def run(plot=True):
## Initialize peak extraction
# Create peak extraction object
ppe = PDFPeakExtraction()
# Load the PDF from a file
ppe.loadpdf("data/Ag_nyquist_qmax30.gr")
## Set up extraction parameters.
# For convenience we add all parameters to a dictionary before passing them
# to the extraction object.
#
# The "rng" (range) parameter defines the region over which peaks will be
# extracted and fit. For the well isolated nearest-neighbor silver peak,
# which occurs near 2.9 angstroms, it is sufficient to perform extraction
# between 2 and 3.5 angstroms.
#
# The "baseline" parameter lets us define the PDF baseline, which is
# linear for a crystal. If a linear baseline is specified without
# numerical parameters diffpy.srmise attempts to estimate them from the
# data, and this is usually sufficient when peaks do not overlap much.
kwds = {}
kwds["rng"] = [2.0, 3.5]
kwds["baseline"] = Polynomial(degree=1)
# Apply peak extraction parameters.
ppe.setvars(**kwds)
## Perform peak extraction
ppe.extract()
## Save output
# The write() method saves a file which preserves all aspects of peak
# extraction and its results, by convention using the .srmise extension,
# and which can later be read by diffpy.srmise.
#
# The writepwa() method saves a file intended as a human-readable summary.
# In particular, it reports the position, width (as full-width at
# half-maximum), and area of of extracted peaks. The reported values
# are for Gaussians in the radial distribution function (RDF) corresponding
# to this PDF.
ppe.write("output/extract_single_peak.srmise")
ppe.writepwa("output/extract_single_peak.pwa")
## Plot results.
# Display plot of extracted peak. It is also possible to plot an existing
# .srmise file from the command line using
# srmise output/Ag_singlepeak.srmise --no-extract --plot
# For additional plotting options, run "srmiseplot --help".
if plot:
makeplot(ppe)
plt.show()
print "Could not save result to '%s'." % options.savefile
if options.pwafile is not None:
try:
ext.writepwa(options.pwafile)
except SrMiseFileError, err:
print err
print "Could not save pwa summary to '%s'." % options.pwafile
print ext
if cov is not None:
print cov
if options.plot:
from diffpy.srmise.applications.plot import makeplot
makeplot(ext)
plt.show()
elif options.liveplot:
plt.show()
def parsepars(mp, parseq):
"""Return actualized model from sequence of strings.
Each item in parseq must be interpretable as a float, or as
a float with the character 'c' appended. If 'c' is appended,
that parameter will be fixed.
Parameters:
mp - A ModelPart instance
parseq - A sequence of string
def run(plot=True):
## Initialize peak extraction
ppe = PDFPeakExtraction()
ppe.loadpdf("data/C60_fine_qmax21.gr")
## Set up interpolated baseline.
# The FromSequence baseline creates an interpolated baseline from provided
# r and G(r) values, either two lists or a file containing (r, G(r)) pairs.
# The baseline has no parameters. This particular baseline was estimated
# by fitting interparticle correlations of an FCC lattice of hollow
# spheres to the PDF.
blf = FromSequence("data/C60baseline.dat")
bl = blf.actualize([])
## Set up fitting parameters
# A summary of how parameters impact fitting is given below.
# "rng" - Same as peak extraction
# "baseline" - Same as peak extraction
# "qmax" and "nyquist" - If qmax > 0 and Nyquist is true, fitting is
# performed on a Nyquist-sampled grid. The data are
# never supersampled first.
# "dg" - Since the model to fit is prespecified, the uncertainty does not
# impact model complexity. Impact on refined parameter values and
# estimated uncertainties as per standard chi-square fitting.
# "pf" - The peak function used when estimating peak parameters given an
# approximate position. Unike peak extraction, peak fitting never
# alters the peak function used by initial peaks.
# "cres" - Estimation of peak parameters given an approximate position uses
# clustering for peak finding. No other effect on peak fitting.
# "supersample" - No effect.
kwds={}
kwds["rng"] = [1., 7.25]
kwds["baseline"] = bl
kwds["cres"] = 0.05
kwds["dg"] = 5000 # ad hoc, but gives each point equal weight in fit.
ppe.setvars(**kwds)
## Set up termination ripples
# Peak fitting never changes the peak function, so termination ripples
# are not applied automatically as they are in peak extraction.
# Termination ripples require setting the underlying peak function and qmax.
# In this case they ared added to the default GaussianOverR peak function.
# TerminationRipples use the estimation methods of the base peak function.
pf = TerminationRipples(ppe.pf[0], ppe.qmax)
ppe.setvars(pf=[pf])
# Specify some initial peaks using approximate positions. These use the
# peak function passed to PDFPeakExtraction instance.
rough_guess = [1.4, 2.4, 2.8, 3.6, 4.1, 4.5, 4.8, 5.2, 5.4, 5.7, 6.1]
for g in rough_guess:
ppe.estimate_peak(g)
# Specify some peaks explicitly. These may be constructed from any peak
# function, or combination of peak functions.
explicit_guess = [[6.7, .3, 100000], [7.0, .15, 50000]]
explicit_peaks = Peaks([pf.actualize(eg, in_format="pwa") \
for eg in explicit_guess])
ppe.add_peaks(explicit_peaks)
# Plot initial peaks
if plot:
makeplot(ppe)
plt.title("Initial Peaks")
# Perform fit.
ppe.fit()
## Save results
ppe.write("output/fit_initial.srmise")
ppe.writepwa("output/fit_initial.pwa")
if plot:
plt.figure()
makeplot(ppe)
plt.show()
def run(plot=True):
# Suppress mundane output
sml.setlevel("warning")
## Create multimodeling object and load diffpy.srmise results from file.
ms = MultimodelSelection()
ms.load("output/known_dG_models.dat")
ms.loadaics("output/known_dG_aics.dat")
