### Clone Repository and Install PyMultifracs

Source: https://www.neurospin.fr/pymultifracs/index.html

Clone the entire PyMultifracs repository to your local machine and then install it in editable mode. This is useful if you plan to modify the package or run examples.

```bash
git clone https://github.com/neurospin/pymultifracs
pip install -e pymultifracs
```

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

### Install PyMultifracs using Pip

Source: https://www.neurospin.fr/pymultifracs/index.html

Install the package directly from the GitHub repository using pip. This method installs the package for import.

```bash
pip install git+https://github.com/neurospin/pymultifracs
```

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

### Query benchmark results

Source: https://www.neurospin.fr/pymultifracs/auto_examples/05-benchmark.html

Query the results dataframe to inspect specific subsets of the benchmark data. This example filters for results from the 'pleader' method and displays the first 10 rows.

```python
bench.results.query('method=="pleader"').head(10)
```

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

### Install PyMultifracs using Conda

Source: https://www.neurospin.fr/pymultifracs/index.html

Use this command to update your conda environment with the necessary files from the env.yml. Ensure you have the correct environment name set.

```bash
wget https://raw.githubusercontent.com/neurospin/pymultifracs/master/env.yml
conda env update -f env.yml --name $ENVNAME
```

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

### Derive statistics from results

Source: https://www.neurospin.fr/pymultifracs/auto_examples/05-benchmark.html

Derive statistics from the benchmark results dataframe by grouping and calculating means. This example calculates the mean of 'c1' and 'c2' for 'fbm' signals with H=0.7, grouped by 'p_exp'.

```python
bench.results.xs(
    ('fbm', 'pleader', .7), level=('model', 'method', 'H')
    ).groupby('p_exp').mean()
```

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

### Get Wavelet Transform Coefficients

Source: https://www.neurospin.fr/pymultifracs/auto_examples/01-simul.html

Extracts multi-resolution quantities from the wavelet transform. `get_leaders()` retrieves wavelet (p-)leaders, and `get_wse()` retrieves wavelet scale-wise energy. `p_exp` in `get_leaders` determines the exponent for the p-norm.

```python
WTL = WT.get_leaders(p_exp=np.inf)
WTpL = WT.get_leaders(p_exp=2)
WSE = WT.get_wse(theta=0.5)
```

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

### Visualize group statistics with Seaborn

Source: https://www.neurospin.fr/pymultifracs/auto_examples/05-benchmark.html

Use Seaborn to visualize group statistics from the benchmark results. This example creates a boxplot comparing the 'c2' values for 'mrw' signals estimated with the 'pleader' method, across different 'p_exp' values.

```python
import seaborn as sns

sns.boxplot(data=bench.results.xs(
    ('mrw','pleader'), level=('model', 'method')), x='p_exp', y='c2')
```

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

### Instantiate and compute benchmark

Source: https://www.neurospin.fr/pymultifracs/auto_examples/05-benchmark.html

Instantiate the Benchmark class with the defined signal and estimation grids, then compute the benchmark to generate results. This process systematically tests all combinations.

```python
bench = Benchmark(
    signal_gen_grid, signal_param_grid, estimation_grid, estimation_param_grid)
bench.compute_benchmark()
```

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

### Method: bootstrap

Source: https://www.neurospin.fr/pymultifracs/_autosummary/pymultifracs.multiresquantity.WaveletWSE.html

Bootstrap the multi-resolution quantity by repeating.

```APIDOC
## bootstrap

### Description
Bootstrap the multi-resolution quantity by repeating.

### Parameters
#### Request Body
- **R** (int) - Required - Number of repetitions of the bootstrap.
- **min_scale** (int) - Optional - Minimum scale that will be kept in the bootstrapped MRQ.
- **idx_reject** (dict[str, np.ndarray] | None) - Optional - Dictionary of rejected values.

### Response
#### Success Response (200)
- **WaveletDec** - MRQ containing the bootstrapped values.
```

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

### Define estimation parameter grids

Source: https://www.neurospin.fr/pymultifracs/auto_examples/05-benchmark.html

Create dictionaries specifying parameter grids for the estimation methods. This allows the benchmark to test different estimation parameters for each method.

```python
estimation_param_grid = {
    'pleader': {
        'p_exp': np.array([0.5, 1, 2, 5]),
        },
    'coef': {}
}
estimation_grid = {
    'pleader': pleader_est,
    'coef': coef_est,
}
```

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

### Import necessary libraries

Source: https://www.neurospin.fr/pymultifracs/auto_examples/05-benchmark.html

Import numpy and required modules from pymultifracs for signal simulation, wavelet analysis, multifractal analysis, and benchmarking.

```python
import numpy as np

from pymultifracs.simul import fbm, mrw
from pymultifracs import wavelet_analysis, mfa
from pymultifracs.robust.benchmark import Benchmark
```

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

### Determine Optimal Scaling Range (Per Moment)

Source: https://www.neurospin.fr/pymultifracs/_downloads/12297bf87b30ae3499b9d7500abaf258/02-bootstrap.ipynb

Finds the optimal scaling range on a per-moment basis using bootstrapping, allowing for moment-specific scale selection.

```python
pwt.cumulants.find_best_range(per_moment=True)
```

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

### Define signal generation parameter grids

Source: https://www.neurospin.fr/pymultifracs/auto_examples/05-benchmark.html

Create dictionaries that specify the parameter grids for generating different types of signals (fbm and mrw). This allows the benchmark to systematically vary signal properties.

```python
signal_param_grid = {
    'fbm': {
        'H': np.array([.7, .8])
    },
    'mrw': {
        'H': np.array([.7, .9])
    }
}

signal_gen_grid = {
    'fbm': fbm_gen,
    'mrw': mrw_gen,
}
```

