def test_confinter_linear():
#=======================================================================
"Check that the confidence intervals of the linear parameters are correct"
# Prepare test data
r = np.linspace(1, 8, 150)
t = np.linspace(0, 4, 200)
lam = 0.25
K = dipolarkernel(t, r, mod=lam)
parin = [3.5, 0.4, 0.6, 4.5, 0.5, 0.4]
P = dd_gauss2(r, *parin)
V = K @ P + whitegaussnoise(t, 0.05, seed=1)
# Non-linear parameters
# nlpar = [lam]
nlpar0 = 0.2
lb = 0
ub = 1
# Linear parameters: non-negativity
lbl = np.zeros(len(r))
ubl = np.full(len(r), np.inf)
# Separable LSQ fit
fit = snlls(V, lambda lam: dipolarkernel(t, r, mod=lam), nlpar0, lb, ub,
lbl)
Pfit = np.round(fit.lin, 6)
uq = fit.linUncert
Pci50 = np.round(uq.ci(50), 6)
Pci95 = np.round(uq.ci(95), 6)
assert_confidence_intervals(Pci50, Pci95, Pfit, lbl, ubl)
def test_multiple_datasets():
# ======================================================================
"Check bootstrapping when using multiple input datasets"
t1 = np.linspace(0, 5, 200)
t2 = np.linspace(-0.5, 3, 300)
r = np.linspace(2, 6, 300)
P = dd_gauss(r, 4, 0.8)
K1 = dipolarkernel(t1, r)
K2 = dipolarkernel(t2, r)
Vexp1 = K1 @ P + whitegaussnoise(t1, 0.01, seed=1)
Vexp2 = K2 @ P + whitegaussnoise(t2, 0.02, seed=2)
def Vmodel(par):
V1 = K1 @ dd_gauss(r, *par)
V2 = K2 @ dd_gauss(r, *par)
return [V1, V2]
par0 = [3, 0.5]
fit = snlls([Vexp1, Vexp2], Vmodel, par0)
Vfit1, Vfit2 = fit.model
def bootfcn(V):
fit = snlls(V, Vmodel, par0)
return fit.nonlin
paruq = bootstrap_analysis(bootfcn, [Vexp1, Vexp2], [Vfit1, Vfit2], 5)
assert all(abs(paruq.mean - fit.nonlin) < 1.5e-2)
def test_globalfit():
# ======================================================================
"Check that global fitting yields correct results"
r = np.linspace(2, 5, 200)
par = np.array([3, 0.2])
P = dd_gauss(r, *par)
t1 = np.linspace(0, 3, 300)
K1 = dipolarkernel(t1, r)
V1 = K1 @ P
t2 = np.linspace(-0.5, 4, 200)
K2 = dipolarkernel(t2, r)
V2 = K2 @ P
def Vmodel(par):
Pfit = dd_gauss(r, *par)
V1 = K1 @ Pfit
V2 = K2 @ Pfit
return [V1, V2]
par0 = [5, 0.5]
lb = [1, 0.1]
ub = [20, 1]
fit = snlls([V1, V2], Vmodel, par0, lb, ub)
assert all(abs(par - fit.nonlin) < 1e-2)
def assert_solver(solver):
# Prepare test data
r = np.linspace(1, 8, 80)
t = np.linspace(0, 4, 200)
lam = 0.25
K = dipolarkernel(t, r, mod=lam)
parin = [3.5, 0.4, 0.6, 4.5, 0.5, 0.4]
P = dd_gauss2(r, *parin)
V = K @ P
# Non-linear parameters
# nlpar = [lam]
nlpar0 = 0.2
lb = 0
ub = 1
# Linear parameters: non-negativity
lbl = np.zeros(len(r))
ubl = []
# Separable LSQ fit
fit = snlls(V,
lambda lam: dipolarkernel(t, r, mod=lam),
nlpar0,
lb,
ub,
lbl,
ubl,
nnlsSolver=solver,
uq=False)
Pfit = fit.lin
assert ovl(P, Pfit) > 0.95
def test_globalfit_scales():
#============================================================
"Check that the global fit with arbitrary amplitudes works."
