### MFDFA Usage Example

Source: https://fathon.readthedocs.io/en/latest/_sources/fun_class/fathon.MFDFA.rst.txt

This example demonstrates the typical workflow for using the MFDFA class. It includes data preparation, initialization, computation of fluctuation functions and exponents, and derivation of the multifractal spectrum. Ensure data is zero-mean cumulative sum before initialization.

```python
import numpy as np
import fathon
from fathon import fathonUtils as fu

#time series
a = np.random.randn(10000)

#zero-mean cumulative sum
a = fu.toAggregated(a)

#initialize mfdfa object
pymfdfa = fathon.MFDFA(a)
#compute fluctuation function and generalized Hurst exponents
wins = fu.linRangeByStep(10, 2000)
n, F = pymfdfa.computeFlucVec(wins, np.arange(-3, 4, 0.1), revSeg=True, polOrd=1)
list_H, list_H_intercept = pymfdfa.fitFlucVec()

#compute mass exponents
tau = pymfdfa.computeMassExponents()

#compute multifractal spectrum
alpha, mfSpect = pymfdfa.computeMultifractalSpectrum()

```

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

### Usage Example

Source: https://fathon.readthedocs.io/en/latest/fun_class/fathon.MFDCCA.html

Example demonstrating how to use the MFDCCA class for time series analysis.

```APIDOC
```python
import numpy as np
import fathon
from fathon import fathonUtils as fu

#time series
a = np.random.randn(10000)
b = np.random.randn(10000)

#zero-mean cumulative sum
a = fu.toAggregated(a)
b = fu.toAggregated(b)

#initialize mfdfa object
pymfdcca = fathon.MFDCCA(a, b)
#compute fluctuation function and generalized Hurst exponents
wins = fu.linRangeByStep(10, 2000)
n, F = pymfdcca.computeFlucVec(wins, np.arange(-3, 4, 0.1), revSeg=True, polOrd=1)
list_H, list_H_intercept = pymfdcca.fitFlucVec()

#compute mass exponents
tau = pymfdcca.computeMassExponents()

#compute multifractal spectrum
alpha, mfSpect = pymfdcca.computeMultifractalSpectrum()

```
```

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

### MFDCCA Usage Example

Source: https://fathon.readthedocs.io/en/latest/fun_class/fathon.MFDCCA.html

This example demonstrates the complete workflow for MFDCCA analysis, including data preparation, object initialization, computation of fluctuation vectors, fitting, mass exponents, and the multifractal spectrum. Ensure time series are preprocessed to be zero-mean and aggregated.

```python
import numpy as np
import fathon
from fathon import fathonUtils as fu

#time series
a = np.random.randn(10000)
b = np.random.randn(10000)

#zero-mean cumulative sum
a = fu.toAggregated(a)
b = fu.toAggregated(b)

#initialize mfdfa object
pymfdcca = fathon.MFDCCA(a, b)
#compute fluctuation function and generalized Hurst exponents
wins = fu.linRangeByStep(10, 2000)
n, F = pymfdcca.computeFlucVec(wins, np.arange(-3, 4, 0.1), revSeg=True, polOrd=1)
list_H, list_H_intercept = pymfdcca.fitFlucVec()

#compute mass exponents
tau = pymfdcca.computeMassExponents()

#compute multifractal spectrum
alpha, mfSpect = pymfdcca.computeMultifractalSpectrum()

```

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

### MFDCCA Usage Example

Source: https://fathon.readthedocs.io/en/latest/_sources/fun_class/fathon.MFDCCA.rst.txt

Demonstrates initializing the MFDCCA object with two time series and computing various multifractal spectrum components.

```python
import numpy as np
import fathon
from fathon import fathonUtils as fu

#time series
a = np.random.randn(10000)
b = np.random.randn(10000)

#zero-mean cumulative sum
a = fu.toAggregated(a)
b = fu.toAggregated(b)

#initialize mfdfa object
pymfdcca = fathon.MFDCCA(a, b)
#compute fluctuation function and generalized Hurst exponents
wins = fu.linRangeByStep(10, 2000)
n, F = pymfdcca.computeFlucVec(wins, np.arange(-3, 4, 0.1), revSeg=True, polOrd=1)
list_H, list_H_intercept = pymfdcca.fitFlucVec()

#compute mass exponents
tau = pymfdcca.computeMassExponents()

#compute multifractal spectrum
alpha, mfSpect = pymfdcca.computeMultifractalSpectrum()
```

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

### Zero-Mean Cumulative Sum with toAggregated

Source: https://fathon.readthedocs.io/en/latest/_sources/fun_class/fu/fathonUtils.toAggregated.rst.txt

This example demonstrates how to compute the zero-mean cumulative sum of a NumPy array using the toAggregated function. Ensure numpy and fathonUtils are imported.

```python
import numpy as np
from fathon import fathonUtils as fu

#time series
a = np.random.randn(10000)

#zero-mean cumulative sum
a = fu.toAggregated(a)
```

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

### DCCA Usage Example

Source: https://fathon.readthedocs.io/en/latest/_sources/fun_class/fathon.DCCA.rst.txt