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

### WaveletDec Class Overview

Source: https://www.neurospin.fr/pymultifracs/_autosummary/pymultifracs.multiresquantity.WaveletDec.html

Provides an overview of the WaveletDec class, its purpose, and how it should be instantiated.

```APIDOC
## Class WaveletDec

### Description
Represents the wavelet coefficients of a signal dX(j,k). This class should not be instantiated directly but rather created using the `wavelet_analysis` function.

### Attributes
- **values** (`dict`[`int`, `ndarray`]): A dictionary where `values[j]` contains the coefficients at scale `j`. The arrays have a shape of (nj, n_rep).
- **n_channel** (`int`): The number of underlying signals in the wavelet decomposition.
- **gamint** (`float`): The fractional integration used in the computation of the MRQ.
- **wt_name** (`str`): The name of the wavelet used for the decomposition.
- **interval_size** (`int`): The width of the coefficient interval over which the MRQ was computed.
- **origin_mrq** (`WaveletDec` | `None`): If the MRQ is derived from another MRQ, this refers to the original MRQ.
- **bootstrapped_obj** (`WaveletDec` | `None`): Stores the bootstrapped version of the MRQ if bootstrapping has been used.
```

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

### fbm()

Source: https://www.neurospin.fr/pymultifracs/_autosummary/pymultifracs.simul.fbm.html

Function to simulate fractional Brownian motion (fBm).

```APIDOC
## fbm()

### Description
Simulate fractional Brownian motion (fBm).

### Endpoint
pymultifracs.simul.fbm(*args, **kwargs)
```

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

### StructureFunction Analysis and Plotting

Source: https://www.neurospin.fr/pymultifracs/_autosummary/pymultifracs.scalingfunction.StructureFunction.html

Methods for finding optimal scaling ranges and visualizing results.

```APIDOC
## POST StructureFunction.find_best_range

### Description
Find the best range among those computed, given bootstrap was already performed.

### Parameters
#### Request Body
- **per_moment** (bool) - Optional - If True, returns the best range for each moment. Otherwise, returns the overall best range.

## GET StructureFunction.plot

### Description
Plots the structure functions.

### Parameters
#### Query Parameters
- **nrow** (int) - Optional - Number of rows in the plot.
- **filename** (str) - Optional - If provided, the file is saved to this path.
- **ignore_q0** (bool) - Optional - Whether to include the structure function for q=0.
- **figsize** (tuple) - Optional - Size of the figure in inches.
- **scaling_range** (int) - Optional - Index of the scaling range to use.
- **plot_scales** (tuple) - Optional - Constrains the x-axis to the interval [j1,j2].
- **plot_CI** (bool) - Optional - If using bootstrap, plot confidence intervals.
- **signal_idx** (int) - Optional - Index of the signal to plot for multivariate signals.
```

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

### Method: get_wse

Source: https://www.neurospin.fr/pymultifracs/_autosummary/pymultifracs.multiresquantity.WaveletWSE.html

Compute weak scaling exponents from the wavelet coefficients.

```APIDOC
## get_wse

### Description
Compute weak scaling exponents from the wavelet coefficients.

### Parameters
#### Request Body
- **theta** (float) - Optional - Parameter controlling cone reach in lower scales.
- **omega** (float) - Optional - Parameter controlling the angle of the cone.
- **gamint** (float) - Optional - Fractional integration coefficient.

### Response
#### Success Response (200)
- **WaveletWSE** - Weak scaling exponent derived from the coefficients.
```

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

### build_q_log Function

Source: https://www.neurospin.fr/pymultifracs/_autosummary/pymultifracs.utils.build_q_log.html

This function constructs a vector of q values for multifractal analysis, including log-spaced values between specified bounds and key integer points.

```APIDOC
## build_q_log()

### Description
Build a convenient vector of q values for multifractal analysis.

### Method
N/A (This is a Python function, not an API endpoint)

### Endpoint
N/A

### Parameters
#### Path Parameters
N/A

#### Query Parameters
N/A

#### Request Body
N/A

### Request Example
N/A

### Response
#### Success Response (200)
- **q** (ndarray of float) - log-spaced values between q_min and q_max, along with their opposites, and accompanied by -2, -1, 0, 1, 2.

#### Response Example
```python
import numpy as np

q_min = 0.1
q_max = 10.0
n = 5
q = build_q_log(q_min, q_max, n)
print(q)
# Expected output might look like: [-10.   -5.   -2.   -1.    0.    1.    2.    5.   10.] (actual values depend on logspace calculation and inclusion of opposites/integers)
```

### Parameters
- **_q_min_** (float) - Lower bound of q, needs to be strictly positive.
- **_q_max_** (float) - Upper value of q, needs to be strictly positive.
- **_n_** (int) - Number of logspaced values to include.
```

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

### Plot Bootstrapped Results

Source: https://www.neurospin.fr/pymultifracs/auto_examples/02-bootstrap.html

Visualizes cumulants and multifractal spectrum with empirical confidence intervals.

```python
pwt.cumulants.plot()
```

```python
pwt.spectrum.plot()
```

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

### Automatically Integrate Time Series

Source: https://www.neurospin.fr/pymultifracs/auto_examples/01-simul.html

Attempts to find a fractional integration coefficient to satisfy regularity conditions. Returns the integrated multi-resolution quantity. This is useful when the minimal regularity is too low.

```python
WTpL = WTpL.auto_integrate(scaling_ranges)
```

