-
Notifications
You must be signed in to change notification settings - Fork 1
/
linearregression.py
431 lines (376 loc) · 13.9 KB
/
linearregression.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
'''
OVERVIEW
========
With the L{LinearRegression} class you can perform linear regressions,
hypothesis testing on the slope and intercept of said regression and
calculate confidence bands.
MAIN CLASSES
============
- L{LinearRegression} is a class which performs linear regression
- L{linearRegression} a function which given values for the response and
independent variable returns corresponding a L{LinearReg } object.
TYPICAL USAGE
=============
For typical usage see class documentation.
THEORY OF OPERATION
===================
A linear model assumes that a response variable, y, is related to an
independent variable, x, by a linear relationship of the form:
y = S{beta}x + S{alpha}
This module solves for the slope, m, and the intercept, c, by ordinary
least squares.
REFERENCES
==========
- Draper, N.R.; Smith, H. (1998). Applied Regression Analysis (3rd ed.).
John Wiley. ISBN 0-471-17082-8.
- W. Hardle, M. Muller, S. Sperlich, A. Werwatz (2004),
Nonparametric and Semiparametric Models, Springer, ISBN 3540207228
This file is part of Simon Wilshin's Regression module.
Simon Wilshin's Regression module is free software: you can redistribute
it and/or modify it under the terms of the GNU General Public License as
published by the Free Software Foundation, either version 3 of the License,
or (at your option) any later version.
This program is distributed in the hope that it will be useful,
but WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
GNU General Public License for more details.
You should have received a copy of the GNU General Public License
along with this program. If not, see <http://www.gnu.org/licenses/>.
@author: Simon Wilshin
@contact: swilshin@gmail.com
@date: Jan 2016
'''
from numpy import mean,sum,sqrt,std
from scipy.stats import t as tdist
class LinearRegression(object):
'''
Computes a linear regression. Typical usage is to instantiate with
L{linearRegression} function, which calls the fit method
with a independent and response variable. Example:
>>> from regression import LinearRegression, linearRegression
>>> from numpy import linspace
>>> from numpy.random import standard_normal,seed
>>>
>>> # Generate data
>>> alphaA = 1.0
>>> betaA = 2.0
>>> sigmaN = 0.5
>>> x = linspace(0,1,20)
>>> xint = linspace(0,1,1000)
>>> # Seed
>>> seed(0)
>>> y = alphaA + betaA*x + sigmaN*standard_normal(x.shape)
>>>
>>> # Fit model
>>> rho = LinearRegression()
>>> rho.fit(x,y)
>>> # Alternatively for identical effect
>>> rho = linearRegression(x,y)
>>>
>>> # Get CI
>>> print "alpha: ", rho.getInterceptCI()
alpha: (1.1982168202431409, 1.9397307002909874)
>>> print "beta: ", rho.getSlopeCI()
beta: (0.79750362486927528, 2.0652705199537369)
>>> # Run Tests
>>> rho.testIntercept()
Testing intercept against hypothesis both found p of 5.28088324181e-08 .
Significant at the 0.05 significance level.
(8.8907076244941443, 5.2808832418094198e-08)
>>> rho.testSlope()
Testing slope against hypothesis both found p of 0.000162012943822 .
Significant at the 0.05 significance level.
(4.7441413089509128, 0.00016201294382156561)
@ivar n: number of points
@type n: int
@ivar dof: number of degrees of freedom in fit
@type dof: int
@ivar mux: average of independent variable
@type mux: float
ivar muy: average of response variable
@type muy: float
@ivar kappa: square difference between indepedent variable and its mean
@type kappa: float
@ivar beta: slope estimate
@type bate: float
@ivar alpha: intercept estimate
@type alpha: float
@ivar r: R statistic
@type r: float
@ivar eps: residuals
@type eps: float
@ivar sbeta: s used for t-test for beta
@type sbeta: float
@ivar salpha: s used for t-test for alpha
@type salpha: float
'''
def __init__(self):
'''
Constructor, currently takes no parameters and does nothing as there
is no initialisation.
'''
pass
def fit(self,x,y):
'''
Given independent variable values, x, and response variable values, y,
perform a linear regression. Once this function is run the regression
object can be used for hypothesis testing, prediction, and calculating
confidence bounds and bands.
