def extract_parameters(d_snm, e_snm, d_pnm=None, e_pnm=None, n0=None):
#
# input:
# d_snm, e_snm: densities and energies of symmetric nuclear matter
# d_pnm, e_pnm: densities and energies of pure neutron matter
# output:
# n0: saturation density
# e0: saturation energy
# K: incompressibility
# S: symmetry energy
# L: slope parameter
# Ksym: parameter
from scipy.interpolate import UnivariateSpline
from scipy.optimize import minimize
snm_spl = UnivariateSpline(d_snm, e_snm, s=0, k=4)
if (n0 == None):
res = minimize(snm_spl, (min(d_snm) + max(d_snm)) / 2,
bounds=((min(d_snm), max(d_snm)), ))
n0 = res.x[0]
e0 = res.fun[0]
else:
e0 = snm_spl(n0)
K = snm_spl.derivatives(n0)[2] * 9 * n0**2
M = snm_spl.derivatives(n0)[3] * 27 * n0**3 # cubic term
if (d_pnm == None and e_pnm == None): return n0, e0, K, M, 0, 0, 0
pnm_spl = UnivariateSpline(d_pnm, e_pnm, s=0, k=4)
S = pnm_spl(n0) - e0
L = pnm_spl.derivatives(n0)[1] * 3 * n0
Ksym = pnm_spl.derivatives(n0)[2] * 9 * n0**2 - K
return n0, e0, K, M, S, L, Ksym
def read_kinematics_data(kinematics_data_file):
"""read kinematics from file"""
kinematics_arr = []
with open(kinematics_data_file) as csv_file:
csv_reader = csv.reader(csv_file, delimiter='(')
line_count = 0
for row in csv_reader:
if line_count <= 1:
line_count += 1
elif row[0] == ')':
line_count += 1
else:
t_datai = row[1]
# print(row)
trans_datai0 = row[3].split()[0]
trans_datai1 = row[3].split()[1]
trans_datai2 = row[3].split()[2].split(')')[0]
rot_datai0 = row[4].split()[0]
rot_datai1 = row[4].split()[1]
rot_datai2 = row[4].split()[2].split(')')[0]
kinematics_arr.append([
float(t_datai),
float(trans_datai0),
float(trans_datai1),
float(trans_datai2),
float(rot_datai0),
float(rot_datai1),
float(rot_datai2)
])
line_count += 1
print(f'Processed {line_count} lines in {kinematics_data_file}')
kinematics_arr = np.array(kinematics_arr)
dkinematics_arr = np.array([kinematics_arr[:, 0]])
ddkinematics_arr = np.array([kinematics_arr[:, 0]])
for i in range(6):
spl = UnivariateSpline(kinematics_arr[:, 0],
kinematics_arr[:, i + 1],
s=0)
dk = []
ddk = []
for t in kinematics_arr[:, 0]:
dki = spl.derivatives(t)[1]
ddki = spl.derivatives(t)[2]
dk.append(dki)
ddk.append(ddki)
dk = np.array([dk])
ddk = np.array([ddk])
dkinematics_arr = np.append(dkinematics_arr, dk, axis=0)
ddkinematics_arr = np.append(ddkinematics_arr, ddk, axis=0)
dkinematics_arr = np.transpose(dkinematics_arr)
ddkinematics_arr = np.transpose(ddkinematics_arr)
return kinematics_arr, dkinematics_arr, ddkinematics_arr
def eos_spline(v, e, tol):
'''
Get volume, energy, pressure, and bulk modulus using spline, given
v in \A^3 and e in eV.
'''
from scipy.interpolate import UnivariateSpline
s = UnivariateSpline(v, e, k=3, s=tol)
vh = np.linspace(v[0], v[-1], 10 * len(v) - 1)
eh = [s.derivatives(i)[0] for i in vh]
ph = [-s.derivatives(i)[1] * units.eVA_GPa for i in vh]
bh = [s.derivatives(i)[2] * vh[i] * units.eVA_GPa for i in vh]
return vh, eh, ph, bh
def calc_conductance_curve(V_list,T,R_T,C_sigma):
#test_voltages = arange(-v_max,v_max,v_step)
test_currents = []
for V in V_list:
test_currents.append(calc_current(V,T,R_T,C_sigma))
#print "V: %g, current %g"%(V,test_currents[-1])
## calc conductances manually
#test_conductances = []
#for idx,V in enumerate (test_currents[1:-2]):
# if idx==0:
# print idx
# test_conductances.append((test_currents[idx+2]-test_currents[idx])/(2.0*v_step))
#
#test_voltages_G = test_voltages[1:-2]
#
# SPLINE
#
spline = UnivariateSpline(V_list,test_currents,s=0)
#print "test_conductances"
#indices = [x for x, y in enumerate(col1) if (y >0.7 or y<-0.7)]
test_conductances = []
for v_iter in V_list:
test_conductances.append(spline.derivatives(v_iter)[1])
return test_conductances
def extract_symmetry_energy_parameters(d_snm, e_snm, d_pnm, e_pnm, n0=None):
#
# input:
# d_snm, e_snm: densities and energies of symmetric nuclear matter
# d_pnm, e_pnm: densities and energies of pure neutron matter
# output:
# n0: saturation density
# e0: saturation energy
# K: incompressibility
# S: symmetry energy
# L: slope parameter
from scipy.interpolate import UnivariateSpline
from scipy.optimize import minimize
snm_spl = UnivariateSpline(d_snm, e_snm, s=0, k=4)
if (n0 == None):
res = minimize(snm_spl, (min(d_snm) + max(d_snm)) / 2,
bounds=((min(d_snm), max(d_snm)), ))
n0 = res.x[0]
e0 = res.fun[0]
else:
e0 = snm_spl(n0)
e_sym = []
for i in range(min(len(e_snm), len(e_pnm))):
dn_snm = d_snm[i]
if (dn_snm != d_pnm[i]): continue
e_sym.append(e_pnm[i] - e_snm[i])
sym_spl = UnivariateSpline(d_pnm, e_sym, s=0, k=4)
S = sym_spl(n0)
L = sym_spl.derivatives(n0)[1] * 3 * n0
K = sym_spl.derivatives(n0)[2] * 9 * n0**2
return n0, S, L, K
def my_curve_fit():
path = [[5, 117], [12, 110], [12, 105], [13, 104], [15, 102], [20, 97], [22, 95], [22, 73], [24, 71], [29, 66], [30, 66], [31, 65], [32, 64], [37, 59], [41, 55], [41, 54], [42, 53], [44, 51], [44, 35], [45, 34], [50, 29], [57, 29], [58, 28], [63, 23], [70, 16], [84, 16], [85, 15]]
points = np.array(path)
x = points[:,0]
x = [float(a) for a in x]
b = set()
for i in range(len(x)):
if x[i] in b:
x[i] = x[i] + 0.1
else:
b.add(x[i])
y = points[:,1]
plt.plot(x, y, "og")
spl = UnivariateSpline(x, points[:,1], s=0)
kn = spl.get_knots()
lst = []
for i in range(4):
lst.append(1/jiecheng(i))
for i in range(len(kn)-1):
cf = lst * spl.derivatives(kn[i])
print("For {0} <= x <= {1}, p(x) = {5}*(x-{0})^3 + {4}*(x-{0})^2 + {3}*(x-{0}) + {2}".format(kn[i], kn[i+1], *cf))
dx = np.linspace(kn[i], kn[i+1], 10)
dy = []
for item in dx:
dy.append(pow((item-kn[i]),3)*cf[3]+pow((item-kn[i]),2)*cf[2]+(item-kn[i])*cf[1]+cf[0])
plt.plot(dx, dy, "-r")
def curie_inflection(self, min_temp: float, max_temp: float)\
-> Tuple[float, UnivariateSpline]:
"""Estimate Curie temperature by inflection point.
Estimate Curie point by determining the inflection point of
the curve segment starting at the Hopkinson peak. The curve
segment must be specified.
:param min_temp: start of curve segment
:param max_temp: end of curve segment
:return: (temp, spline) where
temp is the estimated Curie temperature;
spline is the scipy.interpolate.UnivariateSpline used to fit
the data and determine the inflection point
"""
# Fit a cubic spline to the data. Using the whole dataset gives
# a better approximation at the endpoints of the selected range.
spline = UnivariateSpline(self.data[0][0], self.data[0][1], s=.1)
# Get the data points which lie within the selected range.
temps, _ = MeasurementCycle.chop_data(self.data[0], min_temp, max_temp)
# Evaluate the second derivative of the spline at each selected
# temperature step.
derivs = [spline.derivatives(t)[2] for t in temps]
# Fit a new spline to the derivatives in order to calculate the
# inflection point.
spline2 = UnivariateSpline(temps, derivs, s=3)
# The root of the 2nd-derivative spline gives the inflection point.
return spline2.roots()[0], spline
def QuasiPWeight(ReSE_A): ''' calculating the Fermi-liquid quasiparticle weight (residue) Z ''' N = len(En_A) #M = int(1e-3/dE) if dE < 1e-3 else 1 # very fine grids lead to oscillations # replace 1 with M below to dilute the grid ReSE = UnivariateSpline(En_A[int(N/2-10):int(N/2+10):1],ReSE_A[int(N/2-10):int(N/2+10):1]) dReSEdw = ReSE.derivatives(0.0)[1] Z = 1.0/(1.0-dReSEdw) return sp.array([Z,dReSEdw])
def QuasiPWeight(ReSE_A): ''' calculating the Fermi-liquid quasiparticle weight (residue) Z ''' N = len(En_A) #M = int(1e-3/dE) if dE < 1e-3 else 1 # very fine grids lead to oscillations # replace 1 with M below to dilute the grid ReSE = UnivariateSpline(En_A[int(N/2-10):int(N/2+10):1],ReSE_A[int(N/2-10):int(N/2+10):1]) dReSEdw = ReSE.derivatives(0.0)[1] Z = 1.0/(1.0-dReSEdw) return sp.array([Z,dReSEdw])
def calc_conductance_curve_full(sigma,N,V_list,R_T,C_sigma,T_p,island_volume):
"""
Calculates full conductance curve taking into account self-heating
"""
test_currents = []
for V in V_list:
test_currents.append(calc_current_full(sigma,N,V,R_T,C_sigma,T_p,island_volume))
# SPLINE
spline = UnivariateSpline(V_list,test_currents,s=0)
test_conductances = []
for v_iter in V_list:
test_conductances.append(spline.derivatives(v_iter)[1])
return test_conductances
def read_cfd_data(kinematics_file, cfd_data_file):
"""read cfd results force coefficients data"""
kinematics_arr = []
with open(kinematics_file) as csv_file:
csv_reader = csv.reader(csv_file, delimiter='(')
line_count = 0
for row in csv_reader:
if line_count <= 1:
line_count += 1
elif row[0] == ')':
line_count += 1
else:
t_datai = row[1]
# print(row)
rot_datai0 = row[4].split()[0]
kinematics_arr.append([
float(t_datai),
float(rot_datai0) * np.pi / 180,
])
line_count += 1
print(f'Processed {line_count} lines in {kinematics_file}')
kinematics_arr = np.array(kinematics_arr)
spl = UnivariateSpline(kinematics_arr[:, 0], kinematics_arr[:, 1], s=0)
cf_array = []
with open(cfd_data_file) as csv_file:
csv_reader = csv.reader(csv_file, delimiter='\t')
line_count = 0
for row in csv_reader:
if line_count <= 14:
line_count += 1
else:
ti = float(row[0])
phii = spl(ti)
dphii = -1.0 * spl.derivatives(ti)[1]
cli = float(row[3])
cdi = np.sign(dphii) * (np.sin(phii) * float(row[2]) +
np.cos(phii) * float(row[1]))
csi = np.cos(phii) * float(row[2]) - np.sin(phii) * float(
row[1])
cf_array.append([ti, cli, cdi, csi])
line_count += 1
print(f'Processed {line_count} lines in {cfd_data_file}')
cf_array = np.array(cf_array)
return cf_array
def demo():
x = np.arange(6)
y = np.array([3, 1, 4, 1, 5, 9])
spl = UnivariateSpline(x, y, s=0)
kn = spl.get_knots()
for i in range(len(kn)-1):
cf = [1, 1, 1/2, 1/6] * spl.derivatives(kn[i])
print("For {0} <= x <= {1}, p(x) = {5}*(x-{0})^3 + {4}*(x-{0})^2 + {3}*(x-{0}) + {2}".format(kn[i], kn[i+1], *cf))
dx = np.linspace(kn[i], kn[i+1], 100)
dy = []
for item in dx:
dy.append(pow((item-kn[i]),3)*cf[3]+pow((item-kn[i]),2)*cf[2]+(item-kn[i])*cf[1]+cf[0])
plt.plot(dx, dy, "-r")
plt.plot(x, y, "og")
plt.plot(x, y, "-b")
def _find_peak(x, y, ye, k=3, s=None):
spl = UnivariateSpline(x, y, ye ** -1, k=k, s=s)
f = lambda k : -spl(k)
fprime = np.vectorize(lambda k : - spl.derivatives(k)[1])
xp_best = None
yp_best = -np.inf
bounds = [(x.min(), x.max())]
for i in xrange(5):
x0 = (x.ptp() * np.random.rand() + x.min(),)
xp, nfeval, rc = fmin_tnc(f, x0, fprime=fprime, bounds=bounds,
messages=tnc.MSG_NONE)
xp = xp.item()
yp = spl(xp)
if yp >= yp_best:
xp_best = xp
yp_best = yp
return xp_best, yp_best.item()
def frequency_and_derivative(self,
smth_order=None,
fft_order=None,
spline_derivative=None,
verbose=0):
if (smth_order or fft_order):
if (verbose):
print(
'Cannot assure proper functionality of both order smoothing and low pass filtering.'
