psignifit

https://github.com/wichmann-lab/python-psignifit
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Python toolbox for Bayesian psychometric function estimation using numerical integration to fit...
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### Install psignifit via setup.py

Source: https://github.com/wichmann-lab/python-psignifit/wiki/Install

Run this command in the terminal after unpacking the downloaded ZIP file to install the package.

```bash
python setup.py install
```

--------------------------------

### Install psignifit from source

Source: https://github.com/wichmann-lab/python-psignifit/blob/main/docs/install_guide.md

Use this command within the cloned repository folder to perform an editable installation.

```bash
pip install -e .
```

--------------------------------

### Install psignifit via pip

Source: https://github.com/wichmann-lab/python-psignifit/blob/main/README.md

Use this command to install the package in your Python environment.

```bash
pip install psignifit
```

--------------------------------

### Define example dataset

Source: https://github.com/wichmann-lab/python-psignifit/blob/main/docs/basic-usage.md

Format data as a nx3 matrix with columns for stimulus level, number of correct responses, and total responses. This example is from a signal detection experiment.

```python
#        levels, n-correct,   n-total
data = [[0.0010,   45.0000,   90.0000],
        [0.0015,   50.0000,   90.0000],
        [0.0020,   44.0000,   90.0000],
        [0.0025,   44.0000,   90.0000],
        [0.0030,   52.0000,   90.0000],
        [0.0035,   53.0000,   90.0000],
        [0.0040,   62.0000,   90.0000],
        [0.0045,   64.0000,   90.0000],
        [0.0050,   76.0000,   90.0000],
        [0.0060,   79.0000,   90.0000],
        [0.0070,   88.0000,   90.0000],
        [0.0080,   90.0000,   90.0000],
        [0.0100,   90.0000,   90.0000]]
```

--------------------------------

### Import psignifit modules

Source: https://github.com/wichmann-lab/python-psignifit/blob/main/docs/examples/bias-analysis.md

Initial setup for bias analysis by importing necessary libraries.

```python
import numpy as np

import psignifit as ps
import psignifit.psigniplot as psp

# In this example we eill ignore the warnings from psignifit
```

--------------------------------

### Initialize psignifit with basic options

Source: https://github.com/wichmann-lab/python-psignifit/blob/main/docs/examples/basic-options.md

Demonstrates the basic setup for fitting a psychometric function using a 2AFC experiment type and a normal cumulative distribution function.

```python
import numpy as np

import psignifit as ps

# to have some data we use the data from the first demo.
data = np.array([[0.0010, 45.0000, 90.0000], [0.0015, 50.0000, 90.0000],
                 [0.0020, 44.0000, 90.0000], [0.0025, 44.0000, 90.0000],
                 [0.0030, 52.0000, 90.0000], [0.0035, 53.0000, 90.0000],
                 [0.0040, 62.0000, 90.0000], [0.0045, 64.0000, 90.0000],
                 [0.0050, 76.0000, 90.0000], [0.0060, 79.0000, 90.0000],
                 [0.0070, 88.0000, 90.0000], [0.0080, 90.0000, 90.0000],
                 [0.0100, 90.0000, 90.0000]])

res = ps.psignifit(data, sigmoid='norm', experiment_type='2AFC')
```

--------------------------------

### Fit with Modified Data and Same Options

Source: https://github.com/wichmann-lab/python-psignifit/blob/main/docs/examples/results-object.md

This example shows how to reuse configuration options from a previous fit to analyze modified data. It calculates the difference in the threshold parameter.

```python
# copy the data and introduce a shift in all stimulus values
otherdata = np.copy(data)
otherdata[:, 0] = otherdata[:, 0] + 0.01

# fit with exact same options
other_res = ps.psignifit(otherdata, conf=res.configuration)

# the difference in threshold should return the introduced shift
print(other_res.parameter_estimate['threshold'] - res.parameter_estimate['threshold'])
```

--------------------------------

### Run Psignifit Analysis

Source: https://github.com/wichmann-lab/python-psignifit/blob/main/docs/examples/results-object.md

This code snippet demonstrates how to run the psignifit analysis with sample data. Ensure numpy, matplotlib, and psignifit are installed.

```python
import numpy as np
import matplotlib.pyplot as plt

import psignifit as ps
import psignifit.psigniplot as psp

# to have some data we use the data from demo_001
data = np.array([[0.0010, 45.0000, 90.0000], [0.0015, 50.0000, 90.0000],
                 [0.0020, 44.0000, 90.0000], [0.0025, 44.0000, 90.0000],
                 [0.0030, 52.0000, 90.0000], [0.0035, 53.0000, 90.0000],
                 [0.0040, 62.0000, 90.0000], [0.0045, 64.0000, 90.0000],
                 [0.0050, 76.0000, 90.0000], [0.0060, 79.0000, 90.0000],
                 [0.0070, 88.0000, 90.0000], [0.0080, 90.0000, 90.0000],
                 [0.0100, 90.0000, 90.0000]])

# Run psignifit
res = ps.psignifit(data, sigmoid='norm', experiment_type='2AFC')
```

