Vmodels = [[]] * Nmax
# The basic model for the components (can be e.g. dl.dd_rice)
basisModel = dl.dd_gauss
# Model construction
for n in range(Nmax):
# Construct the n-Gaussian model
Pmodels[n] = dl.lincombine(*[basisModel] * (n + 1))
# Construct the corresponding dipolar signal model
Vmodels[n] = dl.dipolarmodel(t, r, Pmodel=Pmodels[n])
# Fit the models to the data
fits = [[]] * Nmax
for n in range(Nmax):
fits[n] = dl.fit(Vmodels[n], Vexp, reg=False)
#%%
# Extract the values of the Akaike information criterion for each fit
aic = np.array([fit.stats['aic'] for fit in fits])
# Compute the relative difference in AIC
aic -= aic.min()
# Plotting
fig = plt.figure(figsize=[6, 6])
gs = GridSpec(1, 3, figure=fig)
ax1 = fig.add_subplot(gs[0, :-1])
for n in range(Nmax):
# Evaluate the n-Gaussian distance distribution model
Pfit = fits[n].evaluate(Pmodels[n], *[r] * (n + 1))
# Distance vector
r = np.linspace(2,5,150) # nm
# Construct the dipolar models for the individual signals
V1model = dl.dipolarmodel(ts[0],r)
V2model = dl.dipolarmodel(ts[1],r)
# Make the global model by joining the individual models
globalmodel = dl.merge(V1model,V2model)
# Link the distance distribution into a global parameter
globalmodel = dl.link(globalmodel,P=['P_1','P_2'])
# Fit the model to the data
fit = dl.fit(globalmodel,Vs)
# %%
plt.figure(figsize=[10,7])
violet = '#4550e6'
for n in range(len(fit.model)):
# Extract fitted dipolar signal
Vfit = fit.model[n]
Vci = fit.modelUncert[n].ci(95)
# Extract fitted distance distribution
Pfit = fit.P
scale = np.trapz(Pfit,r)
Pci95 = fit.PUncert.ci(95)/scale
r = np.linspace(2, 6, 100)
# 4-pulse DEER can have up to four different dipolar pathways
Nmax = 4
# Create the 4-pulse DEER signal models with increasing number of pathways
experiment = dl.ex_4pdeer(𝜏1, 𝜏2)
Vmodels = [
dl.dipolarmodel(t, r, npathways=n + 1, experiment=experiment)
for n in range(Nmax)
]
# Fit the individual models to the data
fits = [[]] * Nmax
for n, Vmodel in enumerate(Vmodels):
fits[n] = dl.fit(Vmodel, Vexp)
#%%
# Extract the values of the Akaike information criterion for each fit
aic = np.array([fit.stats['aic'] for fit in fits])
# Compute the relative difference in AIC
aic = aic - aic.min() + 1 # ...add plus one for log-scale
# Plotting
colors = ['tab:blue', 'tab:orange', 'tab:green', 'tab:red']
fig = plt.figure(figsize=[8, 9])
gs = GridSpec(1, 3, figure=fig)
ax1 = fig.add_subplot(gs[0, :-1])
for n in range(len(Vmodels)):
# Get the fits of the dipolar signal models
description='Dissociation constant')
titrmodel = dl.relate(titrmodel,
weight_2_1=lambda weight_1_1: 1 - weight_1_1,
weight_1_1=lambda Kdis: chemicalequilibrium(Kdis, L[0]),
weight_2_2=lambda weight_1_2: 1 - weight_1_2,