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

### Load Bivariate Data using pooch and scipy

Source: https://www.neurospin.fr/pymultifracs/auto_examples/10-bivariate.html

Loads a .mat file containing bivariate time series data using pooch for retrieval and scipy for loading. Ensure the 'bivar' directory exists in your cache or adjust the path.

```python
import pooch
from scipy.io import loadmat

url = ('https://github.com/neurospin/pymultifracs/raw/refs/heads/master/tests/'
       'data/DataSet_ssMF.mat')

fname = pooch.retrieve(url=url, known_hash=None, path=pooch.os_cache("bivar"))

X = loadmat(fname)['data'].T

```

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

### Perform Wavelet Analysis for p-Leaders

Source: https://www.neurospin.fr/pymultifracs/auto_examples/10-bivariate.html

Conducts wavelet analysis on the input data to obtain p-leaders, which are then used for bivariate multifractal analysis. The integrate method is used with a specific exponent.

```python
from pymultifracs import wavelet_analysis

WTpL = wavelet_analysis(X).integrate(.75).get_leaders(2)

```

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

### WaveletLeader Methods

Source: https://www.neurospin.fr/pymultifracs/_autosummary/pymultifracs.multiresquantity.WaveletLeader.html

This section details various methods available for the WaveletLeader object, including data access, computation, and visualization.

```APIDOC
## GET /api/values

### Description
Get the values of the MRQ, applying any finite size effects corrections if necessary (Wavelet p-leaders).

### Method
GET

### Endpoint
/api/values

### Parameters
#### Query Parameters
- **_j_** (any) - Required - The scale parameter.
- **_idx_reject_** (dict[str, np.ndarray] | None) - Optional - Dictionary of rejected values.

### Response
#### Success Response (200)
- **values** (any) - The computed values.

## GET /api/wse

### Description
Compute weak scaling exponents from the wavelet coefficients.

### Method
GET

### Endpoint
/api/wse

### Parameters
#### Query Parameters
- **_theta_** (float) - Optional - Parameter controlling to which extent the cone reaches in the lower scales. Practically, (θ,ω)-leaders computed at scale j reach down to j−jθ.
- **_omega_** (float) - Optional - Parameter controlling the angle of the cone over which the weak scaling exponents are computed: practically, for computing the (θ,ω)-leaders at scale j, the width of the cone at scale j′ is (j−j′)omega+1.
- **_gamint_** (float) - Optional - Fractional integration coefficient, defaults to 0 (no integration).

### Response
#### Success Response (200)
- **WaveletWSE** (WaveletWSE) - Weak scaling exponent derived from the coefficients.

## POST /api/integrate

### Description
Re-compute the (p-)leaders on the fractionally integrated wavelet coefficients.

### Method
POST

### Endpoint
/api/integrate

### Parameters
#### Request Body
- **gamint** (float) - Required - Fractional integration coefficient

### Response
#### Success Response (200)
- **WaveletLeader** (WaveletLeader) - The re-computed wavelet leaders.

## GET /api/j2_eff

### Description
Returns the effective maximal scale.

### Method
GET

### Endpoint
/api/j2_eff

## GET /api/max_scale_bootstrap

### Description
Maximum scale at which bootstrapping can be done.

### Method
GET

### Endpoint
/api/max_scale_bootstrap

### Parameters
#### Query Parameters
- **idx_reject** (dict[str, np.ndarray] | None) - Optional - Dictionary of rejected values, by default None which means no rejected values.

### Response
#### Success Response (200)
- **Scale** (int) - The maximum scale for bootstrapping.

## POST /api/plot

### Description
Plot the multi-resolution quantity.

### Method
POST

### Endpoint
/api/plot

### Parameters
#### Request Body
- **j1** (int) - Required - Initial scale from which to display the values.
- **j2** (int) - Required - Maximal scale from which to display the values.
- **ax** (matplotlib.pyplot.axes | None) - Optional - pyplot axes, defaults to None which creates a new figure.
- **vmin** (float | None) - Optional - Minimal value of the colorbar, by default None which uses the minimal value in the data.
- **vmax** (float | None) - Optional - Maximal value of the colorbar, by default None which uses the maximal value in the data.
- **cbar** (bool) - Optional - Whether to display a colorbar.
- **figsize** (tuple) - Optional - Size of the figure, used if ax is None.
- **gamma** (float) - Optional - Exponent of the power-law color normalization, set to 1 for no normalization.
- **nan_idx** (dict[str, ndarray] | None) - Optional - Index of values to highlight (smart indexing), or boolean mask index of values to highlight as output by robust.get_outliers.
- **signal_idx** (int) - Optional - Index of the signal to plot, defaults to the first signal.
- **cbar_kw** (dict | None) - Optional - Arguments to pass to the colorbar function call.
- **cmap** (str) - Optional - Colormap for the plot.

## GET /api/scale2freq

### Description
Get the frequencies associated to scales.

### Method
GET

### Endpoint
/api/scale2freq

### Parameters
#### Query Parameters
- **scale** (int | float | ndarray) - Required - Scales to convert to frequency.
- **sfreq** (float) - Required - Sampling frequency of the signal.

### Response
#### Success Response (200)
- **freq** (float | ndarray) - Frequencies associated to scales.
```

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

### Compute Wavelet Leaders

Source: https://www.neurospin.fr/pymultifracs/_downloads/4f7c425147b815e98b3001323044b973/10-bivariate.ipynb