Demonstrates initializing a DCCA object with time series data, computing fluctuation functions and Hurst exponents, and calculating rho indices. Requires numpy and fathon libraries.

```python
import numpy as np
import fathon
from fathon import fathonUtils as fu

#time series
a = np.random.randn(10000)
b = np.random.randn(10000)

#zero-mean cumulative sum
a = fu.toAggregated(a)
b = fu.toAggregated(b)

#initialize non-empty dcca object
pydcca = fathon.DCCA(a, b)
#compute fluctuation function and Hurst exponent
wins = fu.linRangeByStep(20, 1000, step=50)
n, F = pydcca.computeFlucVec(wins, polOrd=1)
H, H_intercept = pydcca.fitFlucVec()

#compute Hurst exponent in different ranges
limits_list = np.array([[20,120], [220,870]], dtype=int)
list_H, list_H_intercept = pydcca.multiFitFlucVec(limits_list)

#compute rho index
wins = fu.linRangeByStep(20, 100, step=50)
n, rho = pydcca.computeRho(wins, polOrd=1)

#initialize empty dcca object
pythresh = fathon.DCCA()
#compute confidence levels
wins = fu.linRangeByStep(4, 100)
n, cInt1, cInt2 = pythresh.rhoThresholds(300, wins, 100, 0.95, polOrd=1)
```

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

### Generate Linearly Separated Elements

Source: https://fathon.readthedocs.io/en/latest/_sources/fun_class/fu/fathonUtils.linRangeByStep.rst.txt

Use linRangeByStep to create a list of numbers starting from 10, up to (but not including) 100, with a step of 2. Ensure fathonUtils is imported as fu.

```python
from fathon import fathonUtils as fu

#linearly separated elements
a = fu.linRangeByStep(10, 100, step=2)
```

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

### powRangeByStep

Source: https://fathon.readthedocs.io/en/latest/fun_class/fu/fathonUtils.powRangeByStep.html

Generates an array of elements by raising a specified base to linearly separated exponents. The exponents range from `start` to `end` with a given `step`, and the `base` determines the exponential value.

```APIDOC
## powRangeByStep()

### Description
Generates an array of elements by raising a specified base to linearly separated exponents. The exponents range from `start` to `end` with a given `step`, and the `base` determines the exponential value.

### Parameters
#### Path Parameters
- None

#### Query Parameters
- None

#### Request Body
- None

### Function Signature
`fathon.fathonUtils.powRangeByStep(_start_ , _end_ , _step =1_, _base =2_)`

### Parameters
- **start** (_int_) – Smallest exponent.
- **end** (_int_) – Biggest exponent.
- **step** (_int_) – Step between two consecutive exponents (default : 1).
- **base** (_int_) – Base of the exponential (default : 2).

### Returns
Array of elements given by base raised to linearly separated exponents.

### Return Type
numpy ndarray

### Request Example
```python
from fathon import fathonUtils as fu

# elements given by `base` raised to linearly separated exponents
a = fu.powRangeByStep(10, 1000, step=2, base=3)
```
```

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

### linRangeByCount

Source: https://fathon.readthedocs.io/en/latest/_sources/fun_class/fu/fathonUtils.linRangeByCount.rst.txt

Generates a list of numbers linearly spaced between a start and end value, with a specified count of elements.

```APIDOC
## linRangeByCount

### Description
Generates a list of numbers linearly spaced between a start and end value, with a specified count of elements.

### Method
`linRangeByCount(start, end, count)`

### Parameters
#### Path Parameters
- **start** (int) - The starting value of the range.
- **end** (int) - The ending value of the range.
- **count** (int) - The desired number of elements in the generated list.

### Request Example
```python
from fathon import fathonUtils as fu

a = fu.linRangeByCount(10, 100, count=42)
```

### Response
#### Success Response (200)
- **list** (list[int]) - A list of integers representing the linearly spaced elements.
```

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

### powRangeByCount

Source: https://fathon.readthedocs.io/en/latest/_sources/fun_class/fu/fathonUtils.powRangeByCount.rst.txt

Calculates `count` numbers by raising `base` to linearly separated exponents between `start` and `end`.

```APIDOC
## powRangeByCount

### Description
Generates a sequence of numbers by raising a specified `base` to `count` linearly separated exponents. The range of these exponents is determined by `start` and `end` values.

### Parameters
#### Path Parameters
- **start** (int) - Required - The starting value for the range.
- **end** (int) - Required - The ending value for the range.
- **base** (int) - Optional - The base number to be raised to the power. Defaults to 10.
- **count** (int) - Optional - The number of elements to generate in the range. Defaults to 10.

### Request Example
```python
from fathon import fathonUtils as fu

# elements given by `base` raised to linearly separated exponents
a = fu.powRangeByCount(10, 1000, base=3, count=42)
```

### Response
#### Success Response (200)
- **list** (list[int]) - A list of integers representing the generated sequence.
```

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

### Generate Linearly Spaced Elements

Source: https://fathon.readthedocs.io/en/latest/_sources/fun_class/fu/fathonUtils.linRangeByCount.rst.txt

Use linRangeByCount to create a list of numbers evenly distributed between a start and end value. Specify the desired number of elements using the 'count' parameter.

```python
from fathon import fathonUtils as fu

#linearly separated elements
a = fu.linRangeByCount(10, 100, count=42)
```

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

### Generate powers with a step

Source: https://fathon.readthedocs.io/en/latest/_sources/fun_class/fu/fathonUtils.powRangeByStep.rst.txt

Use `powRangeByStep` to create a list of numbers by raising a base to exponents that increase linearly. This is useful for generating sequences with exponential growth.

```python
from fathon import fathonUtils as fu

#elements given by `base` raised to linearly separated exponents
a = fu.powRangeByStep(10, 1000, step=2, base=3)
```