## Use Nyquist sampling
# Standard AIC analysis assumes the data have independent uncertainties.
# Nyquist sampling minimizes correlations in the PDF, which is the closest
# approximation to independence possible for the PDF.
dr = np.pi / ms.ppe.qmax
(r, y, dr2, dy) = ms.ppe.resampledata(dr)
## Classify models
# All models are placed into classes. Models in the same class
# should be essentially identical (same peak parameters, etc.)
# up to a small tolerance determined by comparing individual peaks. The
# best model in each class essentially stands in for all the other models
# in a class in the rest of the analysis. A tolerance of 0 indicates the
# models must be exactly identical. Increasing the tolerance allows
# increasingly different models to be classified as "identical." This step
# reduces a major source of model redundancy, which otherwise weakens
# AIC-based analysis. As a rule of thumb, AIC-based analysis is robust
# to redundant poor models (since they contribute very little to the Akaike
# probabilities in any case), but redundant good models can significantly
# alter how models are ranked. See Granlund (2015) for details.
tolerance = 0.2
ms.classify(r, tolerance)
## Summarize various facts about the analysis.
num_models = len(ms.results)
num_classes = len(ms.classes)
print "------- Multimodeling Summary --------"
print "Models: %i" % num_models
print "Classes: %i (tol=%s)" % (num_classes, tolerance)
print "Range of dgs: %f-%f" % (ms.dgs[0], ms.dgs[-1])
print "Nyquist-sampled data points: %i" % len(r)
## Get dG usable as key in analysis.
# The Akaike probabilities were calculated for many assumed values of the
# experimental uncertainty dG, and each of these assumed dG is used as a
# key when obtaining the corresponding results. Numerical precision can
# make recalculating the exact value difficult, so the dg_key method returns
# the key closest to its argument.
dG = ms.dg_key(np.mean(ms.ppe.dy))
## Find "best" models.
# In short, models with greatest Akaike probability. Akaike probabilities
# can only be validly compared if they were calculated for identical data,
# namely identical PDF values *and* uncertainties, and are only reliable
# with respect to the actual experiment when using a Nyquist-sampled PDF
# with experimentally determined uncertainties.
#
# The present PDF satisifes these conditions, so the rankings below reflect
# an AIC-based estimate of which of the tested models the data best support.
print "\n--------- Model Rankings for dG = %f ---------" % dG
print "Rank Model Class Free AIC Prob File"
for i in range(len(ms.classes)):
## Generate information about best model in ith best class.
# The get(dG, *args, **kwds) method returns a tuple of values
# corresponding to string arguments for the best model in best class at
# given dG. When the corder keyword is given it returns the model from
# the corderth best class (where 0 is best, 1 is next best, etc.)
# "model" -> index of model
# "class" -> index of class
# "nfree" -> number of free parameters in corresponding model
# "aic" -> The AIC for this model given uncertainty dG
# "prob" -> The AIC probability given uncertainty dG
# These all have dedicated getter functions. For example, the model
# index can also be obtained using get_model(dG, corder=i)
(model, cls, nfree, aic, prob) = \
ms.get(dG, "model", "class", "nfree", "aic", "prob", corder=i)
filename_base = "output/known_dG_m" + str(model)
# Print info for this model
print "%4i %5i %5i %4i %10.4e %6.3f %s" \
%(i+1, model, cls, nfree, aic, prob, filename_base + ".pwa")
# A message added as a comment to saved .pwa file.
msg = [
"Multimodeling Summary",
"---------------------",
"Evaluated at dG: %s" % dG,
"Model: %i (of %i)" % (model, num_models),
"Class: %i (of %i, tol=%s)" % (cls, num_classes, tolerance),
"Akaike probability: %g" % prob,
"Rank: %i" % (i + 1),
]
msg = "\n".join(msg)
# Make this the active model
ms.setcurrent(model)
# Save .pwa
ms.ppe.writepwa(filename_base + ".pwa", msg)
# Plot this model
if plot:
plt.figure()
makeplot(ms.ppe, dcif)
plt.title("Model %i/Class %i (Rank %i, AIC prob=%f)" \
%(model, cls, i+1, prob))
# Uncomment line below to save figures.
# plt.savefig(filename_base + ".png", format="png")
## 3D plot of Akaike probabilities
# This plot shows the Akaike probabilities of all classes as a function
# of assumed uncertainty dG. This gives a rough sense of how the models
# selected by an AIC-based analysis would vary if the experimental
# uncertainties contributing to the observed G(r) were different. The
# Akaike probabilities calculated for the actual experimental uncertainty
# are highlighted.
if plot:
plt.figure()
ms.plot3dclassprobs(probfilter=[0.0, 1.], highlight=[dG])
plt.tight_layout()
# Uncomment line below to save figure.
#plt.savefig("output/known_dG_probs.png", format="png", bbox_inches="tight")
if plot:
plt.show()
def run(plot=True):
## Initialize peak extraction
# Create peak extraction object
ppe = PDFPeakExtraction()
# Load the PDF from a file
ppe.loadpdf("data/TiO2_fine_qmax26.gr")
###### Set up extraction parameters.
# In this section we'll examine the major extraction parameters in detail.
# diffpy.srmise strives to provide reasonable default values for these
# parameters. For normal use setting the range, baseline, and uncertainty
# should be sufficient.
kwds = {}
## Range
# Range defaults to the entire PDF if not specified.
kwds["rng"] = [1.5, 10.]
