def test_basis_element_quadratic(self):
xx = np.linspace(-1, 4, 20)
b = BSpline.basis_element(t=[0, 1, 2, 3])
assert_allclose(b(xx),
splev(xx, (b.t, b.c, b.k)), atol=1e-14)
assert_allclose(b(xx),
B_0123(xx), atol=1e-14)
b = BSpline.basis_element(t=[0, 1, 1, 2])
xx = np.linspace(0, 2, 10)
assert_allclose(b(xx),
np.where(xx < 1, xx*xx, (2.-xx)**2), atol=1e-14)
def test_antiderivative_method(self):
b = _make_random_spline()
t, c, k = b.tck
xx = np.linspace(t[k], t[-k-1], 20)
assert_allclose(b.antiderivative().derivative()(xx),
b(xx), atol=1e-14, rtol=1e-14)
# repeat with n-D array for c
c = np.c_[c, c, c]
c = np.dstack((c, c))
b = BSpline(t, c, k)
assert_allclose(b.antiderivative().derivative()(xx),
b(xx), atol=1e-14, rtol=1e-14)
def _sum_basis_elements(x, t, c, k):
n = len(t) - (k+1)
assert n >= k+1
assert len(c) >= n
s = 0.
for i in range(n):
b = BSpline.basis_element(t[i:i+k+2], extrapolate=False)(x)
s += c[i] * np.nan_to_num(b) # zero out out-of-bounds elements
return s
def test_integral(self):
b = BSpline.basis_element([0, 1, 2]) # x for x < 1 else 2 - x
assert_allclose(b.integrate(0, 1), 0.5)
assert_allclose(b.integrate(1, 0), -0.5)
# extrapolate or zeros outside of [0, 2]; default is yes
assert_allclose(b.integrate(-1, 1), 0)
assert_allclose(b.integrate(-1, 1, extrapolate=True), 0)
assert_allclose(b.integrate(-1, 1, extrapolate=False), 0.5)
def test_integral(self):
b = BSpline.basis_element([0, 1, 2]) # x for x < 1 else 2 - x
assert_allclose(b.integrate(0, 1), 0.5)
assert_allclose(b.integrate(1, 0), -1 * 0.5)
assert_allclose(b.integrate(1, 0), -0.5)
# extrapolate or zeros outside of [0, 2]; default is yes
assert_allclose(b.integrate(-1, 1), 0)
assert_allclose(b.integrate(-1, 1, extrapolate=True), 0)
assert_allclose(b.integrate(-1, 1, extrapolate=False), 0.5)
assert_allclose(b.integrate(1, -1, extrapolate=False), -1 * 0.5)
# Test ``_fitpack._splint()``
t, c, k = b.tck
assert_allclose(b.integrate(1, -1, extrapolate=False),
_splint(t, c, k, 1, -1)[0])
# Test ``extrapolate='periodic'``.
b.extrapolate = 'periodic'
i = b.antiderivative()
period_int = i(2) - i(0)
assert_allclose(b.integrate(0, 2), period_int)
assert_allclose(b.integrate(2, 0), -1 * period_int)
assert_allclose(b.integrate(-9, -7), period_int)
assert_allclose(b.integrate(-8, -4), 2 * period_int)
assert_allclose(b.integrate(0.5, 1.5), i(1.5) - i(0.5))
assert_allclose(b.integrate(1.5, 3), i(1) - i(0) + i(2) - i(1.5))
assert_allclose(b.integrate(1.5 + 12, 3 + 12),
i(1) - i(0) + i(2) - i(1.5))
assert_allclose(b.integrate(1.5, 3 + 12),
i(1) - i(0) + i(2) - i(1.5) + 6 * period_int)
assert_allclose(b.integrate(0, -1), i(0) - i(1))
assert_allclose(b.integrate(-9, -10), i(0) - i(1))
assert_allclose(b.integrate(0, -9), i(1) - i(2) - 4 * period_int)
def torsion(
x: np.ndarray,
t: np.ndarray,
c: np.ndarray,
k: np.integer,
aux_outputs: bool = False,
) -> np.ndarray:
r"""Compute the torsion of a B-Spline.
The torsion measures the failure of a curve, `r(u)`, to be planar.
If the curvature `k` of a curve is not zero, then the torsion is defined as
.. math::
\tau = -n \cdot b',
where `n` is the principal normal vector, and `b'` the derivative w.r.t. the
arc length `s` of the binormal vector.
The torsion can also be computed as
.. math::
\tau = \lvert r'(t), r''(t), r'''(t) \rvert / \lVert r'(t) \times r''(t) \rVert^2,
where `r(u)` is the position vector as a function of time.
Arguments:
x: A `1xL` array of parameter values where to evaluate the curve.
It contains the parameter values where the torsion of the B-Spline will
be evaluated. It is required to be non-empty, one-dimensional, and
real-valued.
t: A `1xm` array representing the knots of the B-spline.
It is required to be a non-empty, non-decreasing, and one-dimensional
sequence of real-valued elements. For a B-Spline of degree `k`, at least
`2k + 1` knots are required.
c: A `dxn` array representing the coefficients/control points of the B-spline.
Given `n` real-valued, `d`-dimensional points ::math::`x_k = (x_k(1),...,x_k(d))`,
`c` is the non-empty matrix which columns are ::math::`x_1^T,...,x_N^T`. For a
B-Spline of order `k`, `n` cannot be less than `m-k-1`.
k: A non-negative integer representing the degree of the B-spline.
Returns:
torsion: A `1xL` array containing the torsion of the B-Spline evaluated at `x`
References:
.. [1] Máté Attila, The Frenet–Serret formulas.
http://www.sci.brooklyn.cuny.edu/~mate/misc/frenet_serret.pdf
"""
# convert arguments to desired type
x = np.ascontiguousarray(x)
t = np.ascontiguousarray(t)
c = np.ascontiguousarray(c)
k = operator.index(k)
if k < 0:
raise ValueError("The order of the spline must be non-negative")
check_type(t, np.ndarray)
t_dim = t.ndim
if t_dim != 1:
raise ValueError("t must be one-dimensional")
if len(t) == 0:
raise ValueError("t must be non-empty")
check_iterable_type(t, (np.integer, np.float))
if (np.diff(t) < 0).any():
raise ValueError("t must be a non-decreasing sequence")
check_type(c, np.ndarray)
c_dim = c.ndim
if c_dim > 2:
raise ValueError("c must be 2D max")
if len(c.flatten()) == 0:
raise ValueError("c must be non-empty")
if c_dim == 1:
check_iterable_type(c, (np.integer, np.float))
# expand dims so that we can cycle through a single dimension
c = np.expand_dims(c, axis=0)
if c_dim == 2:
for d in c:
check_iterable_type(d, (np.integer, np.float))
n_dim = len(c)
check_type(x, np.ndarray)
x_dim = x.ndim
if x_dim != 1:
raise ValueError("x must be one-dimensional")
if len(x) == 0:
raise ValueError("x must be non-empty")
check_iterable_type(x, (np.integer, np.float))
L = len(x)
# evaluate first, second, and third derivatives
# deriv, dderiv, ddderiv are (d, L) arrays
deriv = np.empty((n_dim, L))
dderiv = np.empty((n_dim, L))
ddderiv = np.empty((n_dim, L))
for i, dim in enumerate(c):
spl = BSpline(t, dim, k)
deriv[i, :] = spl.derivative(nu=1)(x) if k - 1 >= 0 else np.zeros(L)
dderiv[i, :] = spl.derivative(nu=2)(x) if k - 2 >= 0 else np.zeros(L)
ddderiv[i, :] = spl.derivative(nu=3)(x) if k - 3 >= 0 else np.zeros(L)
# transpose derivs
deriv = deriv.T
dderiv = dderiv.T
ddderiv = ddderiv.T
cross = np.cross(deriv, dderiv)
# Could be more efficient by only computing dot products of corresponding rows
num = np.diag((cross @ ddderiv.T))
denom = np.linalg.norm(cross, axis=1)**2
torsion = np.nan_to_num(num / denom)
if aux_outputs == True:
return torsion, deriv, dderiv, ddderiv
else:
return torsion
class SmoothingSpline(object):
""" Main cubic smoothing spline class. """
def __init__(self, x, y, lam=10., max_knots=250, order=3):
""" Initialize and fit the cubic smoothing spline, using the Bspline basis from
scipy.
x: A numpy array of univariate independent variables.
y: A numpy array of univariate response variables.
lam: smoothing parameter
max_knots: max number of knots (i.e. max number of spline basis functions)."""
## Start by storing the data
assert len(x) == len(
y
), "Independent and response variable vectors must have the same length."
self.x = x
self.y = y
self.order = 3
self.lam = lam
## Then compute the knots, refactoring to evenly
## spaced knots if the number of unique values is too
## large.
self.knots = np.sort(np.unique(self.x))
if len(self.knots) > max_knots:
self.knots = np.linspace(self.knots[0], self.knots[-1], max_knots)
## Construct the spline basis given the knots
self._aug_knots = np.hstack([
self.knots[0] * np.ones((order, )), self.knots,
self.knots[-1] * np.ones((order, ))
])
self.splines = [
BSpline(self._aug_knots, coeffs, order)
for coeffs in np.eye(len(self.knots) + order - 1)
]
## Finally, with the basis functions, we can fit the
## smoother.
self._fit()
def _fit(self):
""" Subroutine for fitting, called by init. """
## Start the fit procedure by constructing the matrix B
B = np.array([sp(self.x) for sp in self.splines]).T
## Then, construct the penalty matrix by first computing second derivatives
## on the knots and then approximating the integral of second derivative products
## with the trapezoid rule.
d2B = np.array([sp.derivative(2)(self.knots) for sp in self.splines])
weights = np.ones(self.knots.shape)
weights[1:-1] = 2.
