def main():
parser = argparse.ArgumentParser()
parser.add_argument("--num-threads", type=int, default=9,
help="Size of the thread pool.")
parser.add_argument("--trials-per-subject", type=int, default=100,
help="Number of trials from each subject to be used in "
"the analysis; if smaller than 1, all trials are used.")
parser.add_argument("--num-simulations", type=int, default=400,
help="Number of simulations to be generated per trial "
"condition.")
parser.add_argument("--range-d", nargs="+", type=float,
default=[0.003, 0.006, 0.009],
help="Search range for parameter d.")
parser.add_argument("--range-sigma", nargs="+", type=float,
default=[0.03, 0.06, 0.09],
help="Search range for parameter sigma.")
parser.add_argument("--range-theta", nargs="+", type=float,
default=[0.3, 0.5, 0.7],
help="Search range for parameter theta.")
parser.add_argument("--expdata-file-name", type=str, default="expdata.csv",
help="Name of experimental data file.")
parser.add_argument("--fixations-file-name", type=str,
default="fixations.csv", help="Name of fixations file.")
parser.add_argument("--use-cis-trials", default=False, action="store_true",
help="Use CIS trials in the analysis.")
parser.add_argument("--use-trans-trials", default=False,
action="store_true", help="Use TRANS trials in the "
"analysis.")
parser.add_argument("--save-simulations", default=False,
action="store_true", help="Save simulations to CSV.")
parser.add_argument("--verbose", default=False, action="store_true",
help="Increase output verbosity.")
args = parser.parse_args()
pool = Pool(args.num_threads)
# Load experimental data from CSV file.
try:
data = load_data_from_csv(
args.expdata_file_name, args.fixations_file_name,
useAngularDists=True)
except Exception as e:
print("An exception occurred while loading the data: " + str(e))
return
choice = data.choice
valueLeft = data.valueLeft
valueRight = data.valueRight
fixItem = data.fixItem
fixTime = data.fixTime
isCisTrial = data.isCisTrial
isTransTrial = data.isTransTrial
# Maximum likelihood estimation.
# Grid search on the parameters of the model using odd trials only.
if args.verbose:
print("Starting grid search...")
models = list()
listParams = list()
for d in args.range_d:
for theta in args.range_theta:
for sigma in args.range_sigma:
models.append((d, theta, sigma))
params = (choice, valueLeft, valueRight, fixItem, fixTime, d,
theta, sigma, args.trials_per_subject, True, False,
isCisTrial, isTransTrial, args.use_cis_trials,
args.use_trans_trials, args.verbose)
listParams.append(params)
results = pool.map(get_model_nll_wrapper, listParams)
# Get optimal parameters.
minNegLogLikeIdx = results.index(min(results))
optimD = models[minNegLogLikeIdx][0]
optimTheta = models[minNegLogLikeIdx][1]
optimSigma = models[minNegLogLikeIdx][2]
if args.verbose:
print("Finished grid search!")
print("Optimal d: " + str(optimD))
print("Optimal theta: " + str(optimTheta))
print("Optimal sigma: " + str(optimSigma))
print("Min NLL: " + str(min(results)))
# Get empirical distributions from even trials only.
try:
dists = get_empirical_distributions(
valueLeft, valueRight, fixItem, fixTime, useOddTrials=False,
useEvenTrials=True, isCisTrial=isCisTrial,
isTransTrial=isTransTrial, useCisTrials=args.use_cis_trials,
useTransTrials=args.use_trans_trials)
except Exception as e:
print("An exception occurred while getting empirical distributions: " +
str(e))
return
probLeftFixFirst = dists.probLeftFixFirst
distLatencies = dists.distLatencies
distTransitions = dists.distTransitions
distFixations = dists.distFixations
# Parameters for generating simulations.
orientations = range(-15,20,5)
trialConditions = list()
for oLeft in orientations:
for oRight in orientations:
vLeft = np.absolute((np.absolute(oLeft) - 15) / 5)
vRight = np.absolute((np.absolute(oRight) - 15) / 5)
if oLeft != oRight and args.use_cis_trials and oLeft * oRight >= 0:
trialConditions.append((vLeft, vRight))
elif (oLeft != oRight and args.use_trans_trials and
oLeft * oRight <= 0):
trialConditions.append((vLeft, vRight))