)
self.deriv = np.zeros_like(self.pos)
for i in range(1, len(self.pos)):
self.deriv[i] = (self.pos[i] - self.pos[i - 1]) / (
self.time[i] - self.time[i - 1])
if (smth_order):
smth_params = np.polyfit(self.time, self.deriv, smth_order)
pos_func = np.poly1d(smth_params)
self.deriv = pos_func(self.time)
if (fft_order):
self.deriv = self.deriv
if (spline_derivative):
# hard set as a cubic spline,
# number is a smoothing factor between knots, see scipy.UnivariateSpline
#
# recommended: 7 for dt=0.002 spacing
spl = UnivariateSpline(self.time,
self.pos,
k=3,
s=spline_derivative)
self.deriv = (spl.derivative())(self.time)
self.dderiv = np.zeros_like(self.deriv)
#
# can also do a second deriv
for indx, timeval in enumerate(self.time):
self.dderiv[indx] = spl.derivatives(timeval)[2]
def _find_peak(x, y, ye, k=3, s=None):
spl = UnivariateSpline(x, y, ye ** -1, k=k, s=s)
f = lambda k : -spl(k)
fprime = np.vectorize(lambda k : - spl.derivatives(k)[1])
xp_best = None
yp_best = -np.inf
bounds = [(x.min(), x.max())]
for i in xrange(5):
x0 = (x.ptp() * np.random.rand() + x.min(),)
xp, nfeval, rc = fmin_tnc(f, x0, fprime=fprime, bounds=bounds,
messages=tnc.MSG_NONE)
xp = xp.item()
yp = spl(xp)
if yp >= yp_best:
xp_best = xp
yp_best = yp
return xp_best, yp_best.item()
def add_Nsquared(self, rhokey="rho", depthkey="z", N2key="N2", s=0.2):
""" Calculate the squared buoyancy frequency, based on in-situ density.
Uses a smoothing spline to compute derivatives.
rhokey::string Data key to use for in-situ density
depthkey::string Data key to use for depth
N2key::string Data key to use for N^2
s::float Spline smoothing factor (smaller values
give a noisier result)
"""
if rhokey not in self.fields:
raise FieldError("add_Nsquared requires in-situ density")
msk = self.nanmask((rhokey, depthkey))
rho = self[rhokey][~msk]
z = self[depthkey][~msk]
rhospl = UnivariateSpline(z, rho, s=s)
drhodz = np.asarray([-rhospl.derivatives(_z)[1] for _z in z])
N2 = np.empty(len(self), dtype=np.float64)
N2[msk] = np.nan
N2[~msk] = -G / rho * drhodz
return self._addkeydata(N2key, N2)
spl = UnivariateSpline(mass, luminosity, k=4, s=0)
print mass, luminosity
lum_deriv = luminosity
#lum_deriv[0] = spl.derivatives(mass[0])[1]
xs = np.linspace(0.5, 100., 1000)
#xs = np.logspace(0.5, 100., 1000)
xs_deriv = np.zeros(len(mass))
#xs_deriv = np.zeros(len(xs))
print len(xs)
for i in range(len(mass)):
# print i, xs[i]
xs_deriv[i] = spl.derivatives(mass[i])[1]
# xs_deriv[i] = spl.derivatives(xs[i])[1]
print mass[i], luminosity[i], xs_deriv[i], xs_deriv[i] * mass[i] / luminosity[i]
#log_lum = np.zeros(len(mass))
#for i in range(len(mass)):#(0,1,2):#range(len(mass)):
# lum_deriv[i] = spl.derivatives(mass[i])[1]
# log_lum[i] = lum_deriv[i] / (mass[i]*mass[i])
# print i, mass[i], luminosity[i], lum_deriv[i], log_lum[i]
#
#print lum_orig
#luminosity = lum_orig
#print mass, luminosity, lum_deriv, log_lum
#print len(xs), len(xs_deriv)
pl.clf()
def estimate(self, observedLC):
"""!
Estimate intrinsicFlux, period, eccentricity, omega, tau, & a2sini
"""
## intrinsicFluxEst
maxPeriodFactor = 10.0
model = LombScargleFast().fit(observedLC.t, observedLC.y, observedLC.yerr)
periods, power = model.periodogram_auto(nyquist_factor = observedLC.numCadences)
model.optimizer.period_range = (2.0*np.mean(observedLC.t[1:] - observedLC.t[:-1]), maxPeriodFactor*observedLC.T)
periodEst = model.best_period
numIntrinsicFlux = 100
lowestFlux = np.min(observedLC.y[np.where(observedLC.mask == 1.0)])
highestFlux = np.max(observedLC.y[np.where(observedLC.mask == 1.0)])
intrinsicFlux = np.linspace(np.min(observedLC.y[np.where(observedLC.mask == 1.0)]), np.max(observedLC.y[np.where(observedLC.mask == 1.0)]), num = numIntrinsicFlux)
intrinsicFluxList = list()
totalIntegralList = list()
for f in xrange(1, numIntrinsicFlux - 1):
beamedLC = observedLC.copy()
beamedLC.x = np.require(np.zeros(beamedLC.numCadences), requirements=['F', 'A', 'W', 'O', 'E'])
for i in xrange(beamedLC.numCadences):
beamedLC.y[i] = observedLC.y[i]/intrinsicFlux[f]
beamedLC.yerr[i] = observedLC.yerr[i]/intrinsicFlux[f]
dopplerLC = beamedLC.copy()
dopplerLC.x = np.require(np.zeros(dopplerLC.numCadences), requirements=['F', 'A', 'W', 'O', 'E'])
for i in xrange(observedLC.numCadences):
dopplerLC.y[i] = math.pow(beamedLC.y[i], 1.0/3.44)
dopplerLC.yerr[i] = (1.0/3.44)*math.fabs(dopplerLC.y[i]*(beamedLC.yerr[i]/beamedLC.y[i]))
dzdtLC = dopplerLC.copy()
dzdtLC.x = np.require(np.zeros(dopplerLC.numCadences), requirements=['F', 'A', 'W', 'O', 'E'])
for i in xrange(observedLC.numCadences):
dzdtLC.y[i] = 1.0 - (1.0/dopplerLC.y[i])
dzdtLC.yerr[i] = math.fabs((-1.0*dopplerLC.yerr[i])/math.pow(dopplerLC.y[i], 2.0))
foldedLC = dzdtLC.fold(periodEst)
foldedLC.x = np.require(np.zeros(foldedLC.numCadences), requirements=['F', 'A', 'W', 'O', 'E'])
integralSpline = UnivariateSpline(foldedLC.t[np.where(foldedLC.mask == 1.0)], foldedLC.y[np.where(foldedLC.mask == 1.0)], 1.0/foldedLC.yerr[np.where(foldedLC.mask == 1.0)], k = 3, s = None, check_finite = True)
totalIntegral = math.fabs(integralSpline.integral(foldedLC.t[0], foldedLC.t[-1]))
intrinsicFluxList.append(intrinsicFlux[f])
totalIntegralList.append(totalIntegral)
intrinsicFluxEst = intrinsicFluxList[np.where(np.array(totalIntegralList) == np.min(np.array(totalIntegralList)))[0][0]]
## periodEst
for i in xrange(beamedLC.numCadences):
beamedLC.y[i] = observedLC.y[i]/intrinsicFluxEst
beamedLC.yerr[i] = observedLC.yerr[i]/intrinsicFluxEst
dopplerLC.y[i] = math.pow(beamedLC.y[i], 1.0/3.44)
dopplerLC.yerr[i] = (1.0/3.44)*math.fabs(dopplerLC.y[i]*(beamedLC.yerr[i]/beamedLC.y[i]))
dzdtLC.y[i] = 1.0 - (1.0/dopplerLC.y[i])
dzdtLC.yerr[i] = math.fabs((-1.0*dopplerLC.yerr[i])/math.pow(dopplerLC.y[i], 2.0))
model = LombScargleFast().fit(dzdtLC.t, dzdtLC.y, dzdtLC.yerr)
periods, power = model.periodogram_auto(nyquist_factor = dzdtLC.numCadences)
model.optimizer.period_range = (2.0*np.mean(dzdtLC.t[1:] - dzdtLC.t[:-1]), maxPeriodFactor*dzdtLC.T)
periodEst = model.best_period
## eccentricityEst & omega2Est
# First find a full period going from rising to falling.
risingSpline = UnivariateSpline(dzdtLC.t[np.where(dzdtLC.mask == 1.0)], dzdtLC.y[np.where(dzdtLC.mask == 1.0)], 1.0/dzdtLC.yerr[np.where(dzdtLC.mask == 1.0)], k = 3, s = None, check_finite = True)
risingSplineRoots = risingSpline.roots()
firstRoot = risingSplineRoots[0]
if risingSpline.derivatives(risingSplineRoots[0])[1] > 0.0:
tRising = risingSplineRoots[0]
else:
tRising = risingSplineRoots[1]
# Now fold the LC starting at tRising and going for a full period.
foldedLC = dzdtLC.fold(periodEst, tStart = tRising)
foldedLC.x = np.require(np.zeros(foldedLC.numCadences), requirements=['F', 'A', 'W', 'O', 'E'])
# Fit the folded LC with a spline to figure out alpha and beta
fitLC = foldedLC.copy()
foldedSpline = UnivariateSpline(foldedLC.t[np.where(foldedLC.mask == 1.0)], foldedLC.y[np.where(foldedLC.mask == 1.0)], 1.0/foldedLC.yerr[np.where(foldedLC.mask == 1.0)], k = 3, s = 2*foldedLC.numCadences, check_finite = True)
for i in xrange(fitLC.numCadences):
fitLC.x[i] = foldedSpline(fitLC.t[i])
# Now get the roots and find the falling root
tZeros = foldedSpline.roots()
if tZeros.shape[0] == 1: # We have found just tFalling
tFalling = tZeros[0]
tRising = fitLC.t[0]
startIndex = 0
tFull = fitLC.t[-1]
stopIndex = fitLC.numCadences
elif tZeros.shape[0] == 2: # We have found tFalling and one of tRising or tFull
if foldedSpline.derivatives(tZeros[0])[1] < 0.0:
tFalling = tZeros[0]
tFull = tZeros[1]
stopIndex = np.where(fitLC.t < tFull)[0][-1]
tRising = fitLC.t[0]
startIndex = 0
elif foldedSpline.derivatives(tZeros[0])[1] > 0.0:
if foldedSpline.derivatives(tZeros[1])[1] < 0.0:
tRising = tZeros[0]
startIndex = np.where(fitLC.t > tRising)[0][0]
tFalling = tZeros[1]
tFull = fitLC.t[-1]
stopIndex = fitLC.numCadences
else:
raise RuntimeError('Could not determine alpha & omega correctly because the first root is rising but the second root is not falling!')
elif tZeros.shape[0] == 3:
tRising = tZeros[0]
startIndex = np.where(fitLC.t > tRising)[0][0]
tFalling = tZeros[1]
tFull = tZeros[2]
stopIndex = np.where(fitLC.t < tFull)[0][-1]
else:
raise RuntimeError('Could not determine alpha & omega correctly because tZeros has %d roots!'%(tZeros.shape[0]))
# One full period now goes from tRising to periodEst. The maxima occurs between tRising and tFalling while the minima occurs between tFalling and tRising + periodEst
# Find the minima and maxima
alpha = math.fabs(fitLC.x[np.where(np.max(fitLC.x[startIndex:stopIndex]) == fitLC.x)[0][0]])
beta = math.fabs(fitLC.x[np.where(np.min(fitLC.x[startIndex:stopIndex]) == fitLC.x)[0][0]])
peakLoc = fitLC.t[np.where(np.max(fitLC.x[startIndex:stopIndex]) == fitLC.x)[0][0]]
troughLoc = fitLC.t[np.where(np.min(fitLC.x[startIndex:stopIndex]) == fitLC.x)[0][0]]
KEst = 0.5*(alpha + beta)
delta2 = (math.fabs(foldedSpline.integral(tRising, peakLoc)) + math.fabs(foldedSpline.integral(troughLoc, tFull)))/2.0
delta1 = (math.fabs(foldedSpline.integral(peakLoc, tFalling)) + math.fabs(foldedSpline.integral(tFalling, troughLoc)))/2.0
eCosOmega2 = (alpha - beta)/(alpha + beta)
eSinOmega2 = ((2.0*math.sqrt(alpha*beta))/(alpha + beta))*((delta2 - delta1)/(delta2 + delta1))
eccentricityEst = math.sqrt(math.pow(eCosOmega2, 2.0) + math.pow(eSinOmega2, 2.0))
tanOmega2 = math.fabs(eSinOmega2/eCosOmega2)
if (eCosOmega2/math.fabs(eCosOmega2) == 1.0) and (eSinOmega2/math.fabs(eSinOmega2) == 1.0):
omega2Est = math.atan(tanOmega2)*(180.0/math.pi)
if (eCosOmega2/math.fabs(eCosOmega2) == -1.0) and (eSinOmega2/math.fabs(eSinOmega2) == 1.0):
omega2Est = 180.0 - math.atan(tanOmega2)*(180.0/math.pi)
if (eCosOmega2/math.fabs(eCosOmega2) == -1.0) and (eSinOmega2/math.fabs(eSinOmega2) == -1.0):
omega2Est = 180.0 + math.atan(tanOmega2)*(180.0/math.pi)
if (eCosOmega2/math.fabs(eCosOmega2) == 1.0) and (eSinOmega2/math.fabs(eSinOmega2) == -1.0):
omega2Est = 360.0 - math.atan(tanOmega2)*(180.0/math.pi)
omega1Est = omega2Est - 180.0
## tauEst
zDot = KEst*(1.0 + eccentricityEst)*(eCosOmega2/eccentricityEst)
zDotLC = dzdtLC.copy()
for i in xrange(zDotLC.numCadences):
zDotLC.y[i] = zDotLC.y[i] - zDot
zDotSpline = UnivariateSpline(zDotLC.t[np.where(zDotLC.mask == 1.0)], zDotLC.y[np.where(zDotLC.mask == 1.0)], 1.0/zDotLC.yerr[np.where(zDotLC.mask == 1.0)], k = 3, s = 2*zDotLC.numCadences, check_finite = True)
for i in xrange(zDotLC.numCadences):
zDotLC.x[i] = zDotSpline(zDotLC.t[i])
zDotZeros = zDotSpline.roots()
zDotFoldedLC = dzdtLC.fold(periodEst)
zDotFoldedSpline = UnivariateSpline(zDotFoldedLC.t[np.where(zDotFoldedLC.mask == 1.0)], zDotFoldedLC.y[np.where(zDotFoldedLC.mask == 1.0)], 1.0/zDotFoldedLC.yerr[np.where(zDotFoldedLC.mask == 1.0)], k = 3, s = 2*zDotFoldedLC.numCadences, check_finite = True)
for i in xrange(zDotFoldedLC.numCadences):
zDotFoldedLC.x[i] = zDotFoldedSpline(zDotFoldedLC.t[i])
tC = zDotFoldedLC.t[np.where(np.max(zDotFoldedLC.x) == zDotFoldedLC.x)[0][0]]
nuC = (360.0 - omega2Est)%360.0
tE = zDotFoldedLC.t[np.where(np.min(zDotFoldedLC.x) == zDotFoldedLC.x)[0][0]]
nuE = (180.0 - omega2Est)%360.0
if math.fabs(360.0 - nuC) < math.fabs(360 - nuE):
tauEst = zDotZeros[np.where(zDotZeros > tC)[0][0]]
else:
tauEst = zDotZeros[np.where(zDotZeros > tE)[0][0]]
## a2sinInclinationEst
a2sinInclinationEst = ((KEst*periodEst*self.Day*self.c*math.sqrt(1.0 - math.pow(eccentricityEst, 2.0)))/self.twoPi)/self.Parsec
return intrinsicFluxEst, periodEst, eccentricityEst, omega1Est, tauEst, a2sinInclinationEst
class ContNormSpline:
def __init__(self,
lmbd,
flux,
ivar,
name=None,
spln_degr=1,
spln_smth=100,
num_lines=9,
wndw_init=100,
wndw_step=0,
crop_rand=0,
mean_wdth=0,
crop_strt=8,
crop_step=25,
spl_2=False):
'''
Initializes all cropping and spline properties.