t1 = np.linspace(0, 5, 300)
t2 = np.linspace(0, 2, 300)
r = np.linspace(3, 5, 100)
P = dd_gauss(r, 4, 0.25)
K1 = dipolarkernel(t1, r)
K2 = dipolarkernel(t2, r)
scales = [1e3, 1e9]
V1 = scales[0] * K1 @ P
V2 = scales[1] * K2 @ P
def Vmodel(par):
Pfit = dd_gauss(r, *par)
V1 = K1 @ Pfit
V2 = K2 @ Pfit
return block_diag(V1, V2).T
par0 = [5, 0.5]
lb = [1, 0.1]
ub = [20, 1]
fit = snlls(np.concatenate([V1, V2]), Vmodel, par0, lb, ub)
assert max(abs(np.asarray(scales) / fit.lin - 1)) < 1e-2
def test_goodness_of_fit():
#============================================================
"Check the goodness-of-fit statistics are correct"
# Prepare test data
r = np.linspace(2, 5, 150)
t = np.linspace(-0.2, 4, 100)
lam = 0.25
K = dipolarkernel(t, r, mod=lam)
parin = [3.5, 0.15, 0.6, 4.5, 0.2, 0.4]
P = dd_gauss2(r, *parin)
sigma = 0.03
V = K @ P + whitegaussnoise(t, sigma, seed=2, rescale=True)
# Non-linear parameters
# nlpar = [lam]
nlpar0 = 0.2
lb = 0
ub = 1
# Linear parameters: non-negativity
lbl = np.zeros(len(r))
ubl = []
# Separable LSQ fit
fit = snlls(V,
lambda lam: dipolarkernel(t, r, mod=lam),
nlpar0,
lb,
ub,
lbl,
ubl,
noiselvl=sigma,
uq=False)
stats = fit.stats
assert abs(stats['chi2red'] - 1) < 0.05
def test_plot():
# ======================================================================
"Check that the plot method works"
# Prepare test data
r = np.linspace(1, 8, 80)
t = np.linspace(0, 4, 200)
lam = 0.25
K = dipolarkernel(t, r, mod=lam)
parin = [3.5, 0.4, 0.6, 4.5, 0.5, 0.4]
P = dd_gauss2(r, *parin)
V = K @ P
# Linear parameters: non-negativity
lbl = np.zeros(len(r))
# Separable LSQ fit
fit = snlls(V,
lambda lam: dipolarkernel(t, r, mod=lam),
par0=0.2,
lb=0,
ub=1,
lbl=lbl,
uq=False)
fig = fit.plot(show=False)
assert str(fig.__class__) == "<class 'matplotlib.figure.Figure'>"
def test_reg_tikhonov():
#============================================================
"Check that Tikhonov regularization of linear problem works"
# Prepare test data
r = np.linspace(1, 8, 80)
t = np.linspace(0, 4, 100)
lam = 0.25
K = dipolarkernel(t, r, mod=lam)
parin = [3.5, 0.4, 0.6, 4.5, 0.5, 0.4]
P = dd_gauss2(r, *parin)
V = K @ P
# Non-linear parameters
# nlpar = [lam]
nlpar0 = 0.2
lb = 0
ub = 1
# Linear parameters: non-negativity
lbl = np.zeros(len(r))
# Separable LSQ fit
fit = snlls(V,
lambda lam: dipolarkernel(t, r, mod=lam),
nlpar0,
lb,
ub,
lbl,
uq=False)
Pfit = fit.lin
assert ovl(P, Pfit) > 0.95
def test_cost_value():
#============================================================
"Check that the cost value is properly returned"