## dg
# diffpy.srmise selects model complexity based primarily on the uncertainty
# of the PDF. Note that very small uncertainties (<1%) can make peak
# extraction excessively slow. In general, the smaller the uncertainty the
# more complex the model. PDFs which report no uncertainty, or report
# unreliable values must be assigned one. By default, a PDF which does not
# report uncertainties uses 5% the maximum minus minimum values. Common
# causes of unreliable uncertainties include oversampling (uncertainties in
# nearby data are strongly correlated, as for this PDF) and/or
# integrated diffraction patterns obtained by a method that also introduces
# correlation to the 1D diffraction pattern. Consequently, the assumption
# of both least-squares fitting and the Akaike Information Criterion that
# the data are at least approximately independently distributed is not
# valid. In this case results obtained by diffpy.srmise may be useful,
# especially when they can be intrepreted in light of prior knowledge, but
# strong statistical conclusions cannot be drawn. For additional
# discussion of this subtle yet important issue see:
# [1] Egami and Billinge. (2012). Underneath the Bragg Peaks: Structural
# Analysis of Complex Materials (2nd ed.). Oxford: Pergamon Press.
# [2] Granlund, et al. (2015) Acta Crystallographica A, 71(4), 392-409.
# doi:10.1107/S2053273315005276
# [3] Yang, et al. (2014). Journal of Applied Crystallography, 47(4),
# 1273-1283. doi:10.1107/S1600576714010516
kwds["dg"] = 0.35 # Play with this value!
## baseline
# As a crystal PDF, a linear baseline crossing the origin is appropriate.
# Here we define the linear baseline B(r) = -.5*r + 0, and explicitly set
# the y-intercept as a fixed parameter which will not be fit. For
# crystal PDFs the theoretical value of the slope is -4*pi*rho0, where
# rho0 is the number density. Nevertheless, imperfect normalization of the
# PDF means the experimental baseline is proportional to that value.
blfunc = Polynomial(degree=1)
slope = -.65 # Play with this value!
y_intercept = 0.
kwds["baseline"] = blfunc.actualize([slope, y_intercept],
free=[True, False])
## pf
# The pf (peakfunction) parameter allows setting the shape of peaks to be
# extracted. Termination effects are added automatically to the peak
# function during extraction. In the harmonic approximation of atomic
# interactions peaks in the PDF are well approximated by a Gaussian/r.
# (Note, however, that the values used for peak parameters -- namely
# position, width, and area -- are for the Gaussian itself). diffpy.srmise
# uses width-limited peaks to reduce the likelihood of extracting
# unphysically wide peaks in regions of high overlap. The parameter
# indicates the max fwhm permitted. By default, diffpy.srmise uses a
# maximum width of 0.7, which is generally reasonable if the r-axis of the
# PDF is given in angstroms. Models where many peaks reach the maximum
# width, and models that are very sensitive to the choice in maximum width,
# are strong signs that diffpy.srmise is having difficulty finding peaks
# which are sufficiently constrained by the data.
pf = GaussianOverR(0.7)
kwds["pf"] = [pf] # Despite the list, only one entry is currently supported.