name - the name of this spline for the generated plots.
spln_degr - the degree of interpolation spline polynomials. spln_degr must be between [1, 5]
spln_smth - the smoothing factor for the interpolation spline.
num_lines - the last Balmer line to directly crop, all data points bluewards of this line
are ignored. num_lines must be an int >= 3.
crop_wdth - the number of indices to crop out on both sides of each Balmer line (note that
2*crop_wdth+1 indices will be ignored). If `crop_wdth` is an array, the crop
width for each line is set individually.
crop_rand - the width of the random window for the center of each cropped index.
mean_wdth - the number of indices to collect on both sides of each selected index to take
the mean of (note that 2*mean_wdth+1 indices will be used). mean_wdth is an int.
crop_strt - the index in the wavelength array to start sampling at. All indices less than
start are ignored.
crop_step - the step size for the sampling of indices. only one index every step_size indies
will be included in the output arrays.
spl_2 - whether a second spline should be run on the initially constructed spline.
'''
self.lmbd = lmbd
self.flux = flux
self.ivar = ivar
self.__xlen = len(lmbd)
self.name = self.__get_name(name)
self.spln_degr = spln_degr
self.spln_smth = spln_smth
self.line_idxs = np.searchsorted(self.lmbd,
self.__get_balmer(num_lines))
self.crop_wdth = self.__crop_wdth(num_lines, wndw_init, wndw_step)
self.crop_rand = crop_rand
self.mean_wdth = mean_wdth
self.crop_strt = crop_strt
self.crop_step = crop_step
self.spl_2 = spl_2
def __get_balmer(self, max_line):
# generate Balmer lines
balmer = (np.arange(3, max_line + 1))**2
balmer = 3645.07 * (balmer / (balmer - 4))
return balmer
def __get_name(self, name):
# name for titles
if name is None:
return ''
else:
return ' - ' + name
def __crop_wdth(self, num_lines, wndw_init, wndw_step):
# convert Balmer line crop window into required format
if wndw_step != 0:
return np.arange(wndw_init, wndw_init + num_lines * wndw_step,
wndw_step)
else:
return np.full((num_lines, ), wndw_init)
def crop(self):
'''
Given the wavelength, flux, and inverse variance arrays, crops out the Balmer lines and
samples a certain number of random points to prepare the arrays for spline fitting.
'''
mask = np.zeros(self.__xlen, dtype=bool)
# crop random
unif = np.arange(self.crop_strt, self.__xlen, self.crop_step)
rand = np.random.randint(-self.crop_rand,
self.crop_rand + 1,
size=len(unif))
idxs = unif + rand
if idxs[-1] >= self.__xlen:
idxs[-1] = self.__xlen - 1
mask[idxs] = True
# crop lines
mask[np.argwhere(self.ivar < 0.1)] = False
mask[:self.line_idxs[-1]] = False
for cdx, wdth in zip(self.line_idxs, self.crop_wdth):
lo = cdx - wdth
hi = cdx + wdth
mask[lo:hi] = False
# crop mean
keep_idxs = np.concatenate(np.argwhere(mask))
self.lmbd_crop = np.zeros(keep_idxs.shape)
self.flux_crop = np.zeros(keep_idxs.shape)
self.ivar_crop = np.zeros(keep_idxs.shape)
for i in range(len(keep_idxs)):
keep = keep_idxs[i]
self.lmbd_crop[i] = np.mean(self.lmbd[keep - self.mean_wdth:keep +
self.mean_wdth + 1])
self.flux_crop[i] = np.mean(self.flux[keep - self.mean_wdth:keep +
self.mean_wdth + 1])
self.ivar_crop[i] = np.sqrt(
np.sum(self.ivar[keep - self.mean_wdth:keep + self.mean_wdth +
1]**2))
def plot_crop(self, legend=True, ylim=(0, 140)):
plt.title('Cropped Regions' + self.name)
plt.errorbar(self.lmbd,
self.flux,
yerr=1 / self.ivar,
label='raw data')
plt.errorbar(self.lmbd_crop,
self.flux_crop,
yerr=1 / self.ivar_crop,
label='cropped')
plt.ylim(ylim)
if legend:
plt.legend()
plt.show()
def construct_spline(self):
self.spline = USpline(self.lmbd_crop,
self.flux_crop,
w=self.ivar_crop,
k=self.spln_degr,
s=self.spln_smth)
self.blue_most = self.lmbd_crop[0]
# get blue extrapolation
frst_derv = self.spline.derivative()
drvs = frst_derv(lmbd)
x_4k, x_5k, x_6k, x_8k, x_9k, x_Tk = np.searchsorted(
lmbd, (4000, 5000, 6000, 8000, 9000, 10000))
dr4k = np.mean(drvs[x_4k:x_5k]) # derivative centered on 4500
dr5k = np.mean(drvs[x_5k:x_6k]) # derivative centered on 5500
lb4k = np.mean(lmbd[x_4k:x_5k]) # exact lambda, roughly 4500
lb5k = np.mean(lmbd[x_5k:x_6k]) # exact lambda, roughly 5500
scnd_derv = (dr4k - dr5k) / (
lb4k - lb5k) # get second derivative between these two points
dist = (self.blue_most -
self.lmbd[0]) / 2 # distance to middle of extrapolated section
b_fl, b_sl = self.spline.derivatives(
self.blue_most) # get flux, slope at blue-most kept point
slop = b_sl - scnd_derv * dist
intr = b_fl - slop * self.blue_most
self.blue_slop = slop
self.blue_intr = intr
if self.spl_2:
self.compute_cont_norm()
self.spline = USpline(self.lmbd,
self.cont,
w=self.ivar,
k=self.spln_degr,
s=self.spln_smth)
self.blue_most = 0
self.cont = None
self.norm = None
def eval(self, x):
return np.where(x < self.blue_most,
self.blue_intr + self.blue_slop * x, self.spline(x))
def compute_cont_norm(self):
self.cont = self.eval(self.lmbd)
self.norm = self.flux / self.cont
self.__clip_norm()
def __clip_norm(self, b_lim=0, u_lim=2):
# clip values beyond b_lim or u_lim to exactly 1
self.norm[np.argwhere(self.norm > u_lim)] = 1
self.norm[np.argwhere(self.norm < b_lim)] = 1
def plot_cont(self, legend=True, ylim=(0, 140)):
plt.title('Continuum' + self.name)
plt.plot(self.lmbd, self.flux, label='data')
plt.plot(self.lmbd, self.cont, label='continuum')
plt.ylim(ylim)
if legend:
plt.legend()
plt.show()
def plot_norm(self, legend=True, ylim=(0, 2)):
plt.title('Normalization' + self.name)
plt.plot(self.lmbd, self.norm, label='normalized')
plt.plot(self.lmbd, [1] * self.__xlen, 'k', label='1')
plt.ylim(ylim)
if legend:
plt.legend()
plt.show()
def run_all(self, plot=True):
# runs all of the requisite functions to compute the continuum and normalized fluxes
self.crop()
self.construct_spline()
self.compute_cont_norm()
if plot:
self.plot_crop()
self.plot_cont()
self.plot_norm()
####
ydelta = Results_a[stage,1]
threslist = Results_a[stage,0]
####
#Intermittency profile
Dv = np.empty((len(threslist),9),dtype=np.double)
epsilon = 2**np.linspace(0,8,9)
for ithres,thres in enumerate(threslist):
sp = UnivariateSpline(np.log10(epsilon),
np.log10(box_counting_vol[ithres,:9]),
k=5)
for j in range(len(epsilon)):
Dv[ithres,j] = sp.derivatives(np.log10(epsilon[j]))[1]
Ds = np.empty((len(threslist),9),dtype=np.double)
epsilon = 2**np.linspace(0,8,9)
for ithres,thres in enumerate(threslist):
sp = UnivariateSpline(np.log10(epsilon),
np.log10(box_counting_sur[ithres,:9]),
k=5)
for j in range(len(epsilon)):
Ds[ithres,j] = sp.derivatives(np.log10(epsilon[j]))[1]
gamma = np.array([r[0] for r in Results_a[stage,2:]])
genusv = np.array([r[1] for r in Results_a[stage,2:]])
genuss = np.array([r[3] for r in Results_a[stage,2:]])
if stage==1:
def get_Paz(self, az_data, R_data, jns):
"""
Computes probability of line of sight acceleration at projected R : P(az|R)
"""
# Under construction !!!
# Return P(az|R)
az_data = abs(az_data) # Consider only positive values
# Assumes units of az [m/s^2] if self.G ==0.004302, else models units
# Conversion factor from [pc (km/s)^2/Msun] -> [m/s^2]
az_fac = 1./3.0857e10 if (self.G==0.004302) else 1
if (R_data < self.rt):
nz = self.nstep # Number of z value equal to number of r values
zt = sqrt(self.rt**2 - R_data**2) # maximum z value at R
z = numpy.logspace(log10(self.r[1]), log10(zt), nz)
spl_Mr = UnivariateSpline(self.r, self.mc, s=0, ext=1) # Spline for enclosed mass
r = sqrt(R_data**2 + z**2) # Local r array
az = self.G*spl_Mr(r)*z/r**3 # Acceleration along los
az[-1] = self.G*spl_Mr(self.rt)*zt/self.rt**3 # Ensure non-zero final data point
az *= az_fac # convert to [m/s^2]
az_spl = UnivariateSpline(z, az, k=4, s=0, ext=1) # 4th order needed to find max (can be done easier?)
zmax = az_spl.derivative().roots() # z where az = max(az), can be done without 4th order spline?