Performs wavelet analysis on the input data X to obtain p-leaders. This step is a prerequisite for bivariate multifractal analysis, integrating the wavelet transform and selecting leaders for a specific order q.

```python
from pymultifracs import wavelet_analysis

WTpL = wavelet_analysis(X).integrate(.75).get_leaders(2)
```

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

### Apply MFA with Bootstrapping

Source: https://www.neurospin.fr/pymultifracs/_downloads/12297bf87b30ae3499b9d7500abaf258/02-bootstrap.ipynb

Applies multifractal analysis (mfa) to the wavelet leaders, performing R bootstrapping repetitions to estimate confidence intervals for the specified scaling ranges.

```python
from pymultifracs import mfa
from pymultifracs.utils import build_q_log

scaling_ranges = [(2, 8), (3, 8)]
pwt = mfa(WTpL, scaling_ranges, weighted=None, q=build_q_log(.1, 4, 15), R=20)
```

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

### Perform Multifractal Analysis (MFA)

Source: https://www.neurospin.fr/pymultifracs/auto_examples/01-simul.html

Conducts the multifractal analysis using the `mfa()` function. It takes a multi-resolution quantity, scaling ranges, and optionally weighting and a list of moments `q`. Regularity check is performed by default.

```python
from pymultifracs import mfa
from pymultifracs.utils import build_q_log

pwt = mfa(WTpL, scaling_ranges, weighted='Nj', q=[-2, -1, 0, 1, 2])
```

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

### Benchmark.compute_benchmark

Source: https://www.neurospin.fr/pymultifracs/_autosummary/pymultifracs.Benchmark.html

Executes the benchmark computation based on the provided signal generation and estimation grids.

```APIDOC
## compute_benchmark

### Description
Compute the benchmark by iterating over the defined signal generation and estimation grids.

### Parameters
- **n_jobs** (int) - Optional - Number of jobs that joblib will start in parallel. Each job handles a different signal configuration.
- **save_load_signals** (bool) - Optional - Whether to save the signals, and load them if found, to speed up computation. Currently not implemented yet.
- **save** (bool) - Optional - Whether to save the final results dataframe to a pickled file.
```

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

### Plot Multifractal Spectrum

Source: https://www.neurospin.fr/pymultifracs/auto_examples/01-simul.html

Visualize the multifractal spectrum using the `spectrum.plot()` method. This requires densely sampled values of q, which can be generated using `build_q_log`.

```python
pwt = mfa(WTpL, scaling_ranges, weighted='Nj', q=build_q_log(.1, 5, 20))
pwt.spectrum.plot()
```

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

### Plot Cumulants with Confidence Intervals

Source: https://www.neurospin.fr/pymultifracs/_downloads/12297bf87b30ae3499b9d7500abaf258/02-bootstrap.ipynb

Plots the cumulants of the multifractal analysis, displaying empirical confidence intervals if bootstrapping was applied.

```python
pwt.cumulants.plot()
```

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

### Method: auto_integrate

Source: https://www.neurospin.fr/pymultifracs/_autosummary/pymultifracs.multiresquantity.WaveletWSE.html

Automatically integrates the signal to match the requirements of the multifractal formalism associated to the multi-resolution quantity.

```APIDOC
## auto_integrate

### Description
Automatically integrates the signal to match the requirements of the multifractal formalism associated to the multi-resolution quantity.

### Parameters
#### Request Body
- **scaling_ranges** (list[tuple[int, int]]) - Required - List of pairs of (j1,j2) ranges of scales for the analysis.
- **weighted** (str | None) - Optional - Weighting mode for the linear regressions. Defaults to None.
- **idx_reject** (dict[int, ndarray of bool]) - Optional - Dictionary associating each scale to a boolean array indicating whether certain coefficients should be removed.
```

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

### Simulation Functions

Source: https://www.neurospin.fr/pymultifracs/reference.html

Functions for simulating signals like fractional Brownian motion.

```APIDOC
## simul.mrw(shape, H, lam, L, sigma, method, z0)

### Description
Create a realization of fractional Brownian motion using circulant matrix embedding.

## simul.fbm(*args, **kwargs)

### Description
Simulate fractional Brownian motion (fBm).
```

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

### Simulate Multifractal Random Walks (MRW)

Source: https://www.neurospin.fr/pymultifracs/auto_examples/01-simul.html

Generates simulated Multifractal Random Walks (MRWs) using specified parameters. Requires matplotlib and numpy. The output is a numpy array representing the time series.

```python
import matplotlib.pyplot as plt
import numpy as np
from pymultifracs.simul import mrw

X = mrw(shape=(2**15, 5), H=0.8, lam=np.sqrt(.05), L=2**15)
plt.plot(X)
plt.show()
```

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

### StructureFunction Methods

Source: https://www.neurospin.fr/pymultifracs/_autosummary/pymultifracs.scalingfunction.StructureFunction.html