Omega = np.dot(d2B, np.dot(np.diag(weights), d2B.T))
## Finally, invert the matrices to construct values of interest.
self.ridge_op = np.linalg.inv(np.dot(B.T, B) + self.lam * Omega)
## From there, we can compute the coefficient matrix H and the
## S matrix (i.e. the hat matrix).
H = np.dot(self.ridge_op, B.T)
self.S = np.dot(B, H)
self.edof = np.trace(self.S)
self.gamma = np.dot(H, self.y)
## Finally, construct the smoothing spline
self.smoother = BSpline(self._aug_knots, self.gamma, self.order)
## And compute the covariance matrix of the coefficients
y_hat = self.smoother(self.x)
self.rss = np.sum((self.y - y_hat)**2)
self.var = self.rss / (len(self.y) - self.edof)
self.cov = self.var * self.ridge_op
return
def __call__(self, x, cov=False):
""" Evaluate it """
## if you want the covariance matrix
if cov:
## Set up the matrix of splines evaluations and similarity transform the
## covariance in the coefficients
F = np.array([BSpline(self._aug_knots, g, self.order)(x) \
for g in np.eye(len(self.knots) + self.order - 1)]).T
covariance_matrix = np.dot(F, np.dot(self.cov, F.T))
## Evaluate the mean using the point estimate
## of the coefficients
point_estimate = np.dot(F, self.gamma)
return point_estimate, covariance_matrix
else:
return self.smoother(x)
def derivative(self, x, degree=1):
""" Evaluate the derivative of degree = degree. """
return self.smoother.derivative(degree)(x)
def correlation_time(self):
""" Use the inferred covariance matrix to compute and estimate of the correlation time
by approximating the width of correlation for a central knot with it's neighbors. """
## Select a central knot
i_mid = int(len(self.knots) / 2)
## Compute a normalized distribution
distribution = np.abs(self.cov[i_mid][1:-1])
distribution = distribution / trapz(distribution, x=self.knots)
## Compute the mean and variance
avg = self.knots[i_mid - 1]
var = trapz(distribution * (self.knots**2), x=self.knots) - avg**2
return np.sqrt(var)
# Construct the linear spline ``x if x < 1 else 2 - x`` on the base # interval :math:`[0, 2]`, and integrate it from scipy.interpolate import BSpline b = BSpline.basis_element([0, 1, 2]) b.integrate(0, 1) # array(0.5) # If the integration limits are outside of the base interval, the result # is controlled by the `extrapolate` parameter b.integrate(-1, 1) # array(0.0) b.integrate(-1, 1, extrapolate=False) # array(0.5) import matplotlib.pyplot as plt fig, ax = plt.subplots() ax.grid(True) ax.axvline(0, c='r', lw=5, alpha=0.5) # base interval ax.axvline(2, c='r', lw=5, alpha=0.5) xx = [-1, 1, 2] ax.plot(xx, b(xx)) plt.show()
def plotDLP(self, order=2, cutoff=0.05, tof=False, DBDplot=False):
if DBDplot == False:
raw_dlp = lplt.get_data(self.fname)
lowpass_dlp = lplt.butter_filter(raw_dlp, order, cutoff)
average_dlp = lplt.butter_avg(lowpass_dlp)
time, density = lplt.density(average_dlp)
ax, fig = plt.subplots()
for key in density.keys():
fig.plot(time, density[key], label=key)
fig.legend(prop={'size': 7})
if tof: # if time of flight is checked
xmaxPos = np.argmax(density[key], axis=0) / 10
yminPos = np.min(density[key])
fig.axvline(x=xmaxPos, color='r', linestyle='--')
textstr = (xmaxPos)
props = dict(boxstyle='square', facecolor='white', alpha=0.0)
fig.text(
xmaxPos,
yminPos,
textstr,
fontsize=9,
verticalalignment='center',
bbox=props)
plt.xlabel(r'Time ($\mu$s)')
plt.ylabel('$n_{e}$ ($m^{-3}$)')
plt.title('Plasma Density')
plt.minorticks_on()
plt.grid(which='major', alpha=0.5)
plt.grid(which='minor', alpha=0.2)
plt.show()
else:
[raw_I_vals, raw_bias_vals] = dlplt.get_data(self.fname)
peak_I_vals_dic = dlplt.get_peak_vals(raw_I_vals, raw_bias_vals)
avg_peak_I_vals = dlplt.peak_avg(peak_I_vals_dic, raw_bias_vals)
data = dlplt.format_data(raw_bias_vals, avg_peak_I_vals)
sectioned_data = dlplt.split_data(data)
regression_data = dlplt.calculate_linear_regressions(sectioned_data)
tol = 1E-8
sat_vals, outside_tols = dlplt.calculate_saturation_values(
regression_data, tol, 2)
V_sat = sat_vals['V sat']
I_sat = sat_vals['I sat']
electron_temp = dlplt.temperature(V_sat)
electron_number_density = dlplt.density(V_sat, I_sat)
# Warning handling
if outside_tols['sat_V_diff'] == True:
message = ('Average saturated voltage value '
+ 'was used because the difference between left '
+ 'and right saturated voltage values was outside the '
+ 'tolerance of %.2E V.' % tol)
warnings.warn(message, RuntimeWarning)
if outside_tols['sat_I_diff'] == True:
message = ('Average saturated current value '
+ 'was used because the difference between left '
+ 'and right saturated voltage values was outside the '
+ 'tolerance of %.2E uA.' % tol)
warnings.warn(message, RuntimeWarning)
ax, fig = plt.subplots()
# The section below still needs to be cleaned, but it will work for now
###############################################################################
x_ion = np.linspace(data[0,0], data[-1,0], num=50)
x_e_ret = np.linspace(data[0,0], data[-1,0], num=50)
x_e_sat = np.linspace(data[0,0], data[-1,0], num=50)
y_ion = np.zeros_like(x_ion)
y_e_ret = np.zeros_like(x_e_ret)
y_e_sat = np.zeros_like(x_e_sat)
i_sat = regression_data['i_sat']
e_ret = regression_data['e_ret']
e_sat = regression_data['e_sat']
index = 0
for val in x_ion:
y_ion[index] = i_sat['slope']*val + i_sat['intercept']
index += 1
index = 0
for val in x_e_ret:
y_e_ret[index] = e_ret['slope']*val + e_ret['intercept']
index += 1
index = 0
for val in x_e_sat:
y_e_sat[index] = e_sat['slope']*val + e_sat['intercept']
index += 1
y_ion = np.array(y_ion)
y_e_ret = np.array(y_e_ret)
y_e_sat = np.array(y_e_sat)
v_fine = np.linspace(data[0,0], data[-1,0], 300)
t, c, k = splrep(
data[:,0], data[:,1], s=0, k=3)
I_func = BSpline(t, c, k, extrapolate=False)
fig.plot(v_fine, I_func(v_fine), color='black',
linestyle='dashed', linewidth=2)
fig.scatter(
data[:,0], data[:,1], color='black', s=10*(2**2))
fig.plot(x_ion, y_ion, color='red', linewidth=2.0)
fig.plot(x_e_ret, y_e_ret, color='magenta', linewidth=2.0)
fig.plot(x_e_sat, y_e_sat, color='green', linewidth=2.0)
# Construct linear regression equations
if i_sat['intercept'] < 0:
i_sat_intercept_negative = True
i_sat_intercept_zero = False
i_sat['intercept'] = np.absolute(i_sat['intercept'])
else:
i_sat_intercept_negative = False
if round(i_sat['intercept'], 4) == 0.0000:
i_sat_intercept_zero = True
else:
i_sat_intercept_zero = False
if e_ret['intercept'] < 0:
e_ret_intercept_negative = True
e_ret['intercept'] = np.absolute(e_ret['intercept'])
else:
e_ret_intercept_negative = False
if round(e_ret['intercept'], 4) == 0.0000:
e_ret_intercept_zero = True
else:
e_ret_intercept_zero = False
if e_sat['intercept'] < 0:
e_sat_intercept_negative = True
e_sat['intercept'] = np.absolute(e_sat['intercept'])
else:
e_sat_intercept_negative = False
if round(e_sat['intercept'], 4) == 0.0000:
e_sat_intercept_zero = True
else:
e_sat_intercept_zero = False
s_ion_str = str("%.4f" % round(i_sat['slope'], 4))
inter_ion_str = str("%.4f" % round(i_sat['intercept'], 4))
s_e_ret_str = str("%.4f" % round(e_ret['slope'], 4))
inter_e_ret_str = str("%.4f" % round(e_ret['intercept'], 4))