--------------------------------

### Fit Data with Fixed Parameters in Psignifit

Source: https://github.com/wichmann-lab/python-psignifit/blob/main/docs/how_to/How-to-Fix-Parameters.md

This example fits psignifit to yes/no data using fixed lambda and gamma parameters. Ensure data and necessary imports are available.

```python
import numpy as np

import psignifit as ps

# to have some data we use the data from the first demo.
data = np.array([[0.0010, 45.0000, 90.0000], [0.0015, 50.0000, 90.0000],
                 [0.0020, 44.0000, 90.0000], [0.0025, 44.0000, 90.0000],
                 [0.0030, 52.0000, 90.0000], [0.0035, 53.0000, 90.0000],
                 [0.0040, 62.0000, 90.0000], [0.0045, 64.0000, 90.0000],
                 [0.0050, 76.0000, 90.0000], [0.0060, 79.0000, 90.0000],
                 [0.0070, 88.0000, 90.0000], [0.0080, 90.0000, 90.0000],
                 [0.0100, 90.0000, 90.0000]])

fixed_parameters = {'lambda': 0.02, 'gamma': 0.5}

res = ps.psignifit(data, sigmoid='norm', experiment_type='yes/no',
                   fixed_parameters=fixed_parameters)
```

--------------------------------

### Retrieve Posterior Samples Programmatically

Source: https://context7.com/wichmann-lab/python-psignifit/llms.txt

Demonstrates how to programmatically retrieve posterior samples for model parameters. This allows for further analysis or custom visualizations. The example prints the mean and standard deviation of the threshold samples.

```python
import numpy as np
import matplotlib.pyplot as plt
import psignifit as ps
import psignifit.psigniplot as psp

data = np.array([
    [0.0010, 45.0, 90.0], [0.0015, 50.0, 90.0],
    [0.0020, 44.0, 90.0], [0.0025, 44.0, 90.0],
    [0.0030, 52.0, 90.0], [0.0035, 53.0, 90.0],
    [0.0040, 62.0, 90.0], [0.0045, 64.0, 90.0],
    [0.0050, 76.0, 90.0], [0.0060, 79.0, 90.0],
    [0.0070, 88.0, 90.0], [0.0080, 90.0, 90.0],
    [0.0100, 90.0, 90.0]
])

# Must use debug=True to enable posterior sampling
result = ps.psignifit(data, experiment_type='2AFC', debug=True)

# Get posterior samples programmatically
samples = result.posterior_samples(n_samples=1000, random_state=np.random.default_rng(42))
print(f"Threshold samples mean: {samples['threshold'].mean():.4f}")
print(f"Threshold samples std: {samples['threshold'].std():.4f}")
```

--------------------------------

### Fit Data with psignifit

Source: https://github.com/wichmann-lab/python-psignifit/blob/main/docs/how_to/How-to-Get-Standard-Parameters.md

This snippet demonstrates fitting data using the psignifit library with a Gaussian sigmoid for a yes/no experiment. Ensure the psignifit library is installed and data is in the correct format.

```python
import numpy as np

import psignifit as ps

# to have some data we use the data from the first demo.
data = np.array([[0.0010, 45.0000, 90.0000], [0.0015, 50.0000, 90.0000],
                 [0.0020, 44.0000, 90.0000], [0.0025, 44.0000, 90.0000],
                 [0.0030, 52.0000, 90.0000], [0.0035, 53.0000, 90.0000],
                 [0.0040, 62.0000, 90.0000], [0.0045, 64.0000, 90.0000],
                 [0.0050, 76.0000, 90.0000], [0.0060, 79.0000, 90.0000],
                 [0.0070, 88.0000, 90.0000], [0.0080, 90.0000, 90.0000],
                 [0.0100, 90.0000, 90.0000]])

res = ps.psignifit(data, sigmoid='norm', experiment_type='yes/no')
```

--------------------------------

### Extract Thresholds and Slopes using Result Object Methods

Source: https://context7.com/wichmann-lab/python-psignifit/llms.txt

Use the threshold() method to get the stimulus level at a given proportion correct, and slope() or slope_at_proportion_correct() to get the slope at a specific stimulus level or proportion correct. Confidence intervals can be returned or omitted.

```python
import numpy as np
import psignifit as ps

data = np.array([
    [0.0010, 45.0, 90.0], [0.0015, 50.0, 90.0],
    [0.0020, 44.0, 90.0], [0.0025, 44.0, 90.0],
    [0.0030, 52.0, 90.0], [0.0035, 53.0, 90.0],
    [0.0040, 62.0, 90.0], [0.0045, 64.0, 90.0],
    [0.0050, 76.0, 90.0], [0.0060, 79.0, 90.0],
    [0.0070, 88.0, 90.0], [0.0080, 90.0, 90.0],
    [0.0100, 90.0, 90.0]
])

result = ps.psignifit(data, experiment_type='2AFC')

# Get threshold at 90% correct (on scaled sigmoid)
threshold_90, ci_90 = result.threshold(0.9)
print(f"Threshold at 90%: {threshold_90:.4f}")
# Threshold at 90%: 0.0058

# Get threshold at 50% on unscaled sigmoid (equals the threshold parameter)
threshold_50, ci_50 = result.threshold(0.5, unscaled=True)
print(f"Threshold at 50% unscaled: {threshold_50:.4f}")
# Should equal result.parameter_estimate['threshold']

# Get threshold without confidence intervals
threshold_only = result.threshold(0.75, return_ci=False)

# Get slope at a specific stimulus level
slope_at_006 = result.slope(0.006)
print(f"Slope at stimulus 0.006: {slope_at_006:.2f}")

# Get slope at a specific proportion correct
slope_at_60pc = result.slope_at_proportion_correct(0.6)
print(f"Slope at 60% correct: {slope_at_60pc:.2f}")

# Get slope at midpoint of unscaled sigmoid
slope_midpoint = result.slope_at_proportion_correct(0.5, unscaled=True)
```