# Generate simulations using the empirical distributions and the
# estimated parameters.
try:
simul = run_simulations(
probLeftFixFirst, distLatencies, distTransitions, distFixations,
args.num_simulations, trialConditions, optimD, optimTheta,
sigma=optimSigma)
except Exception as e:
print("An exception occurred while running simulations: " + str(e))
return
simulRT = simul.RT
simulChoice = simul.choice
simulValueLeft = simul.valueLeft
simulValueRight = simul.valueRight
simulFixItem = simul.fixItem
simulFixTime = simul.fixTime
simulFixRDV = simul.fixRDV
if args.save_simulations:
totalTrials = args.num_simulations * len(trialConditions)
save_simulations_to_csv(
simulChoice, simulRT, simulValueLeft, simulValueRight, simulFixItem,
simulFixTime, simulFixRDV, totalTrials)
def main():
parser = argparse.ArgumentParser()
parser.add_argument("--num-threads", type=int, default=9,
help="Size of the thread pool.")
parser.add_argument("--num-trials", type=int, default=800,
help="Number of artificial data trials to be generated "
"per trial condition.")
parser.add_argument("--d", type=float, default=0.006,
help="aDDM parameter for generating artificial data.")
parser.add_argument("--sigma", type=float, default=0.08,
help="aDDM parameter for generating artificial data.")
parser.add_argument("--theta", type=float, default=0.5,
help="aDDM parameter for generating artificial data.")
parser.add_argument("--range-d", nargs="+", type=float,
default=[0.005, 0.006, 0.007],
help="Search range for parameter d.")
parser.add_argument("--range-sigma", nargs="+", type=float,
default=[0.065, 0.08, 0.095],
help="Search range for parameter sigma.")
parser.add_argument("--range-theta", nargs="+", type=float,
default=[0.4, 0.5, 0.6],
help="Search range for parameter theta.")
parser.add_argument("--expdata-file-name", type=str, default="expdata.csv",
help="Name of experimental data file.")
parser.add_argument("--fixations-file-name", type=str,
default="fixations.csv", help="Name of fixations file.")
parser.add_argument("--verbose", default=False, action="store_true",
help="Increase output verbosity.")
args = parser.parse_args()
pool = Pool(args.num_threads)
# Load experimental data from CSV file.
try:
data = load_data_from_csv(
args.expdata_file_name, args.fixations_file_name,
useAngularDists=True)
except Exception as e:
print("An exception occurred while loading the data: " + str(e))
return
valueLeft = data.valueLeft
valueRight = data.valueRight
fixItem = data.fixItem
fixTime = data.fixTime
# Get empirical distributions.
try:
dists = get_empirical_distributions(valueLeft, valueRight, fixItem,
fixTime)
except Exception as e:
print("An exception occurred while getting empirical distributions: " +
str(e))
return
probLeftFixFirst = dists.probLeftFixFirst
distLatencies = dists.distLatencies
distTransitions = dists.distTransitions
distFixations = dists.distFixations
# Trial conditions for artificial data generation.
orientations = range(-15,20,5)
trialConditions = list()
for oLeft in orientations:
for oRight in orientations:
if oLeft != oRight:
vLeft = np.absolute((np.absolute(oLeft) - 15) / 5)
vRight = np.absolute((np.absolute(oRight) - 15) / 5)
trialConditions.append((vLeft, vRight))
# Generate artificial data.
if args.verbose:
print("Running simulations...")
try:
simul = run_simulations(
probLeftFixFirst, distLatencies, distTransitions, distFixations,
args.num_trials, trialConditions, args.d, args.theta,
sigma=args.sigma)
except Exception as e:
print("An exception occurred while generating artificial data: " +
str(e))
return
simulChoice = simul.choice
simulValueLeft = simul.valueLeft
simulValueRight = simul.valueRight
simulFixItem = simul.fixItem
simulFixTime = simul.fixTime
# Grid search to recover the parameters.
if args.verbose:
print("Starting grid search...")