s_e_sat_str = str("%.4f" % round(e_sat['slope'], 4))
inter_e_sat_str = str("%.4f" % round(e_sat['intercept'], 4))
str_ion = r'$I_{ion \,\,\, sat} = $' + s_ion_str
if i_sat_intercept_negative == False:
if i_sat_intercept_zero == False:
str_ion += r'$\cdot V +$' + inter_ion_str
else:
str_ion += r'$\cdot V$'
else:
str_ion += r'$\cdot V -$' + inter_ion_str
str_e_ret = r'$I_{e \,\,\, ret} = $' + s_e_ret_str
if e_ret_intercept_negative == False:
if e_ret_intercept_zero == False:
str_e_ret += r'$\cdot V +$' + inter_e_ret_str
else:
str_e_ret += r'$\cdot V$'
else:
str_e_ret += r'$\cdot V -$' + inter_e_ret_str
str_e_sat = r'$I_{e \,\,\, sat} = $' + s_e_sat_str
if e_sat_intercept_negative == False:
if e_sat_intercept_zero == False:
str_e_sat += r'$\cdot V +$' + inter_e_sat_str
else:
str_e_sat += r'$\cdot V$'
else:
str_e_sat += r'$\cdot V -$' + inter_e_sat_str
# Construct electron temp and number density output
T_str = (r'$T_e$ $\approx$ '
+ str('%.2f' % round(electron_temp, 2)) + ' eV')
n_e_str = (r'$n_e$ $\approx$ '
+ '%.2E' % Decimal(str(electron_number_density))
+ r' $\mathrm{m}^{-3}$')
# Construct legend
h = []
plot_labels = ['Measured Data',
'Ion Saturation Regression',
'Electron Retarding Regression',
'Electron Saturation Regression']
h.append(mpatches.Patch(color='black', label=plot_labels[0]))
h.append(mpatches.Patch(color='red', label=plot_labels[1]))
h.append(mpatches.Patch(color='magenta', label=plot_labels[2]))
h.append(mpatches.Patch(color='green', label=plot_labels[3]))
fig.legend(loc=2, borderaxespad=0, handles=h, prop={'size': 18})
#############################################################################
ax = plt.gca()
# fig.text(0.55, 0.30, str_ion, transform=ax.transAxes)
# fig.text(0.55, 0.25, str_e_ret, transform=ax.transAxes)
# fig.text(0.55, 0.20, str_e_sat, transform=ax.transAxes)
# fig.text(0.55, 0.10, T_str, transform=ax.transAxes)
# fig.text(0.55, 0.5, n_e_str, transform=ax.transAxes)
txt = (str_ion + "\n" + str_e_ret + "\n" + str_e_sat
+ "\n" + T_str + "\n" + n_e_str)
props = dict(boxstyle='round', facecolor='white', alpha=0.5)
fig.text(0.55, 0.25, txt, size=18,
verticalalignment="top", horizontalalignment="left",
multialignment="left", bbox=props, transform=ax.transAxes)
plt.rcParams.update({'font.size': 22})
for item in ([ax.title, ax.xaxis.label, ax.yaxis.label] +
ax.get_xticklabels() + ax.get_yticklabels()):
item.set_fontsize(20)
plt.xlabel('Voltage (V)')
plt.ylabel(r'Peak Current ($\mu$A)')
plt.title('Bias Voltage vs Peak Current')
plt.minorticks_on()
plt.grid(which='major', alpha=0.5)
plt.grid(which='minor', alpha=0.2)
plt.show()
class Fitting:
"""
Fitting of the curve in the initial days to obtain the estimated values
for day day march, 16th, 2020, of the pandemic.
p: (tau, sigma, rho, delta, gamma1, gamma2).
tyme_varying: definitions about beta and mu bspline (knots, number of parameters and order)
hmax: max value Runge Kutta integration method.
"""
def __init__(
self,
p,
time_varying,
initial_day='2020-03-16',
final_day='2020-07-15',
hmax=0.15,
init_cond={
'x0': [0.8, 0.3, 0.00001, 0.00001, 0.00001],
'bounds': [(0, 1), (0, 1), (0, 0.0001), (0, 0.0001), (0, 0.0001)]
}):
# parameters pre-determined
self.p = np.array(p, dtype=np.float64)
self.hmax = hmax
self.init_cond = init_cond
self.initial_day = initial_day
self.final_day = final_day
# Reading data
ROOT_DIR = os.path.dirname(os.path.abspath(__file__))
filename = os.path.join(ROOT_DIR, "../data/covid_data_organized.csv")
df = pd.read_csv(filename, index_col=0)
self.T = df['confirmed'].loc[initial_day:final_day].to_numpy()
self.D = df['deaths'].loc[initial_day:final_day].to_numpy()
self.tf = len(self.T)
# time-varying hiperparameters
self.sbeta = time_varying['beta']['coefficients']
self.order_beta = time_varying['beta']['bspline_order']
self.smu = time_varying['mu']['coefficients']
self.order_mu = time_varying['mu']['bspline_order']
self.knots_beta = np.linspace(0, self.tf,
self.sbeta + self.order_beta + 1)
self.knots_mu = np.linspace(0, self.tf, self.smu + self.order_mu + 1)
# define the time-varying parameters
self.beta = BSpline(self.knots_beta, np.zeros(self.sbeta),
self.order_beta)
self.mu = BSpline(self.knots_mu, np.zeros(self.smu), self.order_mu)
# Calculate initial conditions
print('Model SEIAQR for Covid-19')
print('-------------------------')
print('Estimating initial Conditions...')
self.initial_conditions()
print('Initiation done!')
def derivative(self, x, t, alpha, beta_, mu_, tau, sigma, rho, delta,
gamma1, gamma2):
"""
System of derivatives simplified.
"""
beta = max(beta_(t), 0)
mu = max(mu_(t), 0)
dx = np.zeros(shape=(len(x), ))
dx[4] = -beta * x[4] * (x[1] + x[2])
dx[0] = -dx[4] - (rho * delta + tau) * x[0]
dx[1] = tau * x[0] - (sigma + rho) * x[1]
dx[2] = sigma * alpha * x[1] - (gamma1 + rho) * x[2]
dx[5] = gamma1 * x[2] + gamma2 * x[3]
dx[6] = mu * x[3]
dx[7] = sigma * (1 - alpha) * x[1] + rho * (delta * x[0] + x[1] + x[2])
dx[3] = dx[7] - gamma2 * x[3] - dx[6]
return dx
def integrate(self, theta, p, time=[]):
"""
Integrate the system given a tuple of parameters.
p is the parameters estimated by the literature. time is a list always
with 0 that indicates where to integrate.
"""
if len(time) == 0:
time = range(self.tf)
self.beta = self.beta.construct_fast(self.knots_beta,
theta[1:1 + self.sbeta],
self.order_beta)
self.mu = self.mu.construct_fast(self.knots_mu, theta[-self.smu:],
self.order_mu)
self.states = odeint(func=self.derivative,
y0=self.y0,
t=time,
args=(theta[0], self.beta, self.mu, *p),
hmax=self.hmax)
return self.states
def rt_calculation(self, theta):
"""
Calculate the reproduction number based on the model.
"""
S = self.states[:, 4]
self.repro_number = np.zeros(shape=(2, self.tf))
beta_ = self.beta.construct_fast(self.knots_beta,
theta[1:1 + self.sbeta],
self.order_beta)
mu_ = self.mu.construct_fast(self.knots_mu, theta[-self.smu:],
self.order_mu)
alpha = theta[0]
tau, sigma, rho, delta, gamma1, _ = self.p
for t in range(self.tf):
beta = max(beta_(t), 0)
mu = max(mu_(t), 0)
varphi = np.array([beta * tau, beta * tau * S[t]
]) # difference between R0 and Rt
varphi /= ((rho * delta + tau) * (sigma + rho))
r0_rt = 1 / 2 * (varphi + np.sqrt(varphi**2 + varphi *
(4 * sigma * alpha) /
(rho + gamma1)))
self.repro_number[:, t] = r0_rt
def initial_conditions(self):
"""
Estimate Initial Conditions
"""
parameters = self.p[[0, 1, 4, 5]]
model = FittingInitial(parameters, self.initial_day, self.hmax)
E0, I0, A0, _, Q0, R0 = model.get_initial_values(
self.init_cond['x0'], self.init_cond['bounds'])
self.initial_phase = model.y
T0 = self.T[0]
D0 = self.D[0]
S0 = 1 - E0 - I0 - A0 - Q0 - R0
self.y0 = [E0, I0, A0, Q0 - D0, S0, R0, D0, T0]
def objective(self, theta, psi):
# theta = (alpha, beta_1, ..., beta_s, mu_1, ..., mu_r)
integrate = self.integrate(theta, self.p)
obj1 = (self.T - integrate[:, 7]) @ self.weights @ (self.T -
integrate[:, 7])
obj2 = (self.D - integrate[:, 6]) @ self.weights @ (self.D -
integrate[:, 6])
obj = 100 * (obj1 + psi * obj2)
return obj
def fit(self, psi, x0, bounds, algorithm='L-BFGS-B'):
"""
Fits the model to the data and recover the estimated parameters.
