def mapestimate(data, nafc=2, sigmoid="logistic", core="ab", priors=None, cuts=None, start=None, gammaislambda=False):
dataset, pmf, nparams = sfu.make_dataset_and_pmf(data, nafc, sigmoid, core, priors, gammaislambda=gammaislambda)
cuts = sfu.get_cuts(cuts)
opt = sfr.PsiOptimizer(pmf, dataset)
estimate = opt.optimize(pmf, dataset, sfu.get_start(start, nparams) if start is not None else None)
H = pmf.ddnegllikeli(estimate, dataset)
thres = [pmf.getThres(estimate, c) for c in cuts]
slope = [pmf.getSlope(estimate, th) for th in thres]
deviance = pmf.deviance(estimate, dataset)
# convert to numpy stuff
estimate = np.array(estimate)
fisher = np.zeros((nparams, nparams))
for (i, j) in ((i, j) for i in xrange(nparams) for j in xrange(nparams)):
fisher[i, j] = sfr.doublep_value(H(i, j))
thres = np.array(thres)
slope = np.array(slope)
deviance = np.array(deviance)
return estimate, fisher, thres, slope, deviance
def mapestimate(data,
nafc=2,
sigmoid='logistic',
core='ab',
priors=None,
cuts=None,
start=None,
gammaislambda=False):
""" MAP or constrained maximum likelihood estimation for a psychometric function.
Parameters
----------
data : A list of lists or an array of data.
The first column should be stimulus intensity, the second column should
be number of correct responses (in 2AFC) or number of yes- responses (in
Yes/No), the third column should be number of trials. See also: the examples
section below.
nafc : int
Number of responses alternatives for nAFC tasks. If nafc==1 a Yes/No task is
assumed.
sigmoid : string
Name of the sigmoid to be fitted. Valid sigmoids include:
logistic
gauss
gumbel_l
gumbel_r
See `swignifit.utility.available_sigmoids()` for all available sigmoids.
core : string
\"core\"-type of the psychometric function. Valid choices include:
ab (x-a)/b
mw%g midpoint and width
linear a+bx
log a+b log(x)
See `swignifit.utility.available_cores()` for all available sigmoids.
priors : sequence of strings length number of parameters
Prior distributions on the parameters of the psychometric function.
These are expressed in the form of a list of prior names.
Valid prior choices include:
Uniform(%g,%g)
Gauss(%g,%g)
Beta(%g,%g)
Gamma(%g,%g)
nGamma(%g,%g)
if an invalid prior or `None` is selected, no constraints are imposed at all.
See `swignifit.utility.available_priors()` for all available sigmoids.
if an invalid prior is selected, no constraints are imposed on that
parameter resulting in an improper prior distribution.
cuts : sequence of floats
Cuts at which thresholds should be determined. That is if cuts =
(.25,.5,.75), thresholds (F^{-1} ( 0.25 ), F^{-1} ( 0.5 ), F^{-1} ( 0.75
)) are returned. Here F^{-1} denotes the inverse of the function
specified by sigmoid. If cuts==None, this is modified to cuts=[0.5].
start : sequence of floats of length number of model parameters
Values at which to start the optimization, if None the starting value is
determined using a coarse grid search.
Output
------
estimate, fisher, thres, slope, deviance
estimate : numpy array length nparams
the map/cml estimate
fisher : numpy array shape (nparams, nparams)
the fisher matrix
thres : numpy array length ncuts
the model prediction at the cuts
slope : numpy array length ncuts
the gradient of the psychometric function at the cuts
deviance : numpy array length 1
the deviance for the estimate
Example
-------
>>> x = [float(2*k) for k in xrange(6)]
>>> k = [34,32,40,48,50,48]
>>> n = [50]*6
>>> d = [[xx,kk,nn] for xx,kk,nn in zip(x,k,n)]
>>> priors = ('flat','flat','Uniform(0,0.1)')
>>> estimate, fisher, thres, slope, deviance = mapestimate ( d, priors=priors )
>>> estimate
array([ 2.75180624, 1.45717745, 0.01555658])
>>> deviance
array(8.0713313642328242)
"""
dataset, pmf, nparams = sfu.make_dataset_and_pmf(
data, nafc, sigmoid, core, priors, gammaislambda=gammaislambda)
cuts = sfu.get_cuts(cuts)
opt = sfr.PsiOptimizer(pmf, dataset)
estimate = opt.optimize(
pmf, dataset,
sfu.get_start(start, nparams) if start is not None else None)
H = pmf.ddnegllikeli(estimate, dataset)
thres = [pmf.getThres(estimate, c) for c in cuts]
slope = [pmf.getSlope(estimate, th) for th in thres]
deviance = pmf.deviance(estimate, dataset)
# convert to numpy stuff
estimate = np.array(estimate)
fisher = np.zeros((nparams, nparams))
for (i, j) in ((i, j) for i in xrange(nparams) for j in xrange(nparams)):
fisher[i, j] = sfr.doublep_value(H(i, j))
thres = np.array(thres)
slope = np.array(slope)
deviance = np.array(deviance)
return estimate, fisher, thres, slope, deviance
def mcmc(data,
start=None,
nsamples=10000,
nafc=2,
sigmoid='logistic',
core='mw0.1',
priors=None,
stepwidths=None,
sampler="MetropolisHastings",
gammaislambda=False):
""" Markov Chain Monte Carlo sampling for a psychometric function.
Parameters
----------
data : A list of lists or an array of data.