This section lists and describes the core methods of the StructureFunction class for multifractal analysis.

```APIDOC
## StructureFunction.find_best_range()

### Description
Finds the optimal range for multifractal analysis.

### Method
Not applicable (method of a class)

### Endpoint
Not applicable (method of a class)

### Parameters
None

### Request Example
```python
# Example usage within a class instance
structure_function_instance.find_best_range()
```

### Response
Returns the determined best range for analysis.

#### Success Response (200)
- **range** (tuple) - The optimal range (min, max).

#### Response Example
```json
{
  "range": [0.1, 10.0]
}
```
```

```APIDOC
## StructureFunction.get_jrange()

### Description
Retrieves the range of 'j' values used in the analysis.

### Method
Not applicable (method of a class)

### Endpoint
Not applicable (method of a class)

### Parameters
None

### Request Example
```python
# Example usage within a class instance
j_range = structure_function_instance.get_jrange()
```

### Response
Returns the 'j' range.

#### Success Response (200)
- **j_range** (list) - A list of 'j' values.

#### Response Example
```json
{
  "j_range": [1, 2, 3, 4, 5]
}
```
```

```APIDOC
## StructureFunction.get_n_bootstrap()

### Description
Gets the number of bootstrap samples used for statistical analysis.

### Method
Not applicable (method of a class)

### Endpoint
Not applicable (method of a class)

### Parameters
None

### Request Example
```python
# Example usage within a class instance
n_bootstrap = structure_function_instance.get_n_bootstrap()
```

### Response
Returns the number of bootstrap samples.

#### Success Response (200)
- **n_bootstrap** (int) - The number of bootstrap samples.

#### Response Example
```json
{
  "n_bootstrap": 100
}
```
```

```APIDOC
## StructureFunction.get_nj_interv()

### Description
Retrieves the interval information for 'nj' values.

### Method
Not applicable (method of a class)

### Endpoint
Not applicable (method of a class)

### Parameters
None

### Request Example
```python
# Example usage within a class instance
nj_intervals = structure_function_instance.get_nj_interv()
```

### Response
Returns the 'nj' intervals.

#### Success Response (200)
- **nj_intervals** (list) - A list of 'nj' interval data.

#### Response Example
```json
{
  "nj_intervals": [[0.5, 1.5], [1.5, 2.5]]
}
```
```

```APIDOC
## StructureFunction.plot()

### Description
Generates a plot of the structure function.

### Method
Not applicable (method of a class)

### Endpoint
Not applicable (method of a class)

### Parameters
None

### Request Example
```python
# Example usage within a class instance
structure_function_instance.plot()
```

### Response
Displays a plot.

#### Success Response (200)
- **plot** (visualization) - The generated plot.

#### Response Example
(No JSON response, displays a plot)
```

```APIDOC
## StructureFunction.plot_scaling()

### Description
Generates a plot of the scaling function.

### Method
Not applicable (method of a class)

### Endpoint
Not applicable (method of a class)

### Parameters
None

### Request Example
```python
# Example usage within a class instance
structure_function_instance.plot_scaling()
```

### Response
Displays a scaling plot.

#### Success Response (200)
- **plot** (visualization) - The generated scaling plot.

#### Response Example
(No JSON response, displays a plot)
```

```APIDOC
## StructureFunction.s_q()

### Description
Calculates the structure function values for a given order 'q'.

### Method
Not applicable (method of a class)

### Endpoint
Not applicable (method of a class)

### Parameters
#### Query Parameters
- **q** (float or list) - The order(s) of the structure function to calculate.

### Request Example
```python
# Example usage within a class instance
sq_values = structure_function_instance.s_q(q=[1, 2, 3])
```

### Response
Returns the calculated structure function values.

#### Success Response (200)
- **s_q** (dict) - A dictionary where keys are 'q' values and values are the corresponding structure function values.

#### Response Example
```json
{
  "s_q": {
    "1": 0.5,
    "2": 0.25,
    "3": 0.125
  }
}
```
```

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

### mrw() - Simulate Multifractal Random Walk

Source: https://www.neurospin.fr/pymultifracs/_autosummary/pymultifracs.simul.mrw.html

Generates a realization of a Multifractal Random Walk (MRW) using fractional Brownian motion and circulant matrix embedding. This function allows for customization of shape, Hurst exponent, intermittency, scale, variance, and the simulation method.

```APIDOC
## mrw()

### Description
Create a realization of fractional Brownian motion using circulant matrix embedding.

### Method
Python Function

### Endpoint
`pymultifracs.simul.mrw`

### Parameters
#### Path Parameters
None

#### Query Parameters
None

#### Request Body
None

### Parameters
- **shape** (`int` | `tuple`(`int`, `int`))
  - Required - If integer, it is the number of samples N. If tuple it is (N, R), the number of samples and realizations, respectively.
- **H** (`float`)
  - Required - Hurst exponent
- **lam** (`float`)
  - Required - Lambda, intermittency parameter
- **L** (`float`)
  - Optional - Integral scale
- **sigma** (`float`)
  - Optional - Variance of process (default: 1)
- **method** (`str`)
  - Optional - Method to use: ‘cme’ selects circulant matrix embedding (default, O(NlogN) in memory), ‘chol’ selects Cholesky decomposition (O(N2) in memory). (default: 'cme')
- **z0** (`tuple`(`ndarray` of `float`, `ndarray` of `float`))
  - Optional - Tuple of white noise values, to fix the random component across simulations. The shape should be (2N−2,R) for ‘cme’ and (N,R).

### Request Example
```python
import numpy as np
from pymultifracs.simul import mrw

# Example with integer shape
N = 1000
H = 0.7
lam = 1.5
realization_int = mrw(shape=N, H=H, lam=lam)

# Example with tuple shape (N, R)
R = 5
realization_tuple = mrw(shape=(N, R), H=H, lam=lam)

# Example with optional parameters
L_scale = 10.0
sigma_val = 0.5
method_chol = 'chol'
realization_custom = mrw(shape=N, H=H, lam=lam, L=L_scale, sigma=sigma_val, method=method_chol)