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

### Compute and Fit Fluctuation Function with DFA

Source: https://fathon.readthedocs.io/en/latest/_sources/fun_class/fathon.DFA.rst.txt

Demonstrates initializing a DFA object with aggregated time series data, computing the fluctuation function over specified window sizes, and fitting the function to obtain the Hurst exponent. Uses `revSeg=True` for reversed segment analysis and `polOrd=3` for cubic polynomial detrending.

```python
import numpy as np
import fathon
from fathon import fathonUtils as fu

#time series
a = np.random.randn(10000)

#zero-mean cumulative sum
a = fu.toAggregated(a)

#initialize dfa object
pydfa = fathon.DFA(a)
#compute fluctuation function and Hurst exponent
wins = fu.linRangeByStep(10, 2000)
n, F = pydfa.computeFlucVec(wins, revSeg=True, polOrd=3)
H, H_intercept = pydfa.fitFlucVec()

#compute Hurst exponent in different ranges
limits_list = np.array([[15,2000], [200,1000]], dtype=int)
list_H, list_H_intercept = pydfa.multiFitFlucVec(limits_list)
```

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

### DFA Class Initialization

Source: https://fathon.readthedocs.io/en/latest/fun_class/fathon.DFA.html

Initializes the DFA class with a time series.

```APIDOC
## DFA Class

### Description
Detrended Fluctuation Analysis class.

### Parameters
#### Parameters
- **tsVec** (iterable) - Time series used for the analysis.
```

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

### linRangeByStep

Source: https://fathon.readthedocs.io/en/latest/_sources/fun_class/fu/fathonUtils.linRangeByStep.rst.txt

Generates a list of linearly separated elements within a given range and step.

```APIDOC
## linRangeByStep

### Description
Generates a list of numbers starting from a given value, up to (but not including) an end value, with a specified step.

### Parameters
#### Path Parameters
None

#### Query Parameters
- **start** (int) - The starting value of the range.
- **end** (int) - The end value of the range (exclusive).
- **step** (int) - The step to increment by.

### Request Example
```python
from fathon import fathonUtils as fu

a = fu.linRangeByStep(10, 100, step=2)
```

### Response
#### Success Response (200)
- **list** (list[int]) - A list of integers representing the generated range.
```

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

### Compute Hurst Exponent with HT Class

Source: https://fathon.readthedocs.io/en/latest/_sources/fun_class/fathon.HT.rst.txt

Initializes the HT object with aggregated time series data and computes the time-dependent Hurst exponent using specified window sizes and polynomial orders.

```python
import numpy as np
import fathon
from fathon import fathonUtils as fu

#time series
a = np.random.randn(10000)

#zero-mean cumulative sum
a = fu.toAggregated(a)

#initialize ht object
pyht = fathon.HT(a)
#compute time-dependent Hurst exponent
ht = pyht.computeHt([100, 200, 1000], mfdfaPolOrd=1, polOrd=1)

```

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

### powRangeByStep

Source: https://fathon.readthedocs.io/en/latest/_sources/fun_class/fu/fathonUtils.powRangeByStep.rst.txt

Generates elements by `base` raised to linearly separated exponents.

```APIDOC
## powRangeByStep

### Description
Generates a sequence of numbers where each number is the result of raising a `base` to a power. The powers are determined by a range starting from an initial value, ending at a specified limit, and incrementing by a given `step`.

### Parameters
#### Path Parameters
None

#### Query Parameters
- **start** (int) - Required - The starting value of the range (inclusive).
- **stop** (int) - Required - The stopping value of the range (inclusive).
- **step** (int) - Optional - The increment between exponents. Defaults to 1.
- **base** (int) - Optional - The base number to be raised to the power. Defaults to 10.

### Request Example
```python
from fathon import fathonUtils as fu

# elements given by `base` raised to linearly separated exponents
a = fu.powRangeByStep(10, 1000, step=2, base=3)
```

### Response
#### Success Response (200)
- **list** (list[int]) - A list of integers representing the calculated powers.
```

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

### MFDFA Class Initialization

Source: https://fathon.readthedocs.io/en/latest/fun_class/fathon.MFDFA.html

Initializes the MFDFA class with a given time series.

```APIDOC
## MFDFA(tsVec)

### Description
Initializes the MultiFractal Detrended Fluctuation Analysis class.

### Parameters
* **tsVec** (iterable) - Time series used for the analysis.
```

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

### DCCA Class Initialization

Source: https://fathon.readthedocs.io/en/latest/fun_class/fathon.DCCA.html

Initializes the DCCA class with two time series.

```APIDOC
## DCCA Class

### Description
Detrended Cross-Correlation Analysis class.

### Parameters
* **tsVec1** (_iterable_) – First time series used for the analysis.
* **tsVec2** (_iterable_) – Second time series used for the analysis.
```

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

### MFDFA Class Methods

Source: https://fathon.readthedocs.io/en/latest/_sources/fun_class/fathon.MFDFA.rst.txt

This section details the core methods available within the MFDFA class for performing multifractal analysis.

```APIDOC
## MFDFA Class

### Description
Provides methods for multifractal detrended fluctuation analysis.

### Methods

- **`computeFlucVec(wins, q, revSeg=False, polOrd=2, fitType='poly')`**
  Computes the fluctuation function and generalized Hurst exponents for a given time series and a range of window sizes and q-orders.

- **`computeMassExponents(q=None, wins=None, revSeg=False, polOrd=2, fitType='poly')`**
  Computes the mass exponents (tau(q)) of the time series.

- **`computeMultifractalSpectrum(q=None, wins=None, revSeg=False, polOrd=2, fitType='poly')`**
  Computes the multifractal spectrum (alpha, f(alpha)) of the time series.

- **`fitFlucVec()`**
  Fits the fluctuation vector data obtained from `computeFlucVec` to determine the generalized Hurst exponents and their intercepts.

- **`saveObject(path)`**
  Saves the current state of the MFDFA object to a specified file path.
```

