>> # 3. create the Dist_rejection distribution using this sum and the original f_f(x) function
>>> dist = qs.Dist_reject(dist_sum, f_f)
>>>
>>> # test result with the normal distribution hypothesis
>>> k2, p = scipy.stats.normaltest(dist.rvs(100000))
>>> print("Normal distribution test = ",p > 1e-3, " with ", "p = {:g}".format(p))
Normal distribution test =  True  with  p = 0.0617403

Methods Summary

F_tot()	Cumulative of the function f() on the whole valid range x
cdf(x)	Cumulative distribution function.
f(x)	function f(x) set initialy f_fx(x)
f_underlying(x)	function dist_ran.f(x): function of the underlying distribution dist_ran set up initialy
name()	Name of the class
pdf(x)	Probability density function
ppf(q)	Percent point function (inverse of cdf) at q
rvs(size)	random numbers in ndarray of lenght size
rvs_xy(size)	random numbers rans in ndarray of lenght size and function(rans)

Methods Documentation

F_tot()

Cumulative of the function f() on the whole valid range x

Returns

Return type

int

Examples

>>> # see example at the top of the class
>>> print(dist.F_tot())
NotImplementedError: Dist_reject.F_tot: not implemented

cdf(x)

Cumulative distribution function.

Parameters

x (array_like of type(values)) –

Returns

Cumulative distribution function evaluated at x

Return type

ndarray

Examples

>>> # see example at the top of the class
>>> x = [-3,1,2,3]
>>> print('cdf(x)=',dist.cdf(x))
NotImplementedError: Dist_reject.cdf: not implemented

f(x)

function f(x) set initialy f_fx(x)

Parameters

x (array_like of type(values)) –

Returns

f(x)

Return type

ndarray

Examples

>>> # see example at the top of the class
>>> x = [-3,1,2,3]
>>> print('f(x)=',dist.f(x))
f(x)= [1.21970898 2.71451225 2.01096014 1.21970898]

f_underlying(x)

function dist_ran.f(x): function of the underlying distribution dist_ran set up initialy

Parameters

x (array_like of type(values)) –

Returns

dist.f(x)

Return type

ndarray

Examples

>>> # see example at the top of the class
>>> x = [-3,1,2,3]
>>> print('f_underlying(x)=',dist.f_underlying(x))
f_underlying(x)= [2.7056004  3.6        2.75102576 2.75102576]
>>> # or similarly
>>> print('f_underlying(x)=',dist.dist_ran.f(x))
f_underlying(x)= [2.7056004  3.6        2.75102576 2.75102576]

name()

Name of the class

Returns

‘Dist_reject’

Return type

string

Examples

>>> # see example at the top of the class
>>> dist.name()
'Dist_reject'

pdf(x)

Probability density function

Parameters

x (array_like of type(values)) –

Returns

f(x)/F_tot()

Return type

ndarray

Examples

>>> # see example at the top of the class
>>> x = [-3,1,2,3]
>>> print('pdf(x)=',dist.pdf(x))
NotImplementedError: Dist_reject.F_tot: not implemented

ppf(q)

Percent point function (inverse of cdf) at q

Parameters

q (array_like of double) – lower tail probability

Returns

quantile corresponding to the lower tail probability q

Return type

ndarray

Examples

>>> # see example at the top of the class
>>> q = [0, 0.3,0.5,0.7,1]
>>> print('ppf(q)=',dist.ppf(q))
NotImplementedError: Dist_reject.ppf: not implemented

rvs(size)

random numbers in ndarray of lenght size

Parameters

size (int) – number of random number

Returns

random numbers

Return type

ndarray

Examples

>>> # see example at the top of the class
>>> print('rans=',dist.rvs(4))
rans= [-1.17789338 -0.69306554  1.1672123  -1.80663854]

rvs_xy(size)

random numbers rans in ndarray of lenght size and function(rans)

Parameters

size (int) – number of random number

Returns

• ndarray – random numbers

• ndarray – f(random numbers)

Examples

>>> # see example at the top of the class
>>> print('rans=',dist.rvs_xy(2))
rans= (array([-2.46439985, -0.96670812]), array([1.63441612, 2.73234395]))