The first column should be stimulus intensity, the second column should
be number of correct responses (in 2AFC) or number of yes- responses (in
Yes/No), the third column should be number of trials. See also: the examples
section below.
start : sequence of floats of length number of model parameters
Starting values for the markov chain. If this is None, the MAP estimate
will be used.
nsamples : int
Number of samples to be taken from the posterior (note that due to
suboptimal sampling, this number may be much lower than the effective
number of samples.
nafc : int
Number of responses alternatives for nAFC tasks. If nafc==1 a Yes/No task is
assumed.
sigmoid : string
Name of the sigmoid to be fitted. Valid sigmoids include:
logistic
gauss
gumbel_l
gumbel_r
See `swignifit.utility.available_sigmoids()` for all available sigmoids.
core : string
\"core\"-type of the psychometric function. Valid choices include:
ab (x-a)/b
mw%g midpoint and width
linear a+bx
log a+b log(x)
See `swignifit.utility.available_cores()` for all available sigmoids.
priors : sequence of strings length number of parameters
Prior distributions on the parameters of the psychometric function.
These are expressed in the form of a list of prior names.
Valid prior choices include:
Uniform(%g,%g)
Gauss(%g,%g)
Beta(%g,%g)
Gamma(%g,%g)
nGamma(%g,%g)
if an invalid prior or `None` is selected, no constraints are imposed at all.
See `swignifit.utility.available_priors()` for all available sigmoids.
if an invalid prior is selected, no constraints are imposed on that
parameter resulting in an improper prior distribution.
stepwidths : sequence of floats of length number of model parameters
Standard deviations of the proposal distribution. The best choice is
sometimes a bit tricky here. However, as a rule of thumb we can
state: if the stepwidths are too small, the samples might not cover
the whole posterior, if the stepwidths are too large, most steps
will leave the area of high posterior density and will therefore be
rejected. Thus, in general stepwidths should be somewhere in the
middle.
sampler : string
The type of MCMC sampler to use.
See: `sw.utility.available_samplers()` for a list of available samplers.
gammaislambda : boolean
Set the gamma == lambda prior.
Output
------
(estimates, deviance, posterior_predictive_data,
posterior_predictive_deviances, posterior_predictive_Rpd,
posterior_predictive_Rkd, logposterior_ratios, accept_rate)
estimates : numpy array, shape: (nsamples, nparameters)
Parameters sampled from the posterior density of parameters given the data.
deviances : numpy array, length: nsamples
Associated deviances for each estimate
posterior_predictive_data : numpy array, shape: (nsamples, nblocks)
Data that are simulated by sampling from the joint posterior of data and
parameters. They are important for model checking.
posterior_predictive_deviances : numpy array, length: nsamples
The deviances that are associated with the posterior predictive data. A
particular way of model checking could be to compare the deviances and the
posterior predicitive deviances. For a good model these should be relatively
similar.
posterior_predictive_Rpd : numpy array, length: nsamples
Correlations between psychometric function and deviance residuals
associated with posterior predictive data
posterior_predictive_Rkd : numpy array, length: nsamples
Correlations between block index and deviance residuals associated with
posterior predictive data.
logposterior_ratios : numpy array, shape: (nsamples, nblocks)
Ratios between the full posetrior and the posterior for a single block
for all samples. Used for calculating the KL-Divergence to detrmine
influential observations in the Bayesian paradigm.
accept_rate : float
The number of proposed MCMC samples that were accepted.
Example
-------
>>> x = [float(2*k) for k in xrange(6)]
>>> k = [34,32,40,48,50,48]
>>> n = [50]*6
>>> d = [[xx,kk,nn] for xx,kk,nn in zip(x,k,n)]
>>> priors = ('Gauss(0,1000)','Gauss(0,1000)','Beta(3,100)')
>>> stepwidths = (1.,1.,0.01)
>>> (estimates, deviance, posterior_predictive_data,
posterior_predictive_deviances, posterior_predictive_Rpd,
posterior_predictive_Rkd, logposterior_ratios, accept_rate) \
= mcmc(d,nsamples=10000,priors=priors,stepwidths=stepwidths)
>>> mean(estimates[:,0])
2.4811791550665272
>>> mean(estimates[:,1])
7.4935217545849184
"""
dataset, pmf, nparams = sfu.make_dataset_and_pmf(
data, nafc, sigmoid, core, priors, gammaislambda=gammaislambda)
if start is not None:
start = sfu.get_start(start, nparams)
else:
# use mapestimate
opt = sfr.PsiOptimizer(pmf, dataset)
start = opt.optimize(pmf, dataset)
proposal = sfr.GaussRandom()
if sampler not in sfu.sampler_dict.keys():
raise sfu.PsignifitException("The sampler: " + sampler +
" is not available.")
else:
sampler = sfu.sampler_dict[sampler](pmf, dataset, proposal)
sampler.setTheta(start)
if stepwidths != None:
stepwidths = np.array(stepwidths)
if len(stepwidths.shape) == 2:
if isinstance(sampler, sfr.GenericMetropolis):
sampler.findOptimalStepwidth(sfu.make_pilotsample(stepwidths))
elif isinstance(sampler, sfr.MetropolisHastings):
sampler.setStepSize(sfr.vector_double(stepwidths.std(0)))
else:
raise sfu.PsignifitException(
"You provided a pilot sample but the selected sampler does not support pilot samples"
)
elif len(stepwidths) != nparams:
raise sfu.PsignifitException("You specified \'"+str(len(stepwidths))+\
"\' stepwidth(s), but there are \'"+str(nparams)+ "\' parameters.")
