Fit a non-linear function to data/observations with pyMCMC/pyMC

Fit a non-linear function to data/observations with pyMCMC/pyMC

To fit a non-linear function to data using the PyMC library for Bayesian inference and Markov Chain Monte Carlo (MCMC) sampling in Python, you'll need to set up a probabilistic model that describes the relationship between your data and the parameters of the non-linear function.

Here's a step-by-step guide to fitting a non-linear function using PyMC:

  • Install PyMC:

If you haven't already, install the pymc3 library:

pip install pymc3
  • Define the Non-Linear Model:

You'll need to define the non-linear model that relates the parameters to the data. For this example, let's assume you want to fit a quadratic function to your data.

import numpy as np import pymc3 as pm import matplotlib.pyplot as plt # Simulated data x = np.linspace(0, 10, 20) true_a, true_b, true_c = 2, 3, 1 true_y = true_a * x**2 + true_b * x + true_c data = true_y + np.random.normal(0, 1, size=len(x)) # Define the PyMC model with pm.Model() as model: a = pm.Normal('a', mu=0, sd=10) b = pm.Normal('b', mu=0, sd=10) c = pm.Normal('c', mu=0, sd=10) y_pred = a * x**2 + b * x + c likelihood = pm.Normal('likelihood', mu=y_pred, sd=1, observed=data) # Perform MCMC sampling with model: trace = pm.sample(1000, tune=1000) # Plot the results pm.traceplot(trace) plt.show()

In this example, the non-linear function is defined as a * x**2 + b * x + c, and you're trying to estimate the parameters a, b, and c that best explain the observed data.

  • Visualize the Results:

After running MCMC sampling, you can use the traceplot to visualize the parameter traces and posterior distributions.

  • Extract and Analyze the Results:

You can extract the posterior samples for the parameters from the trace object and analyze the results. For instance, you can compute mean, median, credible intervals, etc., to understand the uncertainty in your parameter estimates.

Note that the example above assumes a quadratic function, and you'll need to adjust the model structure and prior distributions to fit your specific non-linear function.

Keep in mind that PyMC is a powerful library for Bayesian analysis, but it requires careful consideration of model specification, priors, convergence diagnostics, and other aspects to ensure accurate and reliable results.

Examples

  1. Fitting a non-linear function to data using PyMC3

    • Description: This query seeks information on using PyMC3, a probabilistic programming framework, to fit a non-linear function to observed data.
    # Code Implementation import numpy as np import pymc3 as pm import matplotlib.pyplot as plt # Generate sample data np.random.seed(0) x_data = np.linspace(0, 10, 100) y_data = 3 * np.sin(x_data) + np.random.normal(0, 0.5, size=len(x_data)) # Define non-linear model with pm.Model() as model: # Parameters amp = pm.Normal('amp', mu=0, sigma=10) freq = pm.Normal('freq', mu=0, sigma=10) phase = pm.Normal('phase', mu=0, sigma=10) noise = pm.HalfNormal('noise', sigma=1) # Non-linear model y_pred = amp * np.sin(freq * x_data + phase) # Likelihood likelihood = pm.Normal('y', mu=y_pred, sigma=noise, observed=y_data) # MCMC sampling trace = pm.sample(1000, tune=1000) # Plot posterior distributions pm.traceplot(trace) plt.show()

  2. Fitting a custom-defined non-linear model to data using PyMC3

    • Description: This query focuses on fitting a custom-defined non-linear function to observed data with PyMC3 for more complex modeling scenarios.
    # Code Implementation import numpy as np import pymc3 as pm import matplotlib.pyplot as plt # Custom non-linear model def my_non_linear_model(x, params): a, b, c = params return a * np.sin(b * x + c) # Generate sample data np.random.seed(0) x_data = np.linspace(0, 10, 100) y_data = my_non_linear_model(x_data, [3, 1, 0]) + np.random.normal(0, 0.5, size=len(x_data)) # Define PyMC3 model with pm.Model() as model: params = pm.Normal('params', mu=0, sigma=10, shape=3) y_pred = my_non_linear_model(x_data, params) noise = pm.HalfNormal('noise', sigma=1) likelihood = pm.Normal('y', mu=y_pred, sigma=noise, observed=y_data) trace = pm.sample(1000, tune=1000) # Plot posterior distributions pm.traceplot(trace) plt.show()

  3. Fitting a non-linear logistic function to binary data using PyMC3

    • Description: This query explores using PyMC3 to fit a non-linear logistic function to binary (0/1) data, commonly used in logistic regression.
    # Code Implementation import numpy as np import pymc3 as pm import matplotlib.pyplot as plt # Generate sample binary data np.random.seed(0) x_data = np.linspace(0, 10, 100) p = 1 / (1 + np.exp(-(3 * x_data - 15))) y_data = np.random.binomial(n=1, p=p) # Define non-linear logistic model with pm.Model() as model: # Parameters intercept = pm.Normal('intercept', mu=0, sigma=10) slope = pm.Normal('slope', mu=0, sigma=10) # Non-linear logistic model p = pm.invlogit(intercept + slope * x_data) # Likelihood likelihood = pm.Bernoulli('y', p=p, observed=y_data) # MCMC sampling trace = pm.sample(1000, tune=1000) # Plot posterior distributions pm.traceplot(trace) plt.show()

  4. Fitting a non-linear exponential decay function to time-series data using PyMC3

    • Description: This query investigates using PyMC3 to fit a non-linear exponential decay function to time-series data, commonly encountered in decay processes.
    # Code Implementation import numpy as np import pymc3 as pm import matplotlib.pyplot as plt # Generate sample time-series data np.random.seed(0) x_data = np.linspace(0, 10, 100) y_data = 5 * np.exp(-0.5 * x_data) + np.random.normal(0, 0.1, size=len(x_data)) # Define non-linear exponential decay model with pm.Model() as model: # Parameters amp = pm.Normal('amp', mu=0, sigma=10) rate = pm.Normal('rate', mu=0, sigma=10) # Non-linear exponential decay model y_pred = amp * np.exp(-rate * x_data) # Likelihood likelihood = pm.Normal('y', mu=y_pred, sigma=0.1, observed=y_data) # MCMC sampling trace = pm.sample(1000, tune=1000) # Plot posterior distributions pm.traceplot(trace) plt.show()

  5. Fitting a non-linear growth model to longitudinal data using PyMC3

    • Description: This query focuses on using PyMC3 to fit a non-linear growth model to longitudinal data, where observations are taken at multiple time points.
    # Code Implementation import numpy as np import pymc3 as pm import matplotlib.pyplot as plt # Generate sample longitudinal data np.random.seed(0) x_data = np.linspace(0, 10, 100) y_data = 2 * np.exp(0.3 * x_data) + np.random.normal(0, 0.1, size=len(x_data)) # Define non-linear growth model with pm.Model() as model: # Parameters intercept = pm.Normal('intercept', mu=0, sigma=10) slope = pm.Normal('slope', mu=0, sigma=10) # Non-linear growth model y_pred = intercept * np.exp(slope * x_data) # Likelihood likelihood = pm.Normal('y', mu=y_pred, sigma=0.1, observed=y_data) # MCMC sampling trace = pm.sample(1000, tune=1000) # Plot posterior distributions pm.traceplot(trace) plt.show()

  6. Fitting a non-linear model with hierarchical structure to grouped data using PyMC3

    • Description: This query explores fitting a non-linear model with a hierarchical structure to grouped data, where parameters may vary across groups, using PyMC3.
    # Code Implementation import numpy as np import pymc3 as pm import matplotlib.pyplot as plt # Generate sample grouped data np.random.seed(0) group_size = 50 num_groups = 5 x_data = np.tile(np.linspace(0, 10, group_size), num_groups) group_ids = np.repeat(np.arange(num_groups), group_size) y_data = 3 * np.sin(x_data) + np.random.normal(0, 0.5, size=len(x_data)) # Define non-linear hierarchical model with pm.Model() as model: # Group-level parameters amp_mu = pm.Normal('amp_mu', mu=0, sigma=10) amp_sd = pm.HalfNormal('amp_sd', sigma=10) freq_mu = pm.Normal('freq_mu', mu=0, sigma=10) freq_sd = pm.HalfNormal('freq_sd', sigma=10) phase_mu = pm.Normal('phase_mu', mu=0, sigma=10) phase_sd = pm.HalfNormal('phase_sd', sigma=10) # Individual parameters amp = pm.Normal('amp', mu=amp_mu, sigma=amp_sd, shape=num_groups) freq = pm.Normal('freq', mu=freq_mu, sigma=freq_sd, shape=num_groups) phase = pm.Normal('phase', mu=phase_mu, sigma=phase_sd, shape=num_groups) # Non-linear hierarchical model y_pred = amp[group_ids] * np.sin(freq[group_ids] * x_data + phase[group_ids]) # Likelihood likelihood = pm.Normal('y', mu=y_pred, sigma=0.5, observed=y_data) # MCMC sampling trace = pm.sample(1000, tune=1000) # Plot posterior distributions pm.traceplot(trace) plt.show()

  7. Fitting a non-linear model with autoregressive structure to time-series data using PyMC3

    • Description: This query investigates fitting a non-linear model with autoregressive structure to time-series data, where each observation depends on previous observations.
    # Code Implementation import numpy as np import pymc3 as pm import matplotlib.pyplot as plt # Generate sample time-series data np.random.seed(0) x_data = np.linspace(0, 10, 100) y_data = 3 * np.sin(x_data) + np.random.normal(0, 0.5, size=len(x_data)) # Define non-linear autoregressive model with pm.Model() as model: # Parameters amp = pm.Normal('amp', mu=0, sigma=10) freq = pm.Normal('freq', mu=0, sigma=10) phase = pm.Normal('phase', mu=0, sigma=10) noise = pm.HalfNormal('noise', sigma=1) # Non-linear autoregressive model y_pred = amp * np.sin(freq * x_data + phase) for t in range(1, len(x_data)): y_pred = pm.math.switch(x_data[t] >= 5, amp * np.sin(freq * x_data[t] + phase) + noise * y_pred[t - 1], y_pred) # Likelihood likelihood = pm.Normal('y', mu=y_pred, sigma=noise, observed=y_data) # MCMC sampling trace = pm.sample(1000, tune=1000) # Plot posterior distributions pm.traceplot(trace) plt.show()

  8. Fitting a non-linear model with interaction terms to data using PyMC3

    • Description: This query explores fitting a non-linear model with interaction terms to data using PyMC3, allowing for complex relationships between variables.
    # Code Implementation import numpy as np import pymc3 as pm import matplotlib.pyplot as plt # Generate sample data with interaction terms np.random.seed(0) x1_data = np.linspace(0, 10, 100) x2_data = np.linspace(0, 5, 100) y_data = 3 * np.sin(x1_data) * np.exp(-0.5 * x2_data) + np.random.normal(0, 0.5, size=len(x1_data)) # Define non-linear model with interaction terms with pm.Model() as model: # Parameters amp = pm.Normal('amp', mu=0, sigma=10) decay_rate = pm.Normal('decay_rate', mu=0, sigma=10) noise = pm.HalfNormal('noise', sigma=1) # Non-linear model with interaction terms y_pred = amp * pm.math.sin(x1_data) * pm.math.exp(-decay_rate * x2_data) # Likelihood likelihood = pm.Normal('y', mu=y_pred, sigma=noise, observed=y_data) # MCMC sampling trace = pm.sample(1000, tune=1000) # Plot posterior distributions pm.traceplot(trace) plt.show()

  9. Fitting a non-linear model with latent variables to data using PyMC3

    • Description: This query investigates fitting a non-linear model with latent variables to data using PyMC3, allowing for the incorporation of unobserved factors.
    # Code Implementation import numpy as np import pymc3 as pm import matplotlib.pyplot as plt # Generate sample data np.random.seed(0) x_data = np.linspace(0, 10, 100) y_data = 3 * np.sin(x_data) + np.random.normal(0, 0.5, size=len(x_data)) # Define non-linear model with latent variables with pm.Model() as model: # Parameters amp = pm.Normal('amp', mu=0, sigma=10) freq = pm.Normal('freq', mu=0, sigma=10) phase = pm.Normal('phase', mu=0, sigma=10) noise = pm.HalfNormal('noise', sigma=1) # Latent variable latent_variable = pm.Normal('latent_variable', mu=0, sigma=10) # Non-linear model with latent variable y_pred = amp * np.sin(freq * x_data + phase) + latent_variable # Likelihood likelihood = pm.Normal('y', mu=y_pred, sigma=noise, observed=y_data) # MCMC sampling trace = pm.sample(1000, tune=1000) # Plot posterior distributions pm.traceplot(trace) plt.show()

  10. Fitting a non-linear model with heteroscedastic errors to data using PyMC3

    • Description: This query focuses on fitting a non-linear model with heteroscedastic (varying) errors to data using PyMC3, accommodating situations where the variability of the errors changes across the data range.
    # Code Implementation import numpy as np import pymc3 as pm import matplotlib.pyplot as plt # Generate sample data with heteroscedastic errors np.random.seed(0) x_data = np.linspace(0, 10, 100) y_data = 3 * np.sin(x_data) + np.random.normal(0, 0.5 + 0.1 * x_data, size=len(x_data)) # Define non-linear model with heteroscedastic errors with pm.Model() as model: # Parameters amp = pm.Normal('amp', mu=0, sigma=10) freq = pm.Normal('freq', mu=0, sigma=10) phase = pm.Normal('phase', mu=0, sigma=10) # Non-linear model y_pred = amp * np.sin(freq * x_data + phase) # Likelihood with heteroscedastic errors likelihood = pm.Normal('y', mu=y_pred, sigma=0.5 + 0.1 * x_data, observed=y_data) # MCMC sampling trace = pm.sample(1000, tune=1000) # Plot posterior distributions pm.traceplot(trace) plt.show()

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