azt = az[-1] # acceleration at the max(z) = sqrt(r_t**2 - R**2)
# Setup spline for rho(z)
if jns == 0 and self.nmbin == 1:
rho = self.rho
else:
rho = self.rhoj[jns]
rho_spl = UnivariateSpline(self.r, rho, ext=1, s=0)
rhoz = rho_spl(sqrt(z**2 + R_data**2))
rhoz_spl = UnivariateSpline(z, rhoz, ext=1, s=0)
# Now compute P(a_z|R)
# There are 2 possibilities depending on R:
# (1) the maximum acceleration occurs within the cluster boundary, or
# (2) max(a_z) = a_z,t (this happens when R ~ r_t)
nr, k = nz, 3 # bit of experimenting
# Option (1): zmax < max(z)
if len(zmax)>0:
zmax = zmax[0] # Take first entry for the rare cases with multiple peaks
# Set up 2 splines for the inverse z(a_z) for z < zmax and z > zmax
z1 = numpy.linspace(z[0], zmax, nr)
z2 = (numpy.linspace(zmax, z[-1], nr))[::-1] # Reverse z for ascending az
z1_spl = UnivariateSpline(az_spl(z1), z1, k=k, s=0, ext=1)
z2_spl = UnivariateSpline(az_spl(z2), z2, k=k, s=0, ext=1)
# Option 2: zmax = max(z)
else:
zmax = z[-1]
z1 = numpy.linspace(z[0], zmax, nr)
z1_spl = UnivariateSpline(az_spl(z1), z1, k=k, s=0, ext=1)
# Maximum acceleration along this los
azmax = az_spl(zmax)
# Now determine P(az_data|R)
if (az_data < azmax):
z1 = max([z1_spl(az_data), z[0]]) # first radius where az = az_data
Paz = rhoz_spl(z1)/abs(az_spl.derivatives(z1)[1])
if (az_data> azt):
# Find z where a_z = a_z,t
z2 = z2_spl(az_data)
Paz += rhoz_spl(z2)/abs(az_spl.derivatives(z2)[1])
# Normalize to 1
Paz /= rhoz_spl.integral(0, zt)
self.z = z
self.az = az
self.Paz = Paz
self.azmax = azmax
self.zmax = zmax
else:
self.Paz = 0
else:
self.Paz = 0
return
def read_cfd_data(cfd_data_file, u2, karr, dkarr):
"""read cfd results force coefficients data"""
phi_spl = UnivariateSpline(karr[:, 0], karr[:, 4] * np.pi / 180, s=0)
cf_array = []
with open(cfd_data_file) as csv_file:
csv_reader = csv.reader(csv_file, delimiter='\t')
line_count = 0
for row in csv_reader:
if line_count <= 14:
line_count += 1
else:
ti = float(row[0])
phii = phi_spl(ti)
dphii = -1.0 * phi_spl.derivatives(ti)[1]
cli = float(row[3])
cdi = np.sign(dphii) * (np.sin(phii) * float(row[2]) +
np.cos(phii) * float(row[1]))
csi = np.cos(phii) * float(row[2]) - np.sin(phii) * float(
row[1])
cf_array.append([
ti, cdi, csi, cli,
float(row[4]),
float(row[5]),
float(row[6])
])
line_count += 1
print(f'Processed {line_count} lines in {cfd_data_file}')
cf_array = np.array(cf_array)
cl_spl = UnivariateSpline(cf_array[:, 0], cf_array[:, 3], s=0)
cd_spl = UnivariateSpline(cf_array[:, 0], cf_array[:, 1], s=0)
cmx_spl = UnivariateSpline(cf_array[:, 0], cf_array[:, 6], s=0)
cmy_spl = UnivariateSpline(cf_array[:, 0], -cf_array[:, 5], s=0)
cmz_spl = UnivariateSpline(cf_array[:, 0], cf_array[:, 4], s=0)
omegax_spl = UnivariateSpline(dkarr[:, 0], dkarr[:, 4] * np.pi / 180, s=0)
omegay_spl = UnivariateSpline(
dkarr[:, 0],
np.multiply(dkarr[:, 5], np.cos(karr[:, 4] * np.pi / 180)) * np.pi /
180,
s=0)
omegaz_spl = UnivariateSpline(
dkarr[:, 0],
np.multiply(dkarr[:, 5], np.sin(karr[:, 4] * np.pi / 180)) * np.pi /
180,
s=0)
t_arr = cf_array[:, 0]
cpx_arr = [-1 * cmx_spl(t) * omegax_spl(t) / u2 for t in t_arr]
cpy_arr = [-1 * cmy_spl(t) * omegay_spl(t) / u2 for t in t_arr]
cpz_arr = [-1 * cmz_spl(t) * omegaz_spl(t) / u2 for t in t_arr]
cpx_spl = UnivariateSpline(t_arr, cpx_arr, s=0)
cpy_spl = UnivariateSpline(t_arr, cpy_arr, s=0)
cpz_spl = UnivariateSpline(t_arr, cpz_arr, s=0)
#--------plot axes-------
# plt.plot(t_arr, cpx_spl(t_arr), t_arr, cpy_spl(t_arr) + cpz_spl(t_arr))
# plt.plot(t_arr, cpx_spl(t_arr), t_arr, cpy_spl(t_arr), t_arr,
# cpz_spl(t_arr))
# plt.legend(['x', 'y', 'z'], loc='best')
# plt.xlim([4.0, 5.0])
# plt.show()
#------------------------
mcl = cl_spl.integral(4.0, 5.0)
mcd = cd_spl.integral(4.0, 5.0)
mcp = cpx_spl.integral(4.0, 5.0) + cpy_spl.integral(
4.0, 5.0) + cpz_spl.integral(4.0, 5.0)
mcf_array = [mcl, mcd, mcp]
return mcf_array
mSI = []
for x in range(len(m2Raw)):
r = m2Raw.iloc[x, :]
rh = r.values.reshape(1, -1)
regressor = LinearRegression()
# plt.plot(xH, r)
# plt.title(x)
# plt.show()
regressor.fit(xH, r)
slo = regressor.coef_[0]
intr = regressor.intercept_
slInt = [slo, intr]
s = UnivariateSpline(xBase, r, s=sf)
sm = s(xBase)
sd = s.derivatives(1)
mSI.append(slInt)
mDer.append(sd)
m2Smo.append(sm)
plt.plot(xBase, r)
#regular spline - Model 2
smozip = zip(m2Smo)
df2 = pd.DataFrame(smozip)
df22 = pd.DataFrame(df2[0].values.tolist())
#Derivative spline
derzip = zip(mDer)
dfd = pd.DataFrame(derzip)
dfd2 = pd.DataFrame(dfd[0].values.tolist())
#Slope-Int Model
slzip = zip(mSI)
def estimate(self, observedLC):
"""!
Estimate intrinsicFlux, period, eccentricity, omega, tau, & a2sini
"""
# fluxEst
if observedLC.numCadences > 50:
model = gatspy.periodic.LombScargleFast(optimizer_kwds={"quiet": True}).fit(observedLC.t,
observedLC.y,
observedLC.yerr)
else:
model = gatspy.periodic.LombScargle(optimizer_kwds={"quiet": True}).fit(observedLC.t,
observedLC.y,
observedLC.yerr)
periods, power = model.periodogram_auto(nyquist_factor=observedLC.numCadences)
model.optimizer.period_range = (
2.0*np.mean(observedLC.t[1:] - observedLC.t[:-1]), observedLC.T)
periodEst = model.best_period
numIntrinsicFlux = 100
lowestFlux = np.min(observedLC.y[np.where(observedLC.mask == 1.0)])
highestFlux = np.max(observedLC.y[np.where(observedLC.mask == 1.0)])
intrinsicFlux = np.linspace(np.min(observedLC.y[np.where(observedLC.mask == 1.0)]), np.max(
observedLC.y[np.where(observedLC.mask == 1.0)]), num=numIntrinsicFlux)
intrinsicFluxList = list()
totalIntegralList = list()
for f in xrange(1, numIntrinsicFlux - 1):
beamedLC = observedLC.copy()
beamedLC.x = np.require(np.zeros(beamedLC.numCadences), requirements=['F', 'A', 'W', 'O', 'E'])
for i in xrange(beamedLC.numCadences):
beamedLC.y[i] = observedLC.y[i]/intrinsicFlux[f]
beamedLC.yerr[i] = observedLC.yerr[i]/intrinsicFlux[f]
dopplerLC = beamedLC.copy()
dopplerLC.x = np.require(np.zeros(dopplerLC.numCadences), requirements=['F', 'A', 'W', 'O', 'E'])
for i in xrange(observedLC.numCadences):
dopplerLC.y[i] = math.pow(beamedLC.y[i], 1.0/3.44)
dopplerLC.yerr[i] = (1.0/3.44)*math.fabs(dopplerLC.y[i]*(beamedLC.yerr[i]/beamedLC.y[i]))
dzdtLC = dopplerLC.copy()
dzdtLC.x = np.require(np.zeros(dopplerLC.numCadences), requirements=['F', 'A', 'W', 'O', 'E'])
for i in xrange(observedLC.numCadences):
dzdtLC.y[i] = 1.0 - (1.0/dopplerLC.y[i])
dzdtLC.yerr[i] = math.fabs((-1.0*dopplerLC.yerr[i])/math.pow(dopplerLC.y[i], 2.0))
foldedLC = dzdtLC.fold(periodEst)
foldedLC.x = np.require(np.zeros(foldedLC.numCadences), requirements=['F', 'A', 'W', 'O', 'E'])
integralSpline = UnivariateSpline(
foldedLC.t[np.where(foldedLC.mask == 1.0)], foldedLC.y[np.where(foldedLC.mask == 1.0)],
1.0/foldedLC.yerr[np.where(foldedLC.mask == 1.0)], k=3, s=None, check_finite=True)
totalIntegral = math.fabs(integralSpline.integral(foldedLC.t[0], foldedLC.t[-1]))
intrinsicFluxList.append(intrinsicFlux[f])
totalIntegralList.append(totalIntegral)
fluxEst = intrinsicFluxList[
np.where(np.array(totalIntegralList) == np.min(np.array(totalIntegralList)))[0][0]]
# periodEst
for i in xrange(beamedLC.numCadences):
beamedLC.y[i] = observedLC.y[i]/fluxEst
beamedLC.yerr[i] = observedLC.yerr[i]/fluxEst
dopplerLC.y[i] = math.pow(beamedLC.y[i], 1.0/3.44)
dopplerLC.yerr[i] = (1.0/3.44)*math.fabs(dopplerLC.y[i]*(beamedLC.yerr[i]/beamedLC.y[i]))
dzdtLC.y[i] = 1.0 - (1.0/dopplerLC.y[i])
dzdtLC.yerr[i] = math.fabs((-1.0*dopplerLC.yerr[i])/math.pow(dopplerLC.y[i], 2.0))
if observedLC.numCadences > 50:
model = gatspy.periodic.LombScargleFast(optimizer_kwds={"quiet": True}).fit(dzdtLC.t,
dzdtLC.y,
dzdtLC.yerr)
else:
model = gatspy.periodic.LombScargle(optimizer_kwds={"quiet": True}).fit(dzdtLC.t,
dzdtLC.y,
dzdtLC.yerr)
periods, power = model.periodogram_auto(nyquist_factor=dzdtLC.numCadences)
model.optimizer.period_range = (2.0*np.mean(dzdtLC.t[1:] - dzdtLC.t[:-1]), dzdtLC.T)
periodEst = model.best_period
# eccentricityEst & omega2Est
# First find a full period going from rising to falling.
risingSpline = UnivariateSpline(
dzdtLC.t[np.where(dzdtLC.mask == 1.0)], dzdtLC.y[np.where(dzdtLC.mask == 1.0)],
1.0/dzdtLC.yerr[np.where(dzdtLC.mask == 1.0)], k=3, s=None, check_finite=True)
risingSplineRoots = risingSpline.roots()
firstRoot = risingSplineRoots[0]
if risingSpline.derivatives(risingSplineRoots[0])[1] > 0.0:
tRising = risingSplineRoots[0]
else:
tRising = risingSplineRoots[1]
# Now fold the LC starting at tRising and going for a full period.
foldedLC = dzdtLC.fold(periodEst, tStart=tRising)
foldedLC.x = np.require(np.zeros(foldedLC.numCadences), requirements=['F', 'A', 'W', 'O', 'E'])
# Fit the folded LC with a spline to figure out alpha and beta
fitLC = foldedLC.copy()
foldedSpline = UnivariateSpline(
foldedLC.t[np.where(foldedLC.mask == 1.0)], foldedLC.y[np.where(foldedLC.mask == 1.0)],
1.0/foldedLC.yerr[np.where(foldedLC.mask == 1.0)], k=3, s=2*foldedLC.numCadences,
check_finite=True)
for i in xrange(fitLC.numCadences):
fitLC.x[i] = foldedSpline(fitLC.t[i])
# Now get the roots and find the falling root
tZeros = foldedSpline.roots()
# Find tRising, tFalling, tFull, startIndex, & stopIndex via DBSCAN #######################
# Find the number of clusters
'''dbsObj = DBSCAN(eps = periodEst/10.0, min_samples = 1)
db = dbsObj.fit(tZeros.reshape(-1,1))
core_samples_mask = np.zeros_like(db.labels_, dtype=bool)
core_samples_mask[db.core_sample_indices_] = True
labels = db.labels_
unique_labels = set(labels)
n_clusters_ = len(set(labels)) - (1 if -1 in labels else 0)'''
# Find tRising, tFalling, tFull, startIndex, & stopIndex
if tZeros.shape[0] == 1: # We have found just tFalling
tFalling = tZeros[0]
tRising = fitLC.t[0]
startIndex = 0
tFull = fitLC.t[-1]
stopIndex = fitLC.numCadences
elif tZeros.shape[0] == 2: # We have found tFalling and one of tRising or tFull
if foldedSpline.derivatives(tZeros[0])[1] < 0.0:
tFalling = tZeros[0]
tFull = tZeros[1]
stopIndex = np.where(fitLC.t < tFull)[0][-1]
tRising = fitLC.t[0]
startIndex = 0
elif foldedSpline.derivatives(tZeros[0])[1] > 0.0:
if foldedSpline.derivatives(tZeros[1])[1] < 0.0:
tRising = tZeros[0]
startIndex = np.where(fitLC.t > tRising)[0][0]
tFalling = tZeros[1]
tFull = fitLC.t[-1]
stopIndex = fitLC.numCadences
else:
raise RuntimeError(
'Could not determine alpha & omega correctly because the first root is rising but \
the second root is not falling!')