## qmax
# PDFs typically report the value of qmax (i.e. the maximum momentum
# transfer q in the measurement), but it can be specified explicitly also.
# If the PDF does not report qmax, diffpy.srmise attempts to estimate it
# directly from the data. This estimate can also be used by setting qmax
# to "automatic". An infinite qmax can be specified by setting qmax to 0,
# In that case the Nyquist rate is 0 (infinite resolution), and
# diffpy.srmise does not consider Nyquist sampling or termination effects.
kwds["qmax"] = 26.0
## nyquist
# This parameter governs whether diffpy.srmise attempts to find a model
# on a Nyquist-sampled grid with dr=pi/qmax, which is a grid where data
# uncertainties are least correlated without loss of information. By
# default this parameter is True whenever qmax > 0, and generally it
# should not need to be changed. Setting it to False allows extracted
# models retain more complexity because the data appear to have more
# statistically independent points than they truly do. For a detailed
# discussion of Nyquist sampling and the PDF see:
# [4] Farrow et al. (2011). Physical Review B, 84(13), 134105.
# doi:10.1103/PhysRevB.84.134105
kwds["nyquist"] = True
## supersample
# This parameter dictates the data be oversampled by at least this factor
# (relative to the Nyquist rate) during the early stages of peak
# extraction. If the input PDF is even more finely sampled, that level of
# sampling is used instead. The default value of 4.0 is ad hoc, but has
# been empirically sufficient. Increasing this value may help the peak-
# finding and clustering process, but reduces speed.
kwds["supersample"] = 4.0
## cres
# The cres (clustering resolution) parameter governs the sensitivity of the
# clustering method used by diffpy.srmise. In short, when the data are
# being clustered, data which are further than the clustering resolution
# from any other cluster (measured along the r-axis) are considered to be a
# new cluster rather than a member of an existing one. The default value
# is the Nyquist sampling interval pi/qmax, and on most data it should not
# greatly impact model complexity. In some cases making it smaller may
# help the peak-finding process. Here it is roughly half the Nyquist
# interval.
kwds["cres"] = 0.05
# Apply peak extraction parameters.
ppe.setvars(**kwds)
## initial_peaks
# Initial peaks are peaks which are kept fixed during the early stages of
# peak extraction, effectively condition results upon their values. Since
# initial peaks are sometimes dependent on other SrMise parameters (e.g.
# the peak function used) it is good practice to set them after other
# parameters. Although the "initial_peaks" parameter can be set as with
# the parameters above, SrMise provides helper functions to do so more
# easily. There are two basic ways to quickly specify initial peaks:
# 1) Supplying the approximate position of the peak, and letting
# diffpy.srmise estimate the peak parameters.
# 2) Explicit specification of peak parameters.
## Initial peaks from approximate positions.
# This routine estimates peak parameters by finding the peak-like cluster
# containing the specified point. It does not search for occluded peaks,
# so works best on well-separated peaks. It does, however, take any
# existing initial peaks into account during estimation.
positions = [2.0, 4.5]
for p in positions:
ppe.estimate_peak(p) # adds to initial_peaks
## Initial peaks from explicit parameters.
# Adding initial peaks explicitly is similar to defining a baseline.
# Namely, choosing a peak function and then actualizing it with given
# parameters. For this example peaks are created from the same GaussianOverR
# used during extraction, but one could use a different peak function from
# diffpy.srmise.peaks if desired. The peak parameters are given in terms
# terms of position, width (fwhm), and area, and it is important to specify
# that format is being used so they are correctly changed into the
# internal parameterization. Here two peaks are added in a region of
# overlap, and the width parameter is fixed at a reasonable value to aid
# convergence in this region.
pars = [[6.2, 0.25, 2.6],[6.45, 0.25, 2.7],[7.15, 0.25, 5]]
peaks = []
for p in pars:
peaks.append(pf.actualize(p, free=[True, False, True], in_format="pwa"))
ppe.add_peaks(peaks) # adds to initial_peaks
## Initial peaks and pruning
# While initial peaks condition what other peaks can be extracted, by
# default they can also be pruned if a simpler model appears better. To
# prevent this, they can be set as non-removable.
for ip in ppe.initial_peaks:
ip.removable = False
## Plot initial parameters
if plot:
makeplot(ppe)
plt.title("Initial Peaks")
###### Perform peak extraction
ppe.extract()
## Save output
# The write() method saves a file which preserves all aspects of peak
# extraction and its results, by convention using the .srmise extension,
# and which can later be read by diffpy.srmise.
#
# The writepwa() method saves a file intended as a human-readable summary.
# In particular, it reports the position, width (as full-width
# half-maximum), and area of of extracted peaks. The reported values
# are for Gaussians in the radial distribution function (RDF) corresponding
# to this PDF.
ppe.write("output/parameter_summary.srmise")
ppe.writepwa("output/parameter_summary.pwa")
## Plot results.
# Display plot of extracted peak. It is also possible to plot an existing
# .srmise file from the command line using
# srmise output/TiO2_parameterdetail.srmise --no-extract --plot
if plot:
plt.figure()
makeplot(ppe)
plt.show()
def run(plot=True):
# Suppress mundane output
sml.setlevel("warning")
## Create multimodeling object and load diffpy.srmise results from file.
ms = MultimodelSelection()
ms.load("output/unknown_dG_models.dat")
ms.loadaics("output/unknown_dG_aics.dat")
## Use Nyquist sampling
# Standard AIC analysis assumes the data have independent uncertainties.
# Nyquist sampling minimizes correlations in the PDF, which is the closest
# approximation to independence possible for the PDF.
dr = np.pi/ms.ppe.qmax
(r,y,dr2,dy) = ms.ppe.resampledata(dr)
## Classify models
# All models are placed into classes. Models in the same class
# should be essentially identical (same peak parameters, etc.)
# up to a small tolerance determined by comparing individual peaks. The
# best model in each class essentially stands in for all the other models
# in a class in the rest of the analysis. A tolerance of 0 indicates the
# models must be exactly identical. Increasing the tolerance allows
# increasingly different models to be classified as "identical." This step
# reduces a major source of model redundancy, which otherwise weakens
# AIC-based analysis. As a rule of thumb, AIC-based analysis is robust
# to redundant poor models (since they contribute very little to the Akaike
# probabilities in any case), but redundant good models can significantly
# alter how models are ranked. See Granlund (2015) for details.
tolerance = 0.2
ms.classify(r, tolerance)
## Summarize various facts about the analysis.
num_models = len(ms.results)
num_classes = len(ms.classes)
print "------- Multimodeling Summary --------"
print "Models: %i" %num_models
print "Classes: %i (tol=%s)" %(num_classes, tolerance)
print "Range of dgs: %f-%f" %(ms.dgs[0], ms.dgs[-1])
print "Nyquist-sampled data points: %i" %len(r)