numModels = (len(args.range_d) * len(args.range_sigma) *
len(args.range_theta))
models = list()
posteriors = dict()
for d in args.range_d:
for theta in args.range_theta:
for sigma in args.range_sigma:
model = (d, theta, sigma)
models.append(model)
posteriors[model] = 1. / numModels
trials = simulChoice.keys()
for trial in trials:
listParams = list()
for model in models:
listParams.append(
(simulChoice[trial], simulValueLeft[trial],
simulValueRight[trial], simulFixItem[trial],
simulFixTime[trial], model[0], model[1], model[2]))
try:
likelihoods = pool.map(get_trial_likelihood_wrapper, listParams)
except Exception as e:
print("An exception occurred during the likelihood computation for "
"trial " + str(trial) + ": " + str(e))
return
# Get the denominator for normalizing the posteriors.
i = 0
denominator = 0
for model in models:
denominator += posteriors[model] * likelihoods[i]
i += 1
if denominator == 0:
continue
# Calculate the posteriors after this trial.
i = 0
for model in models:
prior = posteriors[model]
posteriors[model] = likelihoods[i] * prior / denominator
i += 1
if args.verbose and trial % 200 == 0:
for model in posteriors:
print("P" + str(model) + " = " + str(posteriors[model]))
print("Sum: " + str(sum(posteriors.values())))
if args.verbose:
for model in posteriors:
print("P" + str(model) + " = " + str(posteriors[model]))
print("Sum: " + str(sum(posteriors.values())))
def main():
parser = argparse.ArgumentParser()
parser.add_argument("subject", type=str, help="Subject name.")
parser.add_argument("--num-threads",
type=int,
default=9,
help="Size of the thread pool.")
parser.add_argument(
"--num-trials",
type=int,
default=200,
help="Number of artificial data trials to be generated "
"per trial condition.")
parser.add_argument("--d",
type=float,
default=0.006,
help="aDDM parameter for generating artificial data.")
parser.add_argument("--sigma",
type=float,
default=0.08,
help="aDDM parameter for generating artificial data.")
parser.add_argument("--theta",
type=float,
default=0.5,
help="aDDM parameter for generating artificial data.")
parser.add_argument("--range-d",
nargs="+",
type=float,
default=[0.005, 0.006, 0.007],
help="Search range for parameter d.")
parser.add_argument("--range-sigma",
nargs="+",
type=float,
default=[0.065, 0.08, 0.095],
help="Search range for parameter sigma.")
parser.add_argument("--range-theta",
nargs="+",
type=float,
default=[0.4, 0.5, 0.6],
help="Search range for parameter theta.")
parser.add_argument("--expdata-file-name",
type=str,
default="expdata.csv",
help="Name of experimental data file.")
parser.add_argument("--fixations-file-name",
type=str,
default="fixations.csv",
help="Name of fixations file.")
parser.add_argument("--verbose",
default=False,
action="store_true",
help="Increase output verbosity.")
args = parser.parse_args()
pool = Pool(args.num_threads)
valueLeft = dict()
valueRight = dict()
fixItem = dict()
fixTime = dict()
# Load experimental data from CSV file.
try:
data = load_data_from_csv(args.expdata_file_name,
args.fixations_file_name,
useAngularDists=True)
except Exception as e:
print("An exception occurred while loading the data: " + str(e))
return
valueLeft[args.subject] = data.valueLeft[args.subject]
valueRight[args.subject] = data.valueRight[args.subject]
fixItem[args.subject] = data.fixItem[args.subject]
fixTime[args.subject] = data.fixTime[args.subject]
# Get empirical distributions.
try:
dists = get_empirical_distributions(valueLeft, valueRight, fixItem,
fixTime)
except Exception as e:
print("An exception occurred while getting empirical distributions: " +
str(e))
return
probLeftFixFirst = dists.probLeftFixFirst
distLatencies = dists.distLatencies
distTransitions = dists.distTransitions
distFixations = dists.distFixations
# Trial conditions for artificial data generation.
orientations = range(-15, 20, 5)
trialConditions = list()
for oLeft in orientations:
for oRight in orientations:
if oLeft != oRight:
vLeft = np.absolute((np.absolute(oLeft) - 15) / 5)
vRight = np.absolute((np.absolute(oRight) - 15) / 5)
trialConditions.append((vLeft, vRight))
# Generate artificial data.
if args.verbose:
print("Running simulations...")