elif tZeros.shape[0] == 3:
tRising = tZeros[0]
startIndex = np.where(fitLC.t > tRising)[0][0]
tFalling = tZeros[1]
tFull = tZeros[2]
stopIndex = np.where(fitLC.t < tFull)[0][-1]
else:
# More than 3 roots!!! Use K-Means to cluster the roots assuming we have 3 groups
root_groups = KMeans(n_clusters=3).fit_predict(tZeros.reshape(-1, 1))
RisingGroupNumber = root_groups[0]
FullGroupNumber = root_groups[-1]
RisingSet = set(root_groups[np.where(root_groups != RisingGroupNumber)[0]])
FullSet = set(root_groups[np.where(root_groups != FullGroupNumber)[0]])
FallingSet = RisingSet.intersection(FullSet)
FallingGroupNumber = FallingSet.pop()
numRisingRoots = np.where(root_groups == RisingGroupNumber)[0].shape[0]
numFallingRoots = np.where(root_groups == FallingGroupNumber)[0].shape[0]
numFullRoots = np.where(root_groups == FullGroupNumber)[0].shape[0]
if numRisingRoots == 1:
tRising = tZeros[np.where(root_groups == RisingGroupNumber)[0]][0]
else:
RisingRootCands = tZeros[np.where(root_groups == RisingGroupNumber)[0]]
for i in xrange(RisingRootCands.shape[0]):
if foldedSpline.derivatives(RisingRootCands[i])[1] > 0.0:
tRising = RisingRootCands[i]
break
if numFallingRoots == 1:
tFalling = tZeros[np.where(root_groups == FallingGroupNumber)[0]][0]
else:
FallingRootCands = tZeros[np.where(root_groups == FallingGroupNumber)[0]]
for i in xrange(FallingRootCands.shape[0]):
if foldedSpline.derivatives(FallingRootCands[i])[1] < 0.0:
tFalling = FallingRootCands[i]
break
if numFullRoots == 1:
tFull = tZeros[np.where(root_groups == FullGroupNumber)[0]][0]
else:
FullRootCands = tZeros[np.where(root_groups == FullGroupNumber)[0]]
for i in xrange(FullRootCands.shape[0]):
if foldedSpline.derivatives(FullRootCands[i])[1] > 0.0:
tFull = FullRootCands[i]
break
startIndex = np.where(fitLC.t > tRising)[0][0]
stopIndex = np.where(fitLC.t < tFull)[0][-1]
#
# One full period now goes from tRising to periodEst. The maxima occurs between tRising and tFalling
# while the minima occurs between tFalling and tRising + periodEst. Find the minima and maxima
alpha = math.fabs(fitLC.x[np.where(np.max(fitLC.x[startIndex:stopIndex]) == fitLC.x)[0][0]])
beta = math.fabs(fitLC.x[np.where(np.min(fitLC.x[startIndex:stopIndex]) == fitLC.x)[0][0]])
peakLoc = fitLC.t[np.where(np.max(fitLC.x[startIndex:stopIndex]) == fitLC.x)[0][0]]
troughLoc = fitLC.t[np.where(np.min(fitLC.x[startIndex:stopIndex]) == fitLC.x)[0][0]]
KEst = 0.5*(alpha + beta)
delta2 = (math.fabs(foldedSpline.integral(tRising, peakLoc)) + math.fabs(
foldedSpline.integral(troughLoc, tFull)))/2.0
delta1 = (math.fabs(foldedSpline.integral(peakLoc, tFalling)) + math.fabs(
foldedSpline.integral(tFalling, troughLoc)))/2.0
eCosOmega2 = (alpha - beta)/(alpha + beta)
eSinOmega2 = ((2.0*math.sqrt(alpha*beta))/(alpha + beta))*((delta2 - delta1)/(delta2 + delta1))
eccentricityEst = math.sqrt(math.pow(eCosOmega2, 2.0) + math.pow(eSinOmega2, 2.0))
tanOmega2 = math.fabs(eSinOmega2/eCosOmega2)
if (eCosOmega2/math.fabs(eCosOmega2) == 1.0) and (eSinOmega2/math.fabs(eSinOmega2) == 1.0):
omega2Est = math.atan(tanOmega2)*(180.0/math.pi)
if (eCosOmega2/math.fabs(eCosOmega2) == -1.0) and (eSinOmega2/math.fabs(eSinOmega2) == 1.0):
omega2Est = 180.0 - math.atan(tanOmega2)*(180.0/math.pi)
if (eCosOmega2/math.fabs(eCosOmega2) == -1.0) and (eSinOmega2/math.fabs(eSinOmega2) == -1.0):
omega2Est = 180.0 + math.atan(tanOmega2)*(180.0/math.pi)
if (eCosOmega2/math.fabs(eCosOmega2) == 1.0) and (eSinOmega2/math.fabs(eSinOmega2) == -1.0):
omega2Est = 360.0 - math.atan(tanOmega2)*(180.0/math.pi)
if omega2Est >= 180.0:
omega1Est = omega2Est - 180.0
if omega2Est < 180.0:
omega1Est = omega2Est + 180.0
# tauEst
zDot = KEst*(1.0 + eccentricityEst)*(eCosOmega2/eccentricityEst)
zDotLC = dzdtLC.copy()
for i in xrange(zDotLC.numCadences):
zDotLC.y[i] = zDotLC.y[i] - zDot
zDotSpline = UnivariateSpline(
zDotLC.t[np.where(zDotLC.mask == 1.0)], zDotLC.y[np.where(zDotLC.mask == 1.0)],
1.0/zDotLC.yerr[np.where(zDotLC.mask == 1.0)], k=3, s=2*zDotLC.numCadences, check_finite=True)
for i in xrange(zDotLC.numCadences):
zDotLC.x[i] = zDotSpline(zDotLC.t[i])
zDotZeros = zDotSpline.roots()
zDotFoldedLC = dzdtLC.fold(periodEst)
zDotFoldedSpline = UnivariateSpline(
zDotFoldedLC.t[np.where(zDotFoldedLC.mask == 1.0)],
zDotFoldedLC.y[np.where(zDotFoldedLC.mask == 1.0)],
1.0/zDotFoldedLC.yerr[np.where(zDotFoldedLC.mask == 1.0)], k=3, s=2*zDotFoldedLC.numCadences,
check_finite=True)
for i in xrange(zDotFoldedLC.numCadences):
zDotFoldedLC.x[i] = zDotFoldedSpline(zDotFoldedLC.t[i])
tC = zDotFoldedLC.t[np.where(np.max(zDotFoldedLC.x) == zDotFoldedLC.x)[0][0]]
nuC = (360.0 - omega2Est)%360.0
tE = zDotFoldedLC.t[np.where(np.min(zDotFoldedLC.x) == zDotFoldedLC.x)[0][0]]
nuE = (180.0 - omega2Est)%360.0
if math.fabs(360.0 - nuC) < math.fabs(360 - nuE):
tauEst = zDotZeros[np.where(zDotZeros > tC)[0][0]]
else:
tauEst = zDotZeros[np.where(zDotZeros > tE)[0][0]]
tauEst = tauEst%periodEst
# a2sinInclinationEst
a2sinInclinationEst = ((KEst*periodEst*self.Day*self.c*math.sqrt(1.0 - math.pow(
eccentricityEst, 2.0)))/self.twoPi)/self.Parsec
return fluxEst, periodEst, eccentricityEst, omega1Est, tauEst, a2sinInclinationEst
f = interpolate.interp1d(x, y, kind="cubic")
xnew = np.arange(1, 3, 0.001)
ynew = f(xnew)
tab = []
for i in range(4):
tab.append(0)
i = 1
plt.plot(tab) # rysowanie OX
plt.plot(x, y, "o", xnew, ynew, '-')
plt.show()
#Wyliczanie miejsc zerowych funkcji
x0 = fsolve(f, [1.15])
x1 = fsolve(f, [2.2])
x2 = fsolve(f, [2.7])
print("x0 =" + str(x0))
print("x1 =" + str(x1))
print("x2 =" + str(x2))
spl = UnivariateSpline(xnew, ynew, k=1, s =2)
# Ponizszy kod wyrysowuje nam ta sama funckje "plt.plot(x, y, "o", xnew, ynew, '-')"
# spl.set_smoothing_factor(0.1)
# plt.plot(xnew, spl(xnew), 'g', lw=2)
# plt.show()
#zgadywanie pochodnej w punkcie 2.1
print(spl.derivatives(2.1))
import numpy as np
#from sklearn import decomposition
import matplotlib
matplotlib.use("cairo")
import matplotlib.pyplot as plt
plt.style.use('ggplot') # Use the ggplot style
from scipy.interpolate import UnivariateSpline
x = np.linspace(0.0, 1.0, 150)
y = 2 * np.sin((x + 0.5) * 5) / (x + 0.1)
s = UnivariateSpline(x, y, k=3)
print(s.get_coeffs())
derivs = s.derivatives(0.0)
print(derivs)
xs = np.linspace(0.0, 1.0, 1000)
ys = s(xs)
ys2 = derivs[0] + xs * derivs[1] + np.power(xs, 2) * derivs[2] / 2 + np.power(
xs, 3) * derivs[3] / 6
plt.plot(x, y, '.-', label='data')
plt.plot(xs, ys, zorder=10, label='smoothed')
plt.plot(xs, ys2, alpha=0.5, zorder=0, label='taylor')
plt.ylim([-6, 12])
plt.legend()
plt.savefig("spline_test.pdf")
def miller(self, psinorm=None, rova=None, omt_factor=None):
"""
Calculate Miller quantities.
Args:
psinorm (float, default 0.8):
psi-norm for Miller quantities (overrides rova)
rova (float, default None):
r/a for Miller quantities (ignored if psinorm evals to True)
omt_factor (float, default 0.2):
sets omt = omp * omt_factor
and omn = omp * (1-omt_factor)
Returns:
dictionary with Miller quantities and reference values
for GENE simulation
"""
output = {}
if psinorm is None and rova is None:
psinorm = 0.8
if omt_factor is None:
omt_factor = 0.2
# r and psi splines
psi_grid_spl = US(self.r_minor_fs, self.psi_grid, k=self.io, s=self.s)
r_min_spl = US(self.psi_grid, self.r_minor_fs, k=self.io, s=self.s)
if psinorm:
print('using `psinorm`, ignoring `rova`')
self.psinorm = psinorm
psi_poi = self.psinorm * (self.psisep - self.psiax) + self.psiax
r_poi = r_min_spl(psi_poi)[()]
self.rova = r_poi / self.a_lcfs
else:
print('using `rova`, ignoring `psinorm`')
self.rova = rova
r_poi = self.rova * self.a_lcfs # r = r/a * a; FS minor radius
psi_poi = psi_grid_spl(r_poi)[()] # psi at FS
self.psinorm = (psi_poi - self.psiax) / (self.psisep - self.psiax
) # psi-norm at FS
output['gfile'] = self.gfile
output['psinorm'] = self.psinorm
output['rova'] = self.rova
R0_spl = US(self.r_minor_fs, self.R_major_fs, k=self.io, s=self.s)
R0_poi = R0_spl(r_poi)[()] # R_maj of FS
q_spl = US(self.r_minor_fs, self.qpsi_fs, k=self.io, s=self.s)
#q_spl_psi = US(self.psi_grid, self.qpsi_fs, k=self.io, s=self.s)
q_poi = q_spl(r_poi)[()]
drdpsi_poi = float(r_min_spl.derivatives(psi_poi)[1])
eps_poi = r_poi / R0_poi
if not self.quiet:
print('\n*** Flux surfance ***')
print('r_min/a = {:.3f}'.format(self.rova))
print('psinorm = {:.3f}'.format(self.psinorm))
print('r_min = {:.3f} m'.format(r_poi))
print('R_maj = {:.3f} m'.format(R0_poi))
print('eps = {:.3f}'.format(eps_poi))
print('q = {:.3f}'.format(q_poi))
print('psi = {:.3e} Wb/rad'.format(psi_poi))
print('dr/dpsi = {:.3g} m/(Wb/rad)'.format(drdpsi_poi))
F_spl = US(self.r_minor_fs, self.F_fs, k=self.io, s=self.s)
F_poi = F_spl(r_poi)[()] # F of FS
Bref_poi = F_poi / R0_poi
p_spl = US(self.r_minor_fs, self.p_fs, k=self.io, s=self.s)
p_poi = p_spl(r_poi)[()]
t_poi = self.ti
n_poi = p_poi / (t_poi * 1e3 * 1.602e-19) / 1e19
output['Lref'] = self.a_lcfs
output['Bref'] = Bref_poi
output['pref'] = p_poi
output['Tref'] = t_poi
output['nref'] = n_poi
beta = 403.e-5 * n_poi * self.ti / (Bref_poi**2)
coll = 2.3031e-5 * (24-np.log(np.sqrt(n_poi*1e13)/(1e3*t_poi))) \
* self.a_lcfs * n_poi / (t_poi**2)
output['beta'] = beta
output['coll'] = coll
if not self.quiet:
print('\n*** Reference values ***')
print('Lref = {:.3g} m ! for Lref=a convention'.format(
self.a_lcfs))
print('Bref = {:.3g} T'.format(Bref_poi))
print('pref = {:.3g} Pa'.format(p_poi))
print('Tref = {:.3g} keV'.format(t_poi))
print('nref = {:.3g} 1e19/m^3'.format(n_poi))
print('beta = {:.3g}'.format(beta))
print('coll = {:.3g}'.format(coll))
pprime_spl = US(self.r_minor_fs, self.pprime_fs, k=self.io, s=1e-4)
pprime_poi = pprime_spl(r_poi)[()]
pm_poi = Bref_poi**2 / (2 * 4 * 3.14e-7)
dpdr_poi = pprime_poi / drdpsi_poi
dpdx_pm = -dpdr_poi / pm_poi