# Example with z0
z_noise1 = np.random.randn(2*N - 2, R) # for 'cme'
z_noise2 = np.random.randn(N, R)       # for 'chol'
realization_with_z0 = mrw(shape=(N, R), H=H, lam=lam, z0=(z_noise1, z_noise2))
```

### Response
#### Success Response (200)
- **mrw** (`ndarray`)
  - Synthesized mrw realizations. If shape is scalar, fbm is of shape (N,). Otherwise, it is of shape (N, R).

#### Response Example
```json
{
  "mrw": "[numpy array of shape (N,) or (N, R)]"
}
```

### References
[1] Bacry, Delour, Muzy, “Multifractal Random Walk”, Physical Review E, 2001
```

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

### Method: get_leaders

Source: https://www.neurospin.fr/pymultifracs/_autosummary/pymultifracs.multiresquantity.WaveletWSE.html

Compute (p-)leaders from the wavelet coefficients.

```APIDOC
## get_leaders

### Description
Compute (p-)leaders from the wavelet coefficients.

### Parameters
#### Request Body
- **p_exp** (float | numpy.inf) - Required - Value of the p exponent.
- **interval_size** (int) - Optional - Width of the time shift interval.
- **gamint** (int | None) - Optional - Fractional integration coefficient.

### Response
#### Success Response (200)
- **WaveletLeader** - Wavelet (p-)leader derived from the coefficients.
```

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

### Check Regularity of Multi-resolution Quantities

Source: https://www.neurospin.fr/pymultifracs/auto_examples/01-simul.html

Verifies the minimum regularity condition for multifractal analysis using the specified scaling ranges. This method is available for `WaveletDec`, `WaveletLeader`, and `WaveletWSE` objects.

```python
WT.check_regularity(scaling_ranges)
WTL.check_regularity(scaling_ranges)
WTpL.check_regularity(scaling_ranges)
WSE.check_regularity(scaling_ranges)
```

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

### Compute Bivariate Multifractal Analysis (BiMFA)

Source: https://www.neurospin.fr/pymultifracs/auto_examples/10-bivariate.html

Applies the bivariate multifractal analysis function (bimfa) to the p-leaders. It requires the p-leader data for both channels and specifies the scaling ranges and q-values for analysis.

```python
import numpy as np
from pymultifracs.bivariate import bimfa

q1 = np.array([0, 1, 2])
q2 = np.array([0, 1, 2])

pwt = bimfa(WTpL, WTpL, scaling_ranges=[(3, 9)], q1=q1, q2=q2)

```

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

### Determine Optimal Scaling Range (Overall)

Source: https://www.neurospin.fr/pymultifracs/_downloads/12297bf87b30ae3499b9d7500abaf258/02-bootstrap.ipynb

Determines the best overall scaling range from a list of choices using a bootstrapped-based criterion.

```python
pwt = mfa(WTpL, [(2, 8), (3, 8), (2, 7), (3, 7)], weighted=None, R=20)
pwt.cumulants.find_best_range()
```

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

### Define Multifractal Estimation Functions

Source: https://www.neurospin.fr/pymultifracs/_downloads/6968a062e5f822a7836cfdeb15f66b35/05-benchmark.ipynb

Define functions for estimating multifractal properties using different methods like 'pleader' (based on wavelet leaders) and 'coef' (based on wavelet coefficients). These functions process the generated signals.

```python
def pleader_est(X, p_exp):

    WT = wavelet_analysis(X).get_leaders(p_exp=p_exp)
    pwt = mfa(WT, [(3, 7)], estimates='c', weighted='Nj')

    out = pwt.cumulants.log_cumulants.isel(
        scaling_range=0).to_series().unstack('m')
    out.columns = out.columns.to_flat_index().map(lambda x: f'c{x}')

    return out


def coef_est(X):

    WT = wavelet_analysis(X)
    dwt = mfa(WT, [(3, 7)], estimates='c')

    out = dwt.cumulants.log_cumulants.isel(
        scaling_range=0).to_series().unstack('m')
    out.columns = out.columns.to_flat_index().map(lambda x: f'c{x}')

    return out
```

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

### Define multifractal estimation functions

Source: https://www.neurospin.fr/pymultifracs/auto_examples/05-benchmark.html

Define functions for estimating multifractal parameters using the p-leader and coefficient-based methods. These functions process wavelet transform results to extract log-cumulants.

```python
def pleader_est(X, p_exp):

    WT = wavelet_analysis(X).get_leaders(p_exp=p_exp)
    pwt = mfa(WT, [(3, 7)], estimates='c', weighted='Nj')

    out = pwt.cumulants.log_cumulants.isel(
        scaling_range=0).to_series().unstack('m')
    out.columns = out.columns.to_flat_index().map(lambda x: f'c{x}')

    return out


def coef_est(X):

    WT = wavelet_analysis(X)
    dwt = mfa(WT, [(3, 7)], estimates='c')

    out = dwt.cumulants.log_cumulants.isel(scaling_range=0).to_series().unstack('m')
    out.columns = out.columns.to_flat_index().map(lambda x: f'c{x}')

    return out
```

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

### WaveletDec Methods

Source: https://www.neurospin.fr/pymultifracs/_autosummary/pymultifracs.multiresquantity.WaveletDec.html