try:
simul = run_simulations(probLeftFixFirst,
distLatencies,
distTransitions,
distFixations,
args.num_trials,
trialConditions,
args.d,
args.theta,
sigma=args.sigma)
except Exception as e:
print("An exception occurred while generating artificial data: " +
str(e))
return
simulChoice = simul.choice
simulValueLeft = simul.valueLeft
simulValueRight = simul.valueRight
simulFixItem = simul.fixItem
simulFixTime = simul.fixTime
# Grid search to recover the parameters.
if args.verbose:
print("Starting grid search...")
numModels = (len(args.range_d) * len(args.range_theta) *
len(args.range_sigma))
models = list()
posteriors = dict()
for d in args.range_d:
for theta in args.range_theta:
for sigma in args.range_sigma:
model = (d, theta, sigma)
models.append(model)
posteriors[model] = 1. / numModels
trials = simulChoice.keys()
for trial in trials:
listParams = list()
for model in models:
listParams.append(
(simulChoice[trial], simulValueLeft[trial],
simulValueRight[trial], simulFixItem[trial],
simulFixTime[trial], model[0], model[1], model[2]))
try:
likelihoods = pool.map(get_trial_likelihood_wrapper, listParams)
except Exception as e:
print(
"An exception occurred during the likelihood computation for "
"trial " + str(trial) + ": " + str(e))
return
# Get the denominator for normalizing the posteriors.
i = 0
denominator = 0
for model in models:
denominator += posteriors[model] * likelihoods[i]
i += 1
if denominator == 0:
continue
# Calculate the posteriors after this trial.
i = 0
for model in models:
prior = posteriors[model]
posteriors[model] = likelihoods[i] * prior / denominator
i += 1
if args.verbose and trial % 200 == 0:
for model in posteriors:
print("P" + str(model) + " = " + str(posteriors[model]))
print("Sum: " + str(sum(posteriors.values())))
if args.verbose:
for model in posteriors:
print("P" + str(model) + " = " + str(posteriors[model]))
print("Sum: " + str(sum(posteriors.values())))
def main():
parser = argparse.ArgumentParser()
parser.add_argument("subject", type=str, help="Subject name")
parser.add_argument("--num-threads", type=int, default=9,
help="Size of the thread pool.")
parser.add_argument("--num-trials", type=int, default=100,
help="Number of trials to be used in the analysis; if "
"smaller than 1, all trials are used.")
parser.add_argument("--num-simulations", type=int, default=32,
help="Number of simulations to be generated per trial "
"condition.")
parser.add_argument("--range-d", nargs="+", type=float,
default=[0.003, 0.006, 0.009],
help="Search range for parameter d.")
parser.add_argument("--range-sigma", nargs="+", type=float,
default=[0.03, 0.06, 0.09],
help="Search range for parameter sigma.")
parser.add_argument("--range-theta", nargs="+", type=float,
default=[0.3, 0.5, 0.7],
help="Search range for parameter theta.")
parser.add_argument("--expdata-file-name", type=str, default="expdata.csv",
help="Name of experimental data file.")
parser.add_argument("--fixations-file-name", type=str,
default="fixations.csv", help="Name of fixations file.")
parser.add_argument("--save-simulations", default=False,
action="store_true", help="Save simulations to CSV.")
parser.add_argument("--save-figures", default=False,
action="store_true", help="Save figures comparing "
"choice and RT curves for data and simulations.")
parser.add_argument("--verbose", default=False, action="store_true",
help="Increase output verbosity.")
args = parser.parse_args()
pool = Pool(args.num_threads)
choice = dict()
valueLeft = dict()
valueRight = dict()
fixItem = dict()
fixTime = dict()
# Load experimental data from CSV file.
try:
data = load_data_from_csv(
args.expdata_file_name, args.fixations_file_name,
useAngularDists=True)
except Exception as e:
print("An exception occurred while loading the data: " + str(e))
return
choice[args.subject] = data.choice[args.subject]
valueLeft[args.subject] = data.valueLeft[args.subject]
valueRight[args.subject] = data.valueRight[args.subject]
fixItem[args.subject] = data.fixItem[args.subject]
fixTime[args.subject] = data.fixTime[args.subject]
# Maximum likelihood estimation using odd trials only.
# Grid search on the parameters of the model.
if args.verbose:
print("Starting grid search for subject " + args.subject + "...")