## Find "best" models.
# In short, models with greatest Akaike probability. Akaike probabilities
# can only be validly compared if they were calculated for identical data,
# namely identical PDF values *and* uncertainties, and are only reliable
# with respect to the actual experiment when using a Nyquist-sampled PDF
# with experimentally determined uncertainties.
#
# In the present case the PDF uncertainties are not reliable, and so the
# analysis cannot be performed by specifying the experimental uncertainty
# dG. Instead, perform a weaker analysis, calculating the Akaike
# probabilities for a range of assumed dG, and identifying classes which
# have greatest probability at least once. The classes identified in this
# way have no particular information-theoretic relationship, but if the
# actual experimental uncertainty is in the interval tested, the best
# class at the experimental uncertainty is among them.
# Get classes which are best for one or more dG, and the specific dG in that
# interval at which they attain greatest Akaike probability.
best_classes = np.unique([ms.get_class(dG) for dG in ms.dgs])
best_dGs = []
for cls in best_classes:
cls_probs = [ms.get_prob(dG) if ms.get_class(dG) == cls else 0 \
for dG in ms.dgs]
dG = ms.dgs[np.argmax(cls_probs)]
best_dGs.append(dG)
print "\n--------- Best models for at least one dG ---------" %dG
print " Best dG Model Class Free AIC Prob File"
for dG in best_dGs:
## Generate information about best model.
# The get(dG, *args, **kwds) method returns a tuple of values
# corresponding to string arguments for the best model in best class at
# given dG. When the corder keyword is given it returns the model from
# the corderth best class (where 0 is best, 1 is next best, etc.)
# "model" -> index of model
# "class" -> index of class
# "nfree" -> number of free parameters in corresponding model
# "aic" -> The AIC for this model given uncertainty dG
# "prob" -> The AIC probability given uncertainty dG
# These all have dedicated getter functions.
(model, cls, nfree, aic, prob) = \
ms.get(dG, "model", "class", "nfree", "aic", "prob")
filename_base = "output/unknown_dG_m"+str(model)
# Print info for this model
print "%10.4e %5i %5i %4i %10.4e %6.3f %s" \
%(dG, model, cls, nfree, aic, prob, filename_base + ".pwa")
# A message added as a comment to saved .pwa file.
best_from = [dg for dg in ms.dgs if ms.get_class(dg) == cls]
msg = ["Multimodeling Summary",
"---------------------",
"Model: %i (of %i)" %(model, num_models),
"Class: %i (of %i, tol=%s)" %(cls, num_classes, tolerance),
"Best model from dG: %s-%s" %(best_from[0], best_from[-1]),
"Evaluated at dG: %s" %dG,
"Akaike probability: %g" %prob]
msg = "\n".join(msg)
# Make this the active model
ms.setcurrent(model)
# Save .pwa
ms.ppe.writepwa(filename_base + ".pwa", msg)
# Plot this model
if plot:
plt.figure()
makeplot(ms.ppe, dcif)
plt.title("Model %i/Class %i (Best dG=%f, AIC prob=%f)" \
%(model, cls, dG, prob))
# Uncomment line below to save figures.
# plt.savefig(filename_base + ".png", format="png")
## 3D plot of Akaike probabilities
# This plot shows the Akaike probabilities of all classes as a function
# of assumed uncertainty dG. This gives a rough sense of how the models
# selected by an AIC-based analysis would vary if the experimental
# uncertainties contributing to the observed G(r) were different. Models
# are highlighted at the various dG values found above.
if plot:
plt.figure()
ms.plot3dclassprobs(probfilter=[0.1, 1.], highlight=best_dGs)
plt.tight_layout()
# Uncomment line below to save figure.
#plt.savefig("output/unknown_dG_probs.png", format="png", bbox_inches="tight")
if plot:
plt.show()
def run(plot=True):
## Initialize peak extraction
# Create peak extraction object
ppe = PDFPeakExtraction()
# Load the PDF from a file
ppe.loadpdf("data/TiO2_fine_qmax26.gr")
###### Set up extraction parameters.
# In this section we'll examine the major extraction parameters in detail.
# diffpy.srmise strives to provide reasonable default values for these
# parameters. For normal use setting the range, baseline, and uncertainty
# should be sufficient.
kwds = {}
## Range
# Range defaults to the entire PDF if not specified.
kwds["rng"] = [1.5, 10.]
## dg
# diffpy.srmise selects model complexity based primarily on the uncertainty
# of the PDF. Note that very small uncertainties (<1%) can make peak
# extraction excessively slow. In general, the smaller the uncertainty the
# more complex the model. PDFs which report no uncertainty, or report
# unreliable values must be assigned one. By default, a PDF which does not
# report uncertainties uses 5% the maximum minus minimum values. Common
# causes of unreliable uncertainties include oversampling (uncertainties in
# nearby data are strongly correlated, as for this PDF) and/or
# integrated diffraction patterns obtained by a method that also introduces
# correlation to the 1D diffraction pattern. Consequently, the assumption
# of both least-squares fitting and the Akaike Information Criterion that
# the data are at least approximately independently distributed is not
# valid. In this case results obtained by diffpy.srmise may be useful,
# especially when they can be intrepreted in light of prior knowledge, but
# strong statistical conclusions cannot be drawn. For additional
# discussion of this subtle yet important issue see:
# [1] Egami and Billinge. (2012). Underneath the Bragg Peaks: Structural
# Analysis of Complex Materials (2nd ed.). Oxford: Pergamon Press.
# [2] Granlund, et al. (2015) Acta Crystallographica A, 71(4), 392-409.
# doi:10.1107/S2053273315005276
# [3] Yang, et al. (2014). Journal of Applied Crystallography, 47(4),
# 1273-1283. doi:10.1107/S1600576714010516
kwds["dg"] = 0.35 # Play with this value!