"""
self.weights = np.array([[min(i, j) + 15 for i in range(self.tf)]
for j in range(self.tf)])
self.weights = np.linalg.inv(self.weights)
print('Starting estimation!')
t0 = time.time()
res = minimize(fun=self.objective,
x0=x0,
method=algorithm,
bounds=bounds,
args=(psi, ))
self.counter = time.time() - t0
print('Estimation finished. It took {} seconds'.format(self.counter))
curve = self.integrate(res.x, self.p)
# Rt calculation
self.rt_calculation(res.x)
# Store important values
self.obj = res.fun
self.res = res
self.theta = res.x
self.psi = psi
self.x0 = x0
self.bounds = bounds
self.algorithm = algorithm
n = self.tf
K = len(self.theta)
# Estimate variances
self.sigma2_1 = (self.T - curve[:, 7]) @ self.weights @ (
self.T - curve[:, 7]) / (n - K)
self.sigma2_2 = (self.D - curve[:, 6]) @ self.weights @ (
self.D - curve[:, 6]) / (n - K)
# Information Criterion
common = n * np.log(self.obj / n)
self.aic = common + 2 * K
self.bic = common + np.log(n) * K
self.aicc = common + 2 * K * n / (n - K - 1)
return res.x
def check_residuals(self):
"""
Simple residual analysis for the fitting. It must be called after the
function fit.
"""
diary_curves = self.integrate(self.theta, self.p)
T = diary_curves[:, 7]
D = diary_curves[:, 6]
errorT = np.diff(self.T - T)
errorD = np.diff(self.D - D)
return errorT, errorD
def correlation_matrix(self):
"""
Calculate the correlation matrix with an estimated parameter. It must be called after the
function fit.
"""
def f(parameters, time, curve):
theta = parameters[0:len(self.theta)]
#p = parameters[len(self.theta):]
return self.integrate(theta, self.p, [0, time])[1, curve]
K = len(self.theta) #+ len(self.p)
J1 = np.zeros((self.tf, K))
J2 = np.zeros((self.tf, K))
parameters = self.theta #np.concatenate([self.theta, self.p])
for i in range(self.tf):
J1[i, :] = approx_fprime(parameters, f,
np.ones_like(parameters) * 1e-5, i, 7)
J2[i, :] = approx_fprime(parameters, f,
np.ones_like(parameters) * 1e-5, i, 6)
# Fisher Information matrix
FIM = J1.transpose() @ self.weights @ J1 / self.sigma2_1 + J2.transpose(
) @ self.weights @ J2 / self.sigma2_2
# Covariance matrix
C = np.linalg.inv(FIM)
# Correlation matrix
R = [[C[i, j] / np.sqrt(C[i, i] * C[j, j]) for i in range(K)]
for j in range(K)]
return np.array(R)
def _get_exp(self, pathname):
with open(pathname, 'r') as f:
line = f.readline()
while line != '':
lineold = line
line = f.readline()
exp = lineold[:lineold.find(';')]
exp = 1 if exp == 'exp' else int(exp) + 1
return exp
def save_experiment(self, objective_function):
"""
Save information about the experiment.
objective_function: name given to compare, like quadratic and divided.
"""
pathname = '../experiments/' + objective_function + '.csv'
if not os.path.exists(pathname):
with open(pathname, 'w') as f:
f.write(
'exp;tau;sigma;rho;delta;gamma1;gamma2;sbeta;order_beta;smu;order_mu;'
)
f.write('initial_day;final_day;hmax;psi;x0;bounds;algorithm;')
f.write('E0;I0;A0;Q0;R0;D0;T0;alpha;beta;mu;obj;time')
f.write('\n')
else:
with open(pathname, 'a') as f:
exp = self._get_exp(pathname)
tau, sigma, rho, delta, gamma1, gamma2 = self.p
info = [
exp, tau, sigma, rho, delta, gamma1, gamma2, self.sbeta,
self.order_beta, self.smu, self.order_mu
]
info2 = [self.initial_day, self.final_day, self.hmax, self.psi]
f.write(';'.join(map(str, info)))
f.write(';')
f.write(';'.join(map(str, info2)))
f.write(';')
f.write(str(self.x0))
f.write(';')
f.write(str(self.bounds))
f.write(';')
f.write(self.algorithm + ';')
f.write(';'.join(map(str, self.y0)))
f.write(str(self.theta[0]) + ';')
f.write(str(self.theta[1:1 + self.sbeta]))
f.write(';')
f.write(str(self.theta[-self.smu:]))
f.write(';')
f.write(str(self.obj) + ';')
f.write(str(self.counter))
f.write('\n')
# if __name__ == '__main__':
# p = [0.3125, 0.5, 2e-5, 1, 1/9.5, 1/18]
# beta = {'sbeta': 4, 'bspline_order': 3}
# model = Fitting(p, beta)
# psi = 0
# bounds = [(0.7,0.95), (0.05,0.3), (0.05,0.3), (0.05,0.3), (0.05,0.3), (0, 0.2)] # bound the parameters
# x0 = [0.9, 0.1, 0.1, 0.1, 0.1, 0.12/14] # initial guess
# theta = model.fit(psi, x0, bounds)
#take log of V and S
v = [math.log(i[0]) for i in glmv]
s = [math.log(i[1]) for i in glmv]
#reverse order for Bspline()
v.reverse()
s.reverse()
#convert to array for splrep
x = np.array(s)
y = np.array(v)
#find vector knots for Bspline
t, c, k = interpolate.splrep(x=x, y=y, s= 0, k=4)
spl = BSpline(t, c, k)
#plot the Bspline
print('''\
t: {}
c: {}
k: {}
'''.format(t, c, k))
N = 100
xmin, xmax = x.min(), x.max()
xx = np.linspace(xmin, xmax, N)
spline = interpolate.BSpline(t, c, k, extrapolate=False)
plt.plot(x, y, 'bo', label='Original points')
plt.plot(xx, spline(xx), 'r', label='BSpline')
plt.grid()
def interpolate(x, y, n=3):
'''return get order n spline interpolation function fit to array x and array y.'''
return BSpline(x, y, n, extrapolate=True)
# x=[i for i in range(1,len(score)+1)]
#
# # fig = plt.figure()
# #设置X轴标签
# # plt.xlabel('X')
# #设置Y轴标签
# plt.ylabel('Y')
# plt.scatter(x,score,c='orange',marker = 'o',alpha=0.6,label='theta = 0.3')
# #设置图标
# plt.legend('theta')
# #显示所画的图
# # plt.show()
from datetime import datetime
import matplotlib.dates as mdates
import matplotlib.pyplot as plt
# 生成横纵坐标信息
xs = [datetime.strptime(d, '%Y/%m').date() for d in times]
plt.gca().xaxis.set_major_formatter(mdates.DateFormatter('%Y/%m'))
# plt.gca().xaxis.set_major_locator(mdates.DayLocator())
xnew = np.linspace(xs.min(), xs.max(), 300) # 300 represents number of points to make between T.min and T.max
p3 = BSpline(xs, score3, xnew)
plt.plot(xs, p3,label='0.3')
plt.plot(xs, score5)
plt.plot(xs, score7)
plt.gcf().autofmt_xdate() # 自动旋转日期标记
plt.show()
def calc_gamma(mp, shotdat, imp0, spdeg, knots):
"""
Calculate correction value "gamma" in the observation eqs.
Parameters
----------
mp : ndarray
complete model parameter vector.
shotdat : DataFrame
GNSS-A shot dataset.
imp0 : ndarray (len=5)
Indices where the type of model parameters change.
p : int
spline degree (=3).
knots : list of ndarray (len=5)
B-spline knots for each component in "gamma".
Returns
-------
gamma : ndarray
Values of "gamma". Note that scale facter is not applied.
a : 2-d list of ndarray
[a0[<alpha>], a1[<alpha>]] :: a[<alpha>] at transmit/received time.
<alpha> is corresponding to <0>, <1E>, <1N>, <2E>, <2N>.
"""
a0 = []
a1 = []
for k, kn in enumerate(knots):
if len(kn) == 0:
a0.append(0.)
a1.append(0.)
continue
ct = mp[imp0[k]:imp0[k + 1]]
bs = BSpline(kn, ct, spdeg, extrapolate=False)
a0.append(bs(shotdat.ST.values))
a1.append(bs(shotdat.RT.values))
ls = 1000. # m/s/m to m/s/km order for gradient
de0 = shotdat.de0.values
de1 = shotdat.de1.values
dn0 = shotdat.dn0.values
dn1 = shotdat.dn1.values
mte = shotdat.mtde.values
mtn = shotdat.mtdn.values
gamma0_0 = a0[0]
gamma0_1 = (a0[1] * de0 + a0[2] * dn0) / ls
gamma0_2 = (a0[3] * mte + a0[4] * mtn) / ls
gamma1_0 = a1[0]
gamma1_1 = (a1[1] * de1 + a1[2] * dn1) / ls
gamma1_2 = (a1[3] * mte + a1[4] * mtn) / ls
gamma0 = gamma0_0 + gamma0_1 + gamma0_2
gamma1 = gamma1_0 + gamma1_1 + gamma1_2
gamma = (gamma0 + gamma1) / 2.