Details the methods available for the WaveletDec class, including integration, bootstrapping, regularity checks, and data retrieval.

```APIDOC
## Methods

### `auto_integrate(_scaling_ranges_ , _weighted =None_, _idx_reject =None_)`

Automatically integrates the signal to match the requirements of the multifractal formalism associated with the multi-resolution quantity.

**Parameters:**
- **scaling_ranges** (`list`[`tuple`[`int`, `int`]]): List of pairs of (j1, j2) ranges of scales for the analysis.
- **weighted** (`str` | `None`): Weighting mode for the linear regressions. Defaults to None (no weighting). Possible values are ‘Nj’ (weighs by number of coefficients) and ‘bootstrap’ (weighs by bootstrap-derived estimates of variance).
- **idx_reject** (`dict`[`int`, `ndarray` of bool]): Dictionary associating each scale to a boolean array indicating whether certain coefficients should be removed.

### `bootstrap(_R_ , _min_scale =1_, _idx_reject =None_)`

Bootstrap the multi-resolution quantity by repeating.

**Parameters:**
- **R** (`int`): Number of repetitions of the bootstrap.
- **min_scale** (`int`): Minimum scale to be kept in the bootstrapped MRQ.
- **idx_reject** (`dict`[`str`, `np.ndarray`] | `None`): Dictionary of rejected values.

**Returns:**
- `WaveletDec`: MRQ containing the bootstrapped values.

### `bootstrap_multiple(_R_ , _min_scale_ , _mrq_list_)`

Class method to bootstrap multiple MRQs at once.

**Parameters:**
- **R** (`int`): Number of repetitions of the bootstrap.
- **min_scale** (`int`): Minimum scale to be kept in the bootstrapped MRQ.
- **mrq_list** (`list`[`WaveletDec`]): List of MRQs to bootstrap.

**Returns:**
- `list`[`WaveletDec`]: List of bootstrapped MRQs.

### `check_regularity(_scaling_ranges_ , _weighted =None_, _idx_reject =None_, _min_j =1_)`

Verify that the MRQ has enough regularity for analysis.

**Parameters:**
- **scaling_ranges** (`list`[`tuple`[`int`, `int`]]): List of pairs of (j1, j2) ranges of scales for the analysis.
- **weighted** (`str` | `None`): Weighting mode for the linear regressions. Defaults to None.
- **idx_reject** (`Dict`[`int`, `ndarray`]): Dictionary associating each scale to a boolean array indicating whether certain coefficients should be removed.

**Returns:**
- `ndarray`: Estimate of the minimal Hölder exponent in the MRQ.

### `freq2scale(_freq_ , _sfreq_)`

Get the scales associated to frequencies.

**Parameters:**
- **freq** (`float` | `ndarray`): Frequencies to convert to scales.
- **sfreq** (`float`): Sampling frequency of the signal.

**Returns:**
- **scales** (`float` | `ndarray`): Scales associated to freq.

### `get_formalism()`

Obtains the formalism of the multi-resolution quantity.

**Returns:**
- **formalism** (`str`): The formalism of the MRQ.

### `get_leaders(_p_exp_ , _interval_size =3_, _gamint =None_)`

Compute (p-)leaders from the wavelet coefficients.

**Parameters:**
- **p_exp** (`float` | `numpy.inf`): `np.inf` for wavelet leaders, a float for wavelet p-leader formalism.
- **interval_size** (`int`): Width of the time shift interval for leader computation.
- **gamint** (`int` | `None`): Fractional integration coefficient.

**Returns:**
- `WaveletLeader`: Wavelet (p-)leader derived from the coefficients.

### `get_n_bootstrap()`

Returns the number of bootstrapping repetitions, or 0 if no bootstrapping was performed.

**Returns:**
- `int`: The number of bootstrap repetitions.

### `get_nj_interv(_j1 =None_, _j2 =None_, _idx_reject =None_)`

Returns `nj` as an array for scales `j` in the range [j1, j2], optionally considering rejected indices.

**Parameters:**
- **j1** (`int`, optional): The starting scale.
- **j2** (`int`, optional): The ending scale.
- **idx_reject** (`dict`, optional): Dictionary of rejected indices.

**Returns:**
- `ndarray`: Array of `nj` values.

### `get_values(_j_ , _idx_reject =None_)`

Get the values of the MRQ, applying any finite size effects corrections if necessary.

**Parameters:**
- **j** (`int`): The scale for which to retrieve values.
- **idx_reject** (`dict`, optional): Dictionary of rejected indices.

**Returns:**
- `ndarray`: The MRQ values at scale `j`.

### `get_wse(_theta =0.5_, _omega =1_, _gamint =None_)`

Compute weak scaling exponents from the wavelet coefficients.

**Parameters:**
- **theta** (`float`, optional): Parameter for weak scaling exponent calculation. Defaults to 0.5.
- **omega** (`int`, optional): Parameter for weak scaling exponent calculation. Defaults to 1.
- **gamint** (`int` | `None`): Fractional integration coefficient.
```

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

### Visualization Functions

Source: https://www.neurospin.fr/pymultifracs/reference.html

Functions for visualizing multifractal analysis results.

```APIDOC
## viz.plot_psd(signal, fs, wt_name, log_base, ax)

### Description
Plot the superposition of Fourier-based Welch estimation and Wavelet-based estimation of PSD on a log-log graphic.

## viz.plot_cm(cm, ind_m, j1, j2, range_idx, ax)

### Description
Helper function to plot individual Cm(j) functions along with their associated cm fit.
```

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

### MFSpectrum Class Overview

Source: https://www.neurospin.fr/pymultifracs/_autosummary/pymultifracs.scalingfunction.MFSpectrum.html