# Prepare test data
r = np.linspace(1, 8, 80)
t = np.linspace(0, 4, 200)
lam = 0.25
K = dipolarkernel(t, r, mod=lam)
parin = [3.5, 0.4, 0.6, 4.5, 0.5, 0.4]
P = dd_gauss2(r, *parin)
V = K @ P
# Non-linear parameters
nlpar0 = 0.2
lb = 0
ub = 1
# Linear parameters: non-negativity
lbl = np.zeros(len(r))
ubl = np.full(len(r), np.inf)
# Separable LSQ fit
fit = snlls(V, lambda lam: dipolarkernel(t, r, mod=lam), nlpar0, lb, ub,
lbl, ubl)
assert isinstance(fit.cost, float) and np.round(
fit.cost / np.sum(fit.residuals**2), 5) == 1
def test_multiple_penalties():
# ======================================================================
"Check that multiple additional penaltyies can be passed correctly"
t = np.linspace(0, 5, 300)
r = np.linspace(2, 6, 90)
P = dd_gauss(r, 4.5, 0.25)
param = 0.2
K = dipolarkernel(t, r, mod=param)
V = K @ P + whitegaussnoise(t, 0.001, seed=1)
dr = np.mean(np.diff(r))
beta = 0.05
R = 0.5
compactness_penalty = lambda pnonlin, plin: beta * np.sqrt(plin * (
r - np.trapz(plin * r, r))**2 * dr)
radial_penalty = lambda pnonlin, plin: 1 / R**2 * (np.linalg.norm(
(pnonlin - param) / param - R))**2
Kmodel = lambda lam: dipolarkernel(t, r, mod=lam)
fit0 = snlls(V,
Kmodel,
par0=0.2,
lb=0,
ub=1,
lbl=np.zeros_like(r),
extrapenalty=[compactness_penalty])
fitmoved = snlls(V,
Kmodel,
par0=0.2,
lb=0,
ub=1,
lbl=np.zeros_like(r),
extrapenalty=[compactness_penalty, radial_penalty])
assert ovl(P, fit0.lin) > ovl(P, fitmoved.lin)
def test_confinter_model():
#=======================================================================
"Check that the confidence intervals of the fitted model are correct"
# Prepare test data
r = np.linspace(1, 8, 150)
t = np.linspace(0, 4, 200)
lam = 0.25
K = dipolarkernel(t, r, mod=lam)
parin = [3.5, 0.4, 0.6, 4.5, 0.5, 0.4]
P = dd_gauss2(r, *parin)
V = K @ P + whitegaussnoise(t, 0.05, seed=1)
nlpar0 = 0.2
lb = 0
ub = 1
lbl = np.full(len(r), 0)
# Separable LSQ fit
fit = snlls(V, lambda lam: dipolarkernel(t, r, mod=lam), nlpar0, lb, ub,
lbl)
Vfit = fit.model
Vuq = fit.modelUncert
Vci50 = Vuq.ci(50)
Vci95 = Vuq.ci(95)
Vlb = np.full(len(t), -np.inf)
Vub = np.full(len(t), np.inf)
assert_confidence_intervals(Vci50, Vci95, Vfit, Vlb, Vub)
def test_confinter_nonlinear():
#=======================================================================
"Check that the confidence intervals of the non-linear parameters are correct"
# Prepare test data
r = np.linspace(1, 8, 80)
t = np.linspace(0, 4, 200)
lam = 0.25
K = dipolarkernel(t, r, mod=lam)
parin = [3.5, 0.4, 0.6, 4.5, 0.5, 0.4]