else:
if isinstance(sampler, sfr.DefaultMCMC):
for i, p in enumerate(stepwidths):
p = sfu.get_prior(p)
sampler.set_proposal(i, p)
else:
sampler.setStepSize(sfr.vector_double(stepwidths))
post = sampler.sample(nsamples)
nblocks = dataset.getNblocks()
estimates = np.zeros((nsamples, nparams))
deviance = np.zeros(nsamples)
posterior_predictive_data = np.zeros((nsamples, nblocks))
posterior_predictive_deviances = np.zeros(nsamples)
posterior_predictive_Rpd = np.zeros(nsamples)
posterior_predictive_Rkd = np.zeros(nsamples)
logposterior_ratios = np.zeros((nsamples, nblocks))
for i in xrange(nsamples):
for j in xrange(nparams):
estimates[i, j] = post.getEst(i, j)
deviance[i] = post.getdeviance(i)
for j in xrange(nblocks):
posterior_predictive_data[i, j] = post.getppData(i, j)
logposterior_ratios[i, j] = post.getlogratio(i, j)
posterior_predictive_deviances[i] = post.getppDeviance(i)
posterior_predictive_Rpd[i] = post.getppRpd(i)
posterior_predictive_Rkd[i] = post.getppRkd(i)
accept_rate = post.get_accept_rate()
return (estimates, deviance, posterior_predictive_data,
posterior_predictive_deviances, posterior_predictive_Rpd,
posterior_predictive_Rkd, logposterior_ratios, accept_rate)
def bootstrap(data,
start=None,
nsamples=2000,
nafc=2,
sigmoid="logistic",
core="ab",
priors=None,
cuts=None,
parametric=True,
gammaislambda=False):
""" Parametric bootstrap of a psychometric function.
Parameters
----------
data : A list of lists or an array of data.
The first column should be stimulus intensity, the second column should
be number of correct responses (in 2AFC) or number of yes- responses (in
Yes/No), the third column should be number of trials. See also: the examples
section below.
start : sequence of floats of length number of model parameters
Generating values for the bootstrap samples. If this is None, the
generating value will be the MAP estimate. Length should be 4 for Yes/No
and 3 for nAFC.
nsamples : number
Number of bootstrap samples to be drawn.
nafc : int
Number of alternatives for nAFC tasks. If nafc==1 a Yes/No task is
assumed.
sigmoid : string
Name of the sigmoid to be fitted. Valid sigmoids include:
logistic
gauss
gumbel_l
gumbel_r
See `swignifit.utility.available_sigmoids()` for all available sigmoids.
core : string
\"core\"-type of the psychometric function. Valid choices include:
ab (x-a)/b
mw%g midpoint and width
linear a+bx
log a+b log(x)
See `swignifit.utility.available_cores()` for all available sigmoids.
priors : sequence of strings length number of parameters
Constraints on the likelihood estimation. These are expressed in the form of a list of
prior names. Valid prior choices include:
Uniform(%g,%g)
Gauss(%g,%g)
Beta(%g,%g)
Gamma(%g,%g)
nGamma(%g,%g)
if an invalid prior or `None` is selected, no constraints are imposed at all.
See `swignifit.utility.available_priors()` for all available sigmoids.
cuts : a single number or a sequence of numbers.
Cuts indicating the performances that should be considered 'threshold'
performances. This means that in a 2AFC task, cuts==0.5 the 'threshold'
is somewhere around 75%% correct performance, depending on the lapse
rate parametric boolean to indicate whether or not the bootstrap
procedure should be parametric or not.
parametric : boolean
If `True` do parametric, otherwise do a non-parametric bootstrap.
gammaislambda : boolean
Set the gamma == lambda prior.
Returns
-------
(samples,estimates,deviance,
threshold, th_bias, th_acceleration,
slope, slope_bias, slope_accelerateion
Rkd,Rpd,outliers,influential)
samples : numpy array, shape: (nsamples, nblocks)
the bootstrap sampled data
estimates : numpy array, shape: (nsamples, nblocks)
estimated parameters associated with the data sets
deviance : numpy array, length: nsamples
deviances for the bootstraped datasets
threshold : numpy array, shape: (nsamples, ncuts)
thresholds/cuts for each bootstraped datasets
th_bias : numpy array, shape: (ncuts)
the bias term associated with the threshold
th_acc : numpy array, shape: (ncuts)
the acceleration constant associated with the threshold
slope : numpy array, shape: (nsamples, ncuts)
slope at each cuts for each bootstraped datasets
sl_bias : numpy array, shape: (ncuts)
bias term associated with the slope
sl_acc : numpy array, shape: (ncuts)
acceleration term associated with the slope
Rkd : numpy array, length: nsamples
correlations between block index and deviance residuals
Rpd : numpy array, length: nsamples
correlations between model prediction and deviance residuals
outliers : numpy array of booleans, length nblocks
points that are outliers
influential : numpy array of booleans, length nblocks
points that are influential observations
Example
-------
>>> x = [float(2*k) for k in xrange(6)]
>>> k = [34,32,40,48,50,48]
>>> n = [50]*6
>>> d = [[xx,kk,nn] for xx,kk,nn in zip(x,k,n)]
>>> priors = ('flat','flat','Uniform(0,0.1)')