weight_1_2=lambda Kdis: chemicalequilibrium(Kdis, L[1]),
weight_2_3=lambda weight_1_3: 1 - weight_1_3,
weight_1_3=lambda Kdis: chemicalequilibrium(Kdis, L[2]),
weight_2_4=lambda weight_1_4: 1 - weight_1_4,
weight_1_4=lambda Kdis: chemicalequilibrium(Kdis, L[3]),
weight_2_5=lambda weight_1_5: 1 - weight_1_5,
weight_1_5=lambda Kdis: chemicalequilibrium(Kdis, L[4]))
# Fit the model to the data
fit = dl.fit(titrmodel, Vs, regparam=0.5)
# %%
# Evaluate the dose-response curve at the fit with confidence bands
xAfcn = lambda Kdis: np.squeeze(
np.array([chemicalequilibrium(Kdis, Ln) for Ln in L]))
xBfcn = lambda Kdis: np.squeeze(
np.array([1 - chemicalequilibrium(Kdis, Ln) for Ln in L]))
xAfit = xAfcn(fit.Kdis)
xBfit = xBfcn(fit.Kdis)
xAci = fit.propagate(xAfcn, lb=np.zeros_like(L), ub=np.ones_like(L)).ci(95)
xBci = fit.propagate(xBfcn, lb=np.zeros_like(L), ub=np.ones_like(L)).ci(95)
# Plot the dose-reponse curve
plt.plot(L, xAfit, '-o')
# Load the experimental data
t, Vexp = np.load('../data/example_data_#1.npy')
# Pre-process
Vexp = dl.correctphase(Vexp)
Vexp = Vexp / np.max(Vexp)
# Distance vector
r = np.linspace(2, 5.5, 80)
# Construct the 4-pulse DEER dipolar model
Vmodel = dl.dipolarmodel(t, r)
Vmodel.reftime.set(par0=0.5, lb=0.0, ub=1.0)
# Fit the model to the data using covariane-based uncertainty
fit_cm = dl.fit(Vmodel, Vexp)
# Fit the model to the data using bootstrapped uncertainty
fit_bs = dl.fit(Vmodel, Vexp, bootstrap=10)
# Compute the covariance-based uncertainty bands of the distance distribution
Pci50_cm = fit_cm.PUncert.ci(50)
Pci95_cm = fit_cm.PUncert.ci(95)
# Compute the bootstrapped uncertainty bands of the distance distribution
Pci50_bs = fit_bs.PUncert.ci(50)
Pci95_bs = fit_bs.PUncert.ci(95)
#%%
# Plot the results
# %%
# Load the experimental data
t, Vexp = np.load('../data/example_5pdeer_#1.npy')
# Distance vector
r = np.linspace(2, 5, 200) # nm
# Construct dipolar model with two dipolar pathways
Vmodel = dl.dipolarmodel(t, r, npathways=2)
# The refocusing time of the second pathway can be well estimated by visual inspection
Vmodel.reftime2.set(lb=3, ub=4, par0=3.5)
# Fit the model to the data
fit = dl.fit(Vmodel, Vexp)
# %%
# Extract fitted dipolar signal
Vfit = fit.model
Vci = fit.modelUncert.ci(95)
# Extract fitted distance distribution
Pfit = fit.P
scale = np.trapz(Pfit, r)
Pci95 = fit.PUncert.ci(95) / scale
Pci50 = fit.PUncert.ci(50) / scale
Pfit = Pfit / scale
# Extract the unmodulated contribution
import deerlab as dl
# %%
# Load the experimental data
t,Vexp = np.load('../data/example_4pdeer_#1.npy')
# Distance vector
r = np.linspace(2,5,100) # nm
# Construct the model
Vmodel = dl.dipolarmodel(t,r)
# Fit the model to the data
fit = dl.fit(Vmodel,Vexp,bootstrap=20)