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

### HT.saveObject()

Source: https://fathon.readthedocs.io/en/latest/fun_class/fathon.HT.html

Saves the current state of the HT object to a binary file. The '.fathon' extension is automatically appended to the provided file name.

```APIDOC
## HT.saveObject()

### Description
Save current object state to binary file.

### Parameters
#### Path Parameters
None

#### Query Parameters
None

#### Request Body
None

### Parameters
* **outFileName** (str) - Output binary file. .fathon extension will be appended to the file name.

### Returns
None
```

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

### HT.computeHt()

Source: https://fathon.readthedocs.io/en/latest/fun_class/fathon.HT.html

Computes the time-dependent local Hurst exponent at each scale using Ihlen’s approach. It allows for customization of polynomial orders for fitting and provides an option to use pre-computed values for MFDFA's Hurst exponent at q=0.

```APIDOC
## HT.computeHt()

### Description
Computation of the time-dependent local Hurst exponent at each scale, using Ihlen’s approach.

### Parameters
#### Path Parameters
None

#### Query Parameters
None

#### Request Body
None

### Parameters
* **scales** (int or iterable or numpy ndarray) - Window’s sizes used for the computation of the time-dependent local Hurst exponent.
* **polOrd** (int, optional) - Order of the polynomial to be fitted in each window (default : 1).
* **mfdfaPolOrd** (int, optional) - Order of the polynomial to be fitted to MFDFA’s fluctuations at q = 0 (default : 1).
* **q0Fit** (iterable or numpy ndarray of floats, optional) - MFDFA’s Hurst exponent at order q = 0 and the corresponding intercept of the fit, [hq0, hq0_intercept]. These values must come from a log-log fit, with the log base equal to e. If not empty, it will be directly used to compute the time-dependent local Hurst exponent, ignoring mfdfaPolOrd value (default : []).
* **verbose** (bool, optional) - Verbosity (default : False).

### Returns
Time-dependent local Hurst exponent.

### Return Type
numpy ndarray
```

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

### Generate Powers with Count

Source: https://fathon.readthedocs.io/en/latest/_sources/fun_class/fu/fathonUtils.powRangeByCount.rst.txt

Generates elements by raising a base to linearly separated exponents. Use this when you need a sequence of exponentially spaced values.

```python
from fathon import fathonUtils as fu

#elements given by `base` raised to linearly separated exponents
a = fu.powRangeByCount(10, 1000, base=3, count=42)
```

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

### computeMassExponents

Source: https://fathon.readthedocs.io/en/latest/fun_class/fathon.MFDCCA.html

Computes the mass exponents from the fluctuations.

```APIDOC
## MFDCCA.computeMassExponents()

### Description
Computation of the mass exponents.

### Returns
Mass exponents.

### Return type
numpy ndarray
```

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

### saveObject

Source: https://fathon.readthedocs.io/en/latest/fun_class/fathon.DFA.html

Saves the current state of the DFA object to a binary file.

```APIDOC
## saveObject()

### Description
Save current object state to binary file.

### Parameters
#### Parameters
- **outFileName** (str) - Output binary file. .fathon extension will be appended to the file name.
```

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

### HT Class

Source: https://fathon.readthedocs.io/en/latest/_sources/fun_class/fathon.HT.rst.txt

The HT class is the main interface for Hurst exponent calculations. It is initialized with aggregated time series data.

```APIDOC
class HT:
    """The main class for Hurst exponent calculations."""
    def __init__(self, aggregated_time_series):
        """Initializes the HT object.

        Args:
            aggregated_time_series (np.ndarray): The zero-mean cumulative sum of the time series.
        """
        pass

```

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

### saveObject

Source: https://fathon.readthedocs.io/en/latest/fun_class/fathon.DCCA.html

Saves the current DCCA object state to a binary file.

```APIDOC
## saveObject()

### Description
Save current object state to binary file.

### Parameters
* **outFileName** (_str_) – Output binary file. .fathon extension will be appended to the file name.
```

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

### computeFlucVec

Source: https://fathon.readthedocs.io/en/latest/fun_class/fathon.MFDCCA.html

Computes the fluctuations in each window for each q-order.

```APIDOC
## MFDCCA.computeFlucVec(winSizes, qList, polOrd=1, revSeg=False)

### Description
Computation of the fluctuations in each window for each q-order.

### Parameters
* **winSizes** (numpy ndarray) – Array of window’s sizes.
* **qList** (float or iterable or numpy ndarray) – List of q-orders used to compute F.
* **polOrd** (int, optional) – Order of the polynomial to be fitted in each window (default : 1).
* **revSeg** (bool, optional) – If True, the computation of F is repeated starting from the end of the time series (default : False).

### Returns
* numpy ndarray – Array n of window’s sizes.
* numpy ndarray – qxn array F containing the values of the fluctuations in each window for each q-order.
```

