/
morestats.py
2377 lines (1968 loc) · 76.5 KB
/
morestats.py
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
656
657
658
659
660
661
662
663
664
665
666
667
668
669
670
671
672
673
674
675
676
677
678
679
680
681
682
683
684
685
686
687
688
689
690
691
692
693
694
695
696
697
698
699
700
701
702
703
704
705
706
707
708
709
710
711
712
713
714
715
716
717
718
719
720
721
722
723
724
725
726
727
728
729
730
731
732
733
734
735
736
737
738
739
740
741
742
743
744
745
746
747
748
749
750
751
752
753
754
755
756
757
758
759
760
761
762
763
764
765
766
767
768
769
770
771
772
773
774
775
776
777
778
779
780
781
782
783
784
785
786
787
788
789
790
791
792
793
794
795
796
797
798
799
800
801
802
803
804
805
806
807
808
809
810
811
812
813
814
815
816
817
818
819
820
821
822
823
824
825
826
827
828
829
830
831
832
833
834
835
836
837
838
839
840
841
842
843
844
845
846
847
848
849
850
851
852
853
854
855
856
857
858
859
860
861
862
863
864
865
866
867
868
869
870
871
872
873
874
875
876
877
878
879
880
881
882
883
884
885
886
887
888
889
890
891
892
893
894
895
896
897
898
899
900
901
902
903
904
905
906
907
908
909
910
911
912
913
914
915
916
917
918
919
920
921
922
923
924
925
926
927
928
929
930
931
932
933
934
935
936
937
938
939
940
941
942
943
944
945
946
947
948
949
950
951
952
953
954
955
956
957
958
959
960
961
962
963
964
965
966
967
968
969
970
971
972
973
974
975
976
977
978
979
980
981
982
983
984
985
986
987
988
989
990
991
992
993
994
995
996
997
998
999
1000
# Author: Travis Oliphant, 2002
#
# Further updates and enhancements by many SciPy developers.
#
from __future__ import division, print_function, absolute_import
import math
import warnings
import numpy as np
from numpy import (isscalar, r_, log, sum, around, unique, asarray,
zeros, arange, sort, amin, amax, any, atleast_1d, sqrt, ceil,
floor, array, poly1d, compress, not_equal, pi, exp, ravel, angle)
from numpy.testing.decorators import setastest
from scipy.lib.six import string_types
from scipy.lib._numpy_compat import count_nonzero
from scipy import optimize
from scipy import special
from . import statlib
from . import stats
from .stats import find_repeats
from .contingency import chi2_contingency
from . import distributions
from ._distn_infrastructure import rv_generic
__all__ = ['mvsdist',
'bayes_mvs', 'kstat', 'kstatvar', 'probplot', 'ppcc_max', 'ppcc_plot',
'boxcox_llf', 'boxcox', 'boxcox_normmax', 'boxcox_normplot',
'shapiro', 'anderson', 'ansari', 'bartlett', 'levene', 'binom_test',
'fligner', 'mood', 'wilcoxon', 'median_test',
'pdf_fromgamma', 'circmean', 'circvar', 'circstd', 'anderson_ksamp'
]
def bayes_mvs(data, alpha=0.90):
"""
Bayesian confidence intervals for the mean, var, and std.
Parameters
----------
data : array_like
Input data, if multi-dimensional it is flattened to 1-D by `bayes_mvs`.
Requires 2 or more data points.
alpha : float, optional
Probability that the returned confidence interval contains
the true parameter.
Returns
-------
mean_cntr, var_cntr, std_cntr : tuple
The three results are for the mean, variance and standard deviation,
respectively. Each result is a tuple of the form::
(center, (lower, upper))
with `center` the mean of the conditional pdf of the value given the
data, and `(lower, upper)` a confidence interval, centered on the
median, containing the estimate to a probability `alpha`.
Notes
-----
Each tuple of mean, variance, and standard deviation estimates represent
the (center, (lower, upper)) with center the mean of the conditional pdf
of the value given the data and (lower, upper) is a confidence interval
centered on the median, containing the estimate to a probability
`alpha`.
Converts data to 1-D and assumes all data has the same mean and variance.
Uses Jeffrey's prior for variance and std.
