-
Notifications
You must be signed in to change notification settings - Fork 0
/
pdf8.py
427 lines (357 loc) · 12.4 KB
/
pdf8.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
"""Fit of parton distributions functions (PDFs)
Like pdf7, but with more realistic PDFs"""
import time
import warnings
import lsqfitgp as lgp
import numpy as np
from scipy import linalg, interpolate
from matplotlib import pyplot as plt
import gvar
import lsqfit
np.random.seed(20220416)
warnings.filterwarnings('ignore', r'total derivative orders \(\d+, \d+\) greater than kernel minimum \(\d+, \d+\)')
#### DEFINITIONS ####
qtopm = np.array([
# d, db, u, ub, s, sb, c, cb
[ 1, 1, 0, 0, 0, 0, 0, 0], # d+ = d + dbar
[ 1, -1, 0, 0, 0, 0, 0, 0], # d- = d - dbar
[ 0, 0, 1, 1, 0, 0, 0, 0], # u+ = etc.
[ 0, 0, 1, -1, 0, 0, 0, 0], # u-
[ 0, 0, 0, 0, 1, 1, 0, 0], # s+
[ 0, 0, 0, 0, 1, -1, 0, 0], # s-
[ 0, 0, 0, 0, 0, 0, 1, 1], # c+
[ 0, 0, 0, 0, 0, 0, 1, -1], # c-
])
pmtoev = np.array([
# d+, d-, u+, u-, s+, s-, c+, c-
[ 1, 0, 1, 0, 1, 0, 1, 0], # Sigma = sum_q q+
[ 0, 1, 0, 1, 0, 1, 0, 1], # V = sum_q q-
[ 0, -1, 0, 1, 0, 0, 0, 0], # V3 = u- - d-
[ 0, 1, 0, 1, 0, -2, 0, 0], # V8 = u- + d- - 2s-
[ 0, 1, 0, 1, 0, 1, 0, -3], # V15 = u- + d- + s- - 3c-
[ -1, 0, 1, 0, 0, 0, 0, 0], # T3 = u+ - d+
[ 1, 0, 1, 0, -2, 0, 0, 0], # T8 = u+ + d+ - 2s+
[ 1, 0, 1, 0, 1, 0, -3, 0], # T15 = u+ + d+ + s+ - 3c+
])
qnames = ['d' , 'dbar', 'u' , 'ubar', 's' , 'sbar', 'c' , 'cbar']
pmnames = ['d+', 'd-' , 'u+', 'u-' , 's+', 's-' , 'c+', 'c-' ]
evnames = ['Sigma', 'V', 'V3', 'V8', 'V15', 'T3', 'T8', 'T15']
pnames = evnames + ['g']
tpnames = ['xSigma'] + evnames[1:] + ['xg']
nflav = len(pnames)
# linear data
ndata = 10 # number of datapoints
rankmcov = 9 # rank of the covariance matrix of the theory error
# quadratic data
ndata2 = 10
rankmcov2 = 9
# grid used for DGLAP evolution
grid = np.array([
1.9999999999999954e-07, # start logspace
3.034304765867952e-07,
4.6035014748963906e-07,
6.984208530700364e-07,
1.0596094959101024e-06,
1.607585498470808e-06,
2.438943292891682e-06,
3.7002272069854957e-06,
5.613757716930151e-06,
8.516806677573355e-06,
1.292101569074731e-05,
1.9602505002391748e-05,
2.97384953722449e-05,
4.511438394964044e-05,
6.843744918967896e-05,
0.00010381172986576898,
0.00015745605600841445,
0.00023878782918561914,
0.00036205449638139736,
0.0005487795323670796,
0.0008314068836488144,
0.0012586797144272762,
0.0019034634022867384,
0.0028738675812817515,
0.004328500638820811,
0.006496206194633799,
0.009699159574043398,
0.014375068581090129,
0.02108918668378717,
0.030521584007828916,
0.04341491741702269,
0.060480028754447364,
0.08228122126204893,
0.10914375746330703, # end logspace, start linspace
0.14112080644440345,
0.17802566042569432,
0.2195041265003886,
0.2651137041582823,
0.31438740076927585,
0.3668753186482242,
0.4221667753589648,
0.4798989029610255,