P = dd_gauss2(r, *parin)
V = K @ P
# Non-linear parameters
# nlpar = [lam]
nlpar0 = 0.2
lb = 0
ub = 1
# Linear parameters: non-negativity
lbl = np.zeros(len(r))
ubl = np.full(len(r), np.inf)
# Separable LSQ fit
fit = snlls(V, lambda lam: dipolarkernel(t, r, mod=lam), nlpar0, lb, ub,
lbl, ubl)
parfit = fit.nonlin
uq = fit.nonlinUncert
parci50 = np.atleast_2d(uq.ci(50))
parci95 = np.atleast_2d(uq.ci(95))
assert_confidence_intervals(parci50, parci95, parfit, lb, ub)
def test_confinter_scaling():
#============================================================
"Check that the confidence intervals are agnostic w.r.t. scaling"
# Prepare test data
r = np.linspace(1, 8, 80)
t = np.linspace(0, 4, 50)
lam = 0.25
K = dipolarkernel(t, r, mod=lam)
parin = [3.5, 0.4, 0.6, 4.5, 0.5, 0.4]
P = dd_gauss2(r, *parin)
V = K @ P + whitegaussnoise(t, 0.01, seed=1)
# Non-linear parameters
nlpar0 = 0.2
lb = 0
ub = 1
# Linear parameters: non-negativity
lbl = np.zeros(len(r))
V0_1 = 1
V0_2 = 1e8
# Separable LSQ fit
fit1 = snlls(V * V0_1,
lambda lam: dipolarkernel(t, r, mod=lam),
nlpar0,
lb,
ub,
lbl,
nonlin_tol=1e-3)
fit2 = snlls(V * V0_2,
lambda lam: dipolarkernel(t, r, mod=lam),
nlpar0,
lb,
ub,
lbl,
nonlin_tol=1e-3)
# Assess linear parameter uncertainties
ci1 = fit1.linUncert.ci(95)
ci2 = fit2.linUncert.ci(95)
ci1[ci1 == 0] = 1e-16
ci2[ci2 == 0] = 1e-16
assert np.max(abs(ci2 / V0_2 - ci1)) < 1e-6
# Assess nonlinear parameter uncertainties
ci1 = fit1.nonlinUncert.ci(95)
ci2 = fit2.nonlinUncert.ci(95)
ci1[ci1 == 0] = 1e-16
ci2[ci2 == 0] = 1e-16
assert np.max(abs(ci2 - ci1)) < 1e-6
def generate_global_dataset():
t1 = np.linspace(-0.5, 3, 200)
t2 = np.linspace(-0.5, 4, 200)
r = np.linspace(2.5, 5, 80)
P = dd_gauss2(r, 3.7, 4.3, 0.2, 0.1, 0.5, 0.5)
P /= np.trapz(P, r)
K1 = dipolarkernel(t1, r)
K2 = dipolarkernel(t2, r)
np.random.seed(1)
V1 = K1 @ P + whitegaussnoise(t1, 0.01, seed=1, rescale=True)
np.random.seed(2)
V2 = K2 @ P + whitegaussnoise(t2, 0.01, seed=1, rescale=True)
return r, P, V1, V2, K1, K2
def assert_full_output(method):
t = np.linspace(0, 5, 80)
r = np.linspace(2, 5, 80)
P = dd_gauss(r, 3, 0.4)
K = dipolarkernel(t, r)
L = regoperator(r, 2)
V = K @ P
alpha, alphas_evaled, functional, residuals, penalties = selregparam(
V,
K,
cvxnnls,
method='aic',
algorithm=method,
full_output=True,
regop=L)
errors = []
if np.size(alpha) != 1:
errors.append("alphaopt is not a scalar")
if len(functional) != len(alphas_evaled):
errors.append(
"The number of elements of functional values and evaluated alphas are different."
