acf_confint

mdtools.statistics.acf_confint(x, axis=None, alpha=0.05, n=None)

Calculate the confidence intervals of an autocorrelation function.

Parameters:

• x (array_like) – The values of the ACF. Intermediate lag times must not be missing. That means if you used e.g. mdtools.statistics.acf() to compute the ACF, the argument dtau must have been 1.

• axis (int or None, optional) – The axis along which to calculate the confidence interval. By default, the flattened input array is used.

• alpha (scalar, optional) – The significance level (also known as probability of error). The significance level is the maximum probability for rejecting the null hypothesis although its true (Type 1 error). Here, the null hypothesis is that the underlying time series has no autocorrelation. Typical values for the significance level are 0.01 or 0.05. The smaller the significance level (probability for Type 1 error), the higher the probability of a Type 2 error (i.e. the null hypothesis is not rejected although it is wrong).

• n (int, optional) – Sample size (i.e. number of recorded time points) of the underlying time series. By default, n is set to the number of lag times \(\tau\) in the given ACF. This is valid for ACFs that were computed at all possible lag times from the underlying time series. That means if you used e.g. mdtools.statistics.acf() to compute the ACF, the argument dtau must have been 1 and tau_max must have been None or the length of x along the given axis..

Returns:

confint (numpy.ndarray) – Array of the same shape as x containing the upper limit of the confidence interval of the ACF at each lag time. The lower limit is simply given by -confint. The confidence interval is centered at zero. To get the confidence interval around the actual ACF values compute x + confint and x - confint.

See also

mdtools.statistics.acf()

Calculate the autocorrelation function of an array

mdtools.statistics.acf_se()

Calculate the standard errors of an autocorrelation function

Notes

The confidence interval of the autocorrelation function (ACF) \(C_\tau\) at lag time \(\tau\) is estimated according to:[1]

\[B(C_\tau) = z_{1-\alpha/2} SE(C_\tau)\]

with the significance level \(\alpha\), the quantile function \(z_p\) of the standard normal distribution (\(\sigma = 1\)) and the standard error \(SE(C_\tau)\).

The quantile function \(z_p\) is also known as the inverse cumulative distribution function \(F^{-1}(p)\). The cumulative distribution function \(F(X)\) returns the probability \(p\) to gain a value below or equal to \(X\). Hence, the inverse cumulative distribution function \(F^{-1}(p)\) returns the value \(X\) such that the probability of gaining a value below or equal to \(X\) is given by \(p\). Thus, the probability for gaining a value higher than \(X\) is \(1-p\).

A normal random variable \(X\) will exceed the value \(\mu + z_p \sigma\) with probability \(1-p\) and will lie outside the interval \(\mu \pm z_p \sigma\) with probability \(2(1-p)\) (for \(p \ge 0.5\) or \(2p\) for \(p \le 0.5\)). In particular, the quantile \(z_{0.975}\) is \(1.96\). Therefore, a normal random variable will lie outside the interval \(\mu \pm 1.96\sigma\) in only 5% of cases.[2] Here \(\mu\) is the mean of the random variable and \(\sigma\) is its standard deviation. Note that for the normal distribution \(z_p - \mu = -(z_{1-p} - \mu)\) holds, because it is symmetric around the mean.

The standard error of the ACF is estimated according to Bartlett’s formula for MA(l) processes:[1]^,[3]

\[SE(C_\tau) = \sqrt{\frac{1 + 2 \sum_{\tau^{'} = 1}^{\tau - 1} C_{\tau^{'}}}{N}}\]

for \(\tau \gt 1\). For \(\tau = 1\) the standard error is estimated by \(SE(C_1) = \frac{1}{\sqrt{N}}\) and for \(\tau = 0\) the standard error is \(SE(C_0) = 0\).

If the ACF \(C_\tau\) is zero within the confidence interval, the null hypothesis that there is no autocorrelation at the given lag time \(\tau\) is rejected with a significance level of \(\alpha\). This is an approximate test that assumes that the underlying time series is normally distributed.[1]

References

[1] (1,2,3)

Wikipedia Correlogram

[2]

Wikipedia Normal distribution

[3]

Wikipedia Autoregressive-moving-average model

Examples

>>> mdt.stats.acf_confint([3])
array([0.])
>>> a = np.arange(4)
>>> se = mdt.stats.acf_se(a)
>>> ci = mdt.stats.acf_confint(a)
>>> ci
array([0.        , 0.97998199, 1.6973786 , 3.25023257])
>>> np.allclose(ci, 1.959960*se, rtol=0, atol=1e-5)
True
>>> ci = mdt.stats.acf_confint(a, alpha=0.01)
>>> ci
array([0.        , 1.28791465, 2.23073361, 4.27152966])
>>> np.allclose(ci, 2.575830*se, rtol=0, atol=1e-5)
True

>>> b = np.column_stack([a, a])
>>> mdt.stats.acf_confint(b, axis=0)
array([[0.        , 0.        ],
       [0.97998199, 0.97998199],
       [1.6973786 , 1.6973786 ],
       [3.25023257, 3.25023257]])
>>> b = np.row_stack([a, a])
>>> mdt.stats.acf_confint(b, axis=1)
array([[0.        , 0.97998199, 1.6973786 , 3.25023257],
       [0.        , 0.97998199, 1.6973786 , 3.25023257]])
>>> c = np.array([b, b])
>>> mdt.stats.acf_confint(c, axis=2)
array([[[0.        , 0.97998199, 1.6973786 , 3.25023257],
        [0.        , 0.97998199, 1.6973786 , 3.25023257]],

       [[0.        , 0.97998199, 1.6973786 , 3.25023257],
        [0.        , 0.97998199, 1.6973786 , 3.25023257]]])