>>> samples,est,D,thres,thbias,thacc,slope,slbias,slacc,Rkd,Rpd,out,influ \
= bootstrap(d,nsamples=2000,priors=priors)
>>> np.mean(est[:,0])
2.7547034408466811
>>> mean(est[:,1])
1.4057297989923003
"""
dataset, pmf, nparams = sfu.make_dataset_and_pmf(
data, nafc, sigmoid, core, priors, gammaislambda=gammaislambda)
cuts = sfu.get_cuts(cuts)
ncuts = len(cuts)
if start is not None:
start = sfu.get_start(start, nparams)
bs_list = sfr.bootstrap(nsamples, dataset, pmf, cuts, start, True,
parametric)
jk_list = sfr.jackknifedata(dataset, pmf)
nblocks = dataset.getNblocks()
# construct the massive tuple of return values
samples = np.zeros((nsamples, nblocks), dtype=np.int32)
estimates = np.zeros((nsamples, nparams))
deviance = np.zeros((nsamples))
thres = np.zeros((nsamples, ncuts))
slope = np.zeros((nsamples, ncuts))
Rpd = np.zeros((nsamples))
Rkd = np.zeros((nsamples))
for row_index in xrange(nsamples):
samples[row_index] = bs_list.getData(row_index)
estimates[row_index] = bs_list.getEst(row_index)
deviance[row_index] = bs_list.getdeviance(row_index)
thres[row_index] = [
bs_list.getThres_byPos(row_index, j) for j in xrange(ncuts)
]
slope[row_index] = [
bs_list.getSlope_byPos(row_index, j) for j in xrange(ncuts)
]
Rpd[row_index] = bs_list.getRpd(row_index)
Rkd[row_index] = bs_list.getRkd(row_index)
thacc = np.zeros((ncuts))
thbias = np.zeros((ncuts))
slacc = np.zeros((ncuts))
slbias = np.zeros((ncuts))
for cut in xrange(ncuts):
thacc[cut] = bs_list.getAcc_t(cut)
thbias[cut] = bs_list.getBias_t(cut)
slacc[cut] = bs_list.getAcc_s(cut)
slbias[cut] = bs_list.getBias_s(cut)
ci_lower = sfr.vector_double(nparams)
ci_upper = sfr.vector_double(nparams)
for param in xrange(nparams):
ci_lower[param] = bs_list.getPercentile(0.025, param)
ci_upper[param] = bs_list.getPercentile(0.975, param)
outliers = np.zeros((nblocks), dtype=np.bool)
influential = np.zeros((nblocks))
for block in xrange(nblocks):
outliers[block] = jk_list.outlier(block)
influential[block] = jk_list.influential(block, ci_lower, ci_upper)
return samples, estimates, deviance, thres, thbias, thacc, slope, slbias, slacc, Rpd, Rkd, outliers, influential
def mcmc(
data,
start=None,
nsamples=10000,
nafc=2,
sigmoid="logistic",
core="mw0.1",
priors=None,
stepwidths=None,
sampler="MetropolisHastings",
gammaislambda=False,
):
dataset, pmf, nparams = sfu.make_dataset_and_pmf(data, nafc, sigmoid, core, priors, gammaislambda=gammaislambda)
if start is not None:
start = sfu.get_start(start, nparams)
else:
# use mapestimate
opt = sfr.PsiOptimizer(pmf, dataset)
start = opt.optimize(pmf, dataset)
proposal = sfr.GaussRandom()
if sampler not in sfu.sampler_dict.keys():
raise sfu.PsignifitException("The sampler: " + sampler + " is not available.")
else:
sampler = sfu.sampler_dict[sampler](pmf, dataset, proposal)
sampler.setTheta(start)
if stepwidths != None:
stepwidths = np.array(stepwidths)
if len(stepwidths.shape) == 2:
if isinstance(sampler, sfr.GenericMetropolis):
sampler.findOptimalStepwidth(sfu.make_pilotsample(stepwidths))
elif isinstance(sampler, sfr.MetropolisHastings):
sampler.setStepSize(sfr.vector_double(stepwidths.std(0)))
else:
raise sfu.PsignifitException(
"You provided a pilot sample but the selected sampler does not support pilot samples"
)
elif len(stepwidths) != nparams:
raise sfu.PsignifitException(
"You specified '"
+ str(len(stepwidths))
+ "' stepwidth(s), but there are '"
+ str(nparams)
+ "' parameters."
)
else:
if isinstance(sampler, sfr.DefaultMCMC):
for i, p in enumerate(stepwidths):
p = sfu.get_prior(p)
sampler.set_proposal(i, p)
else:
sampler.setStepSize(sfr.vector_double(stepwidths))
post = sampler.sample(nsamples)
nblocks = dataset.getNblocks()
estimates = np.zeros((nsamples, nparams))
deviance = np.zeros(nsamples)
posterior_predictive_data = np.zeros((nsamples, nblocks))
posterior_predictive_deviances = np.zeros(nsamples)
posterior_predictive_Rpd = np.zeros(nsamples)
posterior_predictive_Rkd = np.zeros(nsamples)
logposterior_ratios = np.zeros((nsamples, nblocks))
for i in xrange(nsamples):
for j in xrange(nparams):
estimates[i, j] = post.getEst(i, j)
deviance[i] = post.getdeviance(i)
for j in xrange(nblocks):
posterior_predictive_data[i, j] = post.getppData(i, j)
logposterior_ratios[i, j] = post.getlogratio(i, j)
posterior_predictive_deviances[i] = post.getppDeviance(i)
posterior_predictive_Rpd[i] = post.getppRpd(i)
posterior_predictive_Rkd[i] = post.getppRkd(i)
accept_rate = post.get_accept_rate()
return (
estimates,
deviance,
posterior_predictive_data,
posterior_predictive_deviances,
posterior_predictive_Rpd,
posterior_predictive_Rkd,
logposterior_ratios,
accept_rate,
)
def bootstrap(
data,
start=None,
nsamples=2000,
nafc=2,
sigmoid="logistic",
core="ab",
priors=None,
cuts=None,
parametric=True,
gammaislambda=False,
):
dataset, pmf, nparams = sfu.make_dataset_and_pmf(data, nafc, sigmoid, core, priors, gammaislambda=gammaislambda)
cuts = sfu.get_cuts(cuts)