Equivalent to tuple((x.mean(), x.interval(alpha)) for x in mvsdist(dat))
References
----------
T.E. Oliphant, "A Bayesian perspective on estimating mean, variance, and
standard-deviation from data", http://hdl.handle.net/1877/438, 2006.
"""
res = mvsdist(data)
if alpha >= 1 or alpha <= 0:
raise ValueError("0 < alpha < 1 is required, but alpha=%s was given." % alpha)
return tuple((x.mean(), x.interval(alpha)) for x in res)
def mvsdist(data):
"""
'Frozen' distributions for mean, variance, and standard deviation of data.
Parameters
----------
data : array_like
Input array. Converted to 1-D using ravel.
Requires 2 or more data-points.
Returns
-------
mdist : "frozen" distribution object
Distribution object representing the mean of the data
vdist : "frozen" distribution object
Distribution object representing the variance of the data
sdist : "frozen" distribution object
Distribution object representing the standard deviation of the data
Notes
-----
The return values from bayes_mvs(data) is equivalent to
``tuple((x.mean(), x.interval(0.90)) for x in mvsdist(data))``.
In other words, calling ``<dist>.mean()`` and ``<dist>.interval(0.90)``
on the three distribution objects returned from this function will give
the same results that are returned from `bayes_mvs`.
Examples
--------
>>> from scipy.stats import mvsdist
>>> data = [6, 9, 12, 7, 8, 8, 13]
>>> mean, var, std = mvsdist(data)
We now have frozen distribution objects "mean", "var" and "std" that we can
examine:
>>> mean.mean()
9.0
>>> mean.interval(0.95)
(6.6120585482655692, 11.387941451734431)
>>> mean.std()
1.1952286093343936
"""
x = ravel(data)
n = len(x)
if (n < 2):
raise ValueError("Need at least 2 data-points.")
xbar = x.mean()
C = x.var()
if (n > 1000): # gaussian approximations for large n
mdist = distributions.norm(loc=xbar, scale=math.sqrt(C/n))
sdist = distributions.norm(loc=math.sqrt(C), scale=math.sqrt(C/(2.*n)))
vdist = distributions.norm(loc=C, scale=math.sqrt(2.0/n)*C)
else:
nm1 = n-1
fac = n*C/2.
val = nm1/2.
mdist = distributions.t(nm1,loc=xbar,scale=math.sqrt(C/nm1))
sdist = distributions.gengamma(val,-2,scale=math.sqrt(fac))
vdist = distributions.invgamma(val,scale=fac)
return mdist, vdist, sdist
def kstat(data,n=2):
"""
Return the nth k-statistic (1<=n<=4 so far).
The nth k-statistic is the unique symmetric unbiased estimator of the nth
cumulant kappa_n.
Parameters
----------
data : array_like
Input array.
n : int, {1, 2, 3, 4}, optional
Default is equal to 2.
Returns
-------
kstat : float
The nth k-statistic.
See Also
--------
kstatvar: Returns an unbiased estimator of the variance of the k-statistic.
Notes
-----
The cumulants are related to central moments but are specifically defined
using a power series expansion of the logarithm of the characteristic
function (which is the Fourier transform of the PDF).
In particular let phi(t) be the characteristic function, then::
ln phi(t) = > kappa_n (it)^n / n! (sum from n=0 to inf)
The first few cumulants (kappa_n) in terms of central moments (mu_n) are::
kappa_1 = mu_1
kappa_2 = mu_2
kappa_3 = mu_3
kappa_4 = mu_4 - 3*mu_2**2
kappa_5 = mu_5 - 10*mu_2 * mu_3
References
----------
http://mathworld.wolfram.com/k-Statistic.html
http://mathworld.wolfram.com/Cumulant.html
"""
if n > 4 or n < 1:
raise ValueError("k-statistics only supported for 1<=n<=4")
n = int(n)
S = zeros(n+1,'d')
data = ravel(data)
N = len(data)
for k in range(1,n+1):
S[k] = sum(data**k,axis=0)
if n == 1:
return S[1]*1.0/N
elif n == 2:
return (N*S[2]-S[1]**2.0)/(N*(N-1.0))
elif n == 3:
return (2*S[1]**3 - 3*N*S[1]*S[2]+N*N*S[3]) / (N*(N-1.0)*(N-2.0))
elif n == 4:
return (-6*S[1]**4 + 12*N*S[1]**2 * S[2] - 3*N*(N-1.0)*S[2]**2 -
4*N*(N+1)*S[1]*S[3] + N*N*(N+1)*S[4]) / \
(N*(N-1.0)*(N-2.0)*(N-3.0))
else:
raise ValueError("Should not be here.")
def kstatvar(data,n=2):
"""
Returns an unbiased estimator of the variance of the k-statistic.