0.5397572337880445,
0.601472197967335,
0.6648139482473823,
0.7295868442414312,
0.7956242522922756,
0.8627839323906108,
0.9309440808717544,
1, # end linspace
])
# grid used for data
datagrid = grid[:-1] # exclude 1 since f(1) = 0 and zero errors upset the fit
nx = len(datagrid)
# grid used for plot
gridinterp = interpolate.interp1d(np.linspace(0, 1, len(grid)), grid)
plotgrid = gridinterp(np.linspace(0, 1, 200))
#### GAUSSIAN PROCESS ####
# Ti ~ GP
#
# fi ~ GP with sdev ~ x to compensate the scale ~ 1/x
# Vi = fi'
# f(1) - f(0) = 3
# f3(1) - f3(0) = 1
# f8(1) - f8(0) = 3
# f15(1) - f15(0) = 3
#
# f1 ~ GP (WITHOUT scale compensation)
# tf1(x) = x^(a+1)/(a+2) f1(x) <--- scale comp. is x^(a+1) instead of x^a,
# to avoid doing x^a with a < 0 in x = 0
# Sigma(x) = tf1'(x) / x (such that x Sigma(x) ~ x^a)
# the same with f2, tf2, g
# tf12 = tf1 + tf2
# tf12(0) - tf12(1) = 1
#
# [Sigma, g, V*, T*](1) = 0
# matrix to stack processes
stackplotgrid = np.einsum('ab,ij->abij', np.eye(nflav), np.eye(len(plotgrid)))
stackdatagrid = np.einsum('ab,ij->abij', np.eye(nflav), np.eye(len(datagrid)))
# transformation from evolution to flavor basis
evtoq = linalg.inv(pmtoev @ qtopm)
hyperprior = lgp.copula.makedict({
'log(scale)' : np.log(gvar.gvar(0.5, 0.5)),
'alpha_Sigma': lgp.copula.uniform(-0.5, 0.5),
'alpha_g' : lgp.copula.uniform(-0.5, 0.5),
})
def makegp(hp, quick=False):
gp = lgp.GP(checkpos=False)
eps = grid[0]
scalefun = lambda x: hp['scale'] * (x + eps)
kernel = lgp.Gibbs(scalefun=scalefun)
kernel_prim = kernel.rescale(scalefun, scalefun)
# define Ts and Vs
for suffix in ['', '3', '8', '15']:
if suffix != '':
gp = gp.defproc('T' + suffix, kernel)
gp = gp.defproc('f' + suffix, kernel_prim)
gp = gp.deftransf('V' + suffix, {'f' + suffix: 1}, deriv=1)
# define xSigma
gp = gp.defproc('f1', kernel)
a = hp['alpha_Sigma']
gp = gp.deftransf('tf1', {'f1': lambda x: x ** (a + 1) / (a + 2)})
gp = gp.deftransf('xSigma', {'tf1': 1}, deriv=1)
# define xg
gp = gp.defproc('f2', kernel)
b = hp['alpha_g']
gp = gp.deftransf('tf2', {'f2': lambda x: x ** (b + 1) / (b + 2)})
gp = gp.deftransf('xg', {'tf2': 1}, deriv=1)
# define primitive of xSigma + xg
gp = gp.deftransf('tf12', {'tf1': 1, 'tf2': 1})
# define a matrix of PDF values over the x grid
for proc in tpnames:
gp = gp.addx(datagrid, proc + '-datagrid', proc=proc)
gp = gp.addtransf({
proc + '-datagrid': stackdatagrid[i]
for i, proc in enumerate(tpnames)
}, 'datagrid', axes=1)
# definite integrals
for proc in ['tf12', 'f', 'f3', 'f8', 'f15']:
gp = gp.addx([0, 1], proc + '-endpoints', proc=proc)
gp = gp.addtransf({proc + '-endpoints': [-1, 1]}, proc + '-diff')
# right endpoint
for proc in tpnames:
gp = gp.addx(1, f'{proc}(1)', proc=proc)
if not quick:
# linear data (used for warmup fit)
global M_mean
gp = gp.addtransf({'datagrid': M_mean}, 'data', axes=2)
