如果我的问题不是很简洁，我深表歉意。我正在尝试理解用于估计QuantStart中发布的Hurst指数的通用公式：
https://www.quantstart.com/articles/Basics-of-Statistical-Mean-Reversion-Testing
这是代码：
from numpy import cumsum, log, polyfit, sqrt, std, subtract
from numpy.random import randn
def hurst(ts):
lags = range(10,80)
tau = [sqrt(std(subtract(ts[lag:], ts[:-lag]))) for lag in lags]
poly = polyfit(log(lags), log(tau), 1)
return poly[0]*2.0
print ('> Hurst Exponent:', '%.6s' % hurst(df['Close']))
我已经应用了此代码，并且有效。但是我不太了解这个公式是如何工作的。我特别对滞后感到怀疑。如果有人可以给我一些信息来理解公式的这一部分，我将非常感激。
例如，如果我将滞后范围设置为10和80，这是否意味着将在一个时间序列的10和80个观测值之间计算自相关？
还有另一个重要问题，我需要设置多少个观测值才能达到该滞后范围？我应该至少有80个观测值吗？
正如我之前提到的，如果有人愿意透露一些信息，我将非常感激。我将不胜感激。
谢谢！
Q：我需要设置多少个观测值才能达到滞后范围？
鉴于需要200 bar的滞后，所以观察少于200 bar的确是没有道理的。
Q：如果我将滞后范围设置为10和80，这是否意味着将在一个时间序列的10和80个观测值之间计算自相关？
绝对。这就是赫斯特的工作方式。
[请参阅从QuantFX模块获取的函数实现，原样为v.4.13（Py2.7在大多数地方不会造成麻烦，但是任何xrange（）都应在Py3.x中用range（）代替：]
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.anaconda\lib\site-packages\numpy\lib\polynomial.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.anaconda\lib\site-packages\numpy\core\_methods.py:82: RuntimeWarning: Degrees of freedom <= 0 for slice
warnings.warn("Degrees of freedom <= 0 for slice", RuntimeWarning)
C:\Python27.anaconda\lib\site-packages\numpy\core\_methods.py:94: RuntimeWarning: invalid value encountered in true_divide
arrmean, rcount, out=arrmean, casting='unsafe', subok=False)
C:\Python27.anaconda\lib\site-packages\numpy\core\_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.anaconda\lib\site-packages\numpy\lib\polynomial.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.anaconda\lib\site-packages\numpy\core\_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.anaconda\lib\site-packages\numpy\lib\polynomial.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.anaconda\lib\site-packages\numpy\lib\polynomial.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] )
关于属性的详细文章和讨论是here