## baseline
# As a crystal PDF, a linear baseline crossing the origin is appropriate.
# Here we define the linear baseline B(r) = -.5*r + 0, and explicitly set
# the y-intercept as a fixed parameter which will not be fit. For
# crystal PDFs the theoretical value of the slope is -4*pi*rho0, where
# rho0 is the number density. Nevertheless, imperfect normalization of the
# PDF means the experimental baseline is proportional to that value.
blfunc = Polynomial(degree=1)
slope = -.65 # Play with this value!
y_intercept = 0.
kwds["baseline"] = blfunc.actualize([slope, y_intercept],
free=[True, False])
## pf
# The pf (peakfunction) parameter allows setting the shape of peaks to be
# extracted. Termination effects are added automatically to the peak
# function during extraction. In the harmonic approximation of atomic
# interactions peaks in the PDF are well approximated by a Gaussian/r.
# (Note, however, that the values used for peak parameters -- namely
# position, width, and area -- are for the Gaussian itself). diffpy.srmise
# uses width-limited peaks to reduce the likelihood of extracting
# unphysically wide peaks in regions of high overlap. The parameter
# indicates the max fwhm permitted. By default, diffpy.srmise uses a
# maximum width of 0.7, which is generally reasonable if the r-axis of the
# PDF is given in angstroms. Models where many peaks reach the maximum
# width, and models that are very sensitive to the choice in maximum width,
# are strong signs that diffpy.srmise is having difficulty finding peaks
# which are sufficiently constrained by the data.
pf = GaussianOverR(0.7)
kwds["pf"] = [pf
] # Despite the list, only one entry is currently supported.
## qmax
# PDFs typically report the value of qmax (i.e. the maximum momentum
# transfer q in the measurement), but it can be specified explicitly also.
# If the PDF does not report qmax, diffpy.srmise attempts to estimate it
# directly from the data. This estimate can also be used by setting qmax
# to "automatic". An infinite qmax can be specified by setting qmax to 0,
# In that case the Nyquist rate is 0 (infinite resolution), and
# diffpy.srmise does not consider Nyquist sampling or termination effects.
kwds["qmax"] = 26.0
## nyquist
# This parameter governs whether diffpy.srmise attempts to find a model
# on a Nyquist-sampled grid with dr=pi/qmax, which is a grid where data
# uncertainties are least correlated without loss of information. By
# default this parameter is True whenever qmax > 0, and generally it
# should not need to be changed. Setting it to False allows extracted
# models retain more complexity because the data appear to have more
# statistically independent points than they truly do. For a detailed
# discussion of Nyquist sampling and the PDF see:
# [4] Farrow et al. (2011). Physical Review B, 84(13), 134105.
# doi:10.1103/PhysRevB.84.134105
kwds["nyquist"] = True
## supersample
# This parameter dictates the data be oversampled by at least this factor
# (relative to the Nyquist rate) during the early stages of peak
# extraction. If the input PDF is even more finely sampled, that level of
# sampling is used instead. The default value of 4.0 is ad hoc, but has
# been empirically sufficient. Increasing this value may help the peak-
# finding and clustering process, but reduces speed.
kwds["supersample"] = 4.0
## cres
# The cres (clustering resolution) parameter governs the sensitivity of the
# clustering method used by diffpy.srmise. In short, when the data are
# being clustered, data which are further than the clustering resolution
# from any other cluster (measured along the r-axis) are considered to be a
# new cluster rather than a member of an existing one. The default value
# is the Nyquist sampling interval pi/qmax, and on most data it should not
# greatly impact model complexity. In some cases making it smaller may
# help the peak-finding process. Here it is roughly half the Nyquist
# interval.
kwds["cres"] = 0.05
# Apply peak extraction parameters.
ppe.setvars(**kwds)
## initial_peaks
# Initial peaks are peaks which are kept fixed during the early stages of
# peak extraction, effectively condition results upon their values. Since
# initial peaks are sometimes dependent on other SrMise parameters (e.g.
# the peak function used) it is good practice to set them after other
# parameters. Although the "initial_peaks" parameter can be set as with
# the parameters above, SrMise provides helper functions to do so more
# easily. There are two basic ways to quickly specify initial peaks:
# 1) Supplying the approximate position of the peak, and letting
# diffpy.srmise estimate the peak parameters.
# 2) Explicit specification of peak parameters.
## Initial peaks from approximate positions.
# This routine estimates peak parameters by finding the peak-like cluster
# containing the specified point. It does not search for occluded peaks,
# so works best on well-separated peaks. It does, however, take any
# existing initial peaks into account during estimation.
positions = [2.0, 4.5]
for p in positions:
ppe.estimate_peak(p) # adds to initial_peaks
## Initial peaks from explicit parameters.
# Adding initial peaks explicitly is similar to defining a baseline.
# Namely, choosing a peak function and then actualizing it with given
# parameters. For this example peaks are created from the same GaussianOverR
# used during extraction, but one could use a different peak function from
# diffpy.srmise.peaks if desired. The peak parameters are given in terms
# terms of position, width (fwhm), and area, and it is important to specify
# that format is being used so they are correctly changed into the
# internal parameterization. Here two peaks are added in a region of
# overlap, and the width parameter is fixed at a reasonable value to aid
# convergence in this region.
pars = [[6.2, 0.25, 2.6], [6.45, 0.25, 2.7], [7.15, 0.25, 5]]
peaks = []
for p in pars:
peaks.append(pf.actualize(p, free=[True, False, True],
in_format="pwa"))
ppe.add_peaks(peaks) # adds to initial_peaks
## Initial peaks and pruning
# While initial peaks condition what other peaks can be extracted, by
# default they can also be pruned if a simpler model appears better. To
# prevent this, they can be set as non-removable.
for ip in ppe.initial_peaks:
ip.removable = False
## Plot initial parameters
if plot:
makeplot(ppe)
plt.title("Initial Peaks")
###### Perform peak extraction
ppe.extract()
## Save output
# The write() method saves a file which preserves all aspects of peak
# extraction and its results, by convention using the .srmise extension,
# and which can later be read by diffpy.srmise.
#
# The writepwa() method saves a file intended as a human-readable summary.
# In particular, it reports the position, width (as full-width
# half-maximum), and area of of extracted peaks. The reported values
# are for Gaussians in the radial distribution function (RDF) corresponding
# to this PDF.
ppe.write("output/parameter_summary.srmise")
ppe.writepwa("output/parameter_summary.pwa")
## Plot results.
# Display plot of extracted peak. It is also possible to plot an existing
# .srmise file from the command line using
# srmise output/TiO2_parameterdetail.srmise --no-extract --plot
if plot:
plt.figure()
makeplot(ppe)
plt.show()
def run(plot=True):
# Suppress mundane output
sml.setlevel("warning")
## Create multimodeling object and load diffpy.srmise results from file.
ms = MultimodelSelection()
ms.load("output/known_dG_models.dat")
ms.loadaics("output/known_dG_aics.dat")
## Use Nyquist sampling
# Standard AIC analysis assumes the data have independent uncertainties.
# Nyquist sampling minimizes correlations in the PDF, which is the closest
# approximation to independence possible for the PDF.
dr = np.pi/ms.ppe.qmax
(r,y,dr2,dy) = ms.ppe.resampledata(dr)
## Classify models
# All models are placed into classes. Models in the same class
# should be essentially identical (same peak parameters, etc.)
# up to a small tolerance determined by comparing individual peaks. The
# best model in each class essentially stands in for all the other models
# in a class in the rest of the analysis. A tolerance of 0 indicates the
# models must be exactly identical. Increasing the tolerance allows
# increasingly different models to be classified as "identical." This step
# reduces a major source of model redundancy, which otherwise weakens
# AIC-based analysis. As a rule of thumb, AIC-based analysis is robust
# to redundant poor models (since they contribute very little to the Akaike
# probabilities in any case), but redundant good models can significantly
# alter how models are ranked. See Granlund (2015) for details.
tolerance = 0.2
ms.classify(r, tolerance)
## Summarize various facts about the analysis.
num_models = len(ms.results)
num_classes = len(ms.classes)
print "------- Multimodeling Summary --------"
print "Models: %i" %num_models
print "Classes: %i (tol=%s)" %(num_classes, tolerance)
print "Range of dgs: %f-%f" %(ms.dgs[0], ms.dgs[-1])
print "Nyquist-sampled data points: %i" %len(r)
## Get dG usable as key in analysis.
# The Akaike probabilities were calculated for many assumed values of the
# experimental uncertainty dG, and each of these assumed dG is used as a
# key when obtaining the corresponding results. Numerical precision can
# make recalculating the exact value difficult, so the dg_key method returns
# the key closest to its argument.
dG = ms.dg_key(np.mean(ms.ppe.dy))
## Find "best" models.
# In short, models with greatest Akaike probability. Akaike probabilities
# can only be validly compared if they were calculated for identical data,
# namely identical PDF values *and* uncertainties, and are only reliable
# with respect to the actual experiment when using a Nyquist-sampled PDF
# with experimentally determined uncertainties.
#
# The present PDF satisifes these conditions, so the rankings below reflect
# an AIC-based estimate of which of the tested models the data best support.
print "\n--------- Model Rankings for dG = %f ---------" %dG
print "Rank Model Class Free AIC Prob File"
for i in range(len(ms.classes)):