ncuts = len(cuts)
if start is not None:
start = sfu.get_start(start, nparams)
bs_list = sfr.bootstrap(nsamples, dataset, pmf, cuts, start, True, parametric)
jk_list = sfr.jackknifedata(dataset, pmf)
nblocks = dataset.getNblocks()
# construct the massive tuple of return values
samples = np.zeros((nsamples, nblocks), dtype=np.int32)
estimates = np.zeros((nsamples, nparams))
deviance = np.zeros((nsamples))
thres = np.zeros((nsamples, ncuts))
slope = np.zeros((nsamples, ncuts))
Rpd = np.zeros((nsamples))
Rkd = np.zeros((nsamples))
for row_index in xrange(nsamples):
samples[row_index] = bs_list.getData(row_index)
estimates[row_index] = bs_list.getEst(row_index)
deviance[row_index] = bs_list.getdeviance(row_index)
thres[row_index] = [bs_list.getThres_byPos(row_index, j) for j in xrange(ncuts)]
slope[row_index] = [bs_list.getSlope_byPos(row_index, j) for j in xrange(ncuts)]
Rpd[row_index] = bs_list.getRpd(row_index)
Rkd[row_index] = bs_list.getRkd(row_index)
thacc = np.zeros((ncuts))
thbias = np.zeros((ncuts))
slacc = np.zeros((ncuts))
slbias = np.zeros((ncuts))
for cut in xrange(ncuts):
thacc[cut] = bs_list.getAcc_t(cut)
thbias[cut] = bs_list.getBias_t(cut)
slacc[cut] = bs_list.getAcc_t(cut)
slbias[cut] = bs_list.getBias_t(cut)
ci_lower = sfr.vector_double(nparams)
ci_upper = sfr.vector_double(nparams)
for param in xrange(nparams):
ci_lower[param] = bs_list.getPercentile(0.025, param)
ci_upper[param] = bs_list.getPercentile(0.975, param)
outliers = np.zeros((nblocks), dtype=np.bool)
influential = np.zeros((nblocks))
for block in xrange(nblocks):
outliers[block] = jk_list.outlier(block)
influential[block] = jk_list.influential(block, ci_lower, ci_upper)
return samples, estimates, deviance, thres, thbias, thacc, slope, slbias, slacc, Rpd, Rkd, outliers, influential
def mapestimate ( data, nafc=2, sigmoid='logistic', core='ab', priors=None,
cuts = None, start=None, gammaislambda=False):
""" MAP or constrained maximum likelihood estimation for a psychometric function.
Parameters
----------
data : A list of lists or an array of data.
The first column should be stimulus intensity, the second column should
be number of correct responses (in 2AFC) or number of yes- responses (in
Yes/No), the third column should be number of trials. See also: the examples
section below.
nafc : int
Number of responses alternatives for nAFC tasks. If nafc==1 a Yes/No task is
assumed.
sigmoid : string
Name of the sigmoid to be fitted. Valid sigmoids include:
logistic (1+exp(-x))**-1 [Default]
gauss Phi(x)
gumbel_l 1 - exp(-exp(x))
gumbel_r exp(-exp(-x))
exponential x>0: 1 - exp(-x); else: 0
cauchy atan(x)/pi + 0.5
id x; only useful in conjunction with NakaRushton core
See `swignifit.utility.available_sigmoids()` for all available sigmoids.
core : string
\"core\"-type of the psychometric function. Valid choices include:
ab (x-a)/b [Default]
mw%g midpoint and width, with "%g" a number larger than 0 and less than 0.5.
mw%g corresponds to a parameterization in terms of midpoint and width of
the rising part of the sigmoid. This width is defined as the length of the
interval on which the sigmoidal part reaches from "%g" to 1-"%g".
linear a+b*x
log a+b*log(x)
weibull 2*s*m*(log(x)-log(m))/log(2) + log(log(2))
This will give you a weibull if combined with the gumbel_l sigmoid and a
reverse weibull if combined with the gumbel_r sigmoid.
poly (x/a)**b Will give you a weibull if combined with an exp sigmoid
NakaRushton The Naka-Rushton nonlinearity; should only be used with an id core
See `swignifit.utility.available_cores()` for all available cores.
priors : sequence of strings length number of parameters
Prior distributions on the parameters of the psychometric function.
These are expressed in the form of a list of prior names.
Valid prior choices include:
Uniform(%g,%g) Uniform distribution on an interval
Gauss(%g,%g) Gaussian distribution with mean and standard deviation
Beta(%g,%g) Beta distribution
Gamma(%g,%g) Gamma distribution
nGamma(%g,%g) Gamma distribution on the negative axis
invGamma(%g,%g) inverse Gamma distribution
ninvGamma(%g,%g) inverse Gamma distribution on the negative axis
if an invalid prior or `None` is selected, no constraints are imposed at all.
See `swignifit.utility.available_priors()` for all available sigmoids.
if an invalid prior is selected, no constraints are imposed on that
parameter resulting in an improper prior distribution.
cuts : sequence of floats
Cuts at which thresholds should be determined. That is if cuts =
(.25,.5,.75), thresholds (F^{-1} ( 0.25 ), F^{-1} ( 0.5 ), F^{-1} ( 0.75
)) are returned. Here F^{-1} denotes the inverse of the function
specified by sigmoid. If cuts==None, this is modified to cuts=[0.5].
start : sequence of floats of length number of model parameters
Values at which to start the optimization, if None the starting value is
determined using a coarse grid search.
