Pandas: Apply rolling window on complex function (Hurst Exponent)

In a nutshell: I need to calculate the Hurst Exponent (HE) across a rolling window inside a pandas dataframe and assign the values to its own column.

The HE function I use was lifted from here as it seemed more robust. For convenience it’s posted below:

def HurstEXP( ts = [ None, ] ):                                         
# TESTED: HurstEXP()                Hurst exponent ( Browninan Motion & other observations measure ) 100+ BARs back(!)
        """                                                         __doc__
        USAGE:
                    HurstEXP( ts = [ None, ] )

                    Returns the Hurst Exponent of the time series vector ts[]

        PARAMETERS:
                    ts[,]   a time-series, with 100+ elements
                            ( or [ None, ] that produces a demo run )

        RETURNS:
                    float - a Hurst Exponent approximation,
                            as a real value
                            or
                            an explanatory string on an empty call
        THROWS:
                    n/a
        EXAMPLE:
                    >>> HurstEXP()                                        # actual numbers will vary, as per np.random.randn() generator used
                    HurstEXP( Geometric Browian Motion ):    0.49447454
                    HurstEXP(    Mean-Reverting Series ):   -0.00016013
                    HurstEXP(          Trending Series ):    0.95748937
                    'SYNTH series demo ( on HurstEXP( ts == [ None, ] ) ) # actual numbers vary, as per np.random.randn() generator'

                    >>> HurstEXP( rolling_window( aDSEG[:,idxC], 100 ) )
        REF.s:
                    >>> www.quantstart.com/articles/Basics-of-Statistical-Mean-Reversion-Testing
        """
        #---------------------------------------------------------------------------------------------------------------------------<self-reflective>
        if ( ts[0] == None ):                                       # DEMO: Create a SYNTH Geometric Brownian Motion, Mean-Reverting and Trending Series:

             gbm = np.log( 1000 + np.cumsum(     np.random.randn( 100000 ) ) )  # a Geometric Brownian Motion[log(1000 + rand), log(1000 + rand + rand ), log(1000 + rand + rand + rand ),... log(  1000 + rand + ... )]
             mr  = np.log( 1000 +                np.random.randn( 100000 )   )  # a Mean-Reverting Series    [log(1000 + rand), log(1000 + rand        ), log(1000 + rand               ),... log(  1000 + rand       )]
             tr  = np.log( 1000 + np.cumsum( 1 + np.random.randn( 100000 ) ) )  # a Trending Series          [log(1001 + rand), log(1002 + rand + rand ), log(1003 + rand + rand + rand ),... log(101000 + rand + ... )]

                                                                    # Output the Hurst Exponent for each of the above SYNTH series
             print ( "HurstEXP( Geometric Browian Motion ):   {0: > 12.8f}".format( HurstEXP( gbm ) ) )
             print ( "HurstEXP(    Mean-Reverting Series ):   {0: > 12.8f}".format( HurstEXP( mr  ) ) )
             print ( "HurstEXP(          Trending Series ):   {0: > 12.8f}".format( HurstEXP( tr  ) ) )

             return ( "SYNTH series demo ( on HurstEXP( ts == [ None, ] ) ) # actual numbers vary, as per np.random.randn() generator" )
        """                                                         # FIX:
        ===================================================================================================================
        |
        |>>> QuantFX.HurstEXP( QuantFX.DATA[ :1000,QuantFX.idxH].tolist() )
        0.47537688039105963
        |
        |>>> QuantFX.HurstEXP( QuantFX.DATA[ :101,QuantFX.idxH].tolist() )
        -0.31081076640420308
        |
        |>>> QuantFX.HurstEXP( QuantFX.DATA[ :100,QuantFX.idxH].tolist() )
        nan
        |
        |>>> QuantFX.HurstEXP( QuantFX.DATA[ :99,QuantFX.idxH].tolist() )