--------------------------------

### Generate synthetic psychometric data

Source: https://github.com/wichmann-lab/python-psignifit/blob/main/docs/examples/map_vs_mean.md

Setup parameters and simulate percent correct values for a mixture of sigmoids.

```python
# Parameters of the sigmoids and data generation
stim_range = [0.001, 0.001 + 10 * 1.1]
lambda_ = 0.0232
gamma = 0.1

widths = [1.1, 3.1, 6.3]
threshold = stim_range[1] / 2

sigmoid = sigmoids.Gaussian()
num_trials = 3
nsteps = 20
stimulus_level = np.linspace(stim_range[0], stim_range[1], nsteps)
```

```python
perccorr1 = sigmoid(stimulus_level, threshold, widths[0], gamma, lambda_)
perccorr2 = sigmoid(stimulus_level, threshold, widths[1], gamma, lambda_)
perccorr3 = sigmoid(stimulus_level, threshold, widths[2], gamma, lambda_)
perccorr = np.concatenate((perccorr1, perccorr2, perccorr3))
levels = np.concatenate((stimulus_level, stimulus_level, stimulus_level))
```

```python
random_state = np.random.RandomState(883)

ntrials = np.ones(nsteps * 3, dtype=int) * num_trials
hits = random_state.binomial(ntrials, perccorr)

levels = np.concatenate((stimulus_level, stimulus_level, stimulus_level))
data = np.dstack([levels, hits, ntrials]).squeeze()
```

--------------------------------

### Set Custom Bounds for Parameters

Source: https://github.com/wichmann-lab/python-psignifit/blob/main/docs/examples/advanced-options.md

Provides custom bounds for threshold, lambda, and gamma. The bounds should be a dictionary of parameter names and tuples of start and end ranges.

```python
custom_bounds = {
                 'threshold': (0.0, 0.015),
                 'lambda': (0.0, 0.1),
                 'gamma': (.5, .5),
                }
res = ps.psignifit(data, bounds=custom_bounds)
```

--------------------------------

### Set Custom Threshold Percentage Correct in psignifit

Source: https://github.com/wichmann-lab/python-psignifit/blob/main/docs/how_to/How-to-Change-the-Threshold-Percent-Correct.md

Use the `thresh_PC` argument to specify the desired proportion correct for the threshold. This example sets the threshold to 90% for a 'yes/no' experiment.

```python
import numpy as np

import psignifit as ps

# to have some data we use the data from the first demo.
data = np.array([[0.0010, 45.0000, 90.0000], [0.0015, 50.0000, 90.0000],
                 [0.0020, 44.0000, 90.0000], [0.0025, 44.0000, 90.0000],
                 [0.0030, 52.0000, 90.0000], [0.0035, 53.0000, 90.0000],
                 [0.0040, 62.0000, 90.0000], [0.0045, 64.0000, 90.0000],
                 [0.0050, 76.0000, 90.0000], [0.0060, 79.0000, 90.0000],
                 [0.0070, 88.0000, 90.0000], [0.0080, 90.0000, 90.0000],
                 [0.0100, 90.0000, 90.0000]])

fixed_parameters = {'lambda': 0.02, 'gamma': 0.5}

res = ps.psignifit(data, sigmoid='norm', experiment_type='yes/no', thresh_PC=0.9)
```

--------------------------------

### Import psignifit and numpy

Source: https://github.com/wichmann-lab/python-psignifit/wiki/Basic-Usage

Initialize the environment by importing the necessary libraries.

```python
import numpy as np
import psignifit as ps
```

--------------------------------

### Get specific parameter estimate

Source: https://github.com/wichmann-lab/python-psignifit/blob/main/docs/basic-usage.md

Retrieve the estimated value for a specific parameter, such as the 'threshold'.

```python
print(result.parameter_estimate['threshold'])
```

--------------------------------

### Clone the psignifit repository

Source: https://github.com/wichmann-lab/python-psignifit/wiki/Install

Use this command to download the repository for the first time into your current directory.

```bash
git clone https://github.com/wichmann-lab/python-psignifit.git
```

--------------------------------

### Get specific confidence interval

Source: https://github.com/wichmann-lab/python-psignifit/blob/main/docs/basic-usage.md

Retrieve a specific confidence interval for a parameter, such as the 95% confidence interval for the threshold.

```python
print(result.confidence_intervals['threshold']['0.95'])
```