a = [a0, a1]
return gamma, a
y=y[inz] # iframe1 = np.where(np.diff(y)>0)[0][0] # iframe2 = iframe+iframe1-2 # x, y = data[iframe2:, 0], data[iframe2:, 1] iframe order = 1 dx = 0.001 # dx = np.diff(dx).min()/2 x0, xf = x.min(), x.max() xx = np.arange(x0, xf, dx) # scipy.interpolate.interp1d(x, y, kind='linear', axis=-1, copy=True, bounds_error=None, fill_value=nan, assume_sorted=False) interp = BSpline(x, y, order) yy = interp(xx) # dy = interp(xx,1) # ddy=interp(xx,2) """ iymax, iymin = np.where(dy==dy.max())[0], np.where(dy==dy.min())[0] plt.figure() plt.plot(xx, np.sign(dy)) plt.show() """ inz = yy.nonzero() x0=xx[inz] y0=yy[inz]
def emisFcn(X, tck):
sp = BSpline(tck[0], tck[1], tck[2])
return np.exp(-np.abs(sp(X)))
args=(10, knot_vector1),
order=3)
tmp12 = derivative(func=Ni3,
x0=u1,
dx=h,
n=2,
args=(11, knot_vector1),
order=3)
tmp13 = derivative(func=Ni3,
x0=u1,
dx=h,
n=2,
args=(12, knot_vector1),
order=3)
matrix[-1, -3] = tmp11
matrix[-1, -2] = tmp12
matrix[-1, -1] = tmp13
D = np.array(points)
D = np.concatenate([[(0, 0)], D, [(0, 0)]], axis=0)
P = np.linalg.solve(a=matrix, b=D)
P = np.round(a=P, decimals=8)
print(P)
spl = BSpline(t=knot_vector1, c=P, k=k)
fig, ax = plt.subplots()
xx = np.linspace(1.5, 4.5, 50)
ax.plot(xx, spl(xx), 'b-', lw=4, alpha=0.7, label='BSpline')
ax.grid(True)
ax.legend(loc='best')
plt.show()
a = 1
###########################################
import matplotlib.pyplot as plt
from scipy.interpolate import splprep, BSpline
if __name__ == '__main__':
# Initialize the spline
c = np.array([100, 100])
R = 50
n = 10
phi = np.linspace(0, 2 * np.pi, n)
x_init = c[0] + R * np.cos(phi)
y_init = c[1] + R * np.sin(phi)
x_init[n - 1] = x_init[0]
y_init[n - 1] = y_init[0]
dat = np.array([x_init, y_init])
tck, _ = splprep(dat, s=0, per=0, k=3)
knots = tck[0]
ctrl_pts = np.transpose(tck[1])
deg = tck[2]
spl = BSpline(knots, ctrl_pts, deg)
fig, ax = plt.subplots()
# Compute the value of each basis function
for knot_ind in np.arange(len(knots)):
knots_basis = spl.t[knot_ind:knot_ind + deg + 2]
b = spl.basis_element(knots_basis, extrapolate=False)
x = np.linspace(knots_basis[0], knots_basis[-1], 50)
ax.plot(x, b(x), lw=3)
ax.plot([knots_basis[0]] * 2, [0, 1], 'k--')
plt.xlabel('Noeud')
plt.title('Valeur de fonction de base en fonction de u')
plt.show()
dx = dx[:nx]
# x = np.cumsum(dx)
print(nx, x.shape)
# x0=x[0]
# ymax=y.max()
# y /=ymax
# x-=x[0]
# inz=y.nonzero()[0]
# x=x[inz]
# y=y[inz]
# y /=y.max()
# x-=x[0]
# plt.plot(xx,yy);plt.show()
interp = BSpline(x, y, splorder)
nx = x.shape[0]
icycle = y.shape[0] // npeaks
icycle
icyclef = y.shape[0] / npeaks
print(icycle, '\n', icyclef)
nx2 = nx * dnx
dx2 = dx.min() / dnx
# nx3 = int(nx*icycle / np.gcd(nx, icycle))
# print(nx2,nx3)
# nx2= nx3
x[0]
xx = np.linspace(x[0], dx2 * nx2, nx2, endpoint=True)
print('xx\n', xx)
yy = interp(xx)
def roc_paint(roc_path = '.', roc_files = None, filter = True, output_file = None):
if roc_path is not None and roc_files is not None:
for rf in range(len(roc_files)):
roc_files[rf] = roc_path + '/' + roc_files[rf]
elif roc_path is not None:
roc_files = glob.glob(roc_path+"/*.npy")
lines = []
names = []
for rf in range(len(roc_files)):
roc_file = roc_files[rf]
name = os.path.splitext(os.path.basename(roc_file))[0]
roc = np.load(roc_file)
'''
start = len(roc[0][roc[1]==roc[1][0]])
TPs = [roc[0][0], roc[0][start-1]]
FPs = [roc[1][0], roc[1][start-1]]
for r in range(start, roc.shape[1]-1):
if roc[0][r] != TPs[-1] and roc[1][r] != FPs[-1]:
TPs.append(roc[0][r])
FPs.append(roc[1][r])
TPs.append(roc[0][-1])
FPs.append(roc[1][-1])
xrange = [i for i in range(len(FPs))]
'''
filtering = filter
while filtering:
filtering = False
r = 1
while r < roc.shape[1]-1:
if roc[1][r]!=roc[1][r-1] and (roc[0][r]-roc[0][r-1])/(roc[1][r]-roc[1][r-1])<(roc[0][r+1]-roc[0][r])/(roc[1][r+1]-roc[1][r]):
roc = np.delete(roc, r, axis=1)
filtering = True
else:
r += 1
xstart = len(roc[0][roc[1] == roc[1][0]]) - 1
xnew1 = np.linspace(roc[1][0], roc[1][-1], 80)
xnew2 = np.linspace(roc[1][0], roc[1][-1], 20)
ynew = np.linspace(roc[0][0], roc[0][-1], 40)
#ysmooth = interp1d(roc[1][xstart:], roc[0][xstart:], 'cubic')(xnew)
ysmooth1 = interp1d(roc[1][xstart:], roc[0][xstart:], 'linear')(xnew1)
ysmooth2 = BSpline(xnew1, ysmooth1, 4)(xnew2)
xsmooth = interp1d(roc[0], roc[1], 'linear')(ynew)
xfusion = xsmooth.copy()
yfusion = ynew.copy()
xsidx = 0
xi = 0
#while xi<len(xfusion) and xsidx<len(xnew):
# if xfusion[xi]>=xnew[xsidx] and yfusion[xi]>=ysmooth[xsidx]:
# xfusion = np.insert(xfusion, xi, xnew[xsidx])
# yfusion = np.insert(yfusion, xi, ysmooth[xsidx])
# xsidx += 1
# xi += 1
#xfusion = np.concatenate((xfusion, xnew[xsidx:]), axis=0)
#yfusion = np.concatenate((yfusion, ysmooth[xsidx:]), axis=0)
line, =plt.plot(roc[1], roc[0], alpha=0.8)
#line, = plt.plot(xfusion, yfusion)
#line, = plt.plot(xnew1, ysmooth1)
#line, = plt.plot(xnew2, ysmooth2)
#plt.plot(ynew, xsmooth)
lines.append(line)
names.append(name)
plt.legend(lines, names, loc='lower right')
plt.xlabel("False Positive Rate")
plt.ylabel("True Positive Rate")
plt.grid(False)
if output_file is None:
plt.show()
else:
plt.savefig(output_file, format="pdf")
plt.close()
def actualise_splines(self):
"""
:action: Update the splines re-creating them from the new coefficients
"""
for i in range(self.attitude_splines.shape[0]):
self.attitude_splines[i] = BSpline(self.att_knots, self.att_coeffs[i], k=self.k)
def fit(self, X, y=None):
"""Compute knot positions of splines.
Parameters
----------
X : array-like of shape (n_samples, n_features)
The data.
y : None
Ignored.
Returns
-------
self : object
Fitted transformer.
"""
X = self._validate_data(
X,
reset=True,
accept_sparse=False,
ensure_min_samples=2,
ensure_2d=True,
)
n_samples, n_features = X.shape
if not (isinstance(self.degree, numbers.Integral)
and self.degree >= 0):
raise ValueError(
f"degree must be a non-negative integer, got {self.degree}.")
if isinstance(self.knots, str) and self.knots in [
"uniform",
"quantile",
]:
if not (isinstance(self.n_knots, numbers.Integral)
and self.n_knots >= 2):
raise ValueError(
f"n_knots must be a positive integer >= 2, got: {self.n_knots}"
)
base_knots = self._get_base_knot_positions(X,
n_knots=self.n_knots,
knots=self.knots)
else:
base_knots = check_array(self.knots, dtype=np.float64)
if base_knots.shape[0] < 2:
raise ValueError(
"Number of knots, knots.shape[0], must be >= 2.")
elif base_knots.shape[1] != n_features:
raise ValueError("knots.shape[1] == n_features is violated.")
elif not np.all(np.diff(base_knots, axis=0) > 0):
raise ValueError("knots must be sorted without duplicates.")
if self.extrapolation not in (
"error",
"constant",
"linear",
"continue",
"periodic",
):
raise ValueError("extrapolation must be one of 'error', "
"'constant', 'linear', 'continue' or 'periodic'.")
if not isinstance(self.include_bias, (bool, np.bool_)):
raise ValueError("include_bias must be bool.")
# number of knots for base interval
n_knots = base_knots.shape[0]
if self.extrapolation == "periodic" and n_knots <= self.degree:
raise ValueError(
"Periodic splines require degree < n_knots. Got n_knots="
f"{n_knots} and degree={self.degree}.")