See `kstat` for more details of the k-statistic.
Parameters
----------
data : array_like
Input array.
n : int, {1, 2}, optional
Default is equal to 2.
Returns
-------
kstatvar : float
The nth k-statistic variance.
See Also
--------
kstat
"""
data = ravel(data)
N = len(data)
if n == 1:
return kstat(data,n=2)*1.0/N
elif n == 2:
k2 = kstat(data,n=2)
k4 = kstat(data,n=4)
return (2*k2*k2*N + (N-1)*k4)/(N*(N+1))
else:
raise ValueError("Only n=1 or n=2 supported.")
def _calc_uniform_order_statistic_medians(x):
"""See Notes section of `probplot` for details."""
N = len(x)
osm_uniform = np.zeros(N, dtype=np.float64)
osm_uniform[-1] = 0.5**(1.0 / N)
osm_uniform[0] = 1 - osm_uniform[-1]
i = np.arange(2, N)
osm_uniform[1:-1] = (i - 0.3175) / (N + 0.365)
return osm_uniform
def _parse_dist_kw(dist, enforce_subclass=True):
"""Parse `dist` keyword.
Parameters
----------
dist : str or stats.distributions instance.
Several functions take `dist` as a keyword, hence this utility
function.
enforce_subclass : bool, optional
If True (default), `dist` needs to be a
`_distn_infrastructure.rv_generic` instance.
It can sometimes be useful to set this keyword to False, if a function
wants to accept objects that just look somewhat like such an instance
(for example, they have a ``ppf`` method).
"""
if isinstance(dist, rv_generic):
pass
elif isinstance(dist, string_types):
try:
dist = getattr(distributions, dist)
except AttributeError:
raise ValueError("%s is not a valid distribution name" % dist)
elif enforce_subclass:
msg = ("`dist` should be a stats.distributions instance or a string "
"with the name of such a distribution.")
raise ValueError(msg)
return dist
def probplot(x, sparams=(), dist='norm', fit=True, plot=None):
"""
Calculate quantiles for a probability plot, and optionally show the plot.
Generates a probability plot of sample data against the quantiles of a
specified theoretical distribution (the normal distribution by default).
`probplot` optionally calculates a best-fit line for the data and plots the
results using Matplotlib or a given plot function.
Parameters
----------
x : array_like
Sample/response data from which `probplot` creates the plot.
sparams : tuple, optional
Distribution-specific shape parameters (shape parameters plus location
and scale).
dist : str or stats.distributions instance, optional
Distribution or distribution function name. The default is 'norm' for a
normal probability plot. Objects that look enough like a
stats.distributions instance (i.e. they have a ``ppf`` method) are also
accepted.
fit : bool, optional
Fit a least-squares regression (best-fit) line to the sample data if
True (default).
plot : object, optional
If given, plots the quantiles and least squares fit.
`plot` is an object that has to have methods "plot" and "text".
The `matplotlib.pyplot` module or a Matplotlib Axes object can be used,
or a custom object with the same methods.
Default is None, which means that no plot is created.
Returns
-------
(osm, osr) : tuple of ndarrays
Tuple of theoretical quantiles (osm, or order statistic medians) and
ordered responses (osr). `osr` is simply sorted input `x`.
For details on how `osm` is calculated see the Notes section.
(slope, intercept, r) : tuple of floats, optional
Tuple containing the result of the least-squares fit, if that is
performed by `probplot`. `r` is the square root of the coefficient of
determination. If ``fit=False`` and ``plot=None``, this tuple is not
returned.
