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pdf6.py
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pdf6.py
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"""Fit of parton distributions functions (PDFs)
Like pdf5, but with uncertainties on M and M2"""
import lsqfitgp as lgp
import numpy as np
from matplotlib import pyplot as plt
import gvar
import lsqfit
np.random.seed(20220416)
#### SETTINGS ####
flavor = np.array([
( 1, 'd' ), # 0
(-1, 'dbar' ), # 1
( 2, 'u' ), # 2
(-2, 'ubar' ), # 3
( 3, 's' ), # 4
(-3, 'sbar' ), # 5
( 4, 'c' ), # 6
(-4, 'cbar' ), # 7
(21, 'gluon'), # 8
], 'i8, U16')
indices = dict(
# quark, antiquark
d = [0, 1],
u = [2, 3],
s = [4, 5],
c = [6, 7],
)
pid = flavor['f0']
name = flavor['f1']
nflav = len(flavor)
# linear data
nx = 30 # number of PDF points used for the transformation
ndata = 10 # number of datapoints
rankmcov = 9 # rank of the covariance matrix of the theory error
# quadratic data
nx2 = 30 # must be <= nx
ndata2 = 10
rankmcov2 = 9
#### MODEL ####
# for each PDF:
# h ~ GP
# f = h'' (f is the PDF)
# for the momentum sum rule:
# int_0^1 dx x f(x) = [xh'(x) - h(x)]_0^1
# for the flavor sum rules:
# int_0^1 dx (f_i(x) - f_j(x)) = [h_i'(x) - h_j'(x)]_0^1
xtype = np.dtype([
('x' , float),
('pid', int ),
])
kernel = lgp.ExpQuad(dim='x') * lgp.White(dim='pid')
# grid of points to which we apply the transformation
xdata = np.empty((nflav, nx), xtype)
xdata['pid'] = pid[:, None]
xdata[ 'x'] = np.linspace(0, 1, nx)
# linear map PDF(xdata) -> 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)
# quadratic map PDF(xdata) -> data2
M2comps = np.random.randn(rankmcov2, ndata2, nflav, nx2, nx2)
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)
# endpoints of the integral for each PDF
xinteg = np.empty((nflav, 2), xtype)
xinteg['pid'] = pid[:, None]
xinteg[ 'x'] = [0, 1]
# matrix to subtract the endpoints
suminteg = np.empty(xinteg.shape)
suminteg[:, 0] = -1
suminteg[:, 1] = 1
constraints = {
'momrule': 1,
'uubar' : 2,
'ddbar' : 1,
'ccbar' : 0,
'ssbar' : 0,
}
#### GP OBJECT ####
gp = (lgp.GP()
.defproc('h', kernel)
.deftransf('primitive', {'h': 1}, deriv='x' )
.deftransf('f' , {'h': 1}, deriv=(2, 'x'))
.deftransf('primitive of xf(x)', {
'primitive': lambda x: x['x'],
'h' : -1,
})
.addx(xdata, 'xdata', proc='f')
# linear data (used for warmup fit)
.addtransf({'xdata': M(gvar.mean(Mparams))}, 'data', axes=2)
# total momentum rule
.addx(xinteg, 'xmomrule', proc='primitive of xf(x)')
.addtransf({'xmomrule': suminteg}, 'momrule', axes=2)
)
# quark sum rules
qdiff = np.array([1, -1])[:, None] # vector to subtract two quarks
for quark in 'ducs':
idx = indices[quark] # [quark, antiquark] indices
label = f'{quark}{quark}bar' # the one appearing in `constraints`
xlabel = f'x{label}'
gp = gp.addx(xinteg[idx], xlabel, proc='primitive')
gp = gp.addtransf({xlabel: suminteg[idx] * qdiff}, label, axes=2)
#### NONLINEAR FUNCTION ####
def fcn(params):
xdata = params['xdata']
Mparams = params['Mparams']
M2params = params['M2params']
data = np.tensordot(M(Mparams), xdata, 2)
# data2 = np.einsum('dfxy,fx,fy->d', M2, xdata, xdata)
# np.einsum does not work with gvar
xdata2 = xdata[:, None, :nx2] * xdata[:, :nx2, None]
data2 = np.tensordot(M2(M2params), xdata2, 3)
return dict(data=data, data2=data2)
prior = gp.predfromdata(constraints, ['xdata'])
prior['Mparams'] = Mparams
prior['M2params'] = M2params
#### FAKE DATA ####
trueparams = gvar.sample(prior)
truedata = gvar.BufferDict(fcn(trueparams))
dataerr = np.full_like(truedata.buf, 0.1)
datamean = truedata.buf + dataerr * np.random.randn(*dataerr.shape)
data = gvar.BufferDict(truedata, buf=gvar.gvar(datamean, dataerr))
# check sum rules approximately with trapezoid rule
def check_integrals(x, y):
checksum = np.sum(((y * x)[:, 1:] + (y * x)[:, :-1]) / 2 * np.diff(x, axis=1))
print('sum_i int dx x f_i(x) =', checksum)
for q in 'ducs':
idx = indices[q]
qx = x[idx]
qy = y[idx]
checksum = np.sum(qdiff * (qy[:, 1:] + qy[:, :-1]) / 2 * np.diff(qx, axis=1))
print(f'sum_i={q}{q}bar int dx f_i(x) =', checksum)
print('check integrals in fake data:')
check_integrals(xdata['x'], trueparams['xdata'])
#### FIT ####
# find the minimization starting point with a simplified version of the fit
easyfit = gp.predfromdata(dict(data=data['data'], **constraints), ['xdata'])
p0 = gvar.mean(easyfit)
fit = lsqfit.nonlinear_fit(data, fcn, prior, p0=p0, verbose=2)
print(fit.format(maxline=True, pstyle='v'))
print(fit.format(maxline=-1))
pred = fcn(fit.p)
print('check integrals in fit:')
check_integrals(xdata['x'], fit.p['xdata'])
#### PLOT RESULTS ####
fig, axs = plt.subplots(2, 3, num='pdf6', clear=True, figsize=[13, 8])
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')
ax = axs[0, 0]
for i in range(nflav):
if i >= 4:
ax = axs[1, 0]
x = xdata[i]['x']
ypdf = fit.p['xdata'][i]
ydata = trueparams['xdata'][i]
m = gvar.mean(ypdf)
s = gvar.sdev(ypdf)
color = 'C' + str(i // 2)
if i % 2:
kw = dict(hatch='//////', edgecolor=color, facecolor='none')
kwp = dict(color=color, linestyle='--')
else:
kw = dict(alpha=0.6, facecolor=color)
kwp = dict(color=color)
ax.fill_between(x, m - s, m + s, label=name[i], **kw)
ax.plot(x, ydata, **kwp)
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()