Output
------
estimate, fisher, thres, slope, deviance
estimate : numpy array length nparams
the map/cml estimate
fisher : numpy array shape (nparams, nparams)
the fisher matrix
thres : numpy array length ncuts
the model prediction at the cuts
slope : numpy array length ncuts
the gradient of the psychometric function at the cuts
deviance : numpy array length 1
the deviance for the estimate
Example
-------
>>> x = [float(2*k) for k in xrange(6)]
>>> k = [34,32,40,48,50,48]
>>> n = [50]*6
>>> d = [[xx,kk,nn] for xx,kk,nn in zip(x,k,n)]
>>> priors = ('flat','flat','Uniform(0,0.1)')
>>> estimate, fisher, thres, slope, deviance = mapestimate ( d, priors=priors )
>>> estimate
array([ 2.75180624, 1.45717745, 0.01555658])
>>> deviance
array(8.0713313642328242)
"""
dataset, pmf, nparams = sfu.make_dataset_and_pmf(data,
nafc, sigmoid, core, priors, gammaislambda=gammaislambda)
cuts = sfu.get_cuts(cuts)
opt = sfr.PsiOptimizer(pmf, dataset)
estimate = opt.optimize(pmf, dataset, sfu.get_start(start, nparams) if start is not
None else None)
H = pmf.ddnegllikeli(estimate, dataset)
thres = [pmf.getThres(estimate, c) for c in cuts]
slope = [pmf.getSlope(estimate, th) for th in thres]
deviance = pmf.deviance(estimate, dataset)
# convert to numpy stuff
estimate = np.array(estimate)
fisher = np.zeros((nparams,nparams))
for (i,j) in ((i,j) for i in xrange(nparams) for j in xrange(nparams)):
fisher[i,j] = sfr.doublep_value(H(i,j))
thres = np.array(thres)
slope = np.array(slope)
deviance = np.array(deviance)
return estimate, fisher, thres, slope, deviance
def mcmc( data, start=None, nsamples=10000, nafc=2, sigmoid='logistic',
core='mw0.1', priors=None, stepwidths=None, sampler="MetropolisHastings", gammaislambda=False):
""" Markov Chain Monte Carlo sampling for a psychometric function.
Parameters
----------
data : A list of lists or an array of data.
The first column should be stimulus intensity, the second column should
be number of correct responses (in 2AFC) or number of yes- responses (in
Yes/No), the third column should be number of trials. See also: the examples
section below.
start : sequence of floats of length number of model parameters
Starting values for the markov chain. If this is None, the MAP estimate
will be used.
nsamples : int
Number of samples to be taken from the posterior (note that due to
suboptimal sampling, this number may be much lower than the effective
number of samples.
nafc : int
Number of responses alternatives for nAFC tasks. If nafc==1 a Yes/No task is
assumed.
sigmoid : string
Name of the sigmoid to be fitted. Valid sigmoids include:
logistic (1+exp(-x))**-1 [Default]
gauss Phi(x)
gumbel_l 1 - exp(-exp(x))
gumbel_r exp(-exp(-x))
exponential x>0: 1 - exp(-x); else: 0
cauchy atan(x)/pi + 0.5
id x; only useful in conjunction with NakaRushton core
See `swignifit.utility.available_sigmoids()` for all available sigmoids.
core : string
\"core\"-type of the psychometric function. Valid choices include:
ab (x-a)/b [Default]
mw%g midpoint and width, with "%g" a number larger than 0 and less than 0.5.
mw%g corresponds to a parameterization in terms of midpoint and width of
the rising part of the sigmoid. This width is defined as the length of the
interval on which the sigmoidal part reaches from "%g" to 1-"%g".
linear a+b*x
log a+b*log(x)
weibull 2*s*m*(log(x)-log(m))/log(2) + log(log(2))
This will give you a weibull if combined with the gumbel_l sigmoid and a
reverse weibull if combined with the gumbel_r sigmoid.
poly (x/a)**b Will give you a weibull if combined with an exp sigmoid
NakaRushton The Naka-Rushton nonlinearity; should only be used with an id core
See `swignifit.utility.available_cores()` for all available cores.
priors : sequence of strings length number of parameters
Prior distributions on the parameters of the psychometric function.
These are expressed in the form of a list of prior names.
Valid prior choices include:
Uniform(%g,%g) Uniform distribution on an interval
Gauss(%g,%g) Gaussian distribution with mean and standard deviation
Beta(%g,%g) Beta distribution
Gamma(%g,%g) Gamma distribution
nGamma(%g,%g) Gamma distribution on the negative axis
invGamma(%g,%g) inverse Gamma distribution
ninvGamma(%g,%g) inverse Gamma distribution on the negative axis
if an invalid prior or `None` is selected, no constraints are imposed at all.
See `swignifit.utility.available_priors()` for all available sigmoids.
if an invalid prior is selected, no constraints are imposed on that
parameter resulting in an improper prior distribution.
stepwidths : sequence of floats of length number of model parameters
Standard deviations of the proposal distribution. The best choice is
sometimes a bit tricky here. However, as a rule of thumb we can
state: if the stepwidths are too small, the samples might not cover
the whole posterior, if the stepwidths are too large, most steps
will leave the area of high posterior density and will therefore be
rejected. Thus, in general stepwidths should be somewhere in the
middle.
sampler : string
The type of MCMC sampler to use.
See: `sw.utility.available_samplers()` for a list of available samplers.
gammaislambda : boolean
Set the gamma == lambda prior.
Output
------
(estimates, deviance, posterior_predictive_data,
posterior_predictive_deviances, posterior_predictive_Rpd,
posterior_predictive_Rkd, logposterior_ratios, accept_rate)
estimates : numpy array, shape: (nsamples, nparameters)
Parameters sampled from the posterior density of parameters given the data.
deviances : numpy array, length: nsamples
Associated deviances for each estimate
posterior_predictive_data : numpy array, shape: (nsamples, nblocks)
Data that are simulated by sampling from the joint posterior of data and
parameters. They are important for model checking.
posterior_predictive_deviances : numpy array, length: nsamples
The deviances that are associated with the posterior predictive data. A
particular way of model checking could be to compare the deviances and the
posterior predicitive deviances. For a good model these should be relatively
similar.
posterior_predictive_Rpd : numpy array, length: nsamples
Correlations between psychometric function and deviance residuals
associated with posterior predictive data
posterior_predictive_Rkd : numpy array, length: nsamples
Correlations between block index and deviance residuals associated with
posterior predictive data.
logposterior_ratios : numpy array, shape: (nsamples, nblocks)
Ratios between the full posetrior and the posterior for a single block
for all samples. Used for calculating the KL-Divergence to detrmine
influential observations in the Bayesian paradigm.
accept_rate : float
The number of proposed MCMC samples that were accepted.