Notes
-----
Even if `plot` is given, the figure is not shown or saved by `probplot`;
``plt.show()`` or ``plt.savefig('figname.png')`` should be used after
calling `probplot`.
`probplot` generates a probability plot, which should not be confused with
a Q-Q or a P-P plot. Statsmodels has more extensive functionality of this
type, see ``statsmodels.api.ProbPlot``.
The formula used for the theoretical quantiles (horizontal axis of the
probability plot) is Filliben's estimate::
quantiles = dist.ppf(val), for
0.5**(1/n), for i = n
val = (i - 0.3175) / (n + 0.365), for i = 2, ..., n-1
1 - 0.5**(1/n), for i = 1
where ``i`` indicates the i-th ordered value and ``n`` is the total number
of values.
Examples
--------
>>> from scipy import stats
>>> import matplotlib.pyplot as plt
>>> nsample = 100
>>> np.random.seed(7654321)
A t distribution with small degrees of freedom:
>>> ax1 = plt.subplot(221)
>>> x = stats.t.rvs(3, size=nsample)
>>> res = stats.probplot(x, plot=plt)
A t distribution with larger degrees of freedom:
>>> ax2 = plt.subplot(222)
>>> x = stats.t.rvs(25, size=nsample)
>>> res = stats.probplot(x, plot=plt)
A mixture of two normal distributions with broadcasting:
>>> ax3 = plt.subplot(223)
>>> x = stats.norm.rvs(loc=[0,5], scale=[1,1.5],
... size=(nsample/2.,2)).ravel()
>>> res = stats.probplot(x, plot=plt)
A standard normal distribution:
>>> ax4 = plt.subplot(224)
>>> x = stats.norm.rvs(loc=0, scale=1, size=nsample)
>>> res = stats.probplot(x, plot=plt)
Produce a new figure with a loggamma distribution, using the ``dist`` and
``sparams`` keywords:
>>> fig = plt.figure()
>>> ax = fig.add_subplot(111)
>>> x = stats.loggamma.rvs(c=2.5, size=500)
>>> stats.probplot(x, dist=stats.loggamma, sparams=(2.5,), plot=ax)
>>> ax.set_title("Probplot for loggamma dist with shape parameter 2.5")
Show the results with Matplotlib:
>>> plt.show()
"""
x = np.asarray(x)
osm_uniform = _calc_uniform_order_statistic_medians(x)
dist = _parse_dist_kw(dist, enforce_subclass=False)
if sparams is None:
sparams = ()
if isscalar(sparams):
sparams = (sparams,)
if not isinstance(sparams, tuple):
sparams = tuple(sparams)
osm = dist.ppf(osm_uniform, *sparams)
osr = sort(x)
if fit or (plot is not None):
# perform a linear fit.
slope, intercept, r, prob, sterrest = stats.linregress(osm, osr)
if plot is not None:
plot.plot(osm, osr, 'bo', osm, slope*osm + intercept, 'r-')
try:
if hasattr(plot, 'set_title'):
# Matplotlib Axes instance or something that looks like it
plot.set_title('Probability Plot')
plot.set_xlabel('Quantiles')
plot.set_ylabel('Ordered Values')
else:
# matplotlib.pyplot module
plot.title('Probability Plot')
plot.xlabel('Quantiles')
plot.ylabel('Ordered Values')
except:
# Not an MPL object or something that looks (enough) like it.
# Don't crash on adding labels or title
pass
# Add R^2 value to the plot as text
xmin = amin(osm)
xmax = amax(osm)
ymin = amin(x)
ymax = amax(x)
posx = xmin + 0.70 * (xmax - xmin)
posy = ymin + 0.01 * (ymax - ymin)
plot.text(posx, posy, "$R^2=%1.4f$" % r**2)
if fit:
return (osm, osr), (slope, intercept, r)
else:
return osm, osr
def ppcc_max(x, brack=(0.0,1.0), dist='tukeylambda'):
"""Returns the shape parameter that maximizes the probability plot
correlation coefficient for the given data to a one-parameter
family of distributions.
