v1.0.2.19.05.2019

time_series_arima/airline.csv

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time_series_eda_and_white_noise_test/ljungbox.py

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+
+import numpy as np
+import scipy.stats
+
+
+def sac(x, k=1):
+    """
+    Sample autocorrelation (As used in statistics with normalization)
+
+    http://en.wikipedia.org/wiki/Autocorrelation
+
+    Parameters
+    ----------
+    x : 1d numpy array
+        Signal
+    k : int or list of ints
+        Lags to calculate sample autocorrelation for
+
+    Returns
+    -------
+    res : scalar or np array
+        The sample autocorrelation. A scalar value if k is a scalar, and a
+        numpy array if k is a interable.
+    """
+    try:
+        res = []
+        for ki in k:
+            res.append(sac(x, ki))
+        return np.array(res)
+    except:
+        pass
+    mx = np.mean(x)
+    if k==0:
+        N = np.sum((x-mx)*(x-mx))
+    else:
+        N = np.sum((x[:-k]-mx)*(x[k:]-mx))
+    D = len(x) * np.var(x)
+    return N/D
+
+
+def ljungbox(x, lags, alpha=0.1):
+    """
+    The Ljung-Box test for determining if the data is independently distributed.
+
+    Parameters
+    ----------
+    x : 1d numpy array
+        Signal to test
+    lags : int
+        Number of lags being tested
+       
+    Returns
+    -------
+    Q : float
+        Test statistic
+    """
+    n = len(x)
+    Q = 0
+    for k in range(1, lags+1):
+        Q += (sac(x, k)**2) / (n-k)
+    Q = n*(n+2)*Q
+    return Q
+
+def boxpierce(x, lags, alpha=0.1):
+    """
+    The Box-Pierce test for determining if the data is independently distributed.
+
+    Parameters
+    ----------
+    x : 1d numpy array
+        Signal to test
+    lags : int
+        Number of lags being tested
+       
+    Returns
+    -------
+    Q : float
+        Test statistic
+    """
+    n = len(x)
+    Q = 0
+    for k in range(1, lags+1):
+        Q += (sac(x, k)**2)
+    Q = n*Q
+    return Q
+
+def lbqtest(x, lags, alpha=0.1, method='lb'):
+    """
+    The Ljung-Box test for determining if the data is independently distributed.
+
+    Parameters
+    ----------
+    x : 1d numpy array
+        Signal to test
+    lags : list of ints
+        Lags being tested
+    alpha : float
+        Significance level used for the tests
+    method : string
+        Can be either 'lb' for Ljung-Box, or 'bp' for Box-Pierce
+       
+    Returns
+    -------
+    h : np array
+        Numpy array of bool values, True == H0 hypothesis rejected
+    pV : np array
+        Test statistics p-values
+    Q : np array
+        Test statistics
+    cV : np array
+        Critical values used for determining if H0 should be rejected. The
+        critical values are calculated from the given alpha and lag.
+    """
+    if method=='lb':
+        findq = ljungbox
+    else:
+        findq = boxpierce
+    n = len(x)
+    Q = np.zeros(len(lags))
+    pV = np.zeros(len(lags))
+    cV = np.zeros(len(lags))
+    for i, lag in enumerate(lags):
+        Q[i] = findq(x, lag)
+        pV[i] = 1.0 - scipy.stats.chi2.cdf(Q[i], lag)
+        cV[i] = scipy.stats.chi2.ppf(1-alpha, lag)
+    h = Q>cV
+    return h, pV, Q, cV