# define flavor basis PDFs
gp = gp.deftransf('Sigma', {'xSigma': lambda x: 1 / x})
gp = gp.deftransf('g', {'xg': lambda x: 1 / x})
for qi, qproc in enumerate(qnames):
gp = gp.deftransf(qproc, {
eproc: evtoq[qi, ei]
for ei, eproc in enumerate(evnames)
})
# define a matrix of PDF values over the plot grid
for proc in tpnames:
gp = gp.addx(plotgrid, proc + '-plotgrid', proc=proc)
gp = gp.addtransf({
proc + '-plotgrid': stackplotgrid[i]
for i, proc in enumerate(tpnames)
}, 'plotgrid', axes=1)
return gp
constraints = {
'tf12-diff': 1,
'f-diff' : 3,
'f3-diff' : 1,
'f8-diff' : 3,
'f15-diff' : 3,
'xSigma(1)': 0,
'V(1)' : 0,
'V3(1)' : 0,
'V8(1)' : 0,
'V15(1)' : 0,
'T3(1)' : 0,
'T8(1)' : 0,
'T15(1)' : 0,
'xg(1)' : 0,
}
#### NONLINEAR FUNCTION ####
# linear map PDF(grid) -> data
Mcomps = np.random.randn(rankmcov, ndata, nflav, nx)
Mcomps /= np.sqrt(Mcomps.size / ndata)
Mparams = gvar.gvar(np.random.randn(rankmcov), np.full(rankmcov, 0.1))
M = lambda params: np.tensordot(params, Mcomps, 1)
M_mean = M(gvar.mean(Mparams)) # used in makegp()
# quadratic map PDF(grid) -> data2
M2comps = np.random.randn(rankmcov2, ndata2, nflav, nx, nx)
M2comps /= 2 * np.sqrt(M2comps.size / ndata2)
M2comps = (M2comps + np.swapaxes(M2comps, -1, -2)) / 2
M2params = gvar.gvar(np.random.randn(rankmcov2), np.full(rankmcov2, 0.1))
M2 = lambda params: np.tensordot(params, M2comps, 1)
def fcn(params):
datagrid = params['datagrid']
Mparams = params['Mparams']
M2params = params['M2params']
data = np.tensordot(M(Mparams), datagrid, 2)
# data2 = np.einsum('dfxy,fx,fy->d', M2, datagrid, datagrid)
# np.einsum does not work with gvar
grid2 = datagrid[:, None, :] * datagrid[:, :, None]
data2 = np.tensordot(M2(M2params), grid2, 3)
return dict(data=data, data2=data2)
def makeprior(gp, plot=False):
out = ['datagrid']
if plot:
out += ['plotgrid']
prior = gp.predfromdata(constraints, out)
prior['Mparams'] = Mparams
prior['M2params'] = M2params
return prior
#### FAKE DATA ####
truehp = gvar.sample(hyperprior)
truegp = makegp(truehp)
trueprior = makeprior(truegp, plot=True)
trueparams = gvar.sample(trueprior)
truedata = fcn(trueparams)
dataerr = {
k: np.full_like(v, 0.1 * (np.max(v) - np.min(v)))
for k, v in truedata.items()
}
data = gvar.make_fake_data(gvar.gvar(truedata, dataerr))
def check_constraints(y):
# integrate approximately with trapezoid rule
integ = np.sum((y[:, 1:] + y[:, :-1]) / 2 * np.diff(plotgrid), 1)
print('int dx x (Sigma(x) + g(x)) =', integ[0] + integ[-1])
for i in range(1, 5):
print(f'int dx {tpnames[i]}(x) =', integ[i])
for i, name in enumerate(tpnames):
print(f'{name}(1) =', y[i, -1])
print('\ncheck constraints in fake data:')
check_constraints(trueparams['plotgrid'])
#### FIT ####
# find the minimization starting point with a simplified version of the fit
meangp = makegp(gvar.mean(hyperprior))
easyfit = meangp.predfromdata(dict(data=data['data'], **constraints), ['datagrid'])