See also ppcc_plot
"""
dist = _parse_dist_kw(dist)
osm_uniform = _calc_uniform_order_statistic_medians(x)
osr = sort(x)
# this function computes the x-axis values of the probability plot
# and computes a linear regression (including the correlation)
# and returns 1-r so that a minimization function maximizes the
# correlation
def tempfunc(shape, mi, yvals, func):
xvals = func(mi, shape)
r, prob = stats.pearsonr(xvals, yvals)
return 1-r
return optimize.brent(tempfunc, brack=brack, args=(osm_uniform, osr, dist.ppf))
def ppcc_plot(x,a,b,dist='tukeylambda', plot=None, N=80):
"""Returns (shape, ppcc), and optionally plots shape vs. ppcc
(probability plot correlation coefficient) as a function of shape
parameter for a one-parameter family of distributions from shape
value a to b.
See also ppcc_max
"""
svals = r_[a:b:complex(N)]
ppcc = svals*0.0
k = 0
for sval in svals:
r1,r2 = probplot(x,sval,dist=dist,fit=1)
ppcc[k] = r2[-1]
k += 1
if plot is not None:
plot.plot(svals, ppcc, 'x')
plot.title('(%s) PPCC Plot' % dist)
plot.xlabel('Prob Plot Corr. Coef.')
plot.ylabel('Shape Values')
return svals, ppcc
def boxcox_llf(lmb, data):
r"""The boxcox log-likelihood function.
Parameters
----------
lmb : scalar
Parameter for Box-Cox transformation. See `boxcox` for details.
data : array_like
Data to calculate Box-Cox log-likelihood for. If `data` is
multi-dimensional, the log-likelihood is calculated along the first
axis.
Returns
-------
llf : float or ndarray
Box-Cox log-likelihood of `data` given `lmb`. A float for 1-D `data`,
an array otherwise.
See Also
--------
boxcox, probplot, boxcox_normplot, boxcox_normmax
Notes
-----
The Box-Cox log-likelihood function is defined here as
.. math::
llf = (\lambda - 1) \sum_i(\log(x_i)) -
N/2 \log(\sum_i (y_i - \bar{y})^2 / N),
where ``y`` is the Box-Cox transformed input data ``x``.
Examples
--------
>>> from scipy import stats
>>> import matplotlib.pyplot as plt
>>> from mpl_toolkits.axes_grid1.inset_locator import inset_axes
>>> np.random.seed(1245)
Generate some random variates and calculate Box-Cox log-likelihood values
for them for a range of ``lmbda`` values:
>>> x = stats.loggamma.rvs(5, loc=10, size=1000)
>>> lmbdas = np.linspace(-2, 10)
>>> llf = np.zeros(lmbdas.shape, dtype=np.float)
>>> for ii, lmbda in enumerate(lmbdas):
... llf[ii] = stats.boxcox_llf(lmbda, x)
Also find the optimal lmbda value with `boxcox`:
>>> x_most_normal, lmbda_optimal = stats.boxcox(x)
Plot the log-likelihood as function of lmbda. Add the optimal lmbda as a
horizontal line to check that that's really the optimum:
>>> fig = plt.figure()
>>> ax = fig.add_subplot(111)
>>> ax.plot(lmbdas, llf, 'b.-')
>>> ax.axhline(stats.boxcox_llf(lmbda_optimal, x), color='r')
>>> ax.set_xlabel('lmbda parameter')
>>> ax.set_ylabel('Box-Cox log-likelihood')
Now add some probability plots to show that where the log-likelihood is
maximized the data transformed with `boxcox` looks closest to normal:
>>> locs = [3, 10, 4] # 'lower left', 'center', 'lower right'
>>> for lmbda, loc in zip([-1, lmbda_optimal, 9], locs):
... xt = stats.boxcox(x, lmbda=lmbda)
... (osm, osr), (slope, intercept, r_sq) = stats.probplot(xt)
... ax_inset = inset_axes(ax, width="20%", height="20%", loc=loc)
... ax_inset.plot(osm, osr, 'c.', osm, slope*osm + intercept, 'k-')
... ax_inset.set_xticklabels([])
... ax_inset.set_yticklabels([])
... ax_inset.set_title('$\lambda=%1.2f$' % lmbda)
>>> plt.show()
"""
data = np.asarray(data)
N = data.shape[0]
if N == 0:
return np.nan
y = boxcox(data, lmb)
y_mean = np.mean(y, axis=0)
llf = (lmb - 1) * np.sum(np.log(data), axis=0)
llf -= N / 2.0 * np.log(np.sum((y - y_mean)**2. / N, axis=0))
return llf
def _boxcox_conf_interval(x, lmax, alpha):
# Need to find the lambda for which
# f(x,lmbda) >= f(x,lmax) - 0.5*chi^2_alpha;1
fac = 0.5 * distributions.chi2.ppf(1 - alpha, 1)
target = boxcox_llf(lmax, x) - fac
def rootfunc(lmbda, data, target):
return boxcox_llf(lmbda, data) - target
# Find positive endpoint of interval in which answer is to be found
newlm = lmax + 0.5
N = 0
while (rootfunc(newlm, x, target) > 0.0) and (N < 500):
newlm += 0.1
N += 1
if N == 500:
raise RuntimeError("Could not find endpoint.")