--------------------------------

### Execute the fit

Source: https://github.com/wichmann-lab/python-psignifit/wiki/Basic-Usage

Run the psignifit function using the prepared data and options.

```python
result = ps.psignifit(data,options);
```

--------------------------------

### Get parameter estimates

Source: https://github.com/wichmann-lab/python-psignifit/blob/main/docs/basic-usage.md

Access the estimated parameters from the fit result. The 'parameter_estimate' attribute returns a dictionary of fitted parameters.

```python
print(result.parameter_estimate)
```

--------------------------------

### Print Configuration Options

Source: https://github.com/wichmann-lab/python-psignifit/blob/main/docs/examples/results-object.md

Display the complete set of configuration options used for the psignifit analysis, which are stored within the result object.

```python
print(res.configuration)
```

--------------------------------

### Initialize and Set Options in Python psignifit

Source: https://github.com/wichmann-lab/python-psignifit/wiki/Options-Dictionary

Use Python dictionary syntax to initialize and set options, similar to the MATLAB options struct. Refer to the MATLAB wiki for field meanings.

```python
options = dict()              % Initialization as an empty dictionary
```

```python
options['fieldName'] = value  % Setting individual values
```

--------------------------------

### Import necessary libraries

Source: https://github.com/wichmann-lab/python-psignifit/blob/main/docs/basic-usage.md

Import matplotlib for plotting, psignifit for fitting, and psigniplot for plotting utilities.

```python
import matplotlib.pyplot as plt

import psignifit as ps
import psignifit.psigniplot as psp
```

--------------------------------

### Get confidence intervals for parameters

Source: https://github.com/wichmann-lab/python-psignifit/blob/main/docs/basic-usage.md

Access the confidence intervals for all estimated parameters. The 'confidence_intervals' attribute returns a dictionary containing intervals for default confidences (95%, 90%, 68%).

```python
print(result.confidence_intervals['threshold'])
```

--------------------------------

### Access parameter estimates

Source: https://github.com/wichmann-lab/python-psignifit/blob/main/docs/examples/advanced-options.md

Demonstrates how to retrieve parameter estimates from the result object using attributes or the get_parameter_estimate method.

```python
# This gives the default estimate, in this case the mean estimate since we set it in the options.
print(f"parameter estimate (default=MAP): {res.parameter_estimate}")
print(f"parameter estimate (MAP): {res.parameter_estimate_MAP}")
print(f"parameter estimate (mean){res.parameter_estimate_mean}")
```

```python
print(res.get_parameter_estimate(estimate_type="MAP"))
```

--------------------------------

### Configure Pooling Options in Python

Source: https://github.com/wichmann-lab/python-psignifit/wiki/Pooling-Utility

Sets the pooling parameters within the options dictionary to control block formation and trial grouping.

```python
options['nblocks']        = 25;      # number of blocks required to start pooling
options['poolMaxGap']     = np.inf;  # maximal number of trials with other stimulus levels between pooled trials
options['poolMaxLength']  = np.inf;  # maximal number of trials per block
options['poolxTol']       = 0;       # maximal difference in stimulus level from the first trial in a block
```

--------------------------------

### Print Parameter Estimates for Different Beta Priors

Source: https://github.com/wichmann-lab/python-psignifit/blob/main/docs/examples/priors.md

Prints the parameter estimates for fits performed with different beta prior values (1 and 200). This highlights how the beta-variance parameter (eta) is affected by the prior strength.

```python
print('Fit with beta prior = 1: ')
print(json.dumps(res1.parameter_estimate, indent=2))

print('Fit with beta prior = 200: ')
print(json.dumps(res200.parameter_estimate, indent=2))
```

--------------------------------

### Verify fitted parameters against simulated parameters (log-space)

Source: https://github.com/wichmann-lab/python-psignifit/blob/main/docs/examples/parameter_recovery_in_logspace_demo.md

Asserts that the fitted threshold and width parameters in the result dictionary are close to the original log-space values used for simulation. This confirms the accuracy of the log-space fit.

```python
assert np.isclose(res.parameter_estimate['threshold'], log_threshold, atol=1e-4)
assert np.isclose(res.parameter_estimate['width'], log_width, atol=1e-4)
```

--------------------------------

### Specify Stimulus Range for Adaptive Procedures

Source: https://context7.com/wichmann-lab/python-psignifit/llms.txt

For adaptive procedures, explicitly define the stimulus range to guide the fitting process. This is crucial when the full range of possible stimuli needs to be communicated to the fitting algorithm.

```python
result_adaptive = ps.psignifit(
    data,
    experiment_type='2AFC',
    stimulus_range=(0.0, 0.02)  # Full range of possible stimuli
)
```

--------------------------------

### Get Proportion Correct Values - Python

Source: https://github.com/wichmann-lab/python-psignifit/blob/main/docs/how_to/How-to-Get-Proportion-Correct.md

Calculates the proportion correct for specified stimulus levels using a fitted psignifit model. Accepts a list or NumPy array of stimulus levels. The results are printed to the console.

```python
stimulus_levels = [0.0012, 0.0013]
prop_correct = res.proportion_correct(stimulus_levels)
print(prop_correct)
```

