Features

The IntLevPy package offers the following features for simulating, analyzing, and classifying complex stochastic processes:

Simulation

• Generate synthetic intermittent and Lévy trajectories.

• Simulate realistic intermittent processes using various statistical distributions.

Functions:

1. intermittent3 Description: Simulates an intermittent 2D Lévy walk (diffusion and ballistic motion).

   Parameters: - nt (int): Number of time steps. - dt (float): Time interval between steps. - mean_bal_sac (float): Mean velocity for ballistic regime. - diffusion (float): Diffusion coefficient. - rate21 (float): Transition rate from ballistic to diffusive. - rate12 (float): Transition rate from diffusive to ballistic.

   Returns: Tuple (x, y): Arrays with the x and y coordinates over time.

2. wait_times Description: Generates waiting times following a power-law distribution for Lévy processes.

   Parameters: - taui (float): Scaling factor for waiting times. - lN (int): Number of samples. - lalpha (float): Lévy distribution exponent (1 < lalpha < 3).

   Returns: Array of generated waiting times.

3. levy_flight_2D_Simplified Description: Simulates a 2D Lévy flight using specified parameters.

   Parameters: - n_redirections (int): Number of redirection steps. - n_max (int): Maximum number of measurement points. - alpha (float): Lévy distribution exponent (1 < alpha < 3). - tmin (float): Minimum flight time. - v_mean (float): Mean velocity. - measuring_dt (float): Time interval for measurements.

   Returns: Tuple (x_measured, y_measured): Arrays with the x and y positions at measurement times.

Moments Calculation

• Calculate theoretical and empirical second and fourth moments.

• Allows for detailed statistical analysis of trajectories.

Functions:

1. mom4_serg_log Description: Calculates the theoretical logarithm of the 4th moment for intermittent search processes.

   Parameters: - t (array-like): Time lags. - v0 (float): Mean velocity. - D (float): Diffusion coefficient. - lambdaB (float): Transition rate to ballistic regime. - lambdaD (float): Transition rate to diffusive regime.

   Returns: Array of the logarithm of the 4th moment values.

2. mom2_serg_log Description: Calculates the theoretical logarithm of the 2nd moment for intermittent search processes.

   Parameters: - tau (array-like): Time lags. - v (float): Mean velocity. - D (float): Diffusion coefficient. - lambdaB (float): Transition rate to ballistic regime. - lambdaD (float): Transition rate to diffusive regime.

   Returns: Array of the logarithm of the 2nd moment values.

Optimization for Model Fitting

• Provides optimization routines to fit model parameters to empirical data.

• Enables accurate parameter estimation for intermittent and Lévy models.

Functions:

1. to_optimize_mom4_serg_log Description: Objective function to optimize parameters for the 4th moment in intermittent search processes.

   Parameters: - variables (list): [v0, D, lambdaB, lambdaD] values for velocity, diffusion, and transition rates. - tau_list (array-like): List of time lags. - logdx4 (array-like): Empirical 4th moment data (log scale).

   Returns: Mean squared error between empirical and theoretical fourth moments.

2. to_optimize_mom2_serg_log Description: Objective function to optimize parameters for the second moment in intermittent search processes.

   Parameters: - variables (list): [v0, D, lambdaB, lambdaD] values for velocity, diffusion, and transition rates. - tau_list (array-like): List of time lags. - logdx2 (array-like): Empirical second moment data (log scale).

   Returns: Mean squared error between empirical and theoretical second moment.

Classification

• Classify processes as intermittent or Levy based on statistical properties.

• Uses threshold-based and Fisher’s exact test-based classifications.

Functions:

1. real_k_and_fisher Description: Calculates frequency matrix and detection metrics for a binary sequence, with Fisher’s exact test for correlation.

   Parameters: - binary_vector (array-like): Sequence of binary values.

   Returns: Tuple containing the frequency matrix, detection values, and log of Fisher exact test values.

2. frequency_matrix_2D Description: Creates a 2D frequency matrix based on a threshold, optionally normalizing by row sums.

   Parameters: - d__ss (array-like): Input data sequence. - threshold (float): Threshold to create binary vector from data. - normalized (bool): Whether to normalize by row sums.

   Returns: 2x2 frequency matrix.

3. form_groups Description: Calculates detection and Fisher test metrics across multiple thresholds, with optional plotting.

   Parameters: - vector (array-like): Input data sequence. - threshold_array (array-like): Array of thresholds to evaluate. - graph (bool): If True, plots detection metrics. - x_label (str): X-axis label for plot. - title (str): Title for plot. - x_axis_format (str): Format for x-axis labels.

   Returns: Tuple of lists for detection metrics, Fisher metrics, and minimum values of detection and Fisher test results.

For more detailed usage examples, please see the Usage section or check out the example scripts in the examples/ directory.