lmplus = optimize.brentq(rootfunc, lmax, newlm, args=(x, target))
# Now find negative interval in the same way
newlm = lmax - 0.5
N = 0
while (rootfunc(newlm, x, target) > 0.0) and (N < 500):
newlm -= 0.1
N += 1
if N == 500:
raise RuntimeError("Could not find endpoint.")
lmminus = optimize.brentq(rootfunc, newlm, lmax, args=(x, target))
return lmminus, lmplus
def boxcox(x, lmbda=None, alpha=None):
r"""
Return a positive dataset transformed by a Box-Cox power transformation.
Parameters
----------
x : ndarray
Input array. Should be 1-dimensional.
lmbda : {None, scalar}, optional
If `lmbda` is not None, do the transformation for that value.
If `lmbda` is None, find the lambda that maximizes the log-likelihood
function and return it as the second output argument.
alpha : {None, float}, optional
If `alpha` is not None, return the ``100 * (1-alpha)%`` confidence
interval for `lmbda` as the third output argument.
Must be between 0.0 and 1.0.
Returns
-------
boxcox : ndarray
Box-Cox power transformed array.
maxlog : float, optional
If the `lmbda` parameter is None, the second returned argument is
the lambda that maximizes the log-likelihood function.
(min_ci, max_ci) : tuple of float, optional
If `lmbda` parameter is None and `alpha` is not None, this returned
tuple of floats represents the minimum and maximum confidence limits
given `alpha`.
See Also
--------
probplot, boxcox_normplot, boxcox_normmax, boxcox_llf
Notes
-----
The Box-Cox transform is given by::
y = (x**lmbda - 1) / lmbda, for lmbda > 0
log(x), for lmbda = 0
`boxcox` requires the input data to be positive. Sometimes a Box-Cox
transformation provides a shift parameter to achieve this; `boxcox` does
not. Such a shift parameter is equivalent to adding a positive constant to
`x` before calling `boxcox`.
The confidence limits returned when `alpha` is provided give the interval
where:
.. math::
llf(\hat{\lambda}) - llf(\lambda) < \frac{1}{2}\chi^2(1 - \alpha, 1),
with ``llf`` the log-likelihood function and :math:`\chi^2` the chi-squared
function.
References
----------
G.E.P. Box and D.R. Cox, "An Analysis of Transformations", Journal of the
Royal Statistical Society B, 26, 211-252 (1964).
Examples
--------
>>> from scipy import stats
>>> import matplotlib.pyplot as plt
We generate some random variates from a non-normal distribution and make a
probability plot for it, to show it is non-normal in the tails:
>>> fig = plt.figure()
>>> ax1 = fig.add_subplot(211)
>>> x = stats.loggamma.rvs(5, size=500) + 5
>>> stats.probplot(x, dist=stats.norm, plot=ax1)
>>> ax1.set_xlabel('')
>>> ax1.set_title('Probplot against normal distribution')
We now use `boxcox` to transform the data so it's closest to normal:
>>> ax2 = fig.add_subplot(212)
>>> xt, _ = stats.boxcox(x)
>>> stats.probplot(xt, dist=stats.norm, plot=ax2)
>>> ax2.set_title('Probplot after Box-Cox transformation')
>>> plt.show()
"""
x = np.asarray(x)
if x.size == 0:
return x
if any(x <= 0):
raise ValueError("Data must be positive.")
