def acf(x, unbiased=False, nlags=40, confint=None, qstat=False, fft=False,
alpha=None):
'''
Autocorrelation function for 1d arrays.
Parameters
----------
x : array
Time series data
unbiased : bool
If True, then denominators for autocovariance are n-k, otherwise n
nlags: int, optional
Number of lags to return autocorrelation for.
confint : scalar, optional
The use of confint is deprecated. See `alpha`.
If a number is given, the confidence intervals for the given level are
returned. For instance if confint=95, 95 % confidence intervals are
returned where the standard deviation is computed according to
Bartlett\'s formula.
qstat : bool, optional
If True, returns the Ljung-Box q statistic for each autocorrelation
coefficient. See q_stat for more information.
fft : bool, optional
If True, computes the ACF via FFT.
alpha : scalar, optional
If a number is given, the confidence intervals for the given level are
returned. For instance if alpha=.05, 95 % confidence intervals are
returned where the standard deviation is computed according to
Bartlett\'s formula.
Returns
-------
acf : array
autocorrelation function
confint : array, optional
Confidence intervals for the ACF. Returned if confint is not None.
qstat : array, optional
The Ljung-Box Q-Statistic. Returned if q_stat is True.
pvalues : array, optional
The p-values associated with the Q-statistics. Returned if q_stat is
True.
Notes
-----
The acf at lag 0 (ie., 1) is returned.
This is based np.correlate which does full convolution. For very long time
series it is recommended to use fft convolution instead.
If unbiased is true, the denominator for the autocovariance is adjusted
but the autocorrelation is not an unbiased estimtor.
'''
nobs = len(x)
d = nobs # changes if unbiased
if not fft:
avf = acovf(x, unbiased=unbiased, demean=True)
#acf = np.take(avf/avf[0], range(1,nlags+1))
acf = avf[:nlags + 1] / avf[0]
else:
#JP: move to acovf
x0 = x - x.mean()
# ensure that we always use a power of 2 or 3 for zero-padding,
# this way we'll ensure O(n log n) runtime of the fft.
n = _next_regular(2 * nobs + 1)
Frf = np.fft.fft(x0, n=n) # zero-pad for separability
if unbiased:
d = nobs - np.arange(nobs)
acf = np.fft.ifft(Frf * np.conjugate(Frf))[:nobs] / d
acf /= acf[0]
#acf = np.take(np.real(acf), range(1,nlags+1))
acf = np.real(acf[:nlags + 1]) # keep lag 0
if not (confint or qstat or alpha):
return acf
if not confint is None:
import warnings
warnings.warn("confint is deprecated. Please use the alpha keyword",
FutureWarning)
varacf = np.ones(nlags + 1) / nobs
varacf[0] = 0
varacf[1] = 1. / nobs
varacf[2:] *= 1 + 2 * np.cumsum(acf[1:-1]**2)
interval = stats.norm.ppf(1 - (100 - confint) / 200.) * np.sqrt(varacf)
confint = np.array(zip(acf - interval, acf + interval))
if not qstat:
return acf, confint
if alpha is not None:
varacf = np.ones(nlags + 1) / nobs
varacf[0] = 0
varacf[1] = 1. / nobs
varacf[2:] *= 1 + 2 * np.cumsum(acf[1:-1]**2)
interval = stats.norm.ppf(1 - alpha / 2.) * np.sqrt(varacf)
confint = np.array(zip(acf - interval, acf + interval))
if not qstat:
return acf, confint
if qstat:
qstat, pvalue = q_stat(acf[1:], nobs=nobs) # drop lag 0
if (confint is not None or alpha is not None):
return acf, confint, qstat, pvalue
else:
return acf, qstat, pvalue
def test_next_regular():
hams = {
1: 1, 2: 2, 3: 3, 4: 4, 5: 5, 6: 6, 7: 8, 8: 8, 14: 15, 15: 15,
16: 16, 17: 18, 1021: 1024, 1536: 1536, 51200000: 51200000,
510183360: 510183360, 510183360+1: 512000000, 511000000: 512000000,
854296875: 854296875, 854296875+1: 859963392,
196608000000: 196608000000, 196608000000+1: 196830000000,
8789062500000: 8789062500000, 8789062500000+1: 8796093022208,
206391214080000: 206391214080000, 206391214080000+1: 206624260800000,
470184984576000: 470184984576000, 470184984576000+1: 470715894135000,
7222041363087360: 7222041363087360,
7222041363087360+1: 7230196133913600,
# power of 5 5**23
11920928955078125: 11920928955078125,
11920928955078125-1: 11920928955078125,
# power of 3 3**34
16677181699666569: 16677181699666569,
16677181699666569-1: 16677181699666569,
# power of 2 2**54
18014398509481984: 18014398509481984,
18014398509481984-1: 18014398509481984,
# above this, int(ceil(n)) == int(ceil(n+1))
19200000000000000: 19200000000000000,
19200000000000000+1: 19221679687500000,
288230376151711744: 288230376151711744,
288230376151711744+1: 288325195312500000,
288325195312500000-1: 288325195312500000,
288325195312500000: 288325195312500000,
288325195312500000+1: 288555831593533440,
# power of 3 3**83
3990838394187339929534246675572349035227-1:
3990838394187339929534246675572349035227,
3990838394187339929534246675572349035227:
3990838394187339929534246675572349035227,
# power of 2 2**135
43556142965880123323311949751266331066368-1:
43556142965880123323311949751266331066368,
43556142965880123323311949751266331066368:
43556142965880123323311949751266331066368,
# power of 5 5**57
6938893903907228377647697925567626953125-1:
6938893903907228377647697925567626953125,
6938893903907228377647697925567626953125:
6938893903907228377647697925567626953125,
# http://www.drdobbs.com/228700538
# 2**96 * 3**1 * 5**13
290142196707511001929482240000000000000-1:
290142196707511001929482240000000000000,
290142196707511001929482240000000000000:
290142196707511001929482240000000000000,
290142196707511001929482240000000000000+1:
290237644800000000000000000000000000000,
# 2**36 * 3**69 * 5**7
4479571262811807241115438439905203543080960000000-1:
4479571262811807241115438439905203543080960000000,
4479571262811807241115438439905203543080960000000:
4479571262811807241115438439905203543080960000000,
4479571262811807241115438439905203543080960000000+1:
4480327901140333639941336854183943340032000000000,
# 2**37 * 3**44 * 5**42
30774090693237851027531250000000000000000000000000000000000000-1:
30774090693237851027531250000000000000000000000000000000000000,
30774090693237851027531250000000000000000000000000000000000000:
30774090693237851027531250000000000000000000000000000000000000,
30774090693237851027531250000000000000000000000000000000000000+1:
30778180617309082445871527002041377406962596539492679680000000,
}
for x, y in hams.items():
assert_equal(_next_regular(x), y)