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

### computeFlucVec

Source: https://fathon.readthedocs.io/en/latest/fun_class/fathon.MFDFA.html

Computes the fluctuations in each window for each q-order.

```APIDOC
## computeFlucVec(winSizes, qList, polOrd=1, revSeg=False)

### Description
Computation of the fluctuations in each window for each q-order.

### Parameters
* **winSizes** (numpy ndarray) - Array of window’s sizes.
* **qList** (float or iterable or numpy ndarray) - List of q-orders used to compute F.
* **polOrd** (int, optional) - Order of the polynomial to be fitted in each window (default : 1).
* **revSeg** (bool, optional) - If True, the computation of F is repeated starting from the end of the time series (default : False).

### Returns
* numpy ndarray - Array n of window’s sizes.
* numpy ndarray - qxn array F containing the values of the fluctuations in each window for each q-order.
```

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

### saveObject Method

Source: https://fathon.readthedocs.io/en/latest/_sources/fun_class/fathon.HT.rst.txt

Saves the current HT object to a file.

```APIDOC
def saveObject(self, filename):
    """Saves the HT object to a file.

    Args:
        filename (str): The name of the file to save the object to.
    """
    pass

```

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

### fitFlucVec

Source: https://fathon.readthedocs.io/en/latest/fun_class/fathon.DFA.html

Fits the fluctuations values to determine the Hurst exponent.

```APIDOC
## fitFlucVec()

### Description
Fit of the fluctuations values.

### Parameters
#### Parameters
- **nStart** (int, optional) - Size of the smaller window used to fit F (default : first value of n).
- **nEnd** (int, optional) - Size of the bigger window used to fit F (default : last value of n).
- **logBase** (float, optional) - Base of the logarithm for the log-log fit of n vs F (default : e).
- **verbose** (bool, optional) - Verbosity (default : False).

### Returns
- **float** - Slope of the fit.
- **float** - Intercept of the fit.
```

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

### fitFlucVec

Source: https://fathon.readthedocs.io/en/latest/fun_class/fathon.MFDFA.html

Fits the fluctuations values to determine scaling exponents.

```APIDOC
## fitFlucVec(nStart=-999, nEnd=-999, logBase=2.718281828459045, verbose=False)

### Description
Fit of the fluctuations values.

### Parameters
* **nStart** (int, optional) - Size of the smaller window used to fit F at each q-order (default : first value of n).
* **nEnd** (int, optional) - Size of the bigger window used to fit F at each q-order (default : last value of n).
* **logBase** (float, optional) - Base of the logarithm for the log-log fit of n vs F (default : e).
* **verbose** (bool, optional) - Verbosity (default : False).

### Returns
* numpy ndarray - Slope of the fit for each q-order.
* numpy ndarray - Intercept of the fit for each q-order.
```

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

### fitFlucVec

Source: https://fathon.readthedocs.io/en/latest/fun_class/fathon.MFDCCA.html

Fits the fluctuations values to determine scaling exponents.

```APIDOC
## MFDCCA.fitFlucVec(nStart=-999, nEnd=-999, logBase=2.718281828459045, verbose=False)

### Description
Fit of the fluctuations values.

### Parameters
* **nStart** (int, optional) – Size of the smaller window used to fit F at each q-order (default : first value of n).
* **nEnd** (int, optional) – Size of the bigger window used to fit F at each q-order (default : last value of n).
* **logBase** (float, optional) – Base of the logarithm for the log-log fit of n vs F (default : e).
* **verbose** (bool, optional) – Verbosity (default : False).

### Returns
* numpy ndarray – Slope of the fit for each q-order.
* numpy ndarray – Intercept of the fit for each q-order.
```

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

### computeMassExponents

Source: https://fathon.readthedocs.io/en/latest/fun_class/fathon.MFDFA.html

Computes the mass exponents from the MFDFA results.

```APIDOC
## computeMassExponents()

### Description
Computation of the mass exponents.

### Returns
* numpy ndarray - Mass exponents.
```

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

### DFA Class Methods

Source: https://fathon.readthedocs.io/en/latest/_sources/fun_class/fathon.DFA.rst.txt

This section details the methods available within the DFA class for performing Detrended Fluctuation Analysis.

```APIDOC
## DFA Class

### Description
Provides methods for performing Detrended Fluctuation Analysis (DFA) on time series data.

### Methods

#### computeFlucVec

##### Description
Computes the fluctuation function and related parameters for DFA.

##### Parameters
- **wins** (list or array): A list or array of window sizes to use for the analysis.
- **revSeg** (bool, optional): Whether to use the fully generalized DFA method. Defaults to False.
- **polOrd** (int, optional): The order of the polynomial used for detrending. Defaults to 1.

#### fitFlucVec

##### Description
Fits a line to the fluctuation function data to estimate the Hurst exponent.

##### Parameters
None

##### Returns
- **H** (float): The estimated Hurst exponent.
- **H_intercept** (float): The intercept of the fitted line.

#### multiFitFlucVec

##### Description
Computes the Hurst exponent for multiple specified ranges within the fluctuation function data.

##### Parameters
- **limits_list** (array): A 2D array where each row specifies a range [min_window, max_window] for fitting.

##### Returns
- **list_H** (list): A list of Hurst exponents, one for each specified range.
- **list_H_intercept** (list): A list of intercepts, one for each specified range.

#### saveObject

##### Description
Saves the current DFA object to a file.

##### Parameters
- **filename** (str): The name of the file to save the object to.
```