if lmbda is not None: # single transformation
return special.boxcox(x, lmbda)
# If lmbda=None, find the lmbda that maximizes the log-likelihood function.
lmax = boxcox_normmax(x, method='mle')
y = boxcox(x, lmax)
if alpha is None:
return y, lmax
else:
# Find confidence interval
interval = _boxcox_conf_interval(x, lmax, alpha)
return y, lmax, interval
def boxcox_normmax(x, brack=(-2.0, 2.0), method='pearsonr'):
"""Compute optimal Box-Cox transform parameter for input data.
Parameters
----------
x : array_like
Input array.
brack : 2-tuple, optional
The starting interval for a downhill bracket search with
`optimize.brent`. Note that this is in most cases not critical; the
final result is allowed to be outside this bracket.
method : str, optional
The method to determine the optimal transform parameter (`boxcox`
``lmbda`` parameter). Options are:
'pearsonr' (default)
Maximizes the Pearson correlation coefficient between
``y = boxcox(x)`` and the expected values for ``y`` if `x` would be
normally-distributed.
'mle'
Minimizes the log-likelihood `boxcox_llf`. This is the method used
in `boxcox`.
'all'
Use all optimization methods available, and return all results.
Useful to compare different methods.
Returns
-------
maxlog : float or ndarray
The optimal transform parameter found. An array instead of a scalar
for ``method='all'``.
See Also
--------
boxcox, boxcox_llf, boxcox_normplot
Examples
--------
>>> from scipy import stats
>>> import matplotlib.pyplot as plt
>>> np.random.seed(1234) # make this example reproducible
Generate some data and determine optimal ``lmbda`` in various ways:
>>> x = stats.loggamma.rvs(5, size=30) + 5
>>> y, lmax_mle = stats.boxcox(x)
>>> lmax_pearsonr = stats.boxcox_normmax(x)
>>> lmax_mle
7.177...
>>> lmax_pearsonr
7.916...
>>> stats.boxcox_normmax(x, method='all')
array([ 7.91667384, 7.17718692])
>>> fig = plt.figure()
>>> ax = fig.add_subplot(111)
>>> stats.boxcox_normplot(x, -10, 10, plot=ax)
>>> ax.axvline(lmax_mle, color='r')
>>> ax.axvline(lmax_pearsonr, color='g', ls='--')
>>> plt.show()
"""
def _pearsonr(x, brack):
osm_uniform = _calc_uniform_order_statistic_medians(x)
xvals = distributions.norm.ppf(osm_uniform)
def _eval_pearsonr(lmbda, xvals, samps):
# This function computes the x-axis values of the probability plot
# and computes a linear regression (including the correlation) and
# returns ``1 - r`` so that a minimization function maximizes the
# correlation.
y = boxcox(samps, lmbda)
yvals = np.sort(y)
r, prob = stats.pearsonr(xvals, yvals)
return 1 - r
return optimize.brent(_eval_pearsonr, brack=brack, args=(xvals, x))
def _mle(x, brack):
def _eval_mle(lmb, data):
# function to minimize
return -boxcox_llf(lmb, data)
return optimize.brent(_eval_mle, brack=brack, args=(x,))
def _all(x, brack):
maxlog = np.zeros(2, dtype=np.float)
maxlog[0] = _pearsonr(x, brack)
maxlog[1] = _mle(x, brack)
return maxlog
methods = {'pearsonr': _pearsonr,
'mle': _mle,
'all': _all}
if method not in methods.keys():
raise ValueError("Method %s not recognized." % method)
optimfunc = methods[method]
return optimfunc(x, brack)
def boxcox_normplot(x, la, lb, plot=None, N=80):
"""Compute parameters for a Box-Cox normality plot, optionally show it.
A Box-Cox normality plot shows graphically what the best transformation
parameter is to use in `boxcox` to obtain a distribution that is close
to normal.