omp_poi = -(self.a_lcfs / p_poi) * dpdr_poi
output['dpdx_pm'] = dpdx_pm
output['omp'] = omp_poi
if not self.quiet:
print('\n*** Pressure gradients ***')
print('dp/dpsi = {:.3g} Pa/(Wb/rad)'.format(pprime_poi))
print('p_m = 2mu/Bref**2 = {:.3g} Pa'.format(pm_poi))
print('-(dp/dr)/p_m = {:.3g} 1/m'.format(dpdx_pm))
print('omp = a/p * dp/dr = {:.3g} ! with Lref=a'.format(omp_poi))
omt = omp_poi * omt_factor
omn = omp_poi * (1 - omt_factor)
output['omt'] = omt
output['omt_factor'] = omt_factor
output['omn'] = omn
if not self.quiet:
print('\n*** Temp/dens gradients ***')
print('omt_factor = {:.3g}'.format(omt_factor))
print('omt = a/T * dT/dr = {:.3g}'.format(omt))
print('omn = a/n * dn/dr = {:.3g}'.format(omn))
sgstart = self.nw // 10
subgrid = np.arange(sgstart, self.nw)
nsg = subgrid.size
# calc symmetric R/Z on psi/theta grid
kappa = np.empty(nsg)
delta = np.empty(nsg)
zeta = np.empty(nsg)
drR = np.empty(nsg)
amhd = np.empty(nsg)
bp = np.empty(nsg)
bt = np.empty(nsg)
b = np.empty(nsg)
theta_tmp = np.linspace(-2. * np.pi, 2 * np.pi, 2 * self.ntheta - 1)
stencil_width = self.ntheta // 10
for i, isg in enumerate(subgrid):
R_extended = np.empty(2 * self.ntheta - 1)
Z_extended = np.empty(2 * self.ntheta - 1)
R_extended[0:(self.ntheta - 1) //
2] = self.R_ftgrid[isg, (self.ntheta + 1) // 2:-1]
R_extended[(self.ntheta - 1) // 2:(3 * self.ntheta - 3) //
2] = self.R_ftgrid[isg, :-1]
R_extended[(3 * self.ntheta - 3) //
2:] = self.R_ftgrid[isg, 0:(self.ntheta + 3) // 2]
Z_extended[0:(self.ntheta - 1) //
2] = self.Z_ftgrid[isg, (self.ntheta + 1) // 2:-1]
Z_extended[(self.ntheta - 1) // 2:(3 * self.ntheta - 3) //
2] = self.Z_ftgrid[isg, :-1]
Z_extended[(3 * self.ntheta - 3) //
2:] = self.Z_ftgrid[isg, 0:(self.ntheta + 3) // 2]
theta_mod_ext = np.arctan2(Z_extended - self.Z_avg_fs[isg],
R_extended - self.R_major_fs[isg])
# introduce 2pi shifts to theta_mod_ext
for ind in range(self.ntheta):
if theta_mod_ext[ind + 1] < 0. \
and theta_mod_ext[ind] > 0. \
and abs(theta_mod_ext[ind + 1] - theta_mod_ext[ind]) > np.pi:
lshift_ind = ind
if theta_mod_ext[-ind - 1] > 0. \
and theta_mod_ext[-ind] < 0. \
and abs(theta_mod_ext[-ind - 1] - theta_mod_ext[-ind]) > np.pi:
rshift_ind = ind
theta_mod_ext[-rshift_ind:] += 2. * np.pi
theta_mod_ext[:lshift_ind + 1] -= 2. * np.pi
theta_int = interp1d(theta_mod_ext, theta_tmp, kind=self.io)
R_int = interp1d(theta_mod_ext, R_extended, kind=self.io)
Z_int = interp1d(theta_mod_ext, Z_extended, kind=self.io)
R_tm = R_int(self.theta_grid)
Z_tm = Z_int(self.theta_grid)
Z_sym = 0.5 * (Z_tm[:] - Z_tm[::-1]) + self.Z_avg_fs[isg]
R_sym = 0.5 * (R_tm[:] + R_tm[::-1])
delta_ul = np.empty(2)
for o in range(2):
if o:
ind = np.argmax(Z_sym)
section = np.arange(ind + stencil_width // 2,
ind - stencil_width // 2, -1)
else:
ind = np.argmin(Z_sym)
section = np.arange(ind - stencil_width // 2,
ind + stencil_width // 2)
x = R_sym[section]
y = Z_sym[section]
y_int = interp1d(x, y, kind=self.io)
x_fine = np.linspace(np.amin(x), np.amax(x),
stencil_width * 100)
y_fine = y_int(x_fine)
if o:
x_at_extremum = x_fine[np.argmax(y_fine)]
Z_max = np.amax(y_fine)
else:
x_at_extremum = x_fine[np.argmin(y_fine)]
Z_min = np.amin(y_fine)
delta_ul[o] = (self.R_major_fs[isg] - x_at_extremum) \
/ self.r_minor_fs[isg]
kappa[i] = (Z_max - Z_min) / 2. / self.r_minor_fs[isg]
delta[i] = delta_ul.mean()
# calc zeta
zeta_arr = np.empty(4)
for o in range(4):
if o == 0:
val = np.pi / 4
searchval = np.cos(val + np.arcsin(delta[i]) / np.sqrt(2))
searcharr = (R_sym -
self.R_major_fs[isg]) / self.r_minor_fs[isg]
elif o == 1:
val = 3 * np.pi / 4
searchval = np.cos(val + np.arcsin(delta[i]) / np.sqrt(2))
searcharr = (R_sym -
self.R_major_fs[isg]) / self.r_minor_fs[isg]
elif o == 2:
val = -np.pi / 4
searchval = np.cos(val - np.arcsin(delta[i]) / np.sqrt(2))
searcharr = (R_sym -
self.R_major_fs[isg]) / self.r_minor_fs[isg]
elif o == 3:
val = -3 * np.pi / 4
searchval = np.cos(val - np.arcsin(delta[i]) / np.sqrt(2))
searcharr = (R_sym -
self.R_major_fs[isg]) / self.r_minor_fs[isg]
else:
raise ValueError('out of range')
if o in [0, 1]:
searcharr2 = searcharr[self.ntheta // 2:]
ind = self._find(searchval, searcharr2) + self.ntheta // 2
else:
searcharr2 = searcharr[0:self.ntheta // 2]
ind = self._find(searchval, searcharr2)
section = np.arange(ind - stencil_width // 2,
ind + stencil_width // 2)
theta_sec = self.theta_grid[section]
if o in [0, 1]:
theta_int = interp1d(-searcharr[section],
theta_sec,
kind=self.io)
theta_of_interest = theta_int(-searchval)
else:
theta_int = interp1d(searcharr[section],
theta_sec,
kind=self.io)
theta_of_interest = theta_int(searchval)
Z_sec = Z_sym[section]
Z_sec_int = interp1d(theta_sec, Z_sec, kind=self.io)
Z_val = Z_sec_int(theta_of_interest)
zeta_arg = (Z_val - self.Z_avg_fs[isg]
) / kappa[i] / self.r_minor_fs[isg]
if abs(zeta_arg) >= 1:
zeta_arg = 0.999999 * np.sign(zeta_arg)
zeta_arr[o] = np.arcsin(zeta_arg)
zeta_arr[1] = np.pi - zeta_arr[1]
zeta_arr[3] = -np.pi - zeta_arr[3]
zeta[i] = 0.25 * (np.pi + zeta_arr[0] - zeta_arr[1] - zeta_arr[2] +
zeta_arr[3])
# calc dr/dR, amhd, and derivs
amhd[i] = -self.qpsi_fs[isg]**2 * self.R_major_fs[isg] * self.pprime_fs[isg] * \
8 * np.pi * 1e-7 / Bref_poi**2 / \
r_min_spl.derivatives(self.psi_grid[isg])[1]
drR[i] = R0_spl.derivatives(self.r_minor_fs[isg])[1]
R = self.R_major_fs[isg] + self.r_minor_fs[isg]
Z = self.Z_avg_fs[isg]
Br = -self.psi_spl(Z, R, dx=1, dy=0) / R
Bz = self.psi_spl(Z, R, dx=0, dy=1) / R
bp[i] = np.sqrt(Br**2 + Bz**2)
bt[i] = self.F_fs[isg] / R
b[i] = np.sqrt(bp[i]**2 + bt[i]**2)
amhd_spl = US(self.r_minor_fs[sgstart:], amhd, k=self.io, s=1e-3)
drR_spl = US(self.r_minor_fs[sgstart:], drR, k=self.io, s=self.s)
b_spl = US(self.psinorm_grid[sgstart:], b, k=self.io, s=self.s)
# calc derivatives for kappa, delta, zeta
s_kappa = np.empty(nsg)
s_delta = np.empty(nsg)
s_zeta = np.empty(nsg)
kappa_spl = US(self.r_minor_fs[sgstart:], kappa, k=self.io, s=self.s)
delta_spl = US(self.r_minor_fs[sgstart:], delta, k=self.io, s=self.s)
zeta_spl = US(self.r_minor_fs[sgstart:], zeta, k=self.io, s=self.s)
for i, isg in enumerate(subgrid):
s_kappa[i] = kappa_spl.derivatives(self.r_minor_fs[isg])[1] \
* self.r_minor_fs[isg] / kappa[i]
s_delta[i] = delta_spl.derivatives(self.r_minor_fs[isg])[1] \
* self.r_minor_fs[isg] / np.sqrt(1 - delta[i]**2)
s_zeta[i] = zeta_spl.derivatives(self.r_minor_fs[isg])[1] \
* self.r_minor_fs[isg]
output['trpeps'] = eps_poi
output['q0'] = q_poi
output['shat'] = (r_poi / q_poi) * q_spl.derivatives(r_poi)[1]
output['amhd'] = amhd_spl(r_poi)[()]
output['drR'] = drR_spl(r_poi)[()]
output['kappa'] = kappa_spl(r_poi)[()]
output['s_kappa'] = kappa_spl.derivatives(
r_poi)[1] * r_poi / kappa_spl(r_poi)[()]
output['delta'] = delta_spl(r_poi)[()]
output['s_delta'] = delta_spl.derivatives(r_poi)[1] * r_poi \
/ np.sqrt(1 - delta_spl(r_poi)[()]**2)
output['zeta'] = zeta_spl(r_poi)[()]
output['s_zeta'] = zeta_spl.derivatives(r_poi)[1] * r_poi
output['minor_r'] = 1.0
output['major_R'] = R0_poi / self.a_lcfs
if not self.quiet:
print(
'\n\nShaping parameters for flux surface r=%9.5g, r/a=%9.5g:' %
(r_poi, self.rova))
print('Copy the following block into a GENE parameters file:\n')
print('trpeps = %9.5g' % (eps_poi))
print('q0 = %9.5g' % q_poi)
print('shat = %9.5g !(defined as r/q*dq_dr)' % (r_poi / q_poi \
* q_spl.derivatives(r_poi)[1]))
print('amhd = %9.5g' % amhd_spl(r_poi))
print('drR = %9.5g' % drR_spl(r_poi))
print('kappa = %9.5g' % kappa_spl(r_poi))
print('s_kappa = %9.5g' % (kappa_spl.derivatives(r_poi)[1] \
* r_poi / kappa_spl(r_poi)))
print('delta = %9.5g' % delta_spl(r_poi))
print('s_delta = %9.5g' % (delta_spl.derivatives(r_poi)[1] \
* r_poi / np.sqrt(1 - delta_spl(r_poi)**2)))
print('zeta = %9.5g' % zeta_spl(r_poi))
print('s_zeta = %9.5g' % (zeta_spl.derivatives(r_poi)[1] * r_poi))
print('minor_r = %9.5g' % (1.0))
print('major_R = %9.5g' % (R0_poi / self.a_lcfs))
if self.plot:
plt.figure(figsize=(6, 8))
plt.subplot(4, 2, 1)
plt.plot(self.psinorm_grid[sgstart:], kappa)
plt.title('Elongation')
plt.ylabel(r'$\kappa$', fontsize=14)
plt.subplot(4, 2, 2)
plt.plot(self.psinorm_grid[sgstart:], s_kappa)
plt.title(r'$r/\kappa*(d\kappa/dr)$')
plt.ylabel(r'$s_\kappa$', fontsize=14)
plt.subplot(4, 2, 3)
plt.plot(self.psinorm_grid[sgstart:], delta)
plt.title('Triangularity')
plt.ylabel(r'$\delta$', fontsize=14)
plt.subplot(4, 2, 4)
plt.plot(self.psinorm_grid[sgstart:], s_delta)
plt.title(r'$r/\delta*(d\delta/dr)$')
plt.ylabel(r'$s_\delta$', fontsize=14)
plt.subplot(4, 2, 5)
plt.plot(self.psinorm_grid[sgstart:], zeta)
plt.title('Squareness')
plt.ylabel(r'$\zeta$', fontsize=14)
plt.subplot(4, 2, 6)
plt.plot(self.psinorm_grid[sgstart:], s_zeta)
plt.title(r'$r/\zeta*(d\zeta/dr)$')
plt.ylabel(r'$s_\zeta$', fontsize=14)
plt.subplot(4, 2, 7)
plt.plot(self.psinorm_grid[sgstart:], b)
plt.plot(self.psinorm_grid[sgstart:], bp)
plt.plot(self.psinorm_grid[sgstart:], bt)
plt.title('|B|')
plt.ylabel(r'|B|', fontsize=14)
plt.subplot(4, 2, 8)
plt.plot(self.psinorm_grid[sgstart:],
b_spl(self.psinorm_grid[sgstart:], nu=1))
plt.title('|B| deriv.')
plt.ylabel(r'$d|B|/d\Psi_N$', fontsize=14)
for ax in plt.gcf().axes:
ax.set_xlabel(r'$\Psi_N$', fontsize=14)
ax.axvline(self.psinorm, 0, 1, ls='--', color='k', lw=2)
plt.tight_layout(pad=0.3)
return output
def scale_up_function(x,
y,
method=3,
smoothness=1.0,
bound_low=None,
bound_up=None,
auto_bound=1.3,
intervention_end=None,
intervention_start_date=None):
"""
Given a set of points defined by x and y,
this function fits a cubic spline and returns the interpolated function
Args:
x: The independent variable at each observed point
y: The dependent variable at each observed point
method: Select an interpolation method. Methods 1, 2 and 3 use cubic interpolation defined step by step.