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

### multiFitFlucVec

Source: https://fathon.readthedocs.io/en/latest/fun_class/fathon.DFA.html

Fits the fluctuations values in different intervals simultaneously.

```APIDOC
## multiFitFlucVec()

### Description
Fit of the fluctuations values in different intervals at the same time.

### Parameters
#### Parameters
- **limitsList** (numpy ndarray) - kx2 array with the sizes of k starting and ending windows used to fit F.
- **logBase** (float, optional) - Base of the logarithm for the log-log fit of n vs F (default : e).
- **verbose** (bool, optional) - Verbosity (default : False).

### Returns
- **numpy ndarray** - Slopes of the fits.
- **numpy ndarray** - Intercepts of the fits.
```

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

### computeHt Method

Source: https://fathon.readthedocs.io/en/latest/_sources/fun_class/fathon.HT.rst.txt

Computes the time-dependent Hurst exponent for the given time series data using specified parameters.

```APIDOC
def computeHt(self, window_sizes, mfdfaPolOrd=1, polOrd=1):
    """Computes the time-dependent Hurst exponent.

    Args:
        window_sizes (list[int]): A list of window sizes to use for the computation.
        mfdfaPolOrd (int): The order of the polynomial for Multifractal Detrended Fluctuation Analysis.
        polOrd (int): The order of the polynomial for Detrended Fluctuation Analysis.

    Returns:
        np.ndarray: An array containing the computed Hurst exponent values for each window size.
    """
    pass

```

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

### computeMultifractalSpectrum

Source: https://fathon.readthedocs.io/en/latest/fun_class/fathon.MFDCCA.html

Computes the multifractal spectrum.

```APIDOC
## MFDCCA.computeMultifractalSpectrum()

### Description
Computation of the multifractal spectrum.

### Returns
* numpy ndarray – Singularity strengths.
* numpy ndarray – Multifractal spectrum.
```

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

### computeMultifractalSpectrum

Source: https://fathon.readthedocs.io/en/latest/fun_class/fathon.MFDFA.html

Computes the multifractal spectrum.

```APIDOC
## computeMultifractalSpectrum()

### Description
Computation of the multifractal spectrum.

### Returns
* numpy ndarray - Singularity strengths.
* numpy ndarray - Multifractal spectrum.
```

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

### computeRho

Source: https://fathon.readthedocs.io/en/latest/fun_class/fathon.DCCA.html

Computes the cross-correlation index in each window for DCCA.

```APIDOC
## computeRho()

### Description
Computation of the cross-correlation index in each window.

### Parameters
* **winSizes** (_numpy ndarray_) – Array of window’s sizes.
* **polOrd** (_int_, optional) – Order of the polynomial to be fitted in each window (default : 1).
* **verbose** (_bool_, optional) – Verbosity (default : False).
* **overlap** (_bool_, optional) – If True, computes F using overlapping segments (default : False).
* **revSeg** (_bool_, optional) – If True, the computation of F is repeated starting from the end of the time series, ignored if overlap is True (default : False).

### Returns
* _numpy ndarray_ – Array of window’s sizes.
* _numpy ndarray_ – Array containing the cross-correlation index.
```

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

### powRangeByCount

Source: https://fathon.readthedocs.io/en/latest/fun_class/fu/fathonUtils.powRangeByCount.html

Generates an array of elements given by base raised to linearly separated exponents.

```APIDOC
## powRangeByCount

### Description
Array of elements given by base raised to linearly separated exponents.

### Method
`powRangeByCount(_start_ , _end_ , _count =-1_, _base =2_)`

### Parameters
#### Path Parameters
- **start** (int) - Required - Smallest element.
- **end** (int) - Required - Biggest element.
- **count** (int) - Optional - Number of elements (default : end - start + 1).
- **base** (int) - Optional - Base of the exponential (default : 2).

### Response
#### Success Response
- **return** (numpy ndarray) - Array of elements given by base raised to linearly separated exponents.

### Request Example
```python
from fathon import fathonUtils as fu

#elements given by `base` raised to linearly separated exponents
a = fu.powRangeByCount(10, 1000, base=3, count=42)
```
```

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

### MFDCCA Class Methods

Source: https://fathon.readthedocs.io/en/latest/_sources/fun_class/fathon.MFDCCA.rst.txt

The MFDCCA class provides several methods for performing multifractal detrended cross-correlation analysis. These include computing fluctuation vectors, mass exponents, multifractal spectra, fitting fluctuation vectors, and saving the object.

```APIDOC
## computeFlucVec

### Description
Computes the fluctuation vector for the given time series and parameters.

### Method Signature
`computeFlucVec(self, wins, q, revSeg=False, polOrd=1, fitType='poly')`

### Parameters
- **wins** (list or np.array) - List of window sizes.
- **q** (list or np.array) - List of q-orders.
- **revSeg** (bool, optional) - Whether to consider reversed segments. Defaults to False.
- **polOrd** (int, optional) - Order of the polynomial for detrending. Defaults to 1.
- **fitType** (str, optional) - Type of fit to use. Defaults to 'poly'.

### Returns
- **n** (np.array) - Array of window sizes.
- **F** (np.array) - Fluctuation function values.

## computeMassExponents

### Description
Computes the mass exponents (tau) from the fluctuation function.

### Method Signature
`computeMassExponents(self)`

### Returns
- **tau** (np.array) - Array of mass exponents.

## computeMultifractalSpectrum

### Description
Computes the multifractal spectrum (alpha and mfSpect) from the mass exponents.

### Method Signature
`computeMultifractalSpectrum(self)`

### Returns
- **alpha** (np.array) - Array of singularity spectrum values.
- **mfSpect** (np.array) - Array of multifractal spectrum values.

## fitFlucVec

### Description
Fits the fluctuation vector to determine generalized Hurst exponents and their intercepts.

### Method Signature
`fitFlucVec(self)`

### Returns
- **list_H** (list) - List of generalized Hurst exponents.
- **list_H_intercept** (list) - List of intercepts from the fit.

## saveObject

### Description
Saves the current MFDCCA object to a file.

### Method Signature
`saveObject(self, filename)`

### Parameters
- **filename** (str) - The name of the file to save the object to.
```