models = list()
listParams = list()
for d in args.range_d:
for theta in args.range_theta:
for sigma in args.range_sigma:
models.append((d, theta, sigma))
params = (choice, valueLeft, valueRight, fixItem, fixTime, d,
theta, sigma, args.num_trials, True, False,
args.verbose)
listParams.append(params)
results = pool.map(get_model_nll_wrapper, listParams)
# Get optimal parameters.
minNegLogLikeIdx = results.index(min(results))
optimD = models[minNegLogLikeIdx][0]
optimTheta = models[minNegLogLikeIdx][1]
optimSigma = models[minNegLogLikeIdx][2]
if args.verbose:
print("Finished grid search!")
print("Optimal d: " + str(optimD))
print("Optimal theta: " + str(optimTheta))
print("Optimal sigma: " + str(optimSigma))
print("Min NLL: " + str(min(results)))
# Get empirical distributions from even trials.
try:
dists = get_empirical_distributions(
valueLeft, valueRight, fixItem, fixTime, useOddTrials=False,
useEvenTrials=True)
except Exception as e:
print("An exception occurred while getting empirical distributions: " +
str(e))
return
probLeftFixFirst = dists.probLeftFixFirst
distLatencies = dists.distLatencies
distTransitions = dists.distTransitions
distFixations = dists.distFixations
# Trial conditions for generating simulations.
orientations = range(-15,20,5)
trialConditions = list()
for oLeft in orientations:
for oRight in orientations:
if oLeft != oRight:
vLeft = np.absolute((np.absolute(oLeft) - 15) / 5)
vRight = np.absolute((np.absolute(oRight) - 15) / 5)
trialConditions.append((vLeft, vRight))
# Generate simulations using the even trials distributions and the
# estimated parameters.
try:
simul = run_simulations(
probLeftFixFirst, distLatencies, distTransitions, distFixations,
args.num_simulations, trialConditions, optimD, optimTheta,
sigma=optimSigma)
except Exception as e:
print("An exception occurred while running simulations: " + str(e))
return
simulRT = simul.RT
simulChoice = simul.choice
simulValueLeft = simul.valueLeft
simulValueRight = simul.valueRight
simulFixItem = simul.fixItem
simulFixTime = simul.fixTime
simulFixRDV = simul.fixRDV
totalTrials = args.num_simulations * len(trialConditions)
if args.save_simulations:
save_simulations_to_csv(
simulChoice, simulRT, simulValueLeft, simulValueRight, simulFixItem,
simulFixTime, simulFixRDV, totalTrials)
if args.save_figures:
# Create pdf file to save figures.
pp = PdfPages(
"figures_" + str(optimD) + "_" + str(optimTheta) + "_" +
str(optimSigma) + "_" + str(args.num_simulations) + ".pdf")
# Generate choice and RT curves for real data (odd trials) and
# simulations (generated from even trials).
fig1 = generate_choice_curves(
choice, valueLeft, valueRight, simulChoice, simulValueLeft,
simulValueRight, totalTrials)
pp.savefig(fig1)
fig2 = generate_rt_curves(
RT, valueLeft, valueRight, simulRT, simulValueLeft, simulValueRight,
totalTrials)
pp.savefig(fig2)
pp.close()
def main():
parser = argparse.ArgumentParser()
parser.add_argument("subject", type=str, help="Subject name")
parser.add_argument("--num-threads",
type=int,
default=9,
help="Size of the thread pool.")
parser.add_argument("--num-trials",
type=int,
default=100,
help="Number of trials to be used in the analysis; if "
"smaller than 1, all trials are used.")
parser.add_argument("--num-simulations",
type=int,
default=32,
help="Number of simulations to be generated per trial "
"condition.")
parser.add_argument("--range-d",
nargs="+",
type=float,
default=[0.003, 0.006, 0.009],
help="Search range for parameter d.")
parser.add_argument("--range-sigma",
nargs="+",
type=float,
default=[0.03, 0.06, 0.09],
help="Search range for parameter sigma.")
parser.add_argument("--range-theta",
nargs="+",
type=float,
default=[0.3, 0.5, 0.7],
help="Search range for parameter theta.")
parser.add_argument("--expdata-file-name",
type=str,
default="expdata.csv",
help="Name of experimental data file.")
parser.add_argument("--fixations-file-name",
type=str,
default="fixations.csv",
help="Name of fixations file.")
parser.add_argument("--save-simulations",
default=False,
action="store_true",
help="Save simulations to CSV.")