# number of splines basis functions
if self.extrapolation != "periodic":
n_splines = n_knots + self.degree - 1
else:
# periodic splines have self.degree less degrees of freedom
n_splines = n_knots - 1
degree = self.degree
n_out = n_features * n_splines
# We have to add degree number of knots below, and degree number knots
# above the base knots in order to make the spline basis complete.
if self.extrapolation == "periodic":
# For periodic splines the spacing of the first / last degree knots
# needs to be a continuation of the spacing of the last / first
# base knots.
period = base_knots[-1] - base_knots[0]
knots = np.r_[base_knots[-(degree + 1):-1] - period, base_knots,
base_knots[1:(degree + 1)] + period, ]
else:
# Eilers & Marx in "Flexible smoothing with B-splines and
# penalties" https://doi.org/10.1214/ss/1038425655 advice
# against repeating first and last knot several times, which
# would have inferior behaviour at boundaries if combined with
# a penalty (hence P-Spline). We follow this advice even if our
# splines are unpenalized. Meaning we do not:
# knots = np.r_[
# np.tile(base_knots.min(axis=0), reps=[degree, 1]),
# base_knots,
# np.tile(base_knots.max(axis=0), reps=[degree, 1])
# ]
# Instead, we reuse the distance of the 2 fist/last knots.
dist_min = base_knots[1] - base_knots[0]
dist_max = base_knots[-1] - base_knots[-2]
knots = np.r_[linspace(
base_knots[0] - degree * dist_min,
base_knots[0] - dist_min,
num=degree,
), base_knots,
linspace(
base_knots[-1] + dist_max,
base_knots[-1] + degree * dist_max,
num=degree,
), ]
# With a diagonal coefficient matrix, we get back the spline basis
# elements, i.e. the design matrix of the spline.
# Note, BSpline appreciates C-contiguous float64 arrays as c=coef.
coef = np.eye(n_splines, dtype=np.float64)
if self.extrapolation == "periodic":
coef = np.concatenate((coef, coef[:degree, :]))
extrapolate = self.extrapolation in ["periodic", "continue"]
bsplines = [
BSpline.construct_fast(knots[:, i],
coef,
self.degree,
extrapolate=extrapolate)
for i in range(n_features)
]
self.bsplines_ = bsplines
self.n_features_out_ = n_out - n_features * (1 - self.include_bias)
return self
def __new__(cls, x, y, order=None, s=None, w=None, bbox=[None]*2, k=3,
ext=0, check_finite=False, outlier_func=sigma_clip,
niter=3, grow=0, debug=False, **outlier_kwargs):
# Decide what sort of spline object we're making
spline_kwargs = {'bbox': bbox, 'k': k, 'ext': ext,
'check_finite': check_finite}
if order is None:
cls_ = UnivariateSpline
spline_args = ()
spline_kwargs['s'] = s
elif s is None:
cls_ = LSQUnivariateSpline
else:
raise ValueError("Both t and s have been specified")
# Both spline classes require sorted x, so do that here. We also
# require unique x values, so we're going to deal with duplicates by
# making duplicated values slightly larger. But we have to do this
# iteratively in case of a basket-case scenario like (1, 1, 1, 1+eps, 2)
# which would become (1, 1+eps, 1+2*eps, 1+eps, 2), which still has
# duplicates and isn't sorted!
# I can't think of any better way to cope with this, other than write
# least-squares spline-fitting code that handles duplicates from scratch
epsf = np.finfo(float).eps
orig_mask = np.zeros(y.shape, dtype=bool)
if isinstance(y, np.ma.masked_array):
if y.mask is not np.ma.nomask:
orig_mask = y.mask.astype(bool)
y = y.data
if w is not None:
orig_mask |= (w == 0)
if debug:
print(y)
print(orig_mask)
iter = 0
full_mask = orig_mask # Will include pixels masked because of "grow"
while iter < niter+1:
last_mask = full_mask
x_to_fit = x.astype(float)
if order is not None:
# Determine actual order to apply based on fraction of unmasked
# pixels, and unmask everything if there are too few good pixels
this_order = int(order * (1 - np.sum(full_mask) / len(full_mask)) + 0.5)
if this_order == 0:
full_mask = np.zeros(x.shape, dtype=bool)
if w is not None and not all(w == 0):
full_mask |= (w == 0)
this_order = int(order * (1 - np.sum(full_mask) / len(full_mask)) + 0.5)
if debug:
print("FULL MASK", full_mask)
xgood = x_to_fit[~full_mask]
while True:
xunique, indices = np.unique(xgood, return_index=True)
if len(indices) == len(xgood):
# All unique x values so continue
break
if order is None:
raise ValueError("Must specify spline order when there are "
"duplicate x values")
for i in range(len(xgood)):
if not (last_mask[i] or i in indices):
xgood[i] *= (1.0 + epsf)
# Space knots equally based on density of unique x values
if order is not None:
knots = [xunique[int(xx+0.5)]
for xx in np.linspace(0, len(xunique)-1, this_order+1)[1:-1]]
spline_args = (knots,)
if debug:
print ("KNOTS", knots)
sort_indices = np.argsort(xgood)
# Create appropriate spline object using current mask
try:
spline = cls_(xgood[sort_indices], y[~full_mask][sort_indices],
*spline_args, w=None if w is None else w[~full_mask][sort_indices],
**spline_kwargs)
except ValueError as e:
if this_order == 0:
avg_y = np.average(y[~full_mask],
weights=None if w is None else w[~full_mask])
spline = lambda xx: avg_y
else:
raise e
spline_y = spline(x)
#masked_residuals = outlier_func(spline_y - masked_y, **outlier_kwargs)
#mask = masked_residuals.mask
# When sigma-clipping, only remove the originally-masked points.
# Note that this differs from the astropy.modeling code because
# the sigma-clipping and spline-fitting are done independently here.
d, mask, v = NDStacker.sigclip(spline_y-y, mask=orig_mask, variance=None,
**outlier_kwargs)
if grow > 0:
maskarray = np.zeros((grow * 2 + 1, len(y)), dtype=bool)
for i in range(-grow, grow + 1):
mx1 = max(i, 0)
mx2 = min(len(y), len(y) + i)
maskarray[grow + i, mx1:mx2] = mask[:mx2 - mx1]
grow_mask = np.logical_or.reduce(maskarray, axis=0)
full_mask = np.logical_or(mask, grow_mask)
else:
full_mask = mask.astype(bool)
# Check if the mask is unchanged
if not np.logical_or.reduce(last_mask ^ full_mask):
break
iter += 1
# Create a standard BSpline object
try:
bspline = BSpline(*spline._eval_args)
except AttributeError:
# Create a spline object that's just a constant
bspline = BSpline(np.r_[(x[0],)*4, (x[-1],)*4],
np.r_[(spline(0),)*4, (0.,)*4], 3)
# Attach the mask and model (may be useful)
bspline.mask = full_mask
bspline.data = spline_y
return bspline
def get_price(self, num_dt: int, num_dx: int, center: float,
width: float) -> float:
"""
:param num_dt: represents number of discrete time steps
:param num_dx: represents number of discrete state-space steps
(on each side of the center)
:param center: represents the center of the state space grid. For
the case of lognormal == True, it should be Mean[log(x_{expiry}].
For the case of lognormal == False, it should be Mean[x_{expiry}].
:param width: represents the width of the state space grid. For the
case of lognormal == True, it should be a multiple of
Stdev[log(x_{expiry})].
:return: the price of the American option (this is the discounted
expected payoff at time 0 at current stock price.
"""
dt = self.expiry / num_dt
x_pts = 2 * num_dx + 1
lsp = np.linspace(center - width, center + width, x_pts)
prices = np.exp(lsp) if self.lognormal else lsp
res = np.empty([num_dt, x_pts])
res[-1, :] = [max(self.payoff(self.expiry, p), 0.) for p in prices]
sample_points = 201
final = [(p, max(self.payoff(self.expiry, p), 0.)) for p in prices]
ex_boundary = [max(p for p, e in final if e > 0)]
for i in range(num_dt - 2, -1, -1):
t = (i + 1) * dt
knots, coeffs, order = splrep(prices, res[i + 1, :], k=3)
spline_func = BSpline(knots, coeffs, order)
disc = np.exp(self.ir(t) - self.ir(t + dt))
stprcs = []
cp = []
ep = []
for j in range(x_pts):
m, v = get_future_price_mean_var(prices[j], t, dt,
self.lognormal, self.ir,
self.isig)
stdev = np.sqrt(v)
norm_dist = norm(loc=m, scale=stdev)
# noinspection PyShadowingNames
def integr_func(x: float,
spline_func=spline_func,
norm_dist=norm_dist) -> float:
val = np.exp(x) if self.lognormal else x
return max(spline_func(val), 0.) * norm_dist.pdf(x)
low, high = (m - 4 * stdev, m + 4 * stdev)
disc_exp_payoff = disc * trapz(
np.vectorize(integr_func)(np.linspace(
low, high, sample_points)),
dx=(high - low) /
(sample_points - 1)) / (norm_dist.cdf(high) -
norm_dist.cdf(low))
if prices[j] < 100:
stprcs.append(prices[j])
cp.append(disc_exp_payoff)
ep.append(max(self.payoff(t, prices[j]), 0.))