1: Uses derivative at each point as constrain --> see description of the encapsulated function
derivatives(x, y). At each step, the defined portion of the curve covers the interval [x_i, x_(i+1)]
2: Less constrained interpolation as the curve does not necessarily pass by every point.
At each step, the defined portion of the curve covers the interval [x_i, x_(i+2)] except when the derivative
should be 0 at x_(i+1). In this case, the covered interval is [x_i, x_(i+1)]
3: The curve has to pass by every point. At each step, the defined portion of the curve covers
the interval [x_i, x_(i+1)] but x_(i+2) is still used to obtain the fit on [x_i, x_(i+1)]
4: Uses sigmoidal curves. This is a generalisation of the function make_two_step_curve
5: Uses an adaptive algorithm producing either an interpolation or an approximation, depending on the value
of the smoothness. This method allows for consideration of limits that are defined through bound_low and
bound_up. See detailed description of this method in the code after the line "if method == 5:"
smoothness, bound_up, bound_low, auto_bound are only used when method=5
smoothness: Defines the level of smoothness of the curve. The minimum value is 0. and leads to an interpolation.
bound_low: Defines a potential lower bound that the curve should not overcome.
bound_up: Defines a potential upper bound that the curve should not overcome.
auto_bound: In absence of bound_up or bound_low, sets a strip in which the curve should be contained.
Its value is a multiplier that applies to the amplitude of y to determine the width of the
strip. Set equal to None to delete this constraint.
intervention_end: tuple or list of two elements defining the final time and the final level corresponding to a
potential intervention. If it is not None, an additional portion of curve will be added to define
the intervention through a sinusoidal function
intervention_start_date: If not None, define the date at which intervention should start (must be >= max(x)).
If None, the maximal value of x will be used as a start date for the intervention. If the argument
'intervention_end' is not defined, this argument is not relevant and will not be used
Returns:
interpolation function
"""
assert len(x) == len(y), 'x and y must have the same length'
x = [float(i) for i in x]
y = [float(i) for i in y]
x = np.array(x)
y = np.array(y)
# Check that every x_i is unique
assert len(x) == len(set(x)), 'There are duplicate values in x.'
# Make sure the arrays are ordered
order = x.argsort()
x = x[order]
y = y[order]
# Define a scale-up for a potential intervention
if intervention_end is not None:
if intervention_start_date is not None:
assert intervention_start_date >= max(
x), 'The intervention start date should be >= max(x)'
t_intervention_start = intervention_start_date
else:
t_intervention_start = max(x)
curve_intervention = scale_up_function(
x=[t_intervention_start, intervention_end[0]],
y=[y[-1], intervention_end[1]],
method=4)
if (len(x) == 1) or (max(y) - min(y) == 0):
def curve(t):
if intervention_end is not None:
if t >= t_intervention_start:
return curve_intervention(t)
else:
return y[-1]
else:
return y[-1]
return curve
def derivatives(x, y):
"""
Defines the slopes at each data point
Should be 0 for the first and last points (x_0 and x_n)
Should be 0 when the variation is changing (U or n shapes), i.e. when(y_i - y_(i-1))*(y_i - y_(i+1)) > 0
If the variation is not changing, the slope is defined by a linear combination of the two slopes measured
between the point and its two neighbours. This combination is weighted according to the distance with the
neighbours.
This is only relevant when method=1
"""
v = np.zeros(len(x)) # Initialises all zeros
for i in range(len(x))[1:-1]: # For each interior point
if (y[i] - y[i - 1]) * (y[i] -
y[i + 1]) < 0: # Not changing variation
slope_left = (y[i] - y[i - 1]) / (x[i] - x[i - 1])
slope_right = (y[i + 1] - y[i]) / (x[i + 1] - x[i])
w = (x[i] - x[i - 1]) / (x[i + 1] - x[i - 1])
v[i] = w * slope_right + (1 - w) * slope_left
return v
if method in (1, 2, 3):
# Normalise x to avoid too big or too small numbers when taking x**3
coef = np.exp(np.log10(max(x)))
x = x / coef
vel = derivatives(x, y) # Obtain derivatives conditions
m = np.zeros(
(len(x) - 1, 4)
) # To store the polynomial coefficients for each section [x_i, x_(i+1)]
if method == 1:
for i in range(1, len(x)):
x0, x1 = x[i - 1], x[i]
g = np.array([
[x0**3, x0**2, x0, 1],
[x1**3, x1**2, x1, 1],
[3 * x0**2, 2 * x0, 1, 0],
[3 * x1**2, 2 * x1, 1, 0],
])
# Bound conditions: f(x0) = y0 f(x1) = y1 f'(x0) = v0 f'(x1) = v1
d = np.array([y[i - 1], y[i], vel[i - 1], vel[i]])
m[i - 1, :] = np.linalg.solve(g, d)
elif method == 2:
pass_next = 0
for i in range(len(x))[0:-1]:
if pass_next == 1: # When = 1, the next section [x_(i+1), x_(i+2)] is already defined
pass_next = 0
else:
x0 = x[i]
y0 = y[i]
# Define left velocity condition
if vel[i] == 0:
v = 0
else:
# Get former polynomial to get left velocity condition. Derivative has to be continuous
p = m[i - 1, :]
v = 3 * p[0] * x0**2 + 2 * p[1] * x0 + p[2]
if vel[i + 1] == 0: # Define only one section
x1 = x[i + 1]
y1 = y[i + 1]
g = np.array([
[x0**3, x0**2, x0, 1],
[x1**3, x1**2, x1, 1],
[3 * x0**2, 2 * x0, 1, 0],
[3 * x1**2, 2 * x1, 1, 0],
])
# Bound conditions: f(x0) = y0 f(x1) = y1 f'(x0) = v0 f'(x1) = 0
d = np.array([y0, y1, v, 0])
m[i, :] = np.linalg.solve(g, d)
elif vel[i + 2] == 0: # defines two sections
x1, x2 = x[i + 1], x[i + 2]
y1, y2 = y[i + 1], y[i + 2]
g = np.array([
[x0**3, x0**2, x0, 1],
[x2**3, x2**2, x2, 1],
[3 * x0**2, 2 * x0, 1, 0],
[3 * x2**2, 2 * x2, 1, 0],
])
# Bound conditions: f(x0) = y0 f(x2) = y2 f'(x0) = v0 f'(x2) = 0
d = np.array([y0, y2, v, 0])
sol = np.linalg.solve(g, d)
m[i, :] = sol
m[i + 1, :] = sol
pass_next = 1
else: # v1 and v2 are not null. We define two sections
x1, x2 = x[i + 1], x[i + 2]
y1, y2 = y[i + 1], y[i + 2]
g = np.array([
[x0**3, x0**2, x0, 1],
[x1**3, x1**2, x1, 1],
[x2**3, x2**2, x2, 1],
[3 * x0**2, 2 * x0, 1, 0],
])
# Bound conditions: f(x0) = y0 f(x1) = y1 f(x2) = y2 f'(x0) = v0
d = np.array([y0, y1, y2, v])
sol = np.linalg.solve(g, d)
m[i, :] = sol
m[i + 1, :] = sol
pass_next = 1
elif method == 3:
pass_next = 0
for i in range(len(x))[0:-1]:
if pass_next == 1: # When = 1, the next section [x_(i+1), x_(i+2)] is already defined
pass_next = 0
else:
x0 = x[i]
y0 = y[i]
# Define left velocity condition
if vel[i] == 0:
v = 0
else:
# Get former polynomial to get left velocity condition
p = m[i - 1, :]
v = 3 * p[0] * x0**2 + 2 * p[1] * x0 + p[2]
if vel[i + 1] == 0:
x1 = x[i + 1]
y1 = y[i + 1]
g = np.array([
[x0**3, x0**2, x0, 1],
[x1**3, x1**2, x1, 1],
[3 * x0**2, 2 * x0, 1, 0],
[3 * x1**2, 2 * x1, 1, 0],
])
# Bound conditions: f(x0) = y0 f(x1) = y1 f'(x0) = v0 f'(x1) = 0
d = np.array([y0, y1, v, 0])
m[i, :] = np.linalg.solve(g, d)
else:
x1, x2 = x[i + 1], x[i + 2]
y1, y2 = y[i + 1], y[i + 2]
g = np.array([
[x0**3, x0**2, x0, 1],
[x1**3, x1**2, x1, 1],
[x2**3, x2**2, x2, 1],
[3 * x0**2, 2 * x0, 1, 0],
])
# Bound conditions: f(x0) = y0 f(x1) = y1 f(x2) = y2 f'(x0) = v0
d = np.array([y0, y1, y2, v])
m[i, :] = np.linalg.solve(g, d)
elif method == 4:
functions = [[] for j in range(len(x))[0:-1]
] # Initialises an empty list to store functions
for i in range(len(x))[0:-1]:
y_high = y[i + 1]
y_low = y[i]
x_start = x[i]
x_inflect = 0.5 * (x[i] + x[i + 1])
func = make_sigmoidal_curve(y_high=y[i + 1],
y_low=y[i],
x_start=x[i],
x_inflect=0.5 * (x[i] + x[i + 1]),
multiplier=4)
functions[i] = func
def curve(t):
if t <= min(
x
): # t is before the range defined by x -> takes the initial value
return y[0]
elif t >= max(
x
): # t is after the range defined by x -> takes the last value
if intervention_end is not None:
if t >= t_intervention_start:
return curve_intervention(t)
else:
return y[-1]
else:
return y[-1]
else: # t is in the range defined by x
index_low = len(x[x <= t]) - 1
func = functions[index_low]
return func(t)
return curve
elif method == 5:
"""
This method produces an approximation (or interpolation when smoothness=0.0) using cubic splines. When the
arguments 'bound_low' or 'bound_up' are not None, the approximation is constrained so that the curve does not
overcome the bounds.
A low smoothness value will provide a curve that passes close to every point but leads to more variation changes
and a greater curve energy. Set this argument equal to 0.0 to obtain an interpolation.
A high smoothness value will provide a very smooth curve but its distance to certain points may be large.
We use the following approach:
1. We create a curve (f) made of cubic spline portions that approximates/interpolates the data, without any
constraints
2. We modify the initial and final sections of the curve in order to have:
a. Perfect hits at the extreme points
b. Null gradients at the extreme points
3. If bounds are defined, we detect potential sections where they are overcome and we use the following method
to adjust the fit:
3.1. We detect the narrowest interval [x_i, x_j] which contains the overcoming section
3.2. We fit a cubic spline (g) verifying the following conditions:
a. g(x_i) = f(x_i) and g(x_j) = f(x_j)
b. g'(x_i) = f'(x_i) and g'(x_j) = f'(x_j)
3.3. If g still presents irregularities, we repeat the same process from 3.1. on the interval [x_(i-1), x_j]
or [x_i, x_(j+1)], depending on which side of the interval led to an issue
"""
# Calculate an appropriate smoothness (adjusted by smoothness)
rmserror = 0.05
s = smoothness * len(x) * (rmserror * np.fabs(y).max())**2
# Get rid of first elements when they have same y (same for last elements) as there is no need for fitting
ind_start = 0
while y[ind_start] == y[ind_start + 1]:
ind_start += 1
ind_end = len(x) - 1
while y[ind_end] == y[ind_end - 1]:
ind_end -= 1
x = x[ind_start:(ind_end + 1)]
y = y[ind_start:(ind_end + 1)]
k = min(3, len(x) - 1)
w = np.ones(len(x))
w[0] = 5.
w[-1] = 5.