# In this example, just for the sake of time, we will just use 20 bootstrap samples.
#%%
# Extract fitted dipolar signal
Vfit = fit.model
Vci = fit.modelUncert.ci(95)
# Extract fitted distance distribution
Pfit = fit.P
scale = np.trapz(Pfit,r)
Pci95 = fit.PUncert.ci(95)/scale
Pci50 = fit.PUncert.ci(50)/scale
Pfit = Pfit/scale
# %%
# Load the experimental dataset
t,V = np.load('../data/example_data_#1.npy')
# Pre-process
V = dl.correctphase(V)
V = V/np.max(V)
# Construct the dipolar signal model
r = np.linspace(1,7,100)
Vmodel = dl.dipolarmodel(t,r)
# Fit the model to the data
fit = dl.fit(Vmodel,V)
fit.plot(axis=t)
plt.ylabel('V(t)')
plt.xlabel('Time $t$ (μs)')
plt.show()
# From the fit results, extract the distribution and the covariance matrix
Pfit = fit.P
Pci95 = fit.PUncert.ci(95)
# Select a bimodal Gaussian model for the distance distribution
Pmodel = dl.dd_gauss2
# Fit the Gaussian model to the non-parametric distance distribution
fit = dl.fit(Pmodel,Pfit,r)
V5p = V5p / np.max(V5p)
# Run fit
r = np.linspace(2, 4.5, 100)
# Construct the individual dipolar signal models
V4pmodel = dl.dipolarmodel(t4p, r, npathways=1)
V5pmodel = dl.dipolarmodel(t5p, r, npathways=2)
V5pmodel.reftime2.set(lb=3, ub=3.5, par0=3.2)
# Make the joint model with the distribution as a global parameters
globalmodel = dl.merge(V4pmodel, V5pmodel)
globalmodel = dl.link(globalmodel, P=['P_1', 'P_2'])
# Fit the model to the data (with fixed regularization parameter)
fit = dl.fit(globalmodel, [V4p, V5p], regparam=0.5)
# %%
plt.figure(figsize=[10, 7])
violet = '#4550e6'
# Extract fitted distance distribution
Pfit = fit.P
scale = np.trapz(Pfit, r)
Pci95 = fit.PUncert.ci(95) / scale
Pci50 = fit.PUncert.ci(50) / scale
Pfit = Pfit / scale
for n, (t, V) in enumerate(zip([t4p, t5p], [V4p, V5p])):
# Extract fitted dipolar signal
def profile_analysis(model,y, *args, parameters='all', grids=None, samples=50, noiselvl=None, verbose=False,**kargs):
r"""
Profile likelihood analysis for uncertainty quantification
Parameters
----------
model : :ref:`Model`
Model object describing the data. All non-linear model parameters are profiled by default.
y : array_like or list of array_like
Experimental dataset(s).
args : positional arguments
Any other positional arguments to be passed to the ``fit`` function. See the
documentation of the ``fit`` function for further details.
parameters : string or list thereof
Model parameters to profile. If set to ``'all'`` all non-linear parameters in the model are analyzed.
samples : integer scalar
Number of points to take to estimate the profile function. Ignored if ``grids`` is specified.
grids : dict of array_like
Grids of values on which to evaluate the profile for each parameter. Must be a dictionary of
grids with keys corresponding to the names of the model parameters to be profiled. Overrides the ``samples`` argument.
noiselvl : float, optional
Noise level(s) of the datasets. If set to ``None`` it is determined automatically.
verbose : boolean, optional
Specifies whether to print the progress of the bootstrap analysis on the
command window, the default is false.
kargs : keyword-argument pairs
Any other keyword-argument pairs to be passed to the ``fit`` function. See the
documentation of the ``fit`` function for further details.
Returns
-------
profuq : dict of :ref:`UQResult`
Dictionary containing the profile uncertainty quantification for each profiled non-linear model parameter.
The respective parameter's results can be accessed via the model parameter name.
"""
if noiselvl is None:
noiselvl = noiselevel(y)
# Optimize the whole model to fit the data
if verbose:
print(f'Profile analysis routine started.')
print(f'Performing full model fit...')
fitresult = fit(model, y, *args, **kargs)
# Prepare the statistical threshold function
threshold = lambda coverage: noiselvl**2*chi2.ppf(coverage, df=1) + fitresult.cost
if parameters=='all':
parameters = model._parameter_list()
elif not isinstance(parameters,list):
parameters = [parameters]
# Loop over all parameters in the model
uqresults = {}
for parameter in parameters:
if np.any(getattr(model,parameter).linear):
if verbose:
print(f"Skipping linear parameter '{parameter}'.")
uqresults[parameter] = None
continue
if getattr(model, parameter).frozen:
if verbose:
print(f"Skipping frozen parameter '{parameter}'.")