The MFSpectrum class estimates the Multifractal Spectrum based on Herwig Wendt's thesis. It should not be instantiated directly but obtained from pymultifracs.mfa().

```APIDOC
## Class: MFSpectrum

### Description
Estimates the Multifractal Spectrum based on equations 2.74 - 2.78 of Herwig Wendt’s thesis [1]. This class should not be instantiated directly but obtained from calling `pymultifracs.mfa()`.

### Attributes
- **formalism** (`str`): Formalism used. Can be any of ‘wavelet coefs’, ‘wavelet leaders’, or ‘wavelet p-leaders’.
- **j** (`ndarray`, shape (n_scales,)) : List of the j values (scales), in order presented in the value arrays.
- **nj** (`dict`[`int`, `ndarray`]): Dictionary giving the number of non-NaN values at every scale. Arrays are of shape (n_rep,)
- **gamint** (`float`): Value of gamint used in the computation of the underlying MRQ.
- **Dq** (`ndarray`, shape (`n_exponents`, `n_rep`)): Fractal dimensions : D(q), y-axis of the multifractal spectrum.
- **hq** (`ndarray`, shape (`n_exponents`, `n_rep`)): Hölder exponents : h(q), x-axis of the multifractal spectrum.
- **weighted** (`str` | `None`): Weighting mode for the linear regressions. Defaults to None. Possible values are ‘Nj’ or ‘bootstrap’.
- **weights** (`ndarray`): Weights of the linear regression.
- **bootstrapped_obj** (`MFSpectrum` | `None`): Storing the bootstrapped version of the structure function if bootstraping has been used.

### References
[1] H. Wendt (2008). Contributions of Wavelet Leaders and Bootstrap to Multifractal Analysis... Ph.D thesis, Laboratoire de Physique, Ecole Normale Superieure de Lyon. https://www.irit.fr/~Herwig.Wendt/data/ThesisWendt.pdf
```

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

### Perform Wavelet Analysis

Source: https://www.neurospin.fr/pymultifracs/_downloads/12297bf87b30ae3499b9d7500abaf258/02-bootstrap.ipynb

Performs a wavelet transform on the input data X and extracts the leaders with a specified p-exponent.

```python
from pymultifracs import wavelet_analysis

WT = wavelet_analysis(X, wt_name='db3', j2=None, normalization=1)
WTpL = WT.get_leaders(p_exp=2)
```

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

### Generate Multifractal Random Walks

Source: https://www.neurospin.fr/pymultifracs/_downloads/12297bf87b30ae3499b9d7500abaf258/02-bootstrap.ipynb

Generates a Multifractal Random Walk (MRW) with specified parameters H and lambda, then plots the generated series.

```python
import matplotlib.pyplot as plt
import numpy as np
from pymultifracs.simul import mrw

X = mrw(shape=2**15, H=0.8, lam=np.sqrt(.05), L=2**15)
plt.plot(X)
plt.show()
```

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

### Generate Fractional Brownian Motion with mrw

Source: https://www.neurospin.fr/pymultifracs/_autosummary/pymultifracs.simul.mrw.html

Use this function to create a realization of fractional Brownian motion. Specify the shape, Hurst exponent (H), and intermittency parameter (lam). The method can be 'cme' (default) or 'chol'.

```python
pymultifracs.simul.mrw(_shape_ , _H_ , _lam_ , _L =None_, _sigma =1_, _method ='cme'_, _z0 =(None, None)_)
```

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

### BiStructureFunction Methods

Source: https://www.neurospin.fr/pymultifracs/_autosummary/pymultifracs.bivariate.BiStructureFunction.html

Methods available for the BiStructureFunction class.

```APIDOC
## Methods

### S_qq(q1, q2)

#### Description
Get bivariate Sq1q2 function.

#### Parameters
- **q1** (float) - Required - The first q value.
- **q2** (float) - Required - The second q value.

### get_jrange(j1=None, j2=None, bootstrap=False)

#### Description
Get the range of j values.

#### Parameters
- **j1** (int) - Optional - First j value.
- **j2** (int) - Optional - Second j value.
- **bootstrap** (bool) - Optional - Whether to consider bootstrapping.

### get_n_bootstrap()

#### Description
Returns the number of bootstrapping repetitions, or 0 if no bootstrapping.

### get_nj_interv(j_min, j_max)

#### Description
Get number of coefficients per scale.

#### Parameters
- **j_min** (int) - Required - Minimum j value.
- **j_max** (int) - Required - Maximum j value.

### get_nj_interv_margin(j_min, j_max, margin)

#### Description
Get number of coefficients per scale for the margins.

#### Parameters
- **j_min** (int) - Required - Minimum j value.
- **j_max** (int) - Required - Maximum j value.
- **margin** (int) - Required - Margin value.

### plot(figsize=None, scaling_range=0, signal_idx1=0, signal_idx2=0, plot_CI=True, plot_scales=None, filename=None)

#### Description
Plot bivariate structure function.

#### Parameters
- **figsize** (tuple) - Optional - Figure size.
- **scaling_range** (int) - Optional - Scaling range (default: 0).
- **signal_idx1** (int) - Optional - Index for the first signal (default: 0).
- **signal_idx2** (int) - Optional - Index for the second signal (default: 0).
- **plot_CI** (bool) - Optional - Whether to plot confidence intervals (default: True).
- **plot_scales** (list) - Optional - Scales to plot.
- **filename** (str) - Optional - Filename to save the plot.

### s_qq(q1, q2)

#### Description
Get bivariate sq1q2 exponents.

#### Parameters
- **q1** (float) - Required - The first q value.
- **q2** (float) - Required - The second q value.
```

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

### Generate White Noise for z0

Source: https://www.neurospin.fr/pymultifracs/_autosummary/pymultifracs.simul.mrw.html

Arrays in `z0` can be generated using this command. Ensure the shape matches the requirements of the 'cme' or 'chol' methods.

```python
z = np.random.randn(N, R).
```