parser.add_argument("--save-figures",
default=False,
action="store_true",
help="Save figures comparing "
"choice and RT curves for data and simulations.")
parser.add_argument("--verbose",
default=False,
action="store_true",
help="Increase output verbosity.")
args = parser.parse_args()
pool = Pool(args.num_threads)
choice = dict()
valueLeft = dict()
valueRight = dict()
fixItem = dict()
fixTime = dict()
# Load experimental data from CSV file.
try:
data = load_data_from_csv(args.expdata_file_name,
args.fixations_file_name,
useAngularDists=True)
except Exception as e:
print("An exception occurred while loading the data: " + str(e))
return
choice[args.subject] = data.choice[args.subject]
valueLeft[args.subject] = data.valueLeft[args.subject]
valueRight[args.subject] = data.valueRight[args.subject]
fixItem[args.subject] = data.fixItem[args.subject]
fixTime[args.subject] = data.fixTime[args.subject]
# Maximum likelihood estimation using odd trials only.
# Grid search on the parameters of the model.
if args.verbose:
print("Starting grid search for subject " + args.subject + "...")
models = list()
listParams = list()
for d in args.range_d:
for theta in args.range_theta:
for sigma in args.range_sigma:
models.append((d, theta, sigma))
params = (choice, valueLeft, valueRight, fixItem, fixTime, d,
theta, sigma, args.num_trials, True, False,
args.verbose)
listParams.append(params)
results = pool.map(get_model_nll_wrapper, listParams)
# Get optimal parameters.
minNegLogLikeIdx = results.index(min(results))
optimD = models[minNegLogLikeIdx][0]
optimTheta = models[minNegLogLikeIdx][1]
optimSigma = models[minNegLogLikeIdx][2]
if args.verbose:
print("Finished grid search!")
print("Optimal d: " + str(optimD))
print("Optimal theta: " + str(optimTheta))
print("Optimal sigma: " + str(optimSigma))
print("Min NLL: " + str(min(results)))
# Get empirical distributions from even trials.
try:
dists = get_empirical_distributions(valueLeft,
valueRight,
fixItem,
fixTime,
useOddTrials=False,
useEvenTrials=True)
except Exception as e:
print("An exception occurred while getting empirical distributions: " +
str(e))
return
probLeftFixFirst = dists.probLeftFixFirst
distLatencies = dists.distLatencies
distTransitions = dists.distTransitions
distFixations = dists.distFixations
# Trial conditions for generating simulations.
orientations = range(-15, 20, 5)
trialConditions = list()
for oLeft in orientations:
for oRight in orientations:
if oLeft != oRight:
vLeft = np.absolute((np.absolute(oLeft) - 15) / 5)
vRight = np.absolute((np.absolute(oRight) - 15) / 5)
trialConditions.append((vLeft, vRight))
# Generate simulations using the even trials distributions and the
# estimated parameters.
try:
simul = run_simulations(probLeftFixFirst,
distLatencies,
distTransitions,
distFixations,
args.num_simulations,
trialConditions,
optimD,
optimTheta,
sigma=optimSigma)
except Exception as e:
print("An exception occurred while running simulations: " + str(e))
return
simulRT = simul.RT
simulChoice = simul.choice
simulValueLeft = simul.valueLeft
simulValueRight = simul.valueRight
simulFixItem = simul.fixItem
simulFixTime = simul.fixTime
simulFixRDV = simul.fixRDV
totalTrials = args.num_simulations * len(trialConditions)
if args.save_simulations:
save_simulations_to_csv(simulChoice, simulRT, simulValueLeft,
simulValueRight, simulFixItem, simulFixTime,
simulFixRDV, totalTrials)
if args.save_figures:
# Create pdf file to save figures.
pp = PdfPages("figures_" + str(optimD) + "_" + str(optimTheta) + "_" +
str(optimSigma) + "_" + str(args.num_simulations) +
".pdf")
# Generate choice and RT curves for real data (odd trials) and
# simulations (generated from even trials).
fig1 = generate_choice_curves(choice, valueLeft, valueRight,
simulChoice, simulValueLeft,
simulValueRight, totalTrials)
pp.savefig(fig1)
fig2 = generate_rt_curves(RT, valueLeft, valueRight, simulRT,
simulValueLeft, simulValueRight, totalTrials)
pp.savefig(fig2)
pp.close()