Example
-------
>>> x = [float(2*k) for k in xrange(6)]
>>> k = [34,32,40,48,50,48]
>>> n = [50]*6
>>> d = [[xx,kk,nn] for xx,kk,nn in zip(x,k,n)]
>>> priors = ('Gauss(0,1000)','Gauss(0,1000)','Beta(3,100)')
>>> stepwidths = (1.,1.,0.01)
>>> (estimates, deviance, posterior_predictive_data,
posterior_predictive_deviances, posterior_predictive_Rpd,
posterior_predictive_Rkd, logposterior_ratios, accept_rate) \
= mcmc(d,nsamples=10000,priors=priors,stepwidths=stepwidths)
>>> mean(estimates[:,0])
2.4811791550665272
>>> mean(estimates[:,1])
7.4935217545849184
"""
dataset, pmf, nparams = sfu.make_dataset_and_pmf(data, nafc, sigmoid, core, priors, gammaislambda=gammaislambda)
if start is not None:
start = sfu.get_start(start, nparams)
else:
# use mapestimate
opt = sfr.PsiOptimizer(pmf, dataset)
start = opt.optimize(pmf, dataset)
proposal = sfr.GaussRandom()
if sampler not in sfu.sampler_dict.keys():
raise sfu.PsignifitException("The sampler: " + sampler + " is not available.")
else:
sampler = sfu.sampler_dict[sampler](pmf, dataset, proposal)
sampler.setTheta(start)
if stepwidths != None:
stepwidths = np.array(stepwidths)
if len(stepwidths.shape)==2:
if isinstance ( sampler, sfr.GenericMetropolis ):
sampler.findOptimalStepwidth ( sfu.make_pilotsample ( stepwidths ) )
elif isinstance ( sampler, sfr.MetropolisHastings ):
sampler.setStepSize ( sfr.vector_double( stepwidths.std(0) ) )
else:
raise sfu.PsignifitException("You provided a pilot sample but the selected sampler does not support pilot samples")
elif len(stepwidths) != nparams:
raise sfu.PsignifitException("You specified \'"+str(len(stepwidths))+\
"\' stepwidth(s), but there are \'"+str(nparams)+ "\' parameters.")
else:
if isinstance ( sampler, sfr.DefaultMCMC ):
for i,p in enumerate(stepwidths):
p = sfu.get_prior(p)
sampler.set_proposal(i, p)
else:
sampler.setStepSize(sfr.vector_double(stepwidths))
post = sampler.sample(nsamples)
nblocks = dataset.getNblocks()
estimates = np.zeros((nsamples, nparams))
deviance = np.zeros(nsamples)
posterior_predictive_data = np.zeros((nsamples, nblocks))
posterior_predictive_deviances = np.zeros(nsamples)
posterior_predictive_Rpd = np.zeros(nsamples)
posterior_predictive_Rkd = np.zeros(nsamples)
logposterior_ratios = np.zeros((nsamples, nblocks))
for i in xrange(nsamples):
for j in xrange(nparams):
estimates[i, j] = post.getEst(i, j)
deviance[i] = post.getdeviance(i)
for j in xrange(nblocks):
posterior_predictive_data[i, j] = post.getppData(i, j)
logposterior_ratios[i,j] = post.getlogratio(i,j)
posterior_predictive_deviances[i] = post.getppDeviance(i)
posterior_predictive_Rpd[i] = post.getppRpd(i)
posterior_predictive_Rkd[i] = post.getppRkd(i)
accept_rate = post.get_accept_rate()
return (estimates, deviance, posterior_predictive_data,
posterior_predictive_deviances, posterior_predictive_Rpd,
posterior_predictive_Rkd, logposterior_ratios, accept_rate)
def bootstrap(data, start=None, nsamples=2000, nafc=2, sigmoid="logistic",
core="ab", priors=None, cuts=None, parametric=True, gammaislambda=False ):
""" Parametric bootstrap of a psychometric function.
Parameters
----------
data : A list of lists or an array of data.
The first column should be stimulus intensity, the second column should
be number of correct responses (in 2AFC) or number of yes- responses (in
Yes/No), the third column should be number of trials. See also: the examples
section below.
start : sequence of floats of length number of model parameters
Generating values for the bootstrap samples. If this is None, the
generating value will be the MAP estimate. Length should be 4 for Yes/No
and 3 for nAFC.
nsamples : number
Number of bootstrap samples to be drawn.
nafc : int
Number of alternatives for nAFC tasks. If nafc==1 a Yes/No task is
assumed.
sigmoid : string
Name of the sigmoid to be fitted. Valid sigmoids include:
logistic (1+exp(-x))**-1 [Default]
gauss Phi(x)
gumbel_l 1 - exp(-exp(x))
gumbel_r exp(-exp(-x))
exponential x>0: 1 - exp(-x); else: 0
cauchy atan(x)/pi + 0.5
id x; only useful in conjunction with NakaRushton core
See `swignifit.utility.available_sigmoids()` for all available sigmoids.
core : string
\"core\"-type of the psychometric function. Valid choices include:
ab (x-a)/b [Default]
mw%g midpoint and width, with "%g" a number larger than 0 and less than 0.5.
mw%g corresponds to a parameterization in terms of midpoint and width of
the rising part of the sigmoid. This width is defined as the length of the
interval on which the sigmoidal part reaches from "%g" to 1-"%g".
linear a+b*x
log a+b*log(x)
weibull 2*s*m*(log(x)-log(m))/log(2) + log(log(2))
This will give you a weibull if combined with the gumbel_l sigmoid and a
reverse weibull if combined with the gumbel_r sigmoid.