res[i, j] = max(self.payoff(t, prices[j]), disc_exp_payoff)
ex_boundary.append(
max(p for p, c, e in zip(stprcs, cp, ep) if e > c))
# if i == int(num_dt / 10) or i == num_dt - int(num_dt / 10) \
# or i == int(num_dt / 2):
# # print(list(zip(stprcs, cp, ep)))
# plt.title("Grid Time = %.3f" % t)
# plt.plot(stprcs, cp, 'r', stprcs, ep, 'b')
# plt.show()
# plt.plot([t * dt for t in range(1, num_dt + 1)], ex_boundary[::-1])
# plt.title("Grid Boundary")
# plt.savefig(str(Path.home()) + "/Downloads/GridBoundary.png")
knots, coeffs, order = splrep(prices, res[0, :], k=3)
spline_func = BSpline(knots, coeffs, order)
disc = np.exp(-self.ir(dt))
m, v = get_future_price_mean_var(self.spot_price, 0., dt,
self.lognormal, self.ir, self.isig)
stdev = np.sqrt(v)
norm_dist = norm(loc=m, scale=stdev)
# noinspection PyShadowingNames
def integr_func0(x: float,
spline_func=spline_func,
norm_dist=norm_dist) -> float:
val = np.exp(x) if self.lognormal else x
return max(spline_func(val), 0.) * norm_dist.pdf(x)
low, high = (m - 4 * stdev, m + 4 * stdev)
disc_exp_payoff = disc * trapz(
np.vectorize(integr_func0)(np.linspace(low, high, sample_points)),
dx=(high - low) /
(sample_points - 1)) / (norm_dist.cdf(high) - norm_dist.cdf(low))
return max(self.payoff(0., self.spot_price), disc_exp_payoff)
# Construct a cubic b-spline:
from scipy.interpolate import BSpline
b = BSpline.basis_element([0, 1, 2, 3, 4])
k = b.k
b.t[k:-k]
# array([ 0., 1., 2., 3., 4.])
k
# 3
# Construct a second order b-spline on ``[0, 1, 1, 2]``, and compare
# to its explicit form:
t = [-1, 0, 1, 1, 2]
b = BSpline.basis_element(t[1:])
def f(x):
return np.where(x < 1, x * x, (2. - x)**2)
import matplotlib.pyplot as plt
fig, ax = plt.subplots()
x = np.linspace(0, 2, 51)
ax.plot(x, b(x), 'g', lw=3)
ax.plot(x, f(x), 'r', lw=8, alpha=0.4)
ax.grid(True)
plt.show()
def evaluate(self, x_values_seq: Iterable[X]) -> np.ndarray:
spline_func: Callable[[Sequence[float]], np.ndarray] = BSpline(
self.knots, self.coeffs, self.degree
)
return spline_func(self.get_feature_values(x_values_seq))
return sigma_bar
def Delta_Sigma_model(m, c, r_mpc, Lz):
return (Sigma_nfw_bar(m, c, r_mpc) - Sigma_nfw(m, c, r_mpc)) + (Sigma_nfw_bar(m, c, r_mpc) - Sigma_nfw(m, c, r_mpc)) * Sigma_nfw(m, c, r_mpc) * Lz
################################ off-centered term ##################################
perfil_off_sigma = []
Profs = Table.read("Profiles_lookup_table.fits")
xi = np.logspace(-1,1,len(Profs))
radial_x_bins = np.logspace(-2,2,500)
for i in range(len(Profs)):
perfil_off_sigma.append(list(Profs[i]))
perfil_off_sigma= np.array(perfil_off_sigma)
print("Lookup table loaded!")
Delta_Sigma_NFW_off_x = BSpline(xi,radial_x_bins,perfil_off_sigma,s=0)
del perfil_off_sigma, xi, radial_x_bins
def sigma_off(m, c, s_off, r_mpc):
r = np.power(((m * 3) / (800 * np.pi * rho_c_cl_mpc)), (1 / 3))
r_s = r/c
fact = 2*r_s*delta_c(c)*rho_c_cl_mpc
xi = s_off/r_s
X = r_mpc/r_s
return fact*Delta_Sigma_NFW_off_x(xi, X)[0]
############################### likelihood ################################
if stack == True:
def _make_random_spline(n=35, k=3):
np.random.seed(123)
t = np.sort(np.random.random(n + k + 1))
c = np.random.random(n)
return BSpline.construct_fast(t, c, k)
def table_to_model(table):
"""
Convert a Table instance, as created by model_to_table(), back into a
callable function. Some backward compatibility has been introduced, so
the domain can be specified in the Table, rather than the meta, and a
Chebyshev1D model will be assumed if not found in the meta.
Parameters
----------
table : `~astropy.table.Table` or `~astropy.table.Row`
Table describing the model
Returns
-------
callable : either a `~astropy.modeling.core.Model` or a
`~scipy.interpolate.BSpline` instance
"""
meta = table.meta["header"]
model_class = meta.get("MODEL", "Chebyshev1D")
try:
cls = getattr(models, model_class)
except: # it's a spline
k = int(model_class[-1])
knots, coeffs = table["knots"], table["coefficients"]
model = BSpline(knots.data, coeffs.data, k)
setattr(model, "meta", {"xunit": knots.unit, "yunit": coeffs.unit})
else:
if isinstance(table, Table):
if len(table) != 1:
raise ValueError(
"Can only convert single-row Tables to a model")
else:
table = table[0] # now a Row
ndim = int(model_class[-2])
table_dict = dict(zip(table.colnames, table))
if ndim == 1:
r = re.compile("c([0-9]+)")
param_names = list(filter(r.match, table.colnames))
# Handle cases (e.g., APERTURE tables) where the number of
# columns must be the same for all rows but the degree of
# polynomial might be different
degree = max([
int(r.match(p).groups()[0]) for p in param_names
if table[p] is not np.ma.masked
])
domain = [
table_dict.get("domain_start", meta.get("DOMAIN_START", 0)),
table_dict.get("domain_end", meta.get("DOMAIN_END", 1))
]
model = cls(degree=degree, domain=domain)
elif ndim == 2:
r = re.compile("c([0-9]+)_([0-9]+)")
param_names = list(filter(r.match, table.colnames))
xdegree = max([int(r.match(p).groups()[0]) for p in param_names])
ydegree = max([int(r.match(p).groups()[1]) for p in param_names])
xdomain = [
table_dict.get("xdomain_start", meta.get("XDOMAIN_START", 0)),
table_dict.get("xdomain_end", meta.get("XDOMAIN_END", 1))
]
ydomain = [
table_dict.get("ydomain_start", meta.get("YDOMAIN_START", 0)),
table_dict.get("ydomain_end", meta.get("YDOMAIN_END", 1))
]
model = cls(x_degree=xdegree,
y_degree=ydegree,
x_domain=xdomain,
y_domain=ydomain)
else:
raise ValueError(
f"Invalid dimensionality of model '{model_class}'")
for k, v in table_dict.items():
if k in param_names:
setattr(model, k, v)
elif not ("domain" in k or k in ("ndim", "degree")):
# other columns go in the meta
model.meta[k] = v
for unit in ("xunit", "yunit", "zunit"):
value = meta.get(unit.upper())
if value:
model.meta[unit] = u.Unit(value)
return model
def test_nan(self):
# nan in, nan out.
b = BSpline.basis_element([0, 1, 1, 2])
assert_(np.isnan(b(np.nan)))
def bump_cbs(x):
y = np.zeros_like(x)
index = (-2 < x) & (x < 2)
y[index] = BSpline.basis_element([-2, -1, 0, 1, 2],
extrapolate=False)(x[index])
return y
def test_nan(self):