@param x: values for the independent variable
@type x: float array
@param y: values for the response variable
@type y: float array
'''
# Number of points and degrees of freedom
self.n = x.size
self.dof = self.n-2
# mean x, mean y, and kappa
self.mux = mean(x)
self.muy = mean(y)
self.kappa = sum((x-self.mux)**2)
# Parameter estimates
self.beta = (mean(x*y)-self.mux*self.muy)/(mean(x*x)-self.mux**2)
self.alpha = self.muy-self.beta*self.mux
# R statistic, residuals, standard errors
self.r = self.beta*std(x)/std(y)
self.eps = y-self.alpha-self.beta*x
self.sbeta = sqrt(
(self.RSS()/self.dof)
/
sum((x-self.mux)**2)
)
self.salpha = self.sbeta*sqrt(mean(x**2))
def RSS(self):
'''
Calculate the residual sum of squares (RSS) for the fit. Returns the RSS.
@return: The residual sum of squares for the fit.
@rtype: float
'''
return(sum(self.eps**2))
def gettstat(self,x,sx,x0):
'''
Calculate the t-statistic for x-x0 with sx the appropriate variability,
returns the t statistic as a float.
@param x: sample mean
@type x: float
@param sx: population and sample standard deviation ratio
@type sx: float
@param x0: population mean
@type x0: float
@return: t-statistic
@rtype: float
'''
return((x-x0)/sx)
def test(self,x,sx,gamma,h0,x0):
'''
Performs a t-test with to determine if x-x0 is significantly different
from zero with sx the appropriate variability for a student test
at level gamma (0.05 corresponds to p<0.05) under hypothesis
h0 which can be 'greater', 'less', or 'both'.
Returns to t statistic and the p-value
@param x: sample mean
@type x: float
@param sx: population and sample standard deviation ratio
@type sx: float
@param gamma: alpha value (significance critereon)
@type gamma: float
@param h0: sidedness, can be 'greater', 'less', or 'both'
@type h0: float
@param x0: population mean
@type x0: float
@return: t-statistic and p-value
@rtype: tuple of two floats
'''
t = self.gettstat(x,sx,x0)
# Great and less than p-values
pg = 1-tdist.cdf(t,self.dof)
pl = tdist.cdf(t,self.dof)
if h0=='greater':
p=pg
if h0=='less':
p=pl
if h0=='both':
if t<0:
p=2*pl
else:
p=2*pg
return(t,p)
def testSlope(self,gamma=0.05,h0='both',beta0=0.0,verbose=True):
'''
Perform a t-test to determine significance of the slope term. Here gamma
is the significance level (the default 0.05 corresponds to p<0.05), h0
can be both, greater or less and controls the sidedness of the test,
beta0 is the value to test against (default zero), and if verbose is set
to True (the default) will print test results.
Returns two floats, the test statistic and the p-value.
@param gamma: alpha value (significance critereon)
@type gamma: float
@param h0: sidedness, can be 'greater', 'less', or 'both'
@type h0: float
@param beta0: hypothesized intercept
@type beta0: float
@param verbose: if True print results
@type verbose: bool
@return: t-statistic and p-value
@rtype: tuple of two floats
'''
t,p = self.test(self.beta,self.sbeta,gamma,h0,beta0)
if verbose:
print "Testing slope against hypothesis", h0, "found p of ", p, "."
if p<gamma:
print "Significant at the",gamma,"significance level."
if p>=gamma:
print "Not significant at the",gamma,"significance level."
return(t,p)
def testIntercept(self,gamma=0.05,h0='both',alpha0=0.0,verbose=True):
'''
Perform a t-test to determine significance of the intercept term. Here
gamma is the significance level (the default 0.05 corresponds to p<0.05),
h0 can be 'both', 'greater' or 'less' and controls the sidedness
of the test, alpha0 is the value to test against (default zero), and if
verbose is set to True (the default) will print test results.
Returns two floats, the test statistic and the p-value.
@param gamma: alpha value (significance critereon)
@type gamma: float
@param h0: sidedness, can be 'greater', 'less', or 'both'
@type h0: float
@param alpha0: hypothesized slope
@type alpha0: float
@param verbose: if True print results
@type verbose: bool
@return: t-statistic and p-value
@rtype: tuple of two floats
'''
t,p = self.test(self.alpha,self.salpha,gamma,h0,alpha0)
if verbose:
print "Testing intercept against hypothesis", h0, "found p of ", p, "."
if p<gamma:
print "Significant at the",gamma,"significance level."
if p>=gamma:
print "Not significant at the",gamma,"significance level."
return(t,p)
def getSlopeCI(self,gamma=0.95,sided='both'):
'''
Calculate the confidence interval on the slope. gamma controls the
confidence level, default is 0.95 which gives a 95% CI. sided can be 'both', less or greater for double sided
or single sided confidence intervals.