uqresults[parameter] = None
continue
# Construct the values of the model parameter to profile
if grids is None:
start = np.maximum(getattr(model, parameter).lb, getattr(fitresult,parameter)-10*getattr(fitresult,f'{parameter}Uncert').std)
stop = np.minimum(getattr(model, parameter).ub, getattr(fitresult,parameter)+10*getattr(fitresult,f'{parameter}Uncert').std)
grid = np.linspace(start,stop,samples)
else:
grid = grids[parameter]
if verbose:
tqdm.write(f"Profiling model parameter '{parameter}':",end='')
# Calculate the profile objective function for the parameter
profile = np.zeros(len(grid))
for n,value in enumerate(tqdm(grid, disable=not verbose)):
# Freeze the model parameter at current value
getattr(model, parameter).freeze(value)
# Optimize the rest
with warnings.catch_warnings():
warnings.simplefilter("ignore")
fitresult_ = fit(model, y, *args, **kargs)
# Extract the objective function value
profile[n] = fitresult_.cost
# Unfreeze the parameter
getattr(model, parameter).unfreeze()
profile = {'x':np.squeeze(grid),'y':profile}
uqresults[parameter] = UQResult('profile', data=getattr(fitresult,parameter), profiles=profile, threshold=threshold, noiselvl=noiselvl)
uqresults[parameter].profile = uqresults[parameter].profile[0]
return uqresults
globalModel.eta.set(lb=0.468, ub=0.57, par0=0.520)
globalModel.kdis.set(lb=0.0, ub=0.09, par0=0.01)
globalModel.kld.set(lb=0.0, ub=1, par0=0.12)
globalModel.Dld.set(lb=2, ub=4, par0=2.5)
globalModel.rmean_dis.set(lb=3, ub=6.35, par0=3.7)
globalModel.rmean_ld.set(lb=1, ub=8, par0=2.6)
globalModel.width_dis.set(lb=0.25, ub=0.74, par0=0.44)
globalModel.width_ld.set(lb=0.2, ub=2, par0=0.7)
globalModel.lam1.set(lb=0.3, ub=0.5, par0=0.4)
globalModel.lam2.set(lb=0.0, ub=0.2, par0=0.08)
globalModel.reftime1.set(lb=0.1, ub=0.3, par0=0.2)
globalModel.reftime2_1.set(lb=3.2, ub=3.8, par0=3.4)
globalModel.reftime2_2.set(lb=2.0, ub=2.5, par0=2.2)
# Fit the model to the data
fit = dl.fit(globalModel, Vs, nonlin_tol=1e-3)
# Plot the results
plt.figure(figsize=[9, 9])
violet = '#4550e6'
orange = 'tab:orange'
plt.subplot(3, 2, 1)
plt.plot(ts[0], Vs[0], '.', color='grey', label='Data')
plt.plot(ts[0], fit.model[0], label='Fit')
plt.fill_between(ts[0],
fit.modelUncert[0].ci(95)[:, 0],
fit.modelUncert[0].ci(95)[:, 1],
alpha=0.3)
plt.ylim([0.2, 1])
plt.legend(frameon=False, loc='best')
Vmodels = [[]] * Nsignals
for n in range(Nsignals):
Vmodels[n] = dl.dipolarmodel(ts[n], r, Pmodel)
Vmodels[n].reftime.set(lb=0, ub=0.5, par0=0.2)
# Combine the individual signal models into a single global models
globalmodel = dl.merge(*Vmodels)
# Link the global parameters toghether
globalmodel = dl.link(globalmodel,
meanA=['meanA_1', 'meanA_2', 'meanA_3'],
meanB=['meanB_1', 'meanB_2', 'meanB_3'],
widthA=['widthA_1', 'widthA_2', 'widthA_3'],
widthB=['widthB_1', 'widthB_2', 'widthB_3'])
# Fit the datasets to the model globally
fit = dl.fit(globalmodel, Vexps)
# Extract the fitted fractions
fracAfit = [fit.fracA_1, fit.fracA_2, fit.fracA_3]
fracBfit = [1 - fit.fracA_1, 1 - fit.fracA_2, 1 - fit.fracA_3]
plt.figure(figsize=(10, 8))
for i in range(Nsignals):
# Get the fitted signals and confidence bands
Vfit = fit.model[i]
Vfit_ci = fit.modelUncert[i].ci(95)
# Get the fitted distributions of the two states
PAfit = fracAfit[i] * dl.dd_gauss(r, fit.meanA, fit.widthA)
PBfit = fracBfit[i] * dl.dd_gauss(r, fit.meanB, fit.widthB)