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

### getObjectMember

Source: https://fathon.readthedocs.io/en/latest/fun_class/fu/fathonUtils.getObjectMember.html

Retrieves a specified member from a .fathon file. The member's name corresponds to its original object member name. The 'isComputed' status is not practically usable or retrievable.

```APIDOC
## getObjectMember

### Description
Return member of a previously saved object. Member’s name is the same of the object’s member it refers to. Member isComputed has no practical use and cannot be retrieved.

### Method
`getObjectMember(_fileName_, _memberName_)

### Parameters
#### Path Parameters
- **fileName** (str) - Required - Path to a previously saved .fathon file.
- **memberName** (str) - Required - Desired member’s name.

### Response
#### Success Response
- **Object’s member** (numpy ndarray) - The retrieved member from the .fathon file.
```

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

### fitFlucVec

Source: https://fathon.readthedocs.io/en/latest/fun_class/fathon.DCCA.html

Fits the fluctuations values obtained from DCCA.

```APIDOC
## fitFlucVec()

### Description
Fit of the fluctuations values.

### Parameters
* **nStart** (_int_, optional) – Size of the smaller window used to fit F (default : first value of n).
* **nEnd** (_int_, optional) – Size of the bigger window used to fit F (default : last value of n).
* **logBase** (_float_, optional) – Base of the logarithm for the log-log fit of n vs F (default : e).
* **verbose** (_bool_, optional) – Verbosity (default : False).

### Returns
* _float_ – Slope of the fit.
* _float_ – Intercept of the fit.
```

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

### DCCA Class Methods

Source: https://fathon.readthedocs.io/en/latest/_sources/fun_class/fathon.DCCA.rst.txt

This section details the methods available within the DCCA class for performing Detrended Cross-Correlation Analysis.

```APIDOC
## computeFlucVec

### Description
Computes the fluctuation function and related values for DCCA.

### Method
`computeFlucVec(self, wins, polOrd=1)`

### Parameters
- **wins** (list or numpy.ndarray) - A list or array of window sizes to use for the analysis.
- **polOrd** (int, optional) - The order of the polynomial used for detrending. Defaults to 1.

### Returns
- **n** (numpy.ndarray) - The window sizes used.
- **F** (numpy.ndarray) - The computed fluctuation function values.

## computeRho

### Description
Computes the rho index for DCCA.

### Method
`computeRho(self, wins, polOrd=1)`

### Parameters
- **wins** (list or numpy.ndarray) - A list or array of window sizes to use for the analysis.
- **polOrd** (int, optional) - The order of the polynomial used for detrending. Defaults to 1.

### Returns
- **n** (numpy.ndarray) - The window sizes used.
- **rho** (numpy.ndarray) - The computed rho index values.

## fitFlucVec

### Description
Fits a line to the fluctuation function to estimate the Hurst exponent.

### Method
`fitFlucVec(self)`

### Returns
- **H** (float) - The estimated Hurst exponent.
- **H_intercept** (float) - The intercept of the fitted line.

## multiFitFlucVec

### Description
Fits lines to the fluctuation function in multiple specified ranges to estimate Hurst exponents.

### Method
`multiFitFlucVec(self, limits_list)`

### Parameters
- **limits_list** (numpy.ndarray) - A 2D array where each row specifies the start and end of a range for fitting.

### Returns
- **list_H** (list) - A list of estimated Hurst exponents for each range.
- **list_H_intercept** (list) - A list of intercepts for the fitted lines in each range.

## rhoThresholds

### Description
Computes confidence intervals for the rho index.

### Method
`rhoThresholds(self, n_steps, wins, n_realizations, alpha, polOrd=1)`

### Parameters
- **n_steps** (int) - The number of steps for generating surrogate data.
- **wins** (list or numpy.ndarray) - A list or array of window sizes to use for the analysis.
- **n_realizations** (int) - The number of surrogate data realizations to generate.
- **alpha** (float) - The significance level for the confidence intervals.
- **polOrd** (int, optional) - The order of the polynomial used for detrending. Defaults to 1.

### Returns
- **n** (numpy.ndarray) - The window sizes used.
- **cInt1** (numpy.ndarray) - The lower confidence interval for the rho index.
- **cInt2** (numpy.ndarray) - The upper confidence interval for the rho index.

## saveObject

### Description
Saves the DCCA object to a file.

### Method
`saveObject(self, filename)`

### Parameters
- **filename** (str) - The name of the file to save the object to.