--------------------------------

### Get Proportion Correct Values with Eta - Python

Source: https://github.com/wichmann-lab/python-psignifit/blob/main/docs/how_to/How-to-Get-Proportion-Correct.md

Calculates proportion correct values incorporating noise derived from the estimated overdispersion parameter 'eta'. This provides a more realistic variance estimate, compatible with a beta binomial distribution.

```python
stimulus_levels = [0.0012, 0.0013]
prop_correct = res.proportion_correct(stimulus_levels, with_eta=True)
print(prop_correct)
```

--------------------------------

### Setting simulation parameters

Source: https://github.com/wichmann-lab/python-psignifit/blob/main/docs/examples/parameter_recovery_demo.md

Define ground truth parameters and stimulus levels for data simulation.

```python
# 'ground truth' values
width = 0.3
stim_range = [0.001, 0.001 + width * 1.1]
threshold = stim_range[1]/3
lambda_ = 0.05
gamma = 0.01

sigmoid_name = "norm"
sigmoid = sigmoid_by_name(sigmoid_name)

# 
nsteps = 20
num_trials = 100

stimulus_level = np.linspace(stim_range[0], stim_range[1], nsteps)
```

--------------------------------

### Get Proportion Correct Values using Mean Estimate - Python

Source: https://github.com/wichmann-lab/python-psignifit/blob/main/docs/how_to/How-to-Get-Proportion-Correct.md

Retrieves proportion correct values based on the 'mean' estimate of the psychometric function parameters, rather than the default 'MAP' (Maximum A Posteriori) estimate. Useful for alternative statistical interpretations.

```python
stimulus_levels = [0.0012, 0.0013]
prop_correct = res.proportion_correct(stimulus_levels, estimate_type='mean')
print(prop_correct)
```

--------------------------------

### Import psignifit modules

Source: https://github.com/wichmann-lab/python-psignifit/blob/main/docs/examples/map_vs_mean.md

Initial imports required for data generation, fitting, and plotting.

```python
import warnings

from matplotlib import pyplot as plt
import numpy as np

import psignifit
import psignifit.psigniplot as psp
from psignifit import sigmoids
```

--------------------------------

### Configure fitting options

Source: https://github.com/wichmann-lab/python-psignifit/wiki/Basic-Usage

Initialize and populate the options dictionary to define the sigmoid function and experiment type.

```python
options             = dict();   # initialize as an empty dictionary
```

```python
options['sigmoidName'] = 'norm';   # choose a cumulative Gauss as the sigmoid  
options['expType']     = '2AFC';   # choose 2-AFC as the experiment type  
                                   # this sets the guessing rate to .5 (fixed) and  
                                   # fits the rest of the parameters
```

--------------------------------

### Configure Equal Asymptote experiment type

Source: https://github.com/wichmann-lab/python-psignifit/blob/main/docs/examples/basic-options.md

Sets the experiment_type to 'equal asymptote', which yokes the asymptotes and improves fitting speed.

```python
experiment_type = 'equal asymptote'
```

--------------------------------

### Import necessary libraries and load data

Source: https://github.com/wichmann-lab/python-psignifit/blob/main/docs/examples/plotting.md

Imports matplotlib, numpy, and psignifit modules. Loads sample data for psychometric function fitting.

```python
import matplotlib.pyplot as plt
import numpy as np

import psignifit as ps
import psignifit.psigniplot as psp

data = np.array([[0.0010, 45.0000, 90.0000], [0.0015, 50.0000, 90.0000],
                 [0.0020, 44.0000, 90.0000], [0.0025, 44.0000, 90.0000],
                 [0.0030, 52.0000, 90.0000], [0.0035, 53.0000, 90.0000],
                 [0.0040, 62.0000, 90.0000], [0.0045, 64.0000, 90.0000],
                 [0.0050, 76.0000, 90.0000], [0.0060, 79.0000, 90.0000],
                 [0.0070, 88.0000, 90.0000], [0.0080, 90.0000, 90.0000],
                 [0.0100, 90.0000, 90.0000]])

res = ps.psignifit(data, sigmoid='norm', experiment_type='2AFC')
```

--------------------------------

### Running the psignifit fit

Source: https://github.com/wichmann-lab/python-psignifit/blob/main/docs/examples/parameter_recovery_demo.md

Execute the fitting procedure for a yes/no experiment type.

```python
res = psignifit.psignifit(data, sigmoid=sigmoid, experiment_type='yes/no')
```

--------------------------------

### Constructing the data array

Source: https://github.com/wichmann-lab/python-psignifit/blob/main/docs/examples/parameter_recovery_demo.md

Create the data structure required for psignifit fitting.

```python
ntrials = np.ones(nsteps) * num_trials
hits = (perccorr * ntrials).astype(int)
data = np.dstack([stimulus_level, hits, ntrials]).squeeze()
print(data)
```

--------------------------------

### Define simulation dataset

Source: https://github.com/wichmann-lab/python-psignifit/blob/main/docs/examples/priors.md

Initializes a numpy array representing a 3-down-1-up staircase procedure simulation.

```python
data = np.array([[1.5000, 3.0000, 3.0000], [1.3500, 3.0000, 3.0000],
                 [1.2150, 1.0000, 2.0000], [1.3365, 2.0000, 3.0000],
                 [1.4702, 3.0000, 3.0000], [1.3231, 3.0000, 3.0000],
                 [1.1908, 1.0000, 2.0000], [1.3099, 3.0000, 3.0000],
                 [1.1789, 1.0000, 2.0000], [1.2968, 2.0000, 3.0000],
                 [1.4265, 3.0000, 3.0000], [1.2838, 1.0000, 2.0000],
                 [1.4122, 3.0000, 3.0000], [1.2710, 1.0000, 2.0000],
                 [1.3981, 1.0000, 2.0000], [1.5379, 1.0000, 2.0000],
                 [1.6917, 3.0000, 3.0000], [1.5225, 3.0000, 3.0000],
                 [1.3703, 2.0000, 3.0000]])
```

--------------------------------

### Verify Fixed Parameters in Psignifit Results

Source: https://github.com/wichmann-lab/python-psignifit/blob/main/docs/how_to/How-to-Fix-Parameters.md

Print the parameter estimates from the psignifit result to confirm that the specified parameters were fixed.

```python
print(res.parameter_estimate)
```

--------------------------------

### Execute bias analysis with ps.biasAna

Source: https://github.com/wichmann-lab/python-psignifit/wiki/Interval-Bias

Use this syntax to perform bias analysis on 2AFC task data. Ensure the psignifit module is imported as ps.

```python
ps.biasAna(data1,data2,options)
```

--------------------------------

### Set Fixed Borders for Parameters

Source: https://github.com/wichmann-lab/python-psignifit/wiki/Priors

Use 'borders' to set fixed borders for parameters. This option expects a NumPy array of shape (5, 2).

```python
options['borders'] = np.nan((5,2))          # setting fixed borders for parameters
```

--------------------------------

### Configure optimization grid steps

Source: https://github.com/wichmann-lab/python-psignifit/blob/main/docs/examples/advanced-options.md

Adjusts the number of grid points for specific parameters to improve estimation accuracy.

```python
res = ps.psignifit(data, grid_steps={'lambda': 50})
```

--------------------------------

### Importing psignifit dependencies

Source: https://github.com/wichmann-lab/python-psignifit/blob/main/docs/examples/parameter_recovery_demo.md

Initial imports required for data manipulation, plotting, and psignifit functionality.

```python
import numpy as np
import matplotlib.pyplot as plt

import psignifit
import psignifit.psigniplot as psp
from psignifit.sigmoids import sigmoid_by_name
```

--------------------------------

### Import necessary libraries

Source: https://github.com/wichmann-lab/python-psignifit/blob/main/docs/examples/parameter_recovery_in_logspace_demo.md

Imports numpy for numerical operations, matplotlib for plotting, and psignifit for psychometric function fitting.

```python
import numpy as np
from matplotlib import pyplot as plt

import psignifit
import psignifit.psigniplot as psp
from psignifit.sigmoids import sigmoid_by_name
```

--------------------------------

### Visualize custom priors with psignifit

Source: https://context7.com/wichmann-lab/python-psignifit/llms.txt

Uses the psigniplot module to generate a plot of the prior distribution and saves it as a PNG file.

```python
import psignifit.psigniplot as psp
import matplotlib.pyplot as plt

fig = psp.plot_prior(result)
plt.savefig('custom_priors.png')
```

--------------------------------

### Plotting simulated data

Source: https://github.com/wichmann-lab/python-psignifit/blob/main/docs/examples/parameter_recovery_demo.md

Visualize the simulated stimulus levels and percent correct values.

```python
fig, ax = plt.subplots();
ax.scatter(stimulus_level, perccorr);
ax.set_xlabel("Stimulus Level");
ax.set_ylabel("Percent Correct");
ax.spines[['right', 'top']].set_visible(False);
```

--------------------------------

### Configuration Object for psignifit

Source: https://context7.com/wichmann-lab/python-psignifit/llms.txt

Use the Configuration class to hold and validate fitting options, which can be reused across multiple fits.

```python
import numpy as np
import psignifit as ps
from psignifit import Configuration

data = np.array([
    [0.0010, 45.0, 90.0], [0.0015, 50.0, 90.0],
    [0.0020, 44.0, 90.0], [0.0025, 44.0, 90.0],
    [0.0030, 52.0, 90.0], [0.0035, 53.0, 90.0],
    [0.0040, 62.0, 90.0], [0.0045, 64.0, 90.0],
    [0.0050, 76.0, 90.0], [0.0060, 79.0, 90.0],
    [0.0070, 88.0, 90.0], [0.0080, 90.0, 90.0],
    [0.0100, 90.0, 90.0]
])

# Create configuration object
config = Configuration(
    experiment_type='2AFC',
    sigmoid='norm',
    thresh_PC=0.5,           # Threshold at 50% on unscaled sigmoid
    width_alpha=0.05,        # Width = difference between 5% and 95%
    confP=(0.95, 0.9, 0.68), # Confidence levels
    estimate_type='MAP',     # Use MAP estimate (alternative: 'mean')
    CI_method='percentiles', # CI method (alternative: 'project')
    beta_prior=10,           # Strength of binomial prior
    verbose=True
)

# Use configuration for fitting
result = ps.psignifit(data, conf=config)

# Reuse configuration for another dataset
other_data = data.copy()
other_data[:, 0] += 0.001
result2 = ps.psignifit(other_data, conf=config)

# Access configuration from result
print(result.configuration)
```

--------------------------------

### Configure Yes/No experiment type

Source: https://github.com/wichmann-lab/python-psignifit/blob/main/docs/examples/basic-options.md

Sets the experiment_type to 'yes/no', allowing both asymptotes to vary with a prior favoring values near 0 and 1.

```python
experiment_type = 'yes/no'
```

--------------------------------

### Plotting Priors with plot_prior()

Source: https://context7.com/wichmann-lab/python-psignifit/llms.txt

Visualize prior distributions and their effect on psychometric functions. Requires fitting with debug=True.

```python
import numpy as np
import matplotlib.pyplot as plt
import psignifit as ps
import psignifit.psigniplot as psp

data = np.array([
    [0.0010, 45.0, 90.0], [0.0015, 50.0, 90.0],
    [0.0020, 44.0, 90.0], [0.0025, 44.0, 90.0],
    [0.0030, 52.0, 90.0], [0.0035, 53.0, 90.0],
    [0.0040, 62.0, 90.0], [0.0045, 64.0, 90.0],
    [0.0050, 76.0, 90.0], [0.0060, 79.0, 90.0],
    [0.0070, 88.0, 90.0], [0.0080, 90.0, 90.0],
    [0.0100, 90.0, 90.0]
])

# Must use debug=True to access priors
result = ps.psignifit(data, experiment_type='2AFC', debug=True)

# Plot priors and their effect on psychometric functions
# Upper row: prior distributions for threshold, width, lambda
# Lower row: corresponding psychometric functions at different prior values
fig = psp.plot_prior(result)
plt.savefig('priors_visualization.png')
```

--------------------------------

### Visualize marginal distributions and psychometric functions

Source: https://github.com/wichmann-lab/python-psignifit/blob/main/docs/examples/map_vs_mean.md

Compare marginal plots and psychometric functions using different estimate types.

```python
# Uses the default, in this case `'mean'`
psp.plot_marginal(result, 'width');
```

```python
# Overwrites the default, uses `'MAP'`
psp.plot_marginal(result, 'width', estimate_type='MAP');
```

```python
ax = psp.plot_psychometric_function(result, estimate_type='mean');
psp.plot_psychometric_function(result, estimate_type='MAP', ax=ax);
```

--------------------------------

### Custom Priors for psignifit

Source: https://context7.com/wichmann-lab/python-psignifit/llms.txt

Define custom prior distributions for parameters to incorporate domain knowledge or constrain the fitting space. Requires debug=True to visualize.

```python
import numpy as np
import scipy.stats
import psignifit as ps

data = np.array([
    [0.0010, 45.0, 90.0], [0.0015, 50.0, 90.0],
    [0.0020, 44.0, 90.0], [0.0025, 44.0, 90.0],
    [0.0030, 52.0, 90.0], [0.0035, 53.0, 90.0],
    [0.0040, 62.0, 90.0], [0.0045, 64.0, 90.0],
    [0.0050, 76.0, 90.0], [0.0060, 79.0, 90.0],
    [0.0070, 88.0, 90.0], [0.0080, 90.0, 90.0],
    [0.0100, 90.0, 90.0]
])

# Define custom priors as functions
custom_priors = {
    # Uniform prior on threshold
    'threshold': lambda x: np.ones_like(x),

    # Beta distribution prior on gamma (favoring values near 0.5)
    'gamma': lambda x: scipy.stats.beta.pdf(x, 2, 2),

    # Exponential prior on lambda (favoring small lapse rates)
    'lambda': lambda x: scipy.stats.expon.pdf(x, scale=0.05),
}

result = ps.psignifit(
    data,
    experiment_type='yes/no',
    priors=custom_priors,
    debug=True  # Enable debug to visualize priors
)
```

--------------------------------

### Plot All Pairwise 2D Marginal Distributions

Source: https://context7.com/wichmann-lab/python-psignifit/llms.txt

Generates a grid of all pairwise 2D marginal posterior distributions for the model parameters. This helps visualize relationships between parameters. Requires fitting with debug=True. Saves the plot to 'all_2d_marginals.png'.

```python
import numpy as np
import matplotlib.pyplot as plt
import psignifit as ps
import psignifit.psigniplot as psp

data = np.array([
    [0.0010, 45.0, 90.0], [0.0015, 50.0, 90.0],
    [0.0020, 44.0, 90.0], [0.0025, 44.0, 90.0],
    [0.0030, 52.0, 90.0], [0.0035, 53.0, 90.0],
    [0.0040, 62.0, 90.0], [0.0045, 64.0, 90.0],
    [0.0050, 76.0, 90.0], [0.0060, 79.0, 90.0],
    [0.0070, 88.0, 90.0], [0.0080, 90.0, 90.0],
    [0.0100, 90.0, 90.0]
])

# Must use debug=True
result = ps.psignifit(data, experiment_type='2AFC', debug=True)

# Plot all pairwise 2D marginals
fig = psp.plot_bayes(result)
plt.savefig('all_2d_marginals.png')
```

--------------------------------

### Set simulation parameters

Source: https://github.com/wichmann-lab/python-psignifit/blob/main/docs/examples/parameter_recovery_in_logspace_demo.md

Defines parameters for simulating data, including stimulus range, threshold, sigmoid parameters (lambda, gamma), number of steps, and number of trials. It also specifies the sigmoid type ('weibull') and prepares the stimulus levels spaced logarithmically.

```python
stim_range = [1.0, 1000.0]
threshold = 134

lambda_ = 0.0232
gamma = 0.1
nsteps = 20
num_trials = 50000
# We are going to fit a log-Weibull sigmoid
sigmoid_name = 'weibull'
sigmoid = sigmoid_by_name('weibull')
# We choose levels spaced logarithmically in the stimuli space, because that is often what happens
# in experiments where the fit is done in logarithmic space
stimulus_level = np.logspace(np.log10(stim_range[0]), np.log10(stim_range[1]), nsteps, base=10)
# However, one could just as well space the stimuli linearly
#stimulus_level = np.linspace(stim_range[0], stim_range[1], nsteps)
```

--------------------------------

### Print Confidence Intervals for Different Beta Priors

Source: https://github.com/wichmann-lab/python-psignifit/blob/main/docs/examples/priors.md

Prints the 95% confidence intervals for fits performed with beta prior values of 1 and 200. This shows the effect of prior strength on the width of the confidence intervals.

```python
print('Confidence Intervals for beta prior = 1: ')
print(json.dumps(res1.confidence_intervals, indent=2))

print('Confidence Intervals for beta prior = 200: ')
print(json.dumps(res200.confidence_intervals, indent=2))
```

--------------------------------

### Verify other fitted parameters

Source: https://github.com/wichmann-lab/python-psignifit/blob/main/docs/examples/parameter_recovery_in_logspace_demo.md

Asserts that the fitted gamma, lambda, and eta parameters are close to the values used during data simulation, ensuring the overall integrity of the fitting process.

```python
assert np.isclose(res.parameter_estimate['lambda'], lambda_, atol=1e-4)
assert np.isclose(res.parameter_estimate['gamma'], gamma, atol=1e-4)
assert np.isclose(res.parameter_estimate['eta'], 0, atol=1e-4)
```

--------------------------------

### Import necessary libraries

Source: https://github.com/wichmann-lab/python-psignifit/blob/main/docs/examples/priors.md

Imports the required libraries for plotting, numerical operations, and psignifit functionality.

```python
import matplotlib.pyplot as plt
import numpy as np
import json

import psignifit as ps
import psignifit.psigniplot as psp
```

--------------------------------

### Verifying parameter recovery

Source: https://github.com/wichmann-lab/python-psignifit/blob/main/docs/examples/parameter_recovery_demo.md

Assert that the recovered parameters match the ground truth values.

```python
assert np.isclose(res.parameter_estimate['lambda'], lambda_, atol=1e-2)
```

```python
assert np.isclose(res.parameter_estimate['gamma'], gamma, atol=1e-2)
```

```python
assert np.isclose(res.parameter_estimate['eta'], 0, atol=1e-2)
assert np.isclose(res.parameter_estimate['threshold'], threshold, atol=1e-2)
assert np.isclose(res.parameter_estimate['width'], width, atol=1e-2)
```

--------------------------------

### Configure Experiment Types for Psignifit

Source: https://context7.com/wichmann-lab/python-psignifit/llms.txt

Configure different experiment types to control how asymptotes (gamma and lambda) are handled. This affects prior assumptions about guess and lapse rates. Requires numpy and psignifit imports.

```python
import numpy as np
import psignifit as ps

data = np.array([
    [0.0010, 45.0, 90.0], [0.0015, 50.0, 90.0],
    [0.0020, 44.0, 90.0], [0.0025, 44.0, 90.0],
    [0.0030, 52.0, 90.0], [0.0035, 53.0, 90.0],
    [0.0040, 62.0, 90.0], [0.0045, 64.0, 90.0],
    [0.0050, 76.0, 90.0], [0.0060, 79.0, 90.0],
    [0.0070, 88.0, 90.0], [0.0080, 90.0, 90.0],
    [0.0100, 90.0, 90.0]
])

# nAFC experiments: gamma fixed to 1/n
result_2afc = ps.psignifit(data, experiment_type='2AFC')  # gamma = 0.5
result_3afc = ps.psignifit(data, experiment_type='3AFC')  # gamma = 0.333
result_4afc = ps.psignifit(data, experiment_type='4AFC')  # gamma = 0.25

# Yes/No experiments: both asymptotes free to vary
result_yesno = ps.psignifit(data, experiment_type='yes/no')

# Equal asymptote: gamma and lambda constrained to be equal
result_equal = ps.psignifit(data, experiment_type='equal asymptote')
```
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