        Intel MKL ERROR: Parameter 6 was incorrect on entry to DGELSD.
        C:Python27.anacondalibsite-packagesnumpylibpolynomial.py:594: RankWarning: Polyfit may be poorly conditioned
        warnings.warn(msg, RankWarning)
        0.026867491053098096
        """
        pass;     too_short_list = 101 - len( ts )                  # MUST HAVE 101+ ELEMENTS
        if ( 0 <  too_short_list ):                                 # IF NOT:
             ts = too_short_list * ts[:1] + ts                      #    PRE-PEND SUFFICIENT NUMBER of [ts[0],]-as-list REPLICAS TO THE LIST-HEAD
        #---------------------------------------------------------------------------------------------------------------------------
        lags = range( 2, 100 )                                                              # Create the range of lag values
        tau  = [ np.sqrt( np.std( np.subtract( ts[lag:], ts[:-lag] ) ) ) for lag in lags ]  # Calculate the array of the variances of the lagged differences
        #oly = np.polyfit( np.log( lags ), np.log( tau ), 1 )                               # Use a linear fit to estimate the Hurst Exponent
        #eturn ( 2.0 * poly[0] )                                                            # Return the Hurst exponent from the polyfit output
        """ ********************************************************************************************************************************************************************* DONE:[MS]:ISSUE / FIXED ABOVE
        |>>> QuantFX.HurstEXP( QuantFX.DATA[ : QuantFX.aMinPTR,QuantFX.idxH] )
        C:Python27.anacondalibsite-packagesnumpycore_methods.py:82: RuntimeWarning: Degrees of freedom <= 0 for slice
          warnings.warn("Degrees of freedom <= 0 for slice", RuntimeWarning)
        C:Python27.anacondalibsite-packagesnumpycore_methods.py:94: RuntimeWarning: invalid value encountered in true_divide
          arrmean, rcount, out=arrmean, casting='unsafe', subok=False)
        C:Python27.anacondalibsite-packagesnumpycore_methods.py:114: RuntimeWarning: invalid value encountered in true_divide
          ret, rcount, out=ret, casting='unsafe', subok=False)
        QuantFX.py:23034: RuntimeWarning: divide by zero encountered in log
          return ( 2.0 * np.polyfit( np.log( lags ), np.log( tau ), 1 )[0] )                  # Return the Hurst exponent from the polyfit output ( a linear fit to estimate the Hurst Exponent )

        Intel MKL ERROR: Parameter 6 was incorrect on entry to DGELSD.
        C:Python27.anacondalibsite-packagesnumpylibpolynomial.py:594: RankWarning: Polyfit may be poorly conditioned
          warnings.warn(msg, RankWarning)
        0.028471879418359915
        |
        |
        |# DATA:
        |
        |>>> QuantFX.DATA[ : QuantFX.aMinPTR,QuantFX.idxH]
        memmap([ 1763.31005859,  1765.01000977,  1765.44995117,  1764.80004883,
                 1765.83996582,  1768.91003418,  1771.04003906,  1769.43994141,
                 1771.4699707 ,  1771.61999512,  1774.76000977,  1769.55004883,
                 1773.4699707 ,  1773.32995605,  1770.08996582,  1770.20996094,
                 1768.34997559,  1768.02001953,  1767.59997559,  1767.23999023,
                 1768.41003418,  1769.06994629,  1769.56994629,  1770.7800293 ,
                 1770.56994629,  1769.7800293 ,  1769.90002441,  1770.44995117,
                 1770.9699707 ,  1771.04003906,  1771.16003418,  1769.81005859,
                 1768.76000977,  1769.39001465,  1773.23999023,  1771.91003418,
                 1766.92004395,  1765.56994629,  1762.65002441,  1760.18005371,
                 1755.        ,  1756.67004395,  1753.48999023,  1753.7199707 ,
                 1751.92004395,  1745.44995117,  1745.44995117,  1744.54003906,
                 1744.54003906,  1744.84997559,  1744.84997559,  1744.34997559,
                 1744.34997559,  1743.75      ,  1743.75      ,  1745.23999023,
                 1745.23999023,  1745.15002441,  1745.31005859,  1745.47998047,
                 1745.47998047,  1749.06994629,  1749.06994629,  1748.29003906,
                 1748.29003906,  1747.42004395,  1747.42004395,  1746.98999023,
                 1747.61999512,  1748.79003906,  1748.79003906,  1748.38000488,
                 1748.38000488,  1744.81005859,  1744.81005859,  1736.80004883,
                 1736.80004883,  1735.43005371,  1735.43005371,  1737.9699707
                 ], dtype=float32
                )
        |
        |
        | # CONVERTED .tolist() to avoid .memmap-type artifacts:
        |
        |>>> QuantFX.DATA[ : QuantFX.aMinPTR,QuantFX.idxH].tolist()
        [1763.31005859375, 1765.010009765625, 1765.449951171875, 1764.800048828125, 1765.8399658203125, 1768.9100341796875, 1771.0400390625, 1769.43994140625, 1771.469970703125, 1771.6199951171875, 1774.760
        859375, 1743.75, 1743.75, 1745.239990234375, 1745.239990234375, 1745.1500244140625, 1745.31005859375, 1745.47998046875, 1745.47998046875, 1749.0699462890625, 1749.0699462890625, 1748.2900390625, 174
        |
        |>>> QuantFX.HurstEXP( QuantFX.DATA[ : QuantFX.aMinPTR,QuantFX.idxH].tolist() )
        C:Python27.anacondalibsite-packagesnumpycore_methods.py:116: RuntimeWarning: invalid value encountered in double_scalars
          ret = ret.dtype.type(ret / rcount)