)
if len(residuals) != len(penalties):
errors.append(
"The number of elements of evluated residuals and penalties are different"
)
if not alpha in alphas_evaled:
errors.append("The optimal alpha is not part of the evaluated alphas")
assert not errors, f"Errors occured:\n{chr(10).join(errors)}"
def test_extrapenalty():
# ======================================================================
"Check that custom penalties can be passed and act on the solution"
t = np.linspace(0, 3, 300)
r = np.linspace(2, 5, 200)
P = dd_gauss2(r, 3.5, 0.5, 0.5, 4, 0.1, 0.5)
K = dipolarkernel(t, r)
V = K @ P + whitegaussnoise(t, 0.15, seed=1)
par0 = [2.5, 0.01, 0.1, 4.5, 0.01, 0.6]
lb = [1, 0.01, 0, 1, 0.01, 0]
ub = [20, 1, 1, 20, 1, 1]
# Fit case it fails, stuck at "spicky" Gaussians
model = lambda p: K @ dd_gauss2(r, *p)
fit = snlls(V, model, par0, lb, ub)
# Fit with Tikhonov penalty on the Gaussians model
L = regoperator(r, 2)
alpha = 1e-4
tikhonov = lambda p, _: alpha * L @ dd_gauss2(r, *p)
fit_tikh = snlls(V, model, par0, lb, ub, extrapenalty=tikhonov)
Pfit = dd_gauss2(r, *fit.nonlin)
Pfit_tikh = dd_gauss2(r, *fit_tikh.nonlin)
assert ovl(P, Pfit) < ovl(P, Pfit_tikh)
def test_algorithms():
#=======================================================================
"Check that the value returned by the the grid and Brent algorithms coincide"
t = np.linspace(0, 5, 80)
r = np.linspace(2, 5, 80)
P = dd_gauss(r, 3, 0.2)
K = dipolarkernel(t, r)
L = regoperator(r, 2)
V = K @ P + whitegaussnoise(t, 0.02, seed=1)
alpha_grid = selregparam(V,
K,
cvxnnls,
method='aic',
algorithm='grid',
regop=L)
alpha_brent = selregparam(V,
K,
cvxnnls,
method='aic',
algorithm='brent',
regop=L)
assert abs(1 - alpha_grid / alpha_brent) < 0.15
def assert_multigauss_SNLLS_problem(nonlinearconstr=True, linearconstr=True):
# Prepare test data
r = np.linspace(1, 8, 80)
t = np.linspace(0, 4, 200)
K = dipolarkernel(t, r)
parin = [3.5, 0.4, 0.6, 4.5, 0.5, 0.4]
P = dd_gauss2(r, *parin)
V = K @ P
def Kmodel(p, t, r):
# Unpack parameters
r1, w1, r2, w2 = p
# Generate basic kernel
K0 = dipolarkernel(t, r)
# Get Gauss basis functions
P1 = dd_gauss(r, r1, w1)
P2 = dd_gauss(r, r2, w2)
# Combine all non-linear functions into one
K = np.zeros((len(t), 2))
K[:, 0] = K0 @ P1
K[:, 1] = K0 @ P2
return K
# Non-linear parameters
# nlpar = [r1 w1 r2 w2]
nlpar0 = [3.2, 0.2, 4.2, 0.3]
if nonlinearconstr:
lb = [1, 0.1, 1, 0.1]
ub = [20, 5, 20, 5]
else:
lb = []
ub = []
# Linear parameters
if linearconstr:
lbl = [0, 0]
ubl = [1, 1]
else:
lbl = []
ubl = []
# Separable LSQ fit
fit = snlls(V,
lambda p: Kmodel(p, t, r),
nlpar0,
lb,
ub,
lbl,
ubl,
reg=False,
uq=False)
nonlinfit = fit.nonlin
linfit = fit.lin
parout = [
nonlinfit[0], nonlinfit[1], linfit[0], nonlinfit[2], nonlinfit[3],
linfit[1]
]
parout = np.asarray(parout)
parin = np.asarray(parin)
assert np.max(abs(parout - parin) < 1e-1)
def test_global_weights_default():
