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