## Generate information about best model in ith best class.
# The get(dG, *args, **kwds) method returns a tuple of values
# corresponding to string arguments for the best model in best class at
# given dG. When the corder keyword is given it returns the model from
# the corderth best class (where 0 is best, 1 is next best, etc.)
# "model" -> index of model
# "class" -> index of class
# "nfree" -> number of free parameters in corresponding model
# "aic" -> The AIC for this model given uncertainty dG
# "prob" -> The AIC probability given uncertainty dG
# These all have dedicated getter functions. For example, the model
# index can also be obtained using get_model(dG, corder=i)
(model, cls, nfree, aic, prob) = \
ms.get(dG, "model", "class", "nfree", "aic", "prob", corder=i)
filename_base = "output/known_dG_m"+str(model)
# Print info for this model
print "%4i %5i %5i %4i %10.4e %6.3f %s" \
%(i+1, model, cls, nfree, aic, prob, filename_base + ".pwa")
# A message added as a comment to saved .pwa file.
msg = ["Multimodeling Summary",
"---------------------",
"Evaluated at dG: %s" %dG,
"Model: %i (of %i)" %(model, num_models),
"Class: %i (of %i, tol=%s)" %(cls, num_classes, tolerance),
"Akaike probability: %g" %prob,
"Rank: %i" %(i+1),]
msg = "\n".join(msg)
# Make this the active model
ms.setcurrent(model)
# Save .pwa
ms.ppe.writepwa(filename_base + ".pwa", msg)
# Plot this model
if plot:
plt.figure()
makeplot(ms.ppe, dcif)
plt.title("Model %i/Class %i (Rank %i, AIC prob=%f)" \
%(model, cls, i+1, prob))
# Uncomment line below to save figures.
# plt.savefig(filename_base + ".png", format="png")
## 3D plot of Akaike probabilities
# This plot shows the Akaike probabilities of all classes as a function
# of assumed uncertainty dG. This gives a rough sense of how the models
# selected by an AIC-based analysis would vary if the experimental
# uncertainties contributing to the observed G(r) were different. The
# Akaike probabilities calculated for the actual experimental uncertainty
# are highlighted.
if plot:
plt.figure()
ms.plot3dclassprobs(probfilter=[0.0, 1.], highlight=[dG])
plt.tight_layout()
# Uncomment line below to save figure.
#plt.savefig("output/known_dG_probs.png", format="png", bbox_inches="tight")
if plot:
plt.show()
if options.pwafile is not None:
try:
ext.writepwa(options.pwafile)
except SrMiseFileError, err:
print err
print "Could not save pwa summary to '%s'." %options.pwafile
print ext
if cov is not None:
print cov
if options.plot:
from diffpy.srmise.applications.plot import makeplot
makeplot(ext)
plt.show()
elif options.liveplot:
plt.show()
def parsepars(mp, parseq):
"""Return actualized model from sequence of strings.
Each item in parseq must be interpretable as a float, or as
a float with the character 'c' appended. If 'c' is appended,
that parameter will be fixed.
Parameters:
mp - A ModelPart instance
parseq - A sequence of string
"""
def run(plot=True):
## Initialize peak extraction
ppe = PDFPeakExtraction()
ppe.loadpdf("data/C60_fine_qmax21.gr")
## Set up interpolated baseline.
# The FromSequence baseline creates an interpolated baseline from provided
# r and G(r) values, either two lists or a file containing (r, G(r)) pairs.
# The baseline has no parameters. This particular baseline was estimated
# by fitting interparticle correlations of an FCC lattice of hollow
# spheres to the PDF.
blf = FromSequence("data/C60baseline.dat")
bl = blf.actualize([])
## Set up fitting parameters
# A summary of how parameters impact fitting is given below.
# "rng" - Same as peak extraction
# "baseline" - Same as peak extraction
# "qmax" and "nyquist" - If qmax > 0 and Nyquist is true, fitting is
# performed on a Nyquist-sampled grid. The data are
# never supersampled first.
# "dg" - Since the model to fit is prespecified, the uncertainty does not
# impact model complexity. Impact on refined parameter values and
# estimated uncertainties as per standard chi-square fitting.
# "pf" - The peak function used when estimating peak parameters given an
# approximate position. Unike peak extraction, peak fitting never
# alters the peak function used by initial peaks.
# "cres" - Estimation of peak parameters given an approximate position uses
# clustering for peak finding. No other effect on peak fitting.
# "supersample" - No effect.
kwds = {}
kwds["rng"] = [1., 7.25]
kwds["baseline"] = bl
kwds["cres"] = 0.05
kwds["dg"] = 5000 # ad hoc, but gives each point equal weight in fit.
ppe.setvars(**kwds)
## Set up termination ripples
# Peak fitting never changes the peak function, so termination ripples
# are not applied automatically as they are in peak extraction.
# Termination ripples require setting the underlying peak function and qmax.
# In this case they ared added to the default GaussianOverR peak function.
# TerminationRipples use the estimation methods of the base peak function.
pf = TerminationRipples(ppe.pf[0], ppe.qmax)
ppe.setvars(pf=[pf])
# Specify some initial peaks using approximate positions. These use the
# peak function passed to PDFPeakExtraction instance.
rough_guess = [1.4, 2.4, 2.8, 3.6, 4.1, 4.5, 4.8, 5.2, 5.4, 5.7, 6.1]
for g in rough_guess:
ppe.estimate_peak(g)
# Specify some peaks explicitly. These may be constructed from any peak
# function, or combination of peak functions.
explicit_guess = [[6.7, .3, 100000], [7.0, .15, 50000]]
explicit_peaks = Peaks([pf.actualize(eg, in_format="pwa") \
for eg in explicit_guess])
ppe.add_peaks(explicit_peaks)
# Plot initial peaks
if plot:
makeplot(ppe)
plt.title("Initial Peaks")
# Perform fit.
ppe.fit()
## Save results
ppe.write("output/fit_initial.srmise")
ppe.writepwa("output/fit_initial.pwa")
if plot:
plt.figure()
makeplot(ppe)
plt.show()