# ======================================================================
"Check the correct fit of two signals when one is of very low quality"
t = np.linspace(0, 5, 300)
r = np.linspace(2, 6, 90)
param = [4.5, 0.25]
P = dd_gauss(r, *param)
K = dipolarkernel(t, r, mod=0.2)
scales = [1e3, 1e9]
V1 = scales[0] * (K @ P + whitegaussnoise(t, 0.001, seed=1))
V2 = scales[1] * (K @ P + whitegaussnoise(t, 0.1, seed=1))
Kmodel = lambda lam: [dipolarkernel(t, r, mod=lam)] * 2
fit = snlls([V1, V2], Kmodel, par0=[0.2], lb=0, ub=1, lbl=np.zeros_like(r))
assert ovl(P, fit.lin) > 0.93
def get_alpha_from_method(method):
t = np.linspace(0, 5, 500)
r = np.linspace(2, 5, 80)
P = dd_gauss(r, 3.0, 0.16986436005760383)
K = dipolarkernel(t, r)
L = regoperator(r, 2, includeedges=True)
V = K @ P
alpha = selregparam(V, K, cvxnnls, method=method, noiselvl=0, regop=L)
return np.log10(alpha)
def test_compensate_condition():
#=======================================================================
"Check that alpha compensates for larger condition numbers"
r = np.linspace(2, 6, 100)
P = dd_gauss(r, 3, 0.2)
# Lower condition number
t1 = np.linspace(0, 3, 200)
K1 = dipolarkernel(t1, r)
V1 = K1 @ P
alpha1 = selregparam(V1, K1, cvxnnls, method='aic')
# Larger condition number
t2 = np.linspace(0, 3, 400)
K2 = dipolarkernel(t2, r)
V2 = K2 @ P
alpha2 = selregparam(V2, K2, cvxnnls, method='aic')
assert alpha2 > alpha1
def assert_solver(solver):
#============================================================
np.random.seed(1)
t = np.linspace(-2, 4, 300)
r = np.linspace(2, 6, 100)
P = dd_gauss(r, 3, 0.2)
K = dipolarkernel(t, r)
V = K @ P + whitegaussnoise(t, 0.01)
fit = snlls(V, K, lbl=np.zeros_like(r), nnlsSolver=solver, uq=False)
assert ovl(P, fit.param) > 0.95 # more than 95% overlap
def test_confinter_tikh():
#============================================================
"Check that the confidence intervals are correctly computed using Tikhonov regularization"
t = np.linspace(-2, 4, 300)
r = np.linspace(2, 6, 100)
P = dd_gauss(r, 3, 0.2)
K = dipolarkernel(t, r)
V = K @ P
fit = snlls(V, K, lbl=np.zeros_like(r))
assert_confidence_intervals(fit.paramUncert, fit.param)
def test_confinter_Vfit():
#============================================================
"Check that the confidence intervals are correctly for the fitted signal"
t = np.linspace(-2, 4, 300)
r = np.linspace(2, 6, 100)
P = dd_gauss(r, 3, 0.2)
K = dipolarkernel(t, r, mod=0.2)
V = K @ P + whitegaussnoise(t, 0.05)
fit = snlls(V, K, lbl=np.zeros_like(r))
assert_confidence_intervals(fit.modelUncert, fit.model)
def test_nonuniform_r():
#============================================================
"Check that fit works correctly with non-uniform distance vectors"
t = np.linspace(0, 4, 300)
r = np.sqrt(np.linspace(1, 7**2, 200))
P = dd_gauss(r, 3, 0.2)
K = dipolarkernel(t, r)
V = K @ P
fit = snlls(V, K, lbl=np.zeros_like(r))
Vfit = K @ fit.param