Parameters
----------
x : array_like
Input array.
la, lb : scalar
The lower and upper bounds for the ``lmbda`` values to pass to `boxcox`
for Box-Cox transformations. These are also the limits of the
horizontal axis of the plot if that is generated.
plot : object, optional
If given, plots the quantiles and least squares fit.
`plot` is an object that has to have methods "plot" and "text".
The `matplotlib.pyplot` module or a Matplotlib Axes object can be used,
or a custom object with the same methods.
Default is None, which means that no plot is created.
N : int, optional
Number of points on the horizontal axis (equally distributed from
`la` to `lb`).
Returns
-------
lmbdas : ndarray
The ``lmbda`` values for which a Box-Cox transform was done.
ppcc : ndarray
Probability Plot Correlelation Coefficient, as obtained from `probplot`
when fitting the Box-Cox transformed input `x` against a normal
distribution.
See Also
--------
probplot, boxcox, boxcox_normmax, boxcox_llf, ppcc_max
Notes
-----
Even if `plot` is given, the figure is not shown or saved by
`boxcox_normplot`; ``plt.show()`` or ``plt.savefig('figname.png')``
should be used after calling `probplot`.
Examples
--------
>>> from scipy import stats
>>> import matplotlib.pyplot as plt
Generate some non-normally distributed data, and create a Box-Cox plot:
>>> x = stats.loggamma.rvs(5, size=500) + 5
>>> fig = plt.figure()
>>> ax = fig.add_subplot(111)
>>> stats.boxcox_normplot(x, -20, 20, plot=ax)
Determine and plot the optimal ``lmbda`` to transform ``x`` and plot it in
the same plot:
>>> _, maxlog = stats.boxcox(x)
>>> ax.axvline(maxlog, color='r')
>>> plt.show()
"""
x = np.asarray(x)
if x.size == 0:
return x
if lb <= la:
raise ValueError("`lb` has to be larger than `la`.")
lmbdas = np.linspace(la, lb, num=N)
ppcc = lmbdas * 0.0
for i, val in enumerate(lmbdas):
# Determine for each lmbda the correlation coefficient of transformed x
z = boxcox(x, lmbda=val)
_, r2 = probplot(z, dist='norm', fit=True)
ppcc[i] = r2[-1]
if plot is not None:
plot.plot(lmbdas, ppcc, 'x')
try:
if hasattr(plot, 'set_title'):
# Matplotlib Axes instance or something that looks like it
plot.set_title('Box-Cox Normality Plot')
plot.set_ylabel('Prob Plot Corr. Coef.')
plot.set_xlabel('$\lambda$')
else:
# matplotlib.pyplot module
plot.title('Box-Cox Normality Plot')
plot.ylabel('Prob Plot Corr. Coef.')
plot.xlabel('$\lambda$')
except Exception:
# Not an MPL object or something that looks (enough) like it.
# Don't crash on adding labels or title
pass
return lmbdas, ppcc
def shapiro(x, a=None, reta=False):
"""
Perform the Shapiro-Wilk test for normality.
The Shapiro-Wilk test tests the null hypothesis that the
data was drawn from a normal distribution.
Parameters
----------
x : array_like
Array of sample data.
a : array_like, optional
Array of internal parameters used in the calculation. If these
are not given, they will be computed internally. If x has length
n, then a must have length n/2.
reta : bool, optional
Whether or not to return the internally computed a values. The
default is False.
Returns
-------
W : float
The test statistic.
p-value : float
The p-value for the hypothesis test.
a : array_like, optional
If `reta` is True, then these are the internally computed "a"
values that may be passed into this function on future calls.
See Also
--------
anderson : The Anderson-Darling test for normality
References
----------
.. [1] http://www.itl.nist.gov/div898/handbook/prc/section2/prc213.htm
"""
N = len(x)
if N < 3:
raise ValueError("Data must be at least length 3.")
if a is None:
a = zeros(N,'f')
init = 0
else:
if len(a) != N//2:
raise ValueError("len(a) must equal len(x)/2")
init = 1
y = sort(x)
a, w, pw, ifault = statlib.swilk(y, a[:N//2], init)
if ifault not in [0,2]:
warnings.warn(str(ifault))
if N > 5000:
warnings.warn("p-value may not be accurate for N > 5000.")
if reta:
return w, pw, a
else: