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API Reference

This section contains the auto-generated documentation from the helper functions for FMG–FMG model (Inference_Funcs_FMG.md). Tasks raning from loading data and defining storage and loss moduli, to performing the inference and calculating various metrics and plotting are executed separately by various functions.

Bayesian Inference Module

scripts.Inference_Funcs_FMG

SimpleLoggingCallback

A simple callback to log the progress of PyMC sampling with timing metrics.

Source code in scripts/Inference_Funcs_FMG.py 
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class SimpleLoggingCallback:
    """A simple callback to log the progress of PyMC sampling with timing metrics."""
    def __init__(self, total_draws, log_interval=50_000):
        self.total_draws = total_draws
        self.log_interval = log_interval
        self.current_draws = {}  # Draw counter per chain 
        self.start_times = {}  # start time per chain 
        self.first_draw = True  # Flag for first draw

    def __call__(self, draw, *args, **kwargs):
        chain = draw.chain

        # Initialize per-chain counters and start time
        if chain not in self.current_draws:
            self.current_draws[chain] = 0
            self.start_times[chain] = time.time()
            if self.first_draw:
                logger.info(f"Starting sampling for chain {chain}")
                self.first_draw = False

        # Increment draw counter
        self.current_draws[chain] += 1

        # Log at specified intervals or when done
        if (self.current_draws[chain] % self.log_interval == 0 or 
                self.current_draws[chain] == self.total_draws):
            # Calculate timing metrics
            elapsed_time = time.time() - self.start_times[chain]  # Seconds
            sampling_speed = (self.current_draws[chain] / elapsed_time) if elapsed_time > 0 else 0  # Draws/s
            remaining_draws = self.total_draws - self.current_draws[chain]
            remaining_time = (remaining_draws / sampling_speed) if sampling_speed > 0 else 0  # Seconds

            # Format times as HH:MM:SS
            elapsed_str = str(timedelta(seconds=int(elapsed_time)))
            remaining_str = str(timedelta(seconds=int(remaining_time)))

            # Log message with metrics
            logger.info(
                f"Chain {chain}: {self.current_draws[chain]}/{self.total_draws} samples drawn, "
                f"Speed: {sampling_speed:.2f} draws/s, "
                f"Elapsed: {elapsed_str}, Remaining: {remaining_str}"
            )

            # Log completion for this chain
            if self.current_draws[chain] == self.total_draws:
                logger.info(f"Completed sampling for chain {chain}")

load_data(file_path, sheet_name, rows, cols_opt, cols_GnP, omega_limits, rcParams_plot)

Load experimental data and optimized model parameters, and plot the results.

Parameters:

Name Type Description Default
file_path dict

Dictionary containing file paths for experimental and optimized data.

 required
sheet_name dict

Dictionary containing sheet names for experimental and optimized data.

 required
rows dict

Dictionary containing start and end rows for loading data.

 required
cols_opt list

List of column indices for optimized parameters.

 required
cols_GnP dict

Dictionary mapping GnP keys to their respective column indices.

 required
omega_limits dict

Dictionary containing omega limits for filtering data.

 required
rcParams_plot dict

Dictionary containing plot parameters.

 required

Returns:

Name Type Description
v_obs_Ep dict

Dictionary of observed storage modulus values.
v_obs_Epp dict

Dictionary of observed loss modulus values.
x_data dict

Dictionary of frequency data.
optimized_params_df DataFrame

DataFrame of optimized model parameters.
Source code in scripts/Inference_Funcs_FMG.py 
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def load_data(file_path, sheet_name, rows, cols_opt, cols_GnP, omega_limits, rcParams_plot):
    """Load experimental data and optimized model parameters, and plot the results.

    Args:
        file_path (dict): Dictionary containing file paths for experimental and optimized data.
        sheet_name (dict): Dictionary containing sheet names for experimental and optimized data.
        rows (dict): Dictionary containing start and end rows for loading data.
        cols_opt (list): List of column indices for optimized parameters.
        cols_GnP (dict): Dictionary mapping GnP keys to their respective column indices.
        omega_limits (dict): Dictionary containing omega limits for filtering data.
        rcParams_plot (dict): Dictionary containing plot parameters.

    Returns:
        v_obs_Ep (dict): Dictionary of observed storage modulus values.
        v_obs_Epp (dict): Dictionary of observed loss modulus values.
        x_data (dict): Dictionary of frequency data.
        optimized_params_df (pd.DataFrame): DataFrame of optimized model parameters.
    """
    # Load experimental data
    data_dict = {
        key: pd.read_excel(
            file_path['file_path_exp'], sheet_name=sheet_name['sheet_name_exp'], usecols=cols_GnP[key],
            skiprows=rows['start_row_exp'], nrows=rows['end_row_exp'] - rows['start_row_exp'], header=None
        ).rename(columns={cols_GnP[key][0]: "omega", cols_GnP[key][1]: "Ep", cols_GnP[key][2]: "Epp"})
        for key in cols_GnP.keys()
    }

    v_obs_Ep, v_obs_Epp, x_data = {}, {}, {}
    for i, key in enumerate(data_dict.keys()):
        df = data_dict[key]
        df = df[(df['omega'] >= omega_limits['omega_min'][i]) & (df['omega'] <= omega_limits['omega_max'][i])].reset_index(drop=True)
        v_obs_Ep[key] = df['Ep'].values
        v_obs_Epp[key] = df['Epp'].values
        x_data[key] = df['omega'].values

    # Optimized model parameters
    optimized_params_df = pd.read_excel(
        file_path['file_path_opt'], sheet_name=sheet_name['sheet_name_opt'], usecols=cols_opt,
        skiprows=rows['start_row_opt'], nrows=rows['end_row_opt'] - rows['start_row_opt'], header=None
    )
    optimized_params_df.columns = ['E_c1', 'tau_c1', 'alpha_1', 'E_c2', 'tau_c2', 'alpha_2']

    plt.rcParams.update({
        'font.size': rcParams_plot['font.size'],
        'axes.labelsize': rcParams_plot['axes.labelsize'],
        'xtick.labelsize': rcParams_plot['xtick.labelsize'],
        'ytick.labelsize': rcParams_plot['ytick.labelsize'],
        'legend.fontsize': rcParams_plot['legend.fontsize'],
        'axes.titlesize': rcParams_plot['axes.titlesize'],
        'font.family': rcParams_plot['font.family'],
        'mathtext.fontset': rcParams_plot['mathtext.fontset'],
        'mathtext.rm': rcParams_plot['mathtext.rm'],
        'mathtext.it': rcParams_plot['mathtext.it'],
        'mathtext.bf': rcParams_plot['mathtext.bf']
    })

    _, axes = plt.subplots(1, 2, figsize=(12, 4))
    axes = axes.flatten()
    for i, key in enumerate(cols_GnP.keys()):
        axes[0].loglog(
            x_data[key], v_obs_Ep[key], color=f'C{i}', marker=rcParams_plot['markers'][i], linestyle='None', markersize=6, label=f'{key}')
        axes[1].loglog(
            x_data[key], v_obs_Epp[key], color=f'C{i}', marker=rcParams_plot['markers'][i], linestyle='None', markersize=6, label=f'{key}')

    axes[0].set_xlabel(r'$\omega a_T \ (rad/s)$')
    axes[0].set_xlim([0.5*1e-8, 1.5*1e2])
    axes[0].set_xticks([1e-8, 1e-6, 1e-4, 1e-2, 1e0, 1e2])
    axes[0].set_ylabel(r'$E^{\prime} \ (MPa)$')
    axes[0].legend(["0.0% xGnP", "0.5% xGnP", "1.0% xGnP", "1.5% xGnP"])
    axes[1].set_xlabel(r'$\omega a_T \ (rad/s)$')
    axes[1].set_xlim([0.5*1e-8, 1.5*1e2])
    axes[1].set_xticks([1e-8, 1e-6, 1e-4, 1e-2, 1e0, 1e2])
    axes[1].set_ylabel(r'$E^{\prime\prime} \ (MPa)$')

    plt.tight_layout()

    os.makedirs(file_path['file_path_save'], exist_ok=True)
    plt.savefig(f"{file_path['file_path_save']}/datasets_{sheet_name['sheet_name_exp']}.png", dpi=300, bbox_inches='tight')

    return v_obs_Ep, v_obs_Epp, x_data, optimized_params_df

modulus_func(x, model_params, model)

Calculate the modulus function for a given model.

Parameters:

Name Type Description Default
x float

Shifted frequency.

 required
model_params dict

The model parameters including the characteristic moduli, time scales, and fractional-order derivatives.

 required
model str

The model type ('storage' or 'loss').

 required

Returns:

Name Type Description
Ep float

The storage modulus if model is 'storage'.
Epp float

The loss modulus if model is 'loss'.
Source code in scripts/Inference_Funcs_FMG.py 
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def modulus_func(x, model_params, model):
    """Calculate the modulus function for a given model.

    Args:
        x (float): Shifted frequency.
        model_params (dict): The model parameters including the characteristic moduli, time scales, and fractional-order derivatives.
        model (str): The model type ('storage' or 'loss').

    Returns:
        Ep (float): The storage modulus if model is 'storage'.
        Epp (float): The loss modulus if model is 'loss'.
    """
    E_c1, alpha_1 = model_params['E_c1'], model_params['alpha_1']
    E_c2, alpha_2 = model_params['E_c2'], model_params['alpha_2']
    tau_c1, tau_c2 = model_params['tau_c1'], model_params['tau_c2']

    num1_Ep = (x * tau_c1)**alpha_1 * np.cos(np.pi * alpha_1 / 2) + (x * tau_c1)**(2 * alpha_1)
    num2_Ep = (x * tau_c2)**alpha_2 * np.cos(np.pi * alpha_2 / 2) + (x * tau_c2)**(2 * alpha_2)
    num1_Epp = (x * tau_c1)**alpha_1 * np.sin(np.pi * alpha_1 / 2)
    num2_Epp = (x * tau_c2)**alpha_2 * np.sin(np.pi * alpha_2 / 2)
    denom1 = 1 + (x * tau_c1)**alpha_1 * np.cos(np.pi * alpha_1 / 2) + (x * tau_c1)**(2 * alpha_1)
    denom2 = 1 + (x * tau_c2)**alpha_2 * np.cos(np.pi * alpha_2 / 2) + (x * tau_c2)**(2 * alpha_2)

    Ep = E_c1 * num1_Ep / denom1 + E_c2 * num2_Ep / denom2
    Epp = E_c1 * num1_Epp / denom1 + E_c2 * num2_Epp / denom2

    if model == 'storage':
        return Ep

    elif model == 'loss':
        return Epp
    else:
        raise ValueError("Invalid model type. Choose 'storage' or 'loss'.")

define_bounds(model_params_opt, std_fctr=0.2)

Define the bounds for the uniform priors based on the optimized model parameters.

Parameters:

Name Type Description Default
model_params_opt Series

The optimized model parameters.

 required
std_fctr float

The standard deviation factor for the bounds. Defaults to 0.2.

 0.2

Returns:

Name Type Description
bounds dict

A dictionary containing the bounds for each model parameter.
Source code in scripts/Inference_Funcs_FMG.py 
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def define_bounds(model_params_opt, std_fctr=0.2):
    """Define the bounds for the uniform priors based on the optimized model parameters.

    Args:
        model_params_opt (pd.Series): The optimized model parameters.
        std_fctr (float, optional): The standard deviation factor for the bounds. Defaults to 0.2.

    Returns:
        bounds (dict): A dictionary containing the bounds for each model parameter.
    """
    bounds = {}
    bounds['E_c1'] = [
        model_params_opt['E_c1'] - np.sqrt(3) * (std_fctr * model_params_opt['E_c1']),
        model_params_opt['E_c1'] + np.sqrt(3) * (std_fctr * model_params_opt['E_c1'])
    ]
    bounds['alpha_1'] = [
        model_params_opt['alpha_1'] - np.sqrt(3) * (std_fctr * model_params_opt['alpha_1']),
        model_params_opt['alpha_1'] + np.sqrt(3) * (std_fctr * model_params_opt['alpha_1']),
    ]
    bounds['alpha_2'] = [
        model_params_opt['alpha_2'] - np.sqrt(3) * (std_fctr * model_params_opt['alpha_2']),
        model_params_opt['alpha_2'] + np.sqrt(3) * (std_fctr * model_params_opt['alpha_2']),
    ]
    return bounds

inference_function(model_params_opt, bounds, exp_data, hyperparams)

Perform Bayesian inference using PyMC to estimate model parameters.

Parameters:

Name Type Description Default
model_params_opt Series

The optimized model parameter values.

 required
bounds dict

The bounds for the uniform prior distribution of model parameters.

 required
exp_data dict

The experimental data (both storage and loss moduli).

 required
hyperparams dict

The hyperparameters for the inference (draws, tune, etc.).

 required

Returns:

Name Type Description
idata InferenceData

Inference data (trace) from the sampling.
map_estimate dict

MAP estimate of the model parameters.
Source code in scripts/Inference_Funcs_FMG.py 
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def inference_function(model_params_opt, bounds, exp_data, hyperparams):
    """Perform Bayesian inference using PyMC to estimate model parameters.

    Args:
        model_params_opt (pd.Series): The optimized model parameter values.
        bounds (dict): The bounds for the uniform prior distribution of model parameters.
        exp_data (dict): The experimental data (both storage and loss moduli).
        hyperparams (dict): The hyperparameters for the inference (draws, tune, etc.).

    Returns:
        idata (arviz.InferenceData): Inference data (trace) from the sampling.
        map_estimate (dict): MAP estimate of the model parameters.
    """

    with pm.Model() as _:
        # Priors
        sigma_Ep = pm.Uniform(
            'sigma_Ep',
            lower=bounds['sigma_Ep'][0],
            upper=bounds['sigma_Ep'][1]
        )
        sigma_Epp = pm.Uniform(
            'sigma_Epp',
            lower=bounds['sigma_Epp'][0],
            upper=bounds['sigma_Epp'][1]
        )
        E_c1 = pm.Uniform(
            'E_c1',
            lower=bounds['E_c1'][0],
            upper=bounds['E_c1'][1]
        )
        alpha_1 = pm.Uniform(
            'alpha_1',
            lower=bounds['alpha_1'][0],
            upper=bounds['alpha_1'][1]
        )
        alpha_2 = pm.Uniform(
            'alpha_2',
            lower=bounds['alpha_2'][0],
            upper=bounds['alpha_2'][1]
        )

        # Enforce the DN constraint for E_c2
        E_c2 = pm.Deterministic(
            'E_c2',
            E_c1 * (model_params_opt['tau_c1'] / model_params_opt['tau_c2'])**2
        )

        # Model predictions
        model_params = {
            'E_c1': E_c1,
            'E_c2': E_c2,
            'alpha_1': alpha_1,
            'alpha_2': alpha_2,
            'tau_c1': model_params_opt['tau_c1'],
            'tau_c2': model_params_opt['tau_c2']
        }
        Ep_pred = pm.Deterministic("Ep_pred", modulus_func(exp_data['x_data'], model_params, 'storage'))
        Epp_pred = pm.Deterministic("Epp_pred", modulus_func(exp_data['x_data'], model_params, 'loss'))

        # Likelihood (Log-based squares error)
        _ = pm.Normal("likelihood_Ep", mu=pm.math.log(Ep_pred), sigma=sigma_Ep, observed=pm.math.log(exp_data['v_obs_Ep']))
        _ = pm.Normal("likelihood_Epp", mu=pm.math.log(Epp_pred), sigma=sigma_Epp, observed=pm.math.log(exp_data['v_obs_Epp']))

        # Inference magic
        idata = pm.sample(
            draws=hyperparams['draws'],
            tune=hyperparams['tune'],
            return_inferencedata=True,
            chains=hyperparams['chains'],
            cores=hyperparams['cores'],
            progressbar=False,
            discard_tuned_samples=True,
            nuts={'target_accept': hyperparams['target_accept']},
            idata_kwargs={'log_likelihood': True},
            callback=SimpleLoggingCallback(total_draws=hyperparams['draws'], log_interval=10_000)
        )

        # Maximum A Posteriori (MAP) estimate
        map_estimate = pm.find_MAP(
            start={
                'E_c1': model_params_opt['E_c1'],
                'E_c2': model_params_opt['E_c2'],
                'alpha_1': model_params_opt['alpha_1'],
                'alpha_2': model_params_opt['alpha_2'],
                'sigma_Ep': (bounds['sigma_Ep'][0] + bounds['sigma_Ep'][1]) / 2,
                'sigma_Epp': (bounds['sigma_Epp'][0] + bounds['sigma_Epp'][1]) / 2
            },
            method='L-BFGS-B',
            include_transformed=False,
            progressbar=False
        )
        del map_estimate['Ep_pred'], map_estimate['Epp_pred']

        # Posterior predictive checks
        pm.sample_posterior_predictive(
            idata,
            extend_inferencedata=True,
            return_inferencedata=True,
            progressbar=False
        )

    return idata, map_estimate

calculate_diagnostic_metrics(idata, param_list)

Calculate diagnostic metrics such as R-hat, ESS (bulk, 5% and 95% quantiles), and MCSE for the sampled parameters.

Parameters:

Name Type Description Default
idata InferenceData

The inference data object.

 required
param_list list

List of parameter names for which to calculate diagnostics.

 required

Returns:

Name Type Description
calculated_diagnostic_metrics dict

Dictionary containing the calculated diagnostic metrics.
Source code in scripts/Inference_Funcs_FMG.py 
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def calculate_diagnostic_metrics(idata, param_list):
    """Calculate diagnostic metrics such as R-hat, ESS (bulk, 5% and 95% quantiles), and MCSE for the sampled parameters.

    Args:
        idata (arviz.InferenceData): The inference data object.
        param_list (list): List of parameter names for which to calculate diagnostics.

    Returns:
        calculated_diagnostic_metrics (dict): Dictionary containing the calculated diagnostic metrics.
    """
    # Calculate rhat
    rhat = az.rhat(idata, var_names=param_list, method="rank")    
    rhat_dict = {
        name: rhat.values.item()
        for name, rhat in rhat.data_vars.items()
    }

    # Calculate ESS
    ess_bulk = az.ess(idata, method="bulk").to_dataframe() # .reset_index(inplace=True)
    ess_bulk.reset_index(inplace=True)
    ess_bulk_dict = ess_bulk[param_list].mean(axis=0).to_dict()

    ess_quantile_05 = az.ess(idata, method="quantile", prob=0.05).to_dataframe()
    ess_quantile_05.reset_index(inplace=True)
    ess_quantile_05_dict = ess_quantile_05[param_list].mean(axis=0).to_dict()

    ess_quantile_95 = az.ess(idata, method="quantile", prob=0.95).to_dataframe()
    ess_quantile_95.reset_index(inplace=True)
    ess_quantile_95_dict = ess_quantile_95[param_list].mean(axis=0).to_dict()

    # Calculate MCSE
    mcse = az.mcse(idata, var_names=param_list)
    mcse_dict = {
        name: mcse.values.item()
        for name, mcse in mcse.data_vars.items()
    }

    calculated_diagnostic_metrics = {
        'rhat_dict': rhat_dict,
        'ess_bulk_dict': ess_bulk_dict,
        'ess_quantile_05_dict': ess_quantile_05_dict,
        'ess_quantile_95_dict': ess_quantile_95_dict,
        'mcse_dict': mcse_dict
    }

    return calculated_diagnostic_metrics

error_func(Ep_exp, Epp_exp, Ep_opt, Epp_opt, Ep_MCMC_mean, Epp_MCMC_mean, Ep_MCMC_map, Epp_MCMC_map)

Calculate the error metrics between experimental and model data.

Parameters:

Name Type Description Default
Ep_exp ndarray

Experimental data for Ep.

 required
Epp_exp ndarray

Experimental data for Epp.

 required
Ep_opt ndarray

Optimized model data for Ep.

 required
Epp_opt ndarray

Optimized model data for Epp.

 required
Ep_MCMC_mean ndarray

MCMC mean model data for Ep.

 required
Epp_MCMC_mean ndarray

MCMC mean model data for Epp.

 required
Ep_MCMC_map ndarray

MCMC map model data for Ep.

 required
Epp_MCMC_map ndarray

MCMC map model data for Epp.

 required

Returns:

Name Type Description
error_measures dict

Dictionary containing the calculated error metrics.
Source code in scripts/Inference_Funcs_FMG.py 
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def error_func(Ep_exp, Epp_exp, Ep_opt, Epp_opt, Ep_MCMC_mean, Epp_MCMC_mean, Ep_MCMC_map, Epp_MCMC_map):
    """Calculate the error metrics between experimental and model data.

    Args:
        Ep_exp (np.ndarray): Experimental data for Ep.
        Epp_exp (np.ndarray): Experimental data for Epp.
        Ep_opt (np.ndarray): Optimized model data for Ep.
        Epp_opt (np.ndarray): Optimized model data for Epp.
        Ep_MCMC_mean (np.ndarray): MCMC mean model data for Ep.
        Epp_MCMC_mean (np.ndarray): MCMC mean model data for Epp.
        Ep_MCMC_map (np.ndarray): MCMC map model data for Ep.
        Epp_MCMC_map (np.ndarray): MCMC map model data for Epp.

    Returns:
        error_measures (dict): Dictionary containing the calculated error metrics.
    """
    log_mse_Ep_opt = np.mean((np.log(Ep_exp / Ep_opt))**2)
    log_mse_Epp_opt = np.mean((np.log(Epp_exp / Epp_opt))**2)
    log_mse_opt = np.mean((np.log(Ep_exp / Ep_opt))**2 + (np.log(Epp_exp / Epp_opt))**2)

    log_mse_Ep_MCMC_mean = np.mean((np.log(Ep_exp / Ep_MCMC_mean))**2)
    log_mse_Epp_MCMC_mean = np.mean((np.log(Epp_exp / Epp_MCMC_mean))**2)
    log_mse_MCMC_mean = np.mean((np.log(Ep_exp / Ep_MCMC_mean))**2 + (np.log(Epp_exp / Epp_MCMC_mean))**2)

    log_mse_Ep_MCMC_map = np.mean((np.log(Ep_exp / Ep_MCMC_map))**2)
    log_mse_Epp_MCMC_map = np.mean((np.log(Epp_exp / Epp_MCMC_map))**2)
    log_mse_MCMC_map = np.mean((np.log(Ep_exp / Ep_MCMC_map))**2 + (np.log(Epp_exp / Epp_MCMC_map))**2)

    num_opt = np.sum( (np.log(Ep_exp / Ep_opt))**2 ) + np.sum( (np.log(Epp_exp / Epp_opt))**2 )
    denum_opt = np.sum( (np.log(Ep_exp))**2 ) + np.sum( (np.log(Epp_exp))**2 )
    relative_error_opt = np.sqrt(num_opt/denum_opt) * 100

    num_MCMC_mean = np.sum( (np.log(Ep_exp / Ep_MCMC_mean))**2 ) + np.sum( (np.log(Epp_exp / Epp_MCMC_mean))**2 )
    denum_MCMC_mean = np.sum( (np.log(Ep_exp))**2 ) + np.sum( (np.log(Epp_exp))**2 )
    relative_error_MCMC_mean = np.sqrt(num_MCMC_mean/denum_MCMC_mean) * 100

    num_MCMC_map = np.sum( (np.log(Ep_exp / Ep_MCMC_map))**2 ) + np.sum( (np.log(Epp_exp / Epp_MCMC_map))**2 )
    denum_MCMC_map = np.sum( (np.log(Ep_exp))**2 ) + np.sum( (np.log(Epp_exp))**2 )
    relative_error_MCMC_map = np.sqrt(num_MCMC_map/denum_MCMC_map) * 100

    error_measures = {
        'log_mse_Ep_opt': log_mse_Ep_opt,
        'log_mse_Epp_opt': log_mse_Epp_opt,
        'log_mse_Ep_MCMC_mean': log_mse_Ep_MCMC_mean,
        'log_mse_Epp_MCMC_mean': log_mse_Epp_MCMC_mean,
        'log_mse_Ep_MCMC_map': log_mse_Ep_MCMC_map,
        'log_mse_Epp_MCMC_map': log_mse_Epp_MCMC_map,
        'log_mse_opt': log_mse_opt,
        'log_mse_MCMC_mean': log_mse_MCMC_mean,
        'log_mse_MCMC_map': log_mse_MCMC_map,
        'relative_error_opt': relative_error_opt,
        'relative_error_MCMC_mean': relative_error_MCMC_mean,
        'relative_error_MCMC_map': relative_error_MCMC_map
    }
    return error_measures

save_inference_results(to_be_saved, file_path, HS)

Save the inference results, diagnostic metrics, and error measures to specified file paths.

Parameters:

Name Type Description Default
to_be_saved dict

Dictionary containing the inference results to be saved.

 required
file_path dict

Dictionary containing the file paths for saving the results.

 required
HS int

The number of hidden states in the model.

 required
Source code in scripts/Inference_Funcs_FMG.py 
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def save_inference_results(to_be_saved, file_path, HS):
    """Save the inference results, diagnostic metrics, and error measures to specified file paths.

    Args:
        to_be_saved (dict): Dictionary containing the inference results to be saved.
        file_path (dict): Dictionary containing the file paths for saving the results.
        HS (int): The number of hidden states in the model.
    """
    idata = to_be_saved["idata"]
    idata_mean = to_be_saved["idata_mean"]
    idata_std = to_be_saved["idata_std"]
    map_estimate = to_be_saved["map_estimate"]
    error_measures = to_be_saved["error_measures"]
    rhat = to_be_saved["rhat"]
    ess_bulk = to_be_saved["ess_bulk_dict"]
    ess_quantile_05 = to_be_saved["ess_quantile_05_dict"]
    ess_quantile_95 = to_be_saved["ess_quantile_95_dict"]
    mcse = to_be_saved["mcse_dict"]

    for key in idata.keys():
        nc_filename = f"{file_path['file_path_save']}/EpEpp_FMG_{HS}HS_{key}.nc"
        az.to_netcdf(idata[key], nc_filename)
        print(f"Saved inference data for {key} to {nc_filename}")

    df_mean = pd.DataFrame(idata_mean).T.reset_index().rename(columns={'index': 'GnP'})
    df_mean.to_csv(f"{file_path['file_path_save']}/Mean_Inference_{HS}HS_FMG.csv", index=False)

    df_std = pd.DataFrame(idata_std).T.reset_index().rename(columns={'index': 'GnP'})
    df_std.to_csv(f"{file_path['file_path_save']}/std_Inference_{HS}HS_FMG.csv", index=False)

    df_map = pd.DataFrame(map_estimate).T.reset_index().rename(columns={'index': 'GnP'})
    df_map.to_csv(f"{file_path['file_path_save']}/Map_Estimate_{HS}HS_FMG.csv", index=False)

    df_error_measures = pd.DataFrame(error_measures).T.reset_index().rename(columns={'index': 'GnP'})
    df_error_measures.to_csv(f"{file_path['file_path_save']}/Error_Measures_{HS}HS_FMG.csv", index=False)

    df_rhat = pd.DataFrame(rhat).T.reset_index().rename(columns={'index': 'GnP'})
    df_rhat.to_csv(f"{file_path['file_path_save']}/Rhat_{HS}HS_FMG.csv", index=False)

    df_ess_bulk = pd.DataFrame(ess_bulk).T.reset_index().rename(columns={'index': 'GnP'})
    df_ess_bulk.to_csv(f"{file_path['file_path_save']}/ESS_Bulk_{HS}HS_FMG.csv", index=False)

    df_ess_quantile_05 = pd.DataFrame(ess_quantile_05).T.reset_index().rename(columns={'index': 'GnP'})
    df_ess_quantile_05.to_csv(f"{file_path['file_path_save']}/ESS_Quantile_05_{HS}HS_FMG.csv", index=False)

    df_ess_quantile_95 = pd.DataFrame(ess_quantile_95).T.reset_index().rename(columns={'index': 'GnP'})
    df_ess_quantile_95.to_csv(f"{file_path['file_path_save']}/ESS_Quantile_95_{HS}HS_FMG.csv", index=False)

    df_mcse = pd.DataFrame(mcse).T.reset_index().rename(columns={'index': 'GnP'})
    df_mcse.to_csv(f"{file_path['file_path_save']}/MCSE_{HS}HS_FMG.csv", index=False)

    print("Done! ...")

plot_posterior_distributions(idata, idata_mean, param_list, actual_param_name, plt_set, rcParams_plot)

Plot the posterior distributions of model parameters and compare with experimental data.

Parameters:

Name Type Description Default
idata InferenceData

Inference data from the sampling.

 required
idata_mean dict

Dictionary containing the mean values of the posterior distributions.

 required
param_list list

List of parameter names to plot.

 required
actual_param_name list

List of actual parameter names for labeling.

 required
plt_set dict

Dictionary containing plot settings.

 required
rcParams_plot dict

Dictionary containing plot style parameters.

 required
Source code in scripts/Inference_Funcs_FMG.py 
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def plot_posterior_distributions(idata, idata_mean, param_list, actual_param_name, plt_set, rcParams_plot):
    """Plot the posterior distributions of model parameters and compare with experimental data.

    Args:
        idata (arviz.InferenceData): Inference data from the sampling.
        idata_mean (dict): Dictionary containing the mean values of the posterior distributions.
        param_list (list): List of parameter names to plot.
        actual_param_name (list): List of actual parameter names for labeling.
        plt_set (dict): Dictionary containing plot settings.
        rcParams_plot (dict): Dictionary containing plot style parameters.
    """
    plt.rcParams.update({
        'font.size': rcParams_plot['font.size'],
        'axes.labelsize': rcParams_plot['axes.labelsize'],
        'xtick.labelsize': rcParams_plot['xtick.labelsize'],
        'ytick.labelsize': rcParams_plot['ytick.labelsize'],
        'axes.titlesize': rcParams_plot['axes.titlesize'],
        'legend.fontsize': rcParams_plot['legend.fontsize'],
        'font.family': rcParams_plot['font.family'],
        'mathtext.fontset': rcParams_plot['mathtext.fontset'],
        'mathtext.rm': rcParams_plot['mathtext.rm'],
        'mathtext.it': rcParams_plot['mathtext.it'],
        'mathtext.bf': rcParams_plot['mathtext.bf'],
    })

    hdi_prob = 0.95
    model_param_hdi = {
        'E_c1': az.hdi(idata.posterior['E_c1'].values.flatten(), hdi_prob=hdi_prob),
        'E_c2': az.hdi(idata.posterior['E_c2'].values.flatten(), hdi_prob=hdi_prob),
        'alpha_1': az.hdi(idata.posterior['alpha_1'].values.flatten(), hdi_prob=hdi_prob),
        'alpha_2': az.hdi(idata.posterior['alpha_2'].values.flatten(), hdi_prob=hdi_prob)
    }

    _, axes = plt.subplots(1, 4, figsize=(16, 4))
    axes = axes.flatten()

    for i, param in enumerate(param_list[:-2]):
        az.plot_kde(
            idata.posterior[param].values.flatten(),
            plot_kwargs={'color': 'C0', 'linewidth': 3}, ax=axes[i]
        )
        axes[i].set_xlabel(actual_param_name[i])
        axes[i].set_ylabel(r'$\boldsymbol{\pi}(' + actual_param_name[i][1:-1] + r'|\boldsymbol{D})$')

        axes[i].plot(idata_mean[param], 0, marker='o', linestyle='None', markersize=10, color='C7', label='Mean')  # GnP_idx ...
        axes[i].plot(model_param_hdi[param], [0, 0], marker='^', linestyle='None', markersize=10, color='C9', label='95% HDI')

        axes[i].tick_params()
    # Modifications
    axes[0].set_xlabel(r"$E_{c_1} \ (MPa)$")
    axes[2].set_xlabel(r"$E_{c_2} \ (MPa)$")
    axes[0].legend(loc='center right')

    plt.tight_layout(rect=[0, 0.03, 1, 0.99])
    plt.savefig(plt_set['savefig_path'], dpi=300, bbox_inches='tight')

plot_diagnostic(idata, var_names, xticks, xticklabels, plt_set, rcParams_plot)

Plot the diagnostic plots (rank and trace plots) for the inferred model parameters.

Parameters:

Name Type Description Default
idata InferenceData

Inference data from the sampling.

 required
var_names list

List of variable names to plot.

 required
xticks list

List of x-axis tick positions.

 required
xticklabels list

List of x-axis tick labels.

 required
plt_set dict

Dictionary containing plot settings.

 required
rcParams_plot dict

Dictionary containing plot style parameters.

 required
Source code in scripts/Inference_Funcs_FMG.py 
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def plot_diagnostic(idata, var_names, xticks, xticklabels, plt_set, rcParams_plot):
    """Plot the diagnostic plots (rank and trace plots) for the inferred model parameters.

    Args:
        idata (arviz.InferenceData): Inference data from the sampling.
        var_names (list): List of variable names to plot.
        xticks (list): List of x-axis tick positions.
        xticklabels (list): List of x-axis tick labels.
        plt_set (dict): Dictionary containing plot settings.
        rcParams_plot (dict): Dictionary containing plot style parameters.
    """
    plt.rcParams.update({
        'font.size': rcParams_plot['font.size'],
        'axes.labelsize': rcParams_plot['axes.labelsize'],
        'xtick.labelsize': rcParams_plot['xtick.labelsize'],
        'ytick.labelsize': rcParams_plot['ytick.labelsize'],
        'legend.fontsize': rcParams_plot['legend.fontsize'],
        'axes.titlesize': rcParams_plot['axes.titlesize'],
        'font.family': rcParams_plot['font.family'],
        'mathtext.fontset': rcParams_plot['mathtext.fontset'],
        'mathtext.rm': rcParams_plot['mathtext.rm'],
        'mathtext.it': rcParams_plot['mathtext.it'],
        'mathtext.bf': rcParams_plot['mathtext.bf']
    })

    axes = az.plot_rank(idata, var_names=var_names, grid=(1, 4), figsize=(16, 4))
    axes = axes.flatten()
    axes[0].set_title(r"$E_{c_1}$")
    axes[0].set_xlabel("Rank (All Chains)")
    axes[0].set_xticks(xticks)
    axes[0].set_xticklabels(xticklabels)
    axes[0].set_yticklabels([1, 2, 3, 4])

    axes[1].set_title(r"$\alpha_1$")
    axes[1].set_xlabel("Rank (All Chains)")
    axes[1].set_xticks(xticks)
    axes[1].set_xticklabels(xticklabels)
    axes[1].set_yticklabels([1, 2, 3, 4])

    axes[2].set_title(r"$E_{c_2}$")
    axes[2].set_xlabel("Rank (All Chains)")
    axes[2].set_xticks(xticks)
    axes[2].set_xticklabels(xticklabels)
    axes[2].set_yticklabels([1, 2, 3, 4])

    axes[3].set_title(r"$\alpha_2$")
    axes[3].set_xlabel("Rank (All Chains)")
    axes[3].set_xticks(xticks)
    axes[3].set_xticklabels(xticklabels)
    axes[3].set_yticklabels([1, 2, 3, 4])

    plt.tight_layout()
    plt.savefig(plt_set['savefig_path_rank'], dpi=300, bbox_inches='tight')
    plt.show()

    _, axes = plt.subplots(1, 4, figsize=(16, 3.5))
    axes = axes.flatten()
    for i, param in enumerate(var_names):
        axes[i].plot(idata.posterior[param].values[0, :], color='C0', alpha=0.15, linestyle='-')
        axes[i].plot(idata.posterior[param].values[1, :], color='C0', alpha=0.40, linestyle='--')
        axes[i].plot(idata.posterior[param].values[2, :], color='C0', alpha=0.65, linestyle='-.')
        axes[i].plot(idata.posterior[param].values[3, :], color='C0', alpha=0.90, linestyle=':')
        axes[i].set_xlabel('Sample Draws')
    axes[0].set_ylabel(r'$E_{c_1} \ (MPa)$')
    axes[1].set_ylabel(r'$\alpha_1$')
    axes[2].set_ylabel(r'$E_{c_2} \ (MPa)$')
    axes[3].set_ylabel(r'$\alpha_2$')

    plt.tight_layout()
    plt.savefig(plt_set['savefig_path_trace'], dpi=300, bbox_inches='tight')
    plt.show()

plot_posterior_predictive(idata_dict, v_obs_Ep, v_obs_Epp, x_data, GnPs, GnP_idx, xylims, plt_set, rcParams_plot)

Plot the storage and loss mooduli predictions with uncertainty intervals (HDI) from the posterior predictive distribution, and compare with the observed data.

Parameters:

Name Type Description Default
idata_dict InferenceData

Inference data from the sampling.

 required
v_obs_Ep ndarray

Observed storage modulus data.

 required
v_obs_Epp ndarray

Observed loss modulus data.

 required
x_data ndarray

Frequency data.

 required
GnPs list

List of GnP values.

 required
GnP_idx int

Index of the current GnP value.

 required
xylims dict

Limits for the x and y axes of the insets.

 required
plt_set dict

Description of the plot settings.

 required
rcParams_plot dict

Description of the plot parameters.

 required
Source code in scripts/Inference_Funcs_FMG.py 
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def plot_posterior_predictive(idata_dict, v_obs_Ep, v_obs_Epp, x_data, GnPs, GnP_idx, xylims, plt_set, rcParams_plot):
    """Plot the storage and loss mooduli predictions with uncertainty intervals (HDI) from the posterior predictive distribution, and compare with the observed data.

    Args:
        idata_dict (arviz.InferenceData): Inference data from the sampling.
        v_obs_Ep (np.ndarray): Observed storage modulus data.
        v_obs_Epp (np.ndarray): Observed loss modulus data.
        x_data (np.ndarray): Frequency data.
        GnPs (list): List of GnP values.
        GnP_idx (int): Index of the current GnP value.
        xylims (dict): Limits for the x and y axes of the insets.
        plt_set (dict): Description of the plot settings.
        rcParams_plot (dict): Description of the plot parameters.
    """
    plt.rcParams.update({
        'font.size': rcParams_plot['font.size'],
        'axes.labelsize': rcParams_plot['axes.labelsize'],
        'xtick.labelsize': rcParams_plot['xtick.labelsize'],
        'ytick.labelsize': rcParams_plot['ytick.labelsize'],
        'legend.fontsize': rcParams_plot['legend.fontsize'],
        'axes.titlesize': rcParams_plot['axes.titlesize'],
        'font.family': rcParams_plot['font.family'],
        'mathtext.fontset': rcParams_plot['mathtext.fontset'],
        'mathtext.rm': rcParams_plot['mathtext.rm'],
        'mathtext.it': rcParams_plot['mathtext.it'],
        'mathtext.bf': rcParams_plot['mathtext.bf']
    })
    fig, axes = plt.subplots(1, 2, figsize=(16, 5), constrained_layout=True)
    axes = axes.flatten()

    Ep_pred = idata_dict[GnPs[GnP_idx]].posterior["Ep_pred"]
    Ep_pred_mean = Ep_pred.mean(dim=["draw", "chain"])
    Epp_pred = idata_dict[GnPs[GnP_idx]].posterior["Epp_pred"]
    Epp_pred_mean = Epp_pred.mean(dim=["draw", "chain"])

    # Plot for Ep_pred
    axes[0].scatter(x_data[GnPs[GnP_idx]], v_obs_Ep[GnPs[GnP_idx]], label='Observed', marker='o', s=64, color='C3')
    axes[0].plot(x_data[GnPs[GnP_idx]], Ep_pred_mean, label='Mean Predicted', linestyle='-', color='C0', linewidth=1.1)
    az.plot_hdi(
        x_data[GnPs[GnP_idx]],
        Ep_pred,
        hdi_prob=0.95,
        smooth=False,
        plot_kwargs={'color': 'C0', 'alpha': 0.3},
        fill_kwargs={'color': 'C0', 'alpha': 0.3, 'label': r'95% HDI'},
        ax=axes[0]
    )
    axes[0].set_xscale("log")
    axes[0].set_yscale("log")
    axes[0].set_xlabel(r"$\omega a_T \ (rad/s)$")
    axes[0].set_ylabel(r"$E^{\prime}$ (MPa)")
    axes[0].set_xlim([0.5*1e-8, 1.5*1e2])
    axes[0].set_xticks([1e-8, 1e-6, 1e-4, 1e-2, 1e0, 1e2])
    axes[0].set_ylim([1.1e1, 4e3])
    axes[0].text(
        0.012, 0.92, r'$(a)$',
        transform=axes[0].transAxes,
    )
    axes[0].legend(fontsize=16, loc='lower right')

    # First inset for Ep_pred
    axins1 = zoomed_inset_axes(axes[0], zoom=1.5, loc=xylims['Ep_inset1']['loc'])
    axins1.loglog(x_data[GnPs[GnP_idx]], Ep_pred_mean, color='C0')
    axins1.scatter(x_data[GnPs[GnP_idx]], v_obs_Ep[GnPs[GnP_idx]], color='C3', s=64)
    az.plot_hdi(
        x_data[GnPs[GnP_idx]],
        Ep_pred,
        hdi_prob=0.95,
        smooth=False,
        plot_kwargs={'color': 'C0', 'alpha': 0.3},
        fill_kwargs={'color': 'C0', 'alpha': 0.3},
        ax=axins1
    )

    axins1.set_xlim(xylims['Ep_inset1']['xlim'])
    axins1.set_ylim(xylims['Ep_inset1']['ylim'])
    axins1.xaxis.set_visible(False)
    axins1.yaxis.set_visible(False)
    mark_inset(axes[0], axins1, loc1=2, loc2=4, fc="none", ec="0.5")

    # Second inset for Ep_pred
    axins2 = zoomed_inset_axes(axes[0], zoom=1.5, loc=xylims['Ep_inset2']['loc'])
    axins2.loglog(x_data[GnPs[GnP_idx]], Ep_pred_mean, color='C0')
    axins2.scatter(x_data[GnPs[GnP_idx]], v_obs_Ep[GnPs[GnP_idx]], color='C3', s=64)
    az.plot_hdi(
        x_data[GnPs[GnP_idx]],
        Ep_pred,
        hdi_prob=0.95,
        smooth=False,
        plot_kwargs={'color': 'C0', 'alpha': 0.3},
        fill_kwargs={'color': 'C0', 'alpha': 0.3},
        ax=axins2
    )
    axins2.set_xlim(xylims['Ep_inset2']['xlim'])
    axins2.set_ylim(xylims['Ep_inset2']['ylim'])
    axins2.xaxis.set_visible(False)
    axins2.yaxis.set_visible(False)
    mark_inset(axes[0], axins2, loc1=2, loc2=4, fc="none", ec="0.5")

    # Epp
    axes[1].scatter(x_data[GnPs[GnP_idx]], v_obs_Epp[GnPs[GnP_idx]], label='Observed', marker='o', s=64, color='C3')
    axes[1].plot(x_data[GnPs[GnP_idx]], Epp_pred_mean, label='Mean Predicted', linestyle='-', color='C0', linewidth=1.1)
    az.plot_hdi(
        x_data[GnPs[GnP_idx]],
        Epp_pred,
        hdi_prob=0.95,
        smooth=False,
        plot_kwargs={'color': 'C0', 'alpha': 0.3},
        fill_kwargs={'color': 'C0', 'alpha': 0.3},
        ax=axes[1]
    )
    axes[1].set_xscale("log")
    axes[1].set_yscale("log")
    axes[1].set_xlabel(r"$\omega a_T \ (rad/s)$")
    axes[1].set_ylabel(r"$E^{\prime\prime}$ (MPa)")
    axes[1].set_xlim([0.5*1e-8, 1.5*1e2])
    axes[1].set_xticks([1e-8, 1e-6, 1e-4, 1e-2, 1e0, 1e2])
    axes[1].set_ylim([1.1e0, 1.1e3])
    axes[1].text(
        0.012, 0.92, r'$(b)$',
        transform=axes[1].transAxes,
    )

    # First inset for Epp_pred
    axins3 = zoomed_inset_axes(axes[1], zoom=1.5, loc=xylims['Epp_inset3']['loc'])
    axins3.loglog(x_data[GnPs[GnP_idx]], Epp_pred_mean, color='C0')
    axins3.scatter(x_data[GnPs[GnP_idx]], v_obs_Epp[GnPs[GnP_idx]], color='C3', s=64)
    az.plot_hdi(
        x_data[GnPs[GnP_idx]],
        Epp_pred,
        hdi_prob=0.95,
        smooth=False,
        plot_kwargs={'color': 'C0', 'alpha': 0.3},
        fill_kwargs={'color': 'C0', 'alpha': 0.3},
        ax=axins3
    )
    axins3.set_xlim(xylims['Epp_inset3']['xlim'])
    axins3.set_ylim(xylims['Epp_inset3']['ylim'])
    axins3.xaxis.set_visible(False)
    axins3.yaxis.set_visible(False)
    mark_inset(axes[1], axins3, loc1=2, loc2=4, fc="none", ec="0.5")

    # Second inset for Epp_pred
    axins4 = zoomed_inset_axes(axes[1], zoom=1.25, loc=xylims['Epp_inset4']['loc'])

    axins4.loglog(x_data[GnPs[GnP_idx]], Epp_pred_mean, color='C0')
    axins4.scatter(x_data[GnPs[GnP_idx]], v_obs_Epp[GnPs[GnP_idx]], color='C3', s=64)
    az.plot_hdi(
        x_data[GnPs[GnP_idx]],
        Epp_pred,
        hdi_prob=0.95,
        smooth=False,
        plot_kwargs={'color': 'C0', 'alpha': 0.3},
        fill_kwargs={'color': 'C0', 'alpha': 0.3},
        ax=axins4
    )
    axins4.set_xlim(xylims['Epp_inset4']['xlim'])
    axins4.set_ylim(xylims['Epp_inset4']['ylim'])
    axins4.xaxis.set_visible(False)
    axins4.yaxis.set_visible(False)
    mark_inset(axes[1], axins4, loc1=2, loc2=4, fc="none", ec="0.5")

    plt.savefig(plt_set['savefig_path'], dpi=300, bbox_inches='tight')
    plt.show()

plot_posterior_predictive_checks(idata, range1, range2, plt_set, rcParams_plot)

Plot the posterior predictive checks for the model with the inferred model parameters.

Parameters:

Name Type Description Default
idata InferenceData

Inference data from the sampling.

 required
range1 tuple

The range for the first subplot.

 required
range2 tuple

The range for the second subplot.

 required
rcParams_plot dict

Dictionary containing plot style parameters.

 required
Source code in scripts/Inference_Funcs_FMG.py 
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def plot_posterior_predictive_checks(idata, range1, range2, plt_set, rcParams_plot):
    """Plot the posterior predictive checks for the model with the inferred model parameters.

    Args:
        idata (arviz.InferenceData): Inference data from the sampling.
        range1 (tuple): The range for the first subplot.
        range2 (tuple): The range for the second subplot.
        rcParams_plot (dict): Dictionary containing plot style parameters.
    """
    plt.rcParams.update({
        'font.size': rcParams_plot['font.size'],
        'axes.labelsize': rcParams_plot['axes.labelsize'],
        'xtick.labelsize': rcParams_plot['xtick.labelsize'],
        'ytick.labelsize': rcParams_plot['ytick.labelsize'],
        'axes.titlesize': rcParams_plot['axes.titlesize'],
        'legend.fontsize': rcParams_plot['legend.fontsize'],
        'font.family': rcParams_plot['font.family'],
        'mathtext.fontset': rcParams_plot['mathtext.fontset'],
        'mathtext.rm': rcParams_plot['mathtext.rm'],
        'mathtext.it': rcParams_plot['mathtext.it'],
        'mathtext.bf': rcParams_plot['mathtext.bf'],
    })

    ax = az.plot_ppc(idata, figsize=(16, 5), num_pp_samples=500)

    ax[0].set_xlabel("Likelihood of E\' ")
    ax[1].set_xlabel("Likelihood of E\'\' ")

    ax[0].set_ylabel("Density")
    ax[1].set_ylabel("Density")

    ax[0].set_xlim(range1)
    ax[1].set_xlim(range2)

    ax[0].legend().remove()
    ax[1].legend(loc='upper left')

    plt.savefig(plt_set['savefig_path'], dpi=300, bbox_inches='tight')
    plt.show()

plot_pairplot(idata, var_names, subset_data_number, plt_set, rcParams_plot)

Plot the pairplot of the sampled parameters to visualize their joint distributions and correlations.

Parameters:

Name Type Description Default
idata InferenceData

Inference data from the sampling.

 required
var_names list

List of variable names to plot.

 required
subset_data_number int

Number of samples to include in the subset.

 required
rcParams_plot dict

Dictionary containing plot style parameters.

 required
Source code in scripts/Inference_Funcs_FMG.py 
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def plot_pairplot(idata, var_names, subset_data_number, plt_set, rcParams_plot):
    """Plot the pairplot of the sampled parameters to visualize their joint distributions and correlations.

    Args:
        idata (arviz.InferenceData): Inference data from the sampling.
        var_names (list): List of variable names to plot.
        subset_data_number (int): Number of samples to include in the subset.
        rcParams_plot (dict): Dictionary containing plot style parameters.
    """
    data = az.extract(idata, var_names=var_names).to_dataframe()
    data = data[var_names]
    data.reset_index(inplace=True)
    data.drop(columns=['chain', 'draw'], inplace=True)
    subdata = data.iloc[:subset_data_number]

    plt.rcParams.update({
        'font.size': rcParams_plot['font.size'],
        'axes.labelsize': rcParams_plot['axes.labelsize'],
        'xtick.labelsize': rcParams_plot['xtick.labelsize'],
        'ytick.labelsize': rcParams_plot['ytick.labelsize'],
        'axes.titlesize': rcParams_plot['axes.titlesize'],
        'legend.fontsize': rcParams_plot['legend.fontsize'],
        'font.family': rcParams_plot['font.family'],
        'mathtext.fontset': rcParams_plot['mathtext.fontset'],
        'mathtext.rm': rcParams_plot['mathtext.rm'],
        'mathtext.it': rcParams_plot['mathtext.it'],
        'mathtext.bf': rcParams_plot['mathtext.bf'],
    })

    g = sns.PairGrid(subdata, diag_sharey=False)
    g.map_upper(sns.scatterplot, s=30, alpha=0.99, edgecolor='black', linewidth=0.5, color=sns.color_palette("Paired")[1])
    g.map_lower(sns.kdeplot, fill=False, thresh=0.05, levels=5, cmap=sns.color_palette("viridis", as_cmap=True))  # plt.cm.viridis
    g.map_diag(sns.kdeplot, lw=2, color=sns.color_palette("Paired")[0])

    for ax in g.axes.flat:
        ax.spines['right'].set_visible(True)
        ax.spines['top'].set_visible(True)
        ax.spines['right'].set_color('black')
        ax.spines['top'].set_color('black')
        ax.spines['right'].set_linewidth(0.75)
        ax.spines['top'].set_linewidth(0.75)

    g.axes[0, 0].set_ylabel(r"$E_{c_1}$")
    g.axes[1, 0].set_ylabel(r"$\alpha_1$")
    g.axes[2, 0].set_ylabel(r"$\alpha_2$")
    g.axes[3, 0].set_ylabel(r"$E_{c_2}$")

    g.axes[3, 0].set_xlabel(r"$E_{c_1}$")
    g.axes[3, 1].set_xlabel(r"$\alpha_1$")
    g.axes[3, 2].set_xlabel(r"$\alpha_2$")
    g.axes[3, 3].set_xlabel(r"$E_{c_2}$")
    plt.savefig(plt_set['savefig_path'], dpi=300, bbox_inches='tight')
    plt.show()