assert abs(Vfit[0] - 1) < 1e-6
def test_plot():
#============================================================
"Check the plotting method"
t = np.linspace(0, 3, 200)
r = np.linspace(1, 5, 100)
P = dd_gauss(r, 3, 0.08)
K = dipolarkernel(t, r)
V = K @ P
fit = snlls(V, K, lbl=np.zeros_like(r))
fig = fit.plot(show=False)
assert str(fig.__class__) == "<class 'matplotlib.figure.Figure'>"
def test_convergence_criteria():
#============================================================
"Check that convergence criteria can be specified without crashing"
t = np.linspace(0, 3, 200)
r = np.linspace(1, 5, 100)
P = dd_gauss(r, 3, 0.08)
K = dipolarkernel(t, r)
V = K @ P
fit = snlls(V, K, lin_tol=1e-9, lin_maxiter=2e3, lbl=np.zeros_like(r))
assert ovl(P, fit.param) > 0.90 # more than 80% overlap
def Kmodel(p, t, r):
# Unpack parameters
r1, w1, r2, w2 = p
# Generate basic kernel
K0 = dipolarkernel(t, r)
# Get Gauss basis functions
P1 = dd_gauss(r, r1, w1)
P2 = dd_gauss(r, r2, w2)
# Combine all non-linear functions into one
K = np.zeros((len(t), 2))
K[:, 0] = K0 @ P1
K[:, 1] = K0 @ P2
return K
def test_global_weights():
# ======================================================================
"Check that the global weights properly work when specified"
t = np.linspace(-0.3, 5, 300)
r = np.linspace(2, 6, 150)
P1 = dd_gauss(r, 3, 0.2)
P2 = dd_gauss(r, 5, 0.2)
K = dipolarkernel(t, r, mod=0.2)
scales = [1e3, 1e9]
sigma1 = 0.001
V1 = K @ P1 + whitegaussnoise(t, sigma1, seed=1)
sigma2 = 0.001
V2 = K @ P2 + whitegaussnoise(t, sigma2, seed=1)
V1 = scales[0] * V1
V2 = scales[1] * V2
Kmodel = lambda lam: [dipolarkernel(t, r, mod=lam)] * 2
fit1 = snlls([V1, V2],
Kmodel,
par0=[0.2],
lb=0,
ub=1,
lbl=np.zeros_like(r),
weights=[1, 1e-10])
fit2 = snlls([V1, V2],
Kmodel,
par0=[0.2],
lb=0,
ub=1,
lbl=np.zeros_like(r),
weights=[1e-10, 1])
assert ovl(P1, fit1.lin) > 0.93 and ovl(P2, fit2.lin) > 0.93
def test_regularized_global():
#=======================================================================
"Check global SNLLS of a nonlinear-constrained + linear-regularized problem"
t1 = np.linspace(0, 3, 150)
t2 = np.linspace(0, 4, 200)
r = np.linspace(2.5, 5, 80)
P = dd_gauss2(r, 3.7, 0.5, 0.5, 4.3, 0.3, 0.5)
kappa = 0.50
lam1 = 0.25
lam2 = 0.35
K1 = dipolarkernel(t1, r, mod=lam1, bg=bg_exp(t1, kappa))
K2 = dipolarkernel(t2, r, mod=lam2, bg=bg_exp(t2, kappa))
V1 = K1 @ P
V2 = K2 @ P
# Global non-linear model
def globalKmodel(par):
# Unpack parameters
kappa, lam1, lam2 = par
K1 = dipolarkernel(t1, r, mod=lam1, bg=bg_exp(t1, kappa))
K2 = dipolarkernel(t2, r, mod=lam2, bg=bg_exp(t2, kappa))
return K1, K2
# Non-linear parameters
# [kappa lambda1 lambda2]
par0 = [0.5, 0.5, 0.5]
lb = [0, 0, 0]
ub = [1, 1, 1]
# Linear parameters: non-negativity
lbl = np.zeros(len(r))
ubl = []
# Separable LSQ fit
fit = snlls([V1, V2], globalKmodel, par0, lb, ub, lbl, ubl, uq=False)
Pfit = fit.lin
assert ovl(P, Pfit) > 0.9