f = UnivariateSpline(x, y, k=k, s=s, ext=3,
w=w) # Create a first raw approximation
# Shape the initial and final parts of the curve in order to get null gradients and to hit the external points
x0 = x[0]
x1 = x[1]
x_f = x[-1]
x_a = x[-2]
v1 = f.derivatives(x1)[1] #
v_a = f.derivatives(x_a)[1]
g = np.array([[x0**3, x0**2, x0, 1], [x1**3, x1**2, x1, 1],
[3 * x0**2, 2 * x0, 1, 0], [3 * x1**2, 2 * x1, 1, 0]])
d_init = np.array([y[0], f(x1), 0, v1])
a_init = np.linalg.solve(g, d_init)
a_init = a_init[::-1] # Reverse
h = np.array([[x_a**3, x_a**2, x_a, 1], [x_f**3, x_f**2, x_f, 1],
[3 * x_a**2, 2 * x_a, 1, 0], [3 * x_f**2, 2 * x_f, 1,
0]])
d_f = np.array([f(x_a), y[-1], v_a, 0])
a_f = np.linalg.solve(h, d_f)
a_f = a_f[::-1] # Reverse
# We have to make sure that the obtained fits do not go over/under the bounds
cut_off_dict = {}
amplitude = auto_bound * (max(y) - min(y))
if bound_low is None:
if auto_bound is not None:
bound_low = 0.5 * (max(y) + min(y)) - 0.5 * amplitude
if bound_up is None:
if auto_bound is not None:
bound_up = 0.5 * (max(y) + min(y)) + 0.5 * amplitude
if (bound_low is not None) or (bound_up is not None):
# We adjust the data so that no values go over/under the bounds
if bound_low is not None:
for i in range(len(x) - 1):
if y[i] < bound_low:
y[i] = bound_low
if bound_up is not None:
for i in range(len(x)):
if y[i] > bound_up:
y[i] = bound_up
# Check bounds
def cut_off(index, bound_low, bound_up, sign):
if sign == -1:
bound = bound_low
else:
bound = bound_up
x0 = x[index]
if index == 0:
y0 = y[0]
else:
y0 = f(x0)
# Look for the next knot at which the spline is inside of the bounds
go = 1
k = 0
while go == 1:
k += 1
if (index + k) == 0:
y_k = y[0]
elif (index + k) == len(x) - 2:
y_k = a_f[0] \
+ a_f[1] * x[index + k] + a_f[2] * x[index + k] ** 2 + a_f[3] * x[index + k] ** 3
elif (index + k) == len(x) - 1:
y_k = y[-1]
else:
y_k = f(x[index + k])
if (y_k <= bound_up) and (y_k >= bound_low):
go = 0
next_index = index + k
x1 = x[next_index]
y1 = f(x1)
if y0 == y1:
x_peak = 0.5 * (x0 + x1)
else:
if y0 == bound:
x_peak = x0
elif y1 == bound:
x_peak = x1
else:
# Weighted positioning of the contact with bound
x_peak = x0 + (abs(y0 - bound) /
(abs(y0 - bound) + abs(y1 - bound))) * (
x1 - x0)
if index == 0:
v0 = 0
else:
v0 = f.derivatives(x0)[1]
if index == (len(x) - 2):
v1 = 0
else:
v1 = f.derivatives(x1)[1]
if x0 != x_peak:
g = np.array([[x0**3, x0**2, x0, 1],
[x_peak**3, x_peak**2, x_peak, 1],
[3. * x0**2, 2. * x0, 1, 0],
[3. * x_peak**2, 2. * x_peak, 1, 0]])
d = np.array([y0, bound, v0, 0.0])
a1 = np.linalg.solve(g, d)
a1 = a1[::-1] # Reverse
if x1 != x_peak:
g = np.array([[x_peak**3, x_peak**2, x_peak, 1],
[x1**3, x1**2, x1, 1],
[3. * x_peak**2, 2. * x_peak, 1, 0],
[3. * x1**2, 2. * x1, 1, 0]])
d = np.array([bound, y1, 0.0, v1])
a2 = np.linalg.solve(g, d)
a2 = a2[::-1] # Reverse
if x0 == x_peak:
a1 = a2
if x1 == x_peak:
a2 = a1
indice_first = index
t1 = test_a(a1, x0, x_peak, x_peak, bound_low, bound_up)
if t1 == 0: # There is something wrong here
spare = get_spare_fit(index, x_peak, bound, 'left', f,
cut_off_dict, bound_low, bound_up,
a_init, a_f, x, y)
a1 = spare['a']
indice_first = index - spare['cpt']
t2 = test_a(a2, x_peak, x1, x_peak, bound_low, bound_up)
if t2 == 0: # There is something wrong here
spare = get_spare_fit(index, x_peak, bound, 'right', f,
cut_off_dict, bound_low, bound_up,
a_init, a_f, x, y)
a2 = spare['a']
next_index = index + 1 + spare['cpt']
out = {
'a1': a1,
'a2': a2,
'x_peak': x_peak,
'indice_first': indice_first,
'indice_next': next_index
}
return (out)
t = x[0]
while t < x[-1]:
ok = 1
if t == x[0]:
y_t = y[0]
elif t < x[1]:
y_t = a_init[0] + a_init[1] * t + a_init[
2] * t**2 + a_init[3] * t**3
elif t < x[-2]:
y_t = f(t)
elif t == x[-1]:
y_t = y[-1]
else:
y_t = a_f[0] + a_f[1] * t + a_f[2] * t**2 + a_f[3] * t**3
if bound_low is not None:
if y_t < bound_low:
ok = 0
sign = -1.
if bound_up is not None:
if y_t > bound_up:
ok = 0
sign = 1.
if ok == 0:
indice = len(x[x < t]) - 1
out = cut_off(indice, bound_low, bound_up, sign)
for k in range(out['indice_first'], out['indice_next']):
cut_off_dict[k] = out
t = x[out['indice_next']]
t += (x[-1] - x[0]) / 1000.
def curve(t):
t = float(t)
y_t = 0
if t <= x[0]:
y_t = y[0]
elif t > x[-1]:
if intervention_end is not None:
if t >= t_intervention_start:
y_t = curve_intervention(t)
else:
y_t = y[-1]
else:
y_t = y[-1]
elif x[0] < t < x[1]:
y_t = a_init[
0] + a_init[1] * t + a_init[2] * t**2 + a_init[3] * t**3
elif x[-2] < t < x[-1]:
y_t = a_f[0] + a_f[1] * t + a_f[2] * t**2 + a_f[3] * t**3
else:
y_t = f(t)
if x[0] < t < x[-1]:
indice = len(x[x < t]) - 1
if indice in cut_off_dict.keys():
out = cut_off_dict[indice]
if t < out['x_peak']:
a = out['a1']
else:
a = out['a2']
y_t = a[0] + a[1] * t + a[2] * t**2 + a[3] * t**3
if (bound_low is not None) and (t <= x[-1]):
y_t = max(
(y_t, bound_low)) # Security check. Normally not needed
if (bound_up is not None) and (t <= x[-1]):
y_t = min(
(y_t, bound_up)) # Security check. Normally not needed
return y_t
return curve
else:
raise Exception('method ' + method + 'does not exist.')
def curve(t):
t = t / coef
if t <= x[0]: # Constant before x[0]
return y[0]
elif t >= x[-1]: # Constant after x[0]
if intervention_end is not None:
if t >= t_intervention_start / coef:
return curve_intervention(t * coef)
else:
return y[-1]
else:
return y[-1]
else:
index = len(x[x <= t]) - 1
p = m[index, :] # Corresponding coefficients
y_t = p[0] * t**3 + p[1] * t**2 + p[2] * t + p[3]
return y_t
return curve
def Erep(E_el, E_DFT, x1, Rc): # E_el : DFTB electronic energy
Rcuts = np.linspace(x1, x2, 1e5)
for Rc in Rcuts:
spX = UnivariateSpline(X, (E_DFT(Rcuts) - E_DFT(Rc)) -
(E_el(Rcuts) - E_el(Rc)),
k=4,
s=0)
roots = spX.derivative(n=1).roots()
if abs(spX(roots[0])) < 1E-7:
liste_Rc.append(roots[0])
Rc = np.median(liste_Rc)
Rcmin = Rc * np.sqrt(3.0 / 2) / 2
X = np.linspace(Rcmin, Rc,
200) # x1 et x2 : limite of the a0 range used in DFT
fsp = UnivariateSpline(
X, (E_DFT(X) - E_DFT(Rc)) - (E_el(X) - E_el(Rc)), k=5,
s=0) # Difference DFT DFTB (Energie qui sert a calculer Erep)
plt.plot(X, E_DFT(X) - E_DFT(Rc), '--', label='DFT')
plt.plot(X, E_el(X) - E_el(Rc), label='DFTB')
plt.legend()
plt.savefig('Repulsion1.png')
plt.clf()
a = np.sqrt(3.0) / 2 # rcut*a : rayon de coupure de la fonction de paire
x = []
fx = []
def frep(n, frep, rho_0): # Erep for 1st and 2nd neighbout
return 1 / 8 * (fsp(a**n * rho_0) - 6 * frep)
for rho_0 in np.linspace(a * Rc, Rc - 1e-10, 50):
# Initialize
if a * rho_0 > Rcmin:
x.append(a * rho_0)
fx.append(1 / 8 * fsp(rho_0))
for n in np.arange(1, 10, 1):
if a**(n + 1) * rho_0 > x1:
x.append(rho_0 * a**(n + 1))
fx.append(frep(n, fx[-1], rho_0))
else:
break
plt.clf()
plt.plot(x, fx, '-o', color='C3')
plt.show()
x_EOS = np.array(sorted(x))
fx_EOS = np.array(sorted(fx, reverse=True))
x = np.array(sorted(x)) / 0.5291777249 # Bohr
fx = np.array(sorted(fx, reverse=True)) / 13.605698066 # Ry
NPoints = 120
sp1 = UnivariateSpline(x, fx, k=4, s=0)
X = np.linspace(x[0], x[-1], NPoints)
interval = (x[-1] - x[0]) / (NPoints - 1)
sp2 = UnivariateSpline(X, sp1(X), k=4, s=0) # To obtain the derivatives...
# Reconstruction EOS
for val in np.linspace(np.sqrt(3) / 2 * Rc, Rc, 20):
x_EOS = np.append(x_EOS, val)
fx_EOS = np.append(fx_EOS, 0)
spx_eos = UnivariateSpline(x_EOS, fx_EOS, k=5, s=0)
# 2 - Repulsive equation for x < x1 (exp(-a1 x + a2)+a3)
def func(x, alpha, beta, gamma):
return np.exp(-alpha * x + beta) + gamma
coeff = (sp2(X[1]) - sp2(X[0])) / (X[1] - X[0])
res, cov = curve_fit(func,
X[0:5],
sp2(X[0:5]), [1 / X[0], X[0], -X[0] / sp2(X[0])],
maxfev=1000000)
# 3 - Writing at the good format
Repulsion = ""
Repulsion = Repulsion + ("{} {}\n".format(len(X), X[-1]))
Repulsion = Repulsion + "{} {} {}\n".format(*res)
for i, vec in enumerate(X):
if i < len(X) - 1:
Repulsion = Repulsion + "{0:.8f} {1:.8f} {2:.10f} {3:.10f} {4:.10f} {5:.10f}\n".format(
vec, vec + interval,
sp2.derivatives(vec)[0],
sp2.derivatives(vec)[1],
sp2.derivatives(vec)[2] / 2,
sp2.derivatives(vec)[3] / 6)
else:
Repulsion = Repulsion + "{0:.8f} {1:.8f} {2:.10f} {3:.10f} {4:.10f} {5:.10f} {6} {7} \n".format(
vec, vec,
sp2.derivatives(vec)[0],
sp2.derivatives(vec)[1],
sp2.derivatives(vec)[2] / 2,
sp2.derivatives(vec)[3] / 6, 0, 0)
# Final verification
plt.clf()
plt.plot(X, sp2(X), 'o')
Xe = np.linspace(X[0] - 0.2, X[0] + 0.2, 10)
plt.plot(Xe, func(Xe, *res))
plt.savefig('Erep_skf.png')
plt.clf()
return (Repulsion)
k=interpol_order,
s=1e-5)
zeta_spl = UnivariateSpline(r_avg[psi_stencil], zeta, k=interpol_order, s=1e-5)
amhd = np.empty(pw, dtype=np.float)
amhd_Miller = np.empty(pw, dtype=np.float)
Vprime = np.empty(pw, dtype=np.float)
dV_dr = np.empty(pw, dtype=np.float)
V = np.empty(pw, dtype=np.float)
V_manual = np.empty(pw, dtype=np.float)
r_FS = np.empty(pw, dtype=np.float)
for i in psi_stencil:
imod = i - psi_stencil[0]
Vprime[imod] = np.abs(
np.sum(qpsi[i] * R_sym[imod]**2 / F[i]) * 4 * np.pi**2 / ntheta)
dV_dr[imod] = np.abs(np.sum(qpsi[i]*R_sym[imod] ** 2/F[i])*4*np.pi ** 2/ntheta)/ \
ravg_spl.derivatives(linpsi[i])[1]
# V[imod]=trapz(Vprime[:imod+1],linpsi[psi_stencil])
r_FS[imod] = np.average(np.sqrt((R_sym[imod] - R0[i])**2 +
(Z_sym[imod] - Z_avg[i])**2),
weights=qpsi[i] * R_sym[imod]**2 / F[i])
amhd[imod] = -qpsi[i] ** 2*R0[i]*pprime[i]*8*np.pi*1e-7/Bref_miller ** 2/ \
ravg_spl.derivatives(linpsi[i])[1]
# amhd_Miller[imod]=-2*Vprime[imod]/(2*pi)**2*(V[imod]/2/pi**2/R0[i])**0.5*4e-7*pi*pprime[i]
dq_dr_avg[imod] = q_spl.derivatives(r_avg[i])[1]
dq_dpsi[imod] = q_spl_psi.derivatives(linpsi[i])[1]
drR[imod] = R0_spl.derivatives(r_avg[i])[1]
drZ[imod] = Z0_spl.derivatives(r_avg[i])[1]
s_kappa[imod] = kappa_spl.derivatives(r_avg[i])[1] * r_avg[i] / kappa[imod]
s_delta[imod] = delta_spl.derivatives(
r_avg[i])[1] * r_avg[i] / np.sqrt(1 - delta[imod]**2)
s_zeta[imod] = zeta_spl.derivatives(r_avg[i])[1] * r_avg[i]
plt.ylabel('y')
plt.grid()
plt.legend()
#Cubic spline
print('Cubic Spline')
SP_coeffs = UnivariateSpline(t, y, s=0.0001)
plt.figure()
plt.plot(t, y, 'ok', label='Data Points')
plt.plot(T, SP_coeffs(T), label='Cubic Spline')
plt.title('Cubic Spline')
plt.xlabel('t')
plt.ylabel('y')
plt.grid()
plt.legend()
for z in range(len(t)):
dx_val[z] = SP_coeffs.derivatives(t[z])[1]
print('dx_val =', dx_val)
#smoothing spline
print('Smoothing Spline')
K = np.array([1, 2, 4, 5])
for k1 in K:
print('k =', k1)
SP_coeffs = interpolate.UnivariateSpline(t, y, k=k1, s=0.0001)
for z in range(len(t)):
dx_val[z] = SP_coeffs.derivatives(t[z])[1]
print('dx_val =', dx_val)
plt.figure()
plt.plot(t, y, 'ok', label='Data Points')
plt.plot(T, SP_coeffs(T), label='Smoothing Spline, k = %s' % k1)
plt.title('Smoothing Spline, k = %s' % k1)
plt.xlabel('t')