Returns a tuple with two entries, which are either floats or None. The
entries are the lower and upper confidence intervals respectively and
None indicates that this bound is infinite for a single sided CI.
@param gamma: confidence level
@type gamma: float
@param sided: sidedness, can be 'greater', 'less', or 'both'
@type sided: float
@return: confidence bounds on slope
@rtype: tuple, two floats
'''
if sided == 'both':
tstar = tdist.ppf(1-((1-gamma)/2),self.dof)
return(self.beta-tstar*self.sbeta,self.beta+tstar*self.sbeta)
if sided == 'less':
tstar = tdist.ppf(gamma,self.dof)
return(self.beta-tstar*self.sbeta,None)
if sided == 'greater':
tstar = tdist.ppf(gamma,self.dof)
return(None,self.beta+tstar*self.sbeta)
def getInterceptCI(self,gamma=0.95,sided='both'):
'''
Calculate the confidence interval on the intercept. gamma controls the
confidence level, default is 0.95 which gives a 95% CI. sided can be
'both', 'less ' or 'greater' for double sided or single sided confidence
intervals.
Returns a tuple with two entries, which are either floats or None. The
entries are the lower and upper confidence intervals respectively and
None indicates that this bound is infinite for a single sided CI.
@param gamma: confidence level
@type gamma: float
@param sided: sidedness, can be 'greater', 'less', or 'both'
@type sided: float
@return: confidence bounds on intercept
@rtype: tuple, two floats
'''
if sided == 'both':
tstar = tdist.ppf(1-((1-gamma)/2),self.dof)
return(self.alpha-tstar*self.salpha,self.alpha+tstar*self.salpha)
if sided == 'less':
tstar = tdist.ppf(gamma,self.dof)
return(self.alpha-tstar*self.salpha,None)
if sided == 'greater':
tstar = tdist.ppf(gamma,self.dof)
return(None,self.alpha+tstar*self.salpha)
def predict(self,xn):
'''
Uses the co-efficients of the linear regression to predict the behaviour
of the dependent variable at the locations xn of the independent
variable. Returns an array the same shape as xn.
@param xn: values of independent variable
@type xn: float array
@return: predicted values for response variable
@rtype: float array
'''
return(self.alpha+self.beta*xn)
def confidenceBands(self,xn,gamma=0.95):
'''
Constructs confidence bands at locations xn of the independent variable.
gamma controls the confidence level, default is 0.95 which gives a 95%
CB. Returns two arrays the same size as xn, one the lower confidence
band, the other the upper.
@param xn: values of independent variable
@type xn: float array
@param gamma: confidence level
@type gamma: float
@return: confidence bands for response variable
@rtype: tuple of two float arrays
'''
yn = self.predict(xn)
tstar = tdist.ppf(1-((1-gamma)/2),self.dof)
Q = sqrt((self.RSS()/self.dof)*((1.0/self.n)+((xn-self.mux)**2.0/self.kappa)))
return(yn+tstar*Q,yn-tstar*Q)
def linearRegression(x,y):
'''
Constructs a linear regression for the independent variable x and the
dependent variable y. See the L{LinearRegression} class for more
details.
@param x: values for the independent variable
@type x: float array
@param y: values for the response variable
@type y: float array
@return: a linear regression
@rtype: L{LinearRegression}
'''
rho = LinearRegression()
rho.fit(x,y)
return(rho)
if __name__=="__main__":
import doctest
doctest.testmod()
try:
# Make example figure
from numpy import linspace
from numpy.random import standard_normal,seed
from pylab import (figure,plot,scatter,savefig,xlabel,ylabel,
fill_between,legend)
# Generate data
alphaA = 1.0
betaA = 2.0
sigmaN = 0.5
x = linspace(0,1,20)
xint = linspace(0,1,1000)
# Seed
seed(0)
y = alphaA + betaA*x + sigmaN*standard_normal(x.shape)
rho = linearRegression(x,y)
# Get confidence bandss
yu,yd = rho.confidenceBands(xint)
figure()
plot(xint,rho.predict(xint),'b',label='linear regression')
plot(xint,yu,'r',label='confidence band')
plot(xint,yd,'r')
fill_between(xint,yd,yu,color='r',alpha=0.2)
scatter(x,y,marker='x',color='k',label='sample data')
xlabel("dependent variable, x (arb)")
ylabel("response variable, x (arb)")
legend()
savefig("regressionExample.png")
except ImportError as e:
print "Skipping example figure, this depends on numpy and pylab."