```

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

### computeFlucVec

Source: https://fathon.readthedocs.io/en/latest/fun_class/fathon.DFA.html

Computes the fluctuations in each window of the time series.

```APIDOC
## computeFlucVec()

### Description
Computation of the fluctuations in each window.

### Parameters
#### Parameters
- **winSizes** (numpy ndarray) - Array of window’s sizes.
- **polOrd** (int, optional) - Order of the polynomial to be fitted in each window (default : 1).
- **revSeg** (bool, optional) - If True, the computation of F is repeated starting from the end of the time series (default : False).
- **unbiased** (bool, optional) - If True, the unbiased version of DFA is computed, and revSeg is ignored. To be used on short time series (default : False).

### Returns
- **numpy ndarray** - Array n of window’s sizes.
- **numpy ndarray** - Array F containing the values of the fluctuations in each window.
```

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

### Subtract Mean from Time Series

Source: https://fathon.readthedocs.io/en/latest/_sources/fun_class/fu/fathonUtils.subtractMean.rst.txt

Use this function to transform a time series into a zero-mean series. Ensure numpy and fathonUtils are imported.

```python
import numpy as np
from fathon import fathonUtils as fu

#time series
a = np.random.randn(10000)

#zero-mean time series
a = fu.subtractMean(a)
```

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

### subtractMean

Source: https://fathon.readthedocs.io/en/latest/fun_class/fu/fathonUtils.subtractMean.html

Subtracts the mean of a vector (iterable).

```APIDOC
## subtractMean()

### Description
Subtracts mean of a vector.

### Parameters
#### Path Parameters
- **vec** (iterable) - Required - Python iterable whose mean will be subtracted.

### Returns
- **vec with its mean subtracted** (numpy ndarray) - Description of the return value.

### Request Example
```python
import numpy as np
from fathon import fathonUtils as fu

#time series
a = np.random.randn(10000)

#zero-mean time series
a = fu.subtractMean(a)
```
```

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

### rhoThresholds

Source: https://fathon.readthedocs.io/en/latest/fun_class/fathon.DCCA.html

Computes the confidence levels for the cross-correlation index in each window for DCCA.

```APIDOC
## rhoThresholds()

### Description
Computation of the cross-correlation index’s confidence levels in each window.

### Parameters
* **L** (_int_) – Size of the random time series used to evaluate confidence levels.
* **winSizes** (_numpy ndarray_) – Array of window’s sizes.
* **nSim** (_int_) – Number of times the cross-correlation index between two random time series is computed in order to evaluate the confidence levels.
* **confLvl** (_float_) – Confidence level.
* **polOrd** (_int_, optional) – Order of the polynomial to be fitted in each window (default : 1).
* **verbose** (_bool_, optional) – Verbosity (default : False).

### Returns
* _numpy ndarray_ – Array of window’s sizes.
* _numpy ndarray_ – Array containing the first confidence interval.
* _numpy ndarray_ – Array containing the second confidence interval.
```

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

### getObjectMember

Source: https://fathon.readthedocs.io/en/latest/_sources/fun_class/fu/fathonUtils.getObjectMember.rst.txt

Retrieves a member from an object.

```APIDOC
## getObjectMember

### Description
Retrieves a member from an object.

### Signature
`getObjectMember(obj, name)`

### Parameters
* **obj** (object) - The object from which to retrieve the member.
* **name** (string) - The name of the member to retrieve.

### Returns
* The member of the object if found, otherwise None.
```

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

### subtractMean Function

Source: https://fathon.readthedocs.io/en/latest/_sources/fun_class/fu/fathonUtils.subtractMean.rst.txt

The subtractMean function takes a NumPy array representing a time series and returns a new array with the mean subtracted, effectively making the series zero-mean.

```APIDOC
## subtractMean

### Description
Subtracts the mean from a time series.

### Usage
```python
import numpy as np
from fathon import fathonUtils as fu

# time series
a = np.random.randn(10000)

# zero-mean time series
a = fu.subtractMean(a)
```

### Parameters
This function does not explicitly list parameters in the provided documentation, but it is used with a NumPy array as input.
```

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

### toAggregated Function

Source: https://fathon.readthedocs.io/en/latest/_sources/fun_class/fu/fathonUtils.toAggregated.rst.txt

The toAggregated function transforms a numerical array into its zero-mean cumulative sum. This is often used in time series analysis to create a stationary representation of the data.

```APIDOC
## toAggregated

### Description
Computes the zero-mean cumulative sum of a time series.

### Parameters
This function does not explicitly list parameters in the provided documentation, but it is expected to take a numerical array as input.

### Usage Example
```python
import numpy as np
from fathon import fathonUtils as fu

# time series
a = np.random.randn(10000)

# zero-mean cumulative sum
a = fu.toAggregated(a)
```

### Returns
The function returns the zero-mean cumulative sum of the input array.
```

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

### toAggregated

Source: https://fathon.readthedocs.io/en/latest/fun_class/fu/fathonUtils.toAggregated.html

Subtracts the mean of a vector and computes the cumulative sum. This function is useful for time series analysis where a zero-mean cumulative sum is required.

```APIDOC
## toAggregated()

### Description
Subtracts mean of a vector and computes the cumulative sum.

### Parameters
#### Path Parameters
- **vec** (iterable) - The vector to be integrated.

### Returns
Cumulative sum of vec after subtraction of its mean.

### Return Type
numpy ndarray

### Usage Example
```python
import numpy as np
from fathon import fathonUtils as fu

# time series
a = np.random.randn(10000)

# zero-mean cumulative sum
a = fu.toAggregated(a)
```
```

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

### MFDCCA Class

Source: https://fathon.readthedocs.io/en/latest/fun_class/fathon.MFDCCA.html

The MFDCCA class is used for MultiFractal Detrended Cross-Correlation Analysis. It takes two time series as input.

```APIDOC
## class fathon.MFDCCA(tsVec1, tsVec2=[])

### Description
MultiFractal Detrended Cross-Correlation Analysis class.

### Parameters
* **tsVec1** (iterable) – First time series used for the analysis.
* **tsVec2** (iterable) – Second time series used for the analysis.
```

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