        Intel MKL ERROR: Parameter 6 was incorrect on entry to DGELSD.
        C:Python27.anacondalibsite-packagesnumpylibpolynomial.py:594: RankWarning: Polyfit may be poorly conditioned
          warnings.warn(msg, RankWarning)
        0.028471876494884543
        ===================================================================================================================
        |
        |>>> QuantFX.HurstEXP( QuantFX.DATA[ :1000,QuantFX.idxH].tolist() )
        0.47537688039105963
        |
        |>>> QuantFX.HurstEXP( QuantFX.DATA[ :101,QuantFX.idxH].tolist() )
        -0.31081076640420308
        |
        |>>> QuantFX.HurstEXP( QuantFX.DATA[ :100,QuantFX.idxH].tolist() )
        nan
        |
        |>>> QuantFX.HurstEXP( QuantFX.DATA[ :99,QuantFX.idxH].tolist() )

        Intel MKL ERROR: Parameter 6 was incorrect on entry to DGELSD.
        C:Python27.anacondalibsite-packagesnumpylibpolynomial.py:594: RankWarning: Polyfit may be poorly conditioned
        warnings.warn(msg, RankWarning)
        0.026867491053098096
        """
        return ( 2.0 * np.polyfit( np.log( lags ), np.log( tau ), 1 )[0] )  

Now in order to test the function let’s grab some TSLA data from Yahoo finance:

import pandas as pd
import yfinance as yf
from datetime import datetime
from dateutil.relativedelta import relativedelta

years = 5
today = datetime.today().strftime('%Y-%m-%d')
lastyeartoday = (datetime.today() - relativedelta(years=years)).strftime('%Y-%m-%d')

df = yf.download('TSLA', 
                      start=lastyeartoday, 
                      end=today, 
                      progress=False)
df = df.dropna()
df = df[[u'Close']]
df

Output:

Date        Close
2016-02-16  31.034000
2016-02-17  33.736000
2016-02-18  33.354000
2016-02-19  33.316002
2016-02-22  35.548000
... ...
2021-02-08  863.419983
2021-02-09  849.460022
2021-02-10  804.820007
2021-02-11  811.659973
2021-02-12  816.119995
1259 rows × 1 columns

So far so good. We have the function and the data to test it with. Now let’s do a sanity test, i.e. run the function against a subsample of the data:

import numpy as np
window = 20

hurst = lambda x: (HurstEXP(ts = df[u'Close'][:-x].to_numpy()))
hurst(window)

Output:

0.5163981260143369

Excellent.

Now to the meaty part. Applying the lambda across a rolling window and assigning the result to its own column. I’ve pretty much tried every trick I was able to dig up but cannot make it work.

The vanilla approach:

df.Close.rolling(window).apply(hurst, engine='cython', raw=True)

Gives me the following error:

TypeError: Cannot convert input [[-31.0340004  -33.73600006 -33.35400009 -33.31600189 -35.54800034
 -35.44200134 -35.79999924 -37.48600006 -38.06800079 -38.38600159
 -37.27000046 -37.66799927 -39.14799881 -40.20800018 -41.05799866
 -40.52000046 -41.74399948 -41.0359993  -41.5        -43.02999878]] of type <class 'numpy.ndarray'> to Timestamp

Then I tried to get clever:

hurst = df.apply(lambda x: pd.Series(x.index).rolling(window).agg({'Hurst': lambda window: HurstEXP(x.loc[window])})).reset_index()['Hurst']
df.assign(Hurst=hurst) 

Also failed ignominiously. So at this point – half a day later – I’m pretty much stumped. Do any of you hardcore python aficionados know of a way to do this?

Thanks a lot in advance for any insights and pointers.

Answer

I think your problem is that your window is too short. It says in the docstring that the window length has to be 100+ elements, and the Hurst code isn’t handling it properly, resulting in a failure of the SVD.

Separately, your test is actually slicing everything but the last 20 elements, so is actually a long array, which is why it didn’t fail:

tmp = df[u'Close'][:-20].to_numpy()

print(tmp.shape, HurstEXP(ts = tmp))
(1239,) 0.5163981260143368

If you test a window < 100 length, it throws a LinAlg exception:

tmp = df[u'Close'][:20].to_numpy()

print(tmp.shape, HurstEXP(ts = tmp))
(fails)

It should work if increase your rolling window length or repair the code in the Hurst function to pad out the array if it’s too short.

window = 500
df.Close.rolling(window).apply(lambda x: HurstEXP(ts = x), raw=True)

The code in the HurstEXP function for handling lists shorter than 100 elements won’t work for values of ts that are np.ndarray objects like those being provided from the .rolling(raw=True).

You could modify the function to start with the following, and it will work for windows under 100 elements:

def HurstEXP( ts= [ None, ] ):   
        if isinstance(ts, np.ndarray):
            ts = ts.tolist()
        

…alternatively, if you’re always going to have numpy arrays, you could change the line that fixes it:

     ts = too_short_list * ts[:1] + ts                      #    PRE-PEND SUFFICIENT NUMBER of [ts[0],]-as-list REPLICAS TO THE LIST-HEAD

to

     ts = np.pad(ts, pad_width=(too_short_list,0), mode='edge')