# nan in, nan out.
b = BSpline.basis_element([0, 1, 1, 2])
assert_(np.isnan(b(np.nan)))
def d_bump_cbs(x):
dy = np.zeros_like(x)
index = (-2 < x) & (x < 2)
dy[index] = BSpline.basis_element([-2, -1, 0, 1, 2],
extrapolate=False).derivative()(x[index])
return dy
def CrystalBragg_plot_data_vs_fit(xi,
xj,
bragg,
lamb,
phi,
data,
mask=None,
lambfit=None,
phifit=None,
spect1d=None,
dfit1d=None,
dfit2d=None,
lambfitbins=None,
cmap=None,
vmin=None,
vmax=None,
fs=None,
dmargin=None,
angunits='deg',
dmoments=None):
# Check inputs
# ------------
if fs is None:
fs = (16, 9)
if cmap is None:
cmap = plt.cm.viridis
if dmargin is None:
dmargin = {
'left': 0.03,
'right': 0.99,
'bottom': 0.05,
'top': 0.92,
'wspace': None,
'hspace': 0.4
}
assert angunits in ['deg', 'rad']
if angunits == 'deg':
bragg = bragg * 180. / np.pi
phi = phi * 180. / np.pi
phifit = phifit * 180. / np.pi
# pre-compute
# ------------
# extent
extent = (xi.min(), xi.max(), xj.min(), xj.max())
extent2 = (lambfit.min(), lambfit.max(), phifit.min(), phifit.max())
ind = np.digitize(lamb[mask].ravel(), lambfitbins)
spect2dmean = np.zeros((lambfitbins.size + 1, ))
for ii in range(lambfitbins.size + 1):
indi = ind == ii
if np.any(indi):
spect2dmean[ii] = np.nanmean(dfit2d['fit'][indi])
# Plot
# ------------
fig = plt.figure(figsize=fs)
gs = gridspec.GridSpec(4, 6, **dmargin)
ax0 = fig.add_subplot(gs[:3, 0], aspect='equal', adjustable='datalim')
ax1 = fig.add_subplot(gs[:3, 1],
aspect='equal',
adjustable='datalim',
sharex=ax0,
sharey=ax0)
axs1 = fig.add_subplot(gs[3, 1], sharex=ax0)
ax2 = fig.add_subplot(gs[:3, 2])
axs2 = fig.add_subplot(gs[3, 2], sharex=ax2, sharey=axs1)
ax3 = fig.add_subplot(gs[:3, 3], sharex=ax2, sharey=ax2)
axs3 = fig.add_subplot(gs[3, 3], sharex=ax2) #, sharey=axs1)
ax4 = fig.add_subplot(gs[:3, 4], sharex=ax2, sharey=ax2)
axs4 = fig.add_subplot(gs[3, 3], sharex=ax2) #, sharey=axs1)
ax5 = fig.add_subplot(gs[:3, 5], sharey=ax2)
ax0.set_title('Coordinates transform')
ax1.set_title('Camera image')
ax2.set_title('Camera image transformed')
ax3.set_title('2d spectral fit')
ax4.set_title('2d error')
ax5.set_title('Moments')
ax4.set_xlabel('%s' % angunits)
ax0.set_ylabel(r'incidence angle ($deg$)')
ax0.contour(xi, xj, bragg, 10, cmap=cmap)
ax0.contour(xi, xj, phi, 10, cmap=cmap, ls='--')
ax1.imshow(data,
extent=extent,
aspect='equal',
origin='lower',
vmin=vmin,
vmax=vmax)
axs1.plot(xi, np.nanmean(data, axis=0), c='k', ls='-')
ax2.scatter(lamb.ravel(),
phi.ravel(),
c=data.ravel(),
s=1,
marker='s',
edgecolors='None',
cmap=cmap,
vmin=vmin,
vmax=vmax)
axs2.plot(lambfit, spect1d, c='k', ls='None', marker='.', ms=4)
axs2.plot(lambfit, dfit1d['fit'].ravel(), c='r', ls='-', label='fit')
for ll in dfit1d['lamb0']:
axs2.axvline(ll, c='k', ls='--')
# dfit2d
ax3.scatter(lamb[mask].ravel(),
phi[mask].ravel(),
c=dfit2d['fit'],
s=1,
marker='s',
edgecolors='None',
cmap=cmap,
vmin=vmin,
vmax=vmax)
axs3.plot(lambfit, spect1d, c='k', ls='None', marker='.')
axs3.plot(lambfit, spect2dmean, c='b', ls='-')
err = dfit2d['fit'] - data[mask].ravel()
errmax = np.max(np.abs(err))
ax4.scatter(lamb[mask].ravel(),
phi[mask].ravel(),
c=err,
s=1,
marker='s',
edgecolors='None',
cmap=plt.cm.seismic,
vmin=-errmax,
vmax=errmax)
# Moments
if dmoments is not None:
if dmoments.get('ratio') is not None:
ind = dmoments['ratio'].get('ind')
if ind is None:
ind = [
np.argmin(np.abs(dfit2d['lamb0'] - ll))
for ll in dmoments['ratio']['lamb']
]
for indi in ind:
axs3.axvline(dfit2d['lamb0'][indi], c='k', ls='--')
amp0 = BSpline(dfit2d['knots'], dfit2d['camp'][ind[0], :],
dfit2d['deg'])(phifit)
amp1 = BSpline(dfit2d['knots'], dfit2d['camp'][ind[1], :],
dfit2d['deg'])(phifit)
lab = dmoments['ratio']['name'] + '{} / {}'
ratio = (amp0 / amp1) / np.nanmax(amp0 / amp1)
ax5.plot(amp0 / amp1, phifit, ls='-', c='k', label=lab)
if dmoments.get('sigma') is not None:
ind = dmoments['sigma'].get('ind')
if ind is None:
ind = np.argmin(
np.abs(dfit2d['lamb0'] - dmoments['sigma']['lamb']))
axs3.axvline(dfit2d['lamb0'][ind], c='b', ls='--')
sigma = BSpline(dfit2d['knots'], dfit2d['csigma'][ind, :],
dfit2d['deg'])(phifit)
lab = r'$\sigma({} A)$'.format(
np.round(dfit2d['lamb0'][ind] * 1.e10), 4)
ax5.plot(sigma / np.nanmax(sigma),
phifit,
ls='-',
c='b',
label=lab)
ax2.set_xlim(extent2[0], extent2[1])
ax2.set_ylim(extent2[2], extent2[3])
return [ax0, ax1]
def speed(
x: np.ndarray,
t: np.ndarray,
c: np.ndarray,
k: np.integer,
aux_outputs: bool = False,
) -> np.ndarray:
r"""Compute the speed of a B-Spline.
The speed is the norm of the first derivative of the B-Spline.
Arguments:
x: A `1xL` array of parameter values where to evaluate the curve.
It contains the parameter values where the speed of the B-Spline will
be evaluated. It is required to be non-empty, one-dimensional, and
real-valued.
t: A `1xm` array representing the knots of the B-spline.
It is required to be a non-empty, non-decreasing, and one-dimensional
sequence of real-valued elements. For a B-Spline of degree `k`, at least
`2k + 1` knots are required.
c: A `dxn` array representing the coefficients/control points of the B-spline.
Given `n` real-valued, `d`-dimensional points ::math::`x_k = (x_k(1),...,x_k(d))`,
`c` is the non-empty matrix which columns are ::math::`x_1^T,...,x_N^T`. For a
B-Spline of order `k`, `n` cannot be less than `m-k-1`.
k: A non-negative integer representing the degree of the B-spline.
Returns:
speed: A `1xL` array containing the speed of the B-Spline evaluated at `x`
References:
.. [1] Kouba, Parametric Equations.
https://www.math.ucdavis.edu/~kouba/Math21BHWDIRECTORY/ArcLength.pdf
"""
# convert arguments to desired type
x = np.ascontiguousarray(x)
t = np.ascontiguousarray(t)
c = np.ascontiguousarray(c)
k = operator.index(k)
if k < 0:
raise ValueError("The order of the spline must be non-negative")
check_type(t, np.ndarray)
t_dim = t.ndim
if t_dim != 1:
raise ValueError("t must be one-dimensional")
if len(t) == 0:
raise ValueError("t must be non-empty")
check_iterable_type(t, (np.integer, np.float))
if (np.diff(t) < 0).any():
raise ValueError("t must be a non-decreasing sequence")
check_type(c, np.ndarray)
c_dim = c.ndim
if c_dim > 2:
raise ValueError("c must be 2D max")
if len(c.flatten()) == 0:
raise ValueError("c must be non-empty")
if c_dim == 1:
check_iterable_type(c, (np.integer, np.float))
# expand dims so that we can cycle through a single dimension
c = np.expand_dims(c, axis=0)
if c_dim == 2:
for d in c:
check_iterable_type(d, (np.integer, np.float))
n_dim = len(c)
check_type(x, np.ndarray)
x_dim = x.ndim
if x_dim != 1:
raise ValueError("x must be one-dimensional")
if len(x) == 0:
raise ValueError("x must be non-empty")
check_iterable_type(x, (np.integer, np.float))
L = len(x)
# evaluate first and second derivatives
# deriv, dderiv are (d, L) arrays
deriv = np.empty((n_dim, L))
for i, dim in enumerate(c):
spl = BSpline(t, dim, k)
deriv[i, :] = spl.derivative(nu=1)(x) if k - 1 >= 0 else np.zeros(L)
# tranpose deriv
deriv = deriv.T
speed = np.linalg.norm(deriv, axis=1)
if aux_outputs == False:
return speed
else:
return speed, deriv
def _make_random_spline(n=35, k=3):
np.random.seed(123)
t = np.sort(np.random.random(n+k+1))
c = np.random.random(n)
return BSpline.construct_fast(t, c, k)
import numpy as np
import scipy.special
import pickle
from scipy.interpolate import BSpline
import atmosphere as atm
import os
dir_path = os.path.dirname(os.path.realpath(__file__))
with open(os.path.join(dir_path, "pickle/geo_rcut_b_splines.pickle"),
"r") as fin:
spl_rcut_geo_params, spl_b_geo_params = pickle.load(fin)
t, c, k = spl_rcut_geo_params
t = np.append(np.append(np.ones(k) * t[0], t), np.ones(k) * t[-1])
spl_rcut_geo = BSpline(t, c, k)
t, c, k = spl_b_geo_params
t = np.append(np.append(np.ones(k) * t[0], t), np.ones(k) * t[-1])
spl_b_geo = BSpline(t, c, k)
with open(os.path.join(dir_path, "pickle/geo_sigmaR_spl.pickle"), "r") as fin:
data = pickle.load(fin)
t, c, k = data['geo_R_0m']
t = np.append(np.append(np.ones(k) * t[0], t), np.ones(k) * t[-1])
spl_geo_R_0m = BSpline(t, c, k)
t, c, k = data['geo_R_1564m']
t = np.append(np.append(np.ones(k) * t[0], t), np.ones(k) * t[-1])
spl_geo_R_1564m = BSpline(t, c, k)
t, c, k = data['geo_sigma_0m']