poly (x/a)**b Will give you a weibull if combined with an exp sigmoid
NakaRushton The Naka-Rushton nonlinearity; should only be used with an id core
See `swignifit.utility.available_cores()` for all available cores.
priors : sequence of strings length number of parameters
Constraints on the likelihood estimation. These are expressed in the form of a list of
prior names. Valid prior choices include:
Uniform(%g,%g) Uniform distribution on an interval
Gauss(%g,%g) Gaussian distribution with mean and standard deviation
Beta(%g,%g) Beta distribution
Gamma(%g,%g) Gamma distribution
nGamma(%g,%g) Gamma distribution on the negative axis
invGamma(%g,%g) inverse Gamma distribution
ninvGamma(%g,%g) inverse Gamma distribution on the negative axis
if an invalid prior or `None` is selected, no constraints are imposed at all.
See `swignifit.utility.available_priors()` for all available sigmoids.
cuts : a single number or a sequence of numbers.
Cuts indicating the performances that should be considered 'threshold'
performances. This means that in a 2AFC task, cuts==0.5 the 'threshold'
is somewhere around 75%% correct performance, depending on the lapse
rate parametric boolean to indicate whether or not the bootstrap
procedure should be parametric or not.
parametric : boolean
If `True` do parametric, otherwise do a non-parametric bootstrap.
gammaislambda : boolean
Set the gamma == lambda prior.
Returns
-------
(samples,estimates,deviance,
threshold, th_bias, th_acceleration,
slope, slope_bias, slope_accelerateion
Rkd,Rpd,outliers,influential)
samples : numpy array, shape: (nsamples, nblocks)
the bootstrap sampled data
estimates : numpy array, shape: (nsamples, nblocks)
estimated parameters associated with the data sets
deviance : numpy array, length: nsamples
deviances for the bootstraped datasets
threshold : numpy array, shape: (nsamples, ncuts)
thresholds/cuts for each bootstraped datasets
th_bias : numpy array, shape: (ncuts)
the bias term associated with the threshold
th_acc : numpy array, shape: (ncuts)
the acceleration constant associated with the threshold
slope : numpy array, shape: (nsamples, ncuts)
slope at each cuts for each bootstraped datasets
sl_bias : numpy array, shape: (ncuts)
bias term associated with the slope
sl_acc : numpy array, shape: (ncuts)
acceleration term associated with the slope
Rkd : numpy array, length: nsamples
correlations between block index and deviance residuals
Rpd : numpy array, length: nsamples
correlations between model prediction and deviance residuals
outliers : numpy array of booleans, length nblocks
points that are outliers
influential : numpy array of booleans, length nblocks
points that are influential observations
Example
-------
>>> x = [float(2*k) for k in xrange(6)]
>>> k = [34,32,40,48,50,48]
>>> n = [50]*6
>>> d = [[xx,kk,nn] for xx,kk,nn in zip(x,k,n)]
>>> priors = ('flat','flat','Uniform(0,0.1)')
>>> samples,est,D,thres,thbias,thacc,slope,slbias,slacc,Rkd,Rpd,out,influ \
= bootstrap(d,nsamples=2000,priors=priors)
>>> np.mean(est[:,0])
2.7547034408466811
>>> mean(est[:,1])
1.4057297989923003
"""
dataset, pmf, nparams = sfu.make_dataset_and_pmf(data, nafc, sigmoid, core, priors, gammaislambda=gammaislambda)
cuts = sfu.get_cuts(cuts)
ncuts = len(cuts)
if start is not None:
start = sfu.get_start(start, nparams)
bs_list = sfr.bootstrap(nsamples, dataset, pmf, cuts, start, True, parametric)
jk_list = sfr.jackknifedata(dataset, pmf)
nblocks = dataset.getNblocks()
# construct the massive tuple of return values
samples = np.zeros((nsamples, nblocks), dtype=np.int32)
estimates = np.zeros((nsamples, nparams))
deviance = np.zeros((nsamples))
thres = np.zeros((nsamples, ncuts))
slope = np.zeros((nsamples, ncuts))
Rpd = np.zeros((nsamples))
Rkd = np.zeros((nsamples))
for row_index in xrange(nsamples):
samples[row_index] = bs_list.getData(row_index)
estimates[row_index] = bs_list.getEst(row_index)
deviance[row_index] = bs_list.getdeviance(row_index)
thres[row_index] = [bs_list.getThres_byPos(row_index, j) for j in xrange(ncuts)]
slope[row_index] = [bs_list.getSlope_byPos(row_index, j) for j in xrange(ncuts)]
Rpd[row_index] = bs_list.getRpd(row_index)
Rkd[row_index] = bs_list.getRkd(row_index)
thacc = np.zeros((ncuts))
thbias = np.zeros((ncuts))
slacc = np.zeros((ncuts))
slbias = np.zeros((ncuts))
for cut in xrange(ncuts):
thacc[cut] = bs_list.getAcc_t(cut)
thbias[cut] = bs_list.getBias_t(cut)
slacc[cut] = bs_list.getAcc_s(cut)
slbias[cut] = bs_list.getBias_s(cut)
ci_lower = sfr.vector_double(nparams)
ci_upper = sfr.vector_double(nparams)
for param in xrange(nparams):
ci_lower[param] = bs_list.getPercentile(0.025, param)
ci_upper[param] = bs_list.getPercentile(0.975, param)
outliers = np.zeros((nblocks), dtype=np.bool)
influential = np.zeros((nblocks))
for block in xrange(nblocks):
outliers[block] = jk_list.outlier(block)
influential[block] = jk_list.influential(block, ci_lower, ci_upper)
return samples, estimates, deviance, thres, thbias, thacc, slope, slbias, slacc, Rpd, Rkd, outliers, influential