p0 = gvar.mean(easyfit)
hyperprior = gvar.BufferDict(hyperprior)
hypermean = gvar.mean(hyperprior.buf)
hypercov = gvar.evalcov(hyperprior.buf)
hyperchol = linalg.cholesky(hypercov)
i = 0
lasttime = time.time()
def analyzer(hp):
global i, lasttime
i += 1
now = time.time()
interval = now - lasttime
print(f'iteration {i:3d} ({interval:#.2g} s): {hp}')
lasttime = now
def fitargs(hp):
analyzer(hp)
gp = makegp(hp, quick=True)
prior = makeprior(gp)
args = dict(
data = data,
fcn = fcn,
prior = prior,
p0 = p0,
)
residuals = linalg.solve_triangular(hyperchol, hp.buf - hypermean)
plausibility = -1/2 * (residuals @ residuals)
return args, plausibility
print('\ntrue hyperparameters:')
print(truehp)
print('\nfit:')
z0 = gvar.sample(hyperprior)
fit, fithp = lsqfit.empbayes_fit(z0, fitargs)
print(fit.format(maxline=True, pstyle='v'))
print(fit.format(maxline=-1))
gp = makegp(fithp)
prior = makeprior(gp)
pred = fcn(fit.p)
fitgrid = gp.predfromfit(dict(datagrid=fit.p['datagrid'], **constraints), 'plotgrid')
print('\ncheck constraints in fit:')
check_constraints(fitgrid)
def allkeys(d):
for k in d:
yield k
m = d.extension_pattern.match(k)
if m and d.has_distribution(m.group(1)):
yield m.group(2)
print('\nhyperparameters (true, fitted):')
for k in allkeys(fithp):
print(f'{k}: {truehp[k]:#.2g}\t{fithp[k]:#.2g}')
#### PLOT RESULTS ####
fig, axs = plt.subplots(2, 3, num='pdf8', clear=True, figsize=[10.5, 7], gridspec_kw=dict(width_ratios=[2, 1, 1]))
axs[0, 0].set_title('PDFs')
axs[1, 0].set_title('PDFs')
axs[0, 1].set_title('Data')
axs[1, 1].set_title('Data (quadratic)')
axs[0, 2].set_title('M parameters')
axs[1, 2].set_title('M2 parameters')
for i in range(nflav):
if i < 5:
ax = axs[0, 0]
else:
ax = axs[1, 0]
ypdf = fitgrid[i]
m = gvar.mean(ypdf)
s = gvar.sdev(ypdf)
ax.fill_between(plotgrid, m - s, m + s, label=tpnames[i], alpha=0.6, facecolor=f'C{i}')
ax.plot(plotgrid, trueparams['plotgrid'][i], color=f'C{i}')
# ax.set_xscale('log')
for ax in axs[:, 0]:
ax.legend(fontsize='small')
for ax, label in zip(axs[:, 1], ['data', 'data2']):
d = pred[label]
m = gvar.mean(d)
s = gvar.sdev(d)
x = np.arange(len(m))
ax.fill_between(x, m - s, m + s, step='mid', color='lightgray', label='fit')
d = data[label]
ax.errorbar(x, gvar.mean(d), gvar.sdev(d), color='black', linestyle='', capsize=2, label='data')
ax.plot(x, truedata[label], drawstyle='steps-mid', color='black', label='truth')
for ax, label in zip(axs[:, 2], ['Mparams', 'M2params']):
p = fit.p[label]
m = gvar.mean(p)
s = gvar.sdev(p)
x = np.arange(len(m))
ax.fill_between(x, m - s, m + s, step='mid', color='lightgray', label='fit')
p = prior[label]
ax.errorbar(x, gvar.mean(p), gvar.sdev(p), color='black', linestyle='', capsize=2, label='data')
ax.plot(x, trueparams[label], drawstyle='steps-mid', color='black', label='truth')
for ax in axs[:, 1:].flat:
ax.legend()
fig.tight_layout()
fig.show()