• Introduction.
• The statistical characterisation of time series.
• The change-point search in time-series analysis.
• Final change-point determination and test of randomness of the derived homogeneous sequences.
• Example. The annual averages of the air temperature 1908-1995 at Casablanca (Cuba).
• Conclusion.
• References.
• Tables.
• Annex.

Testing randomness. The use of ranks

Transposed from sample to time-series, the definition of randomness assumes an identical and independent distribution of the elements of the series (Sneyers 1975). In particular, for a series of n elements x_i, with i = 1, 2,.. , n, having the common continuous distribution function F(x), identity and independence are ensured if, for the joint distribution F_n of x_i, we have

    F_n(x₁, x₂,.. , x_n) = F(x₁).F(x₂).. .F(x_n).                                                (1)

It follows that, if x_(i) = i are the corresponding ranks of the elements x_i ranged in increasing order, we have the mean values (Gumbel 1958)

    E[F(x_(i))] = i/(n+1).                                                                             (2)

Replacing in the time series x_i by x_(i) = X_i, keeps then unchanged all the inequalities between any couple of elements x_i and x_j. Moreover, with (1), the n! permutations of the elements of the series have all the same probability with, for whatever i and j,

    Prob (X_i > X_j) = p = Prob (X_i < X_j) = q, with p = q = 1/2.                 (3)

Thus, for n trials, we have the binomial distribution of f(m)

    f(m) = Prob [nb (X_i > X_j) = m] = C^m_np^mq^n-m; with S _0,n f(m) = (p + q)ⁿ = 1. (4)

Moreover, for the set of (X_i, X_j), we have the relations

    var X_i = (n² - 1)/12,

    E(X_i - X_j) = 0, E[cov (X_i, X_j)] = 0, var (X_i - X_j) = n(n+1)/6,             (5)

while, for the correlation coefficient r = r (X_i, X_j) of the series X_i with i + j = 1, 2,.. , n, with relations (5) we have (Kendall and Stuart, 1967)

    E(r) = 0 and var r = 1/(n - 1).                                                              (6)

Continuous distribution functions excluding ties as x_i = x_j for different i and j, when such ties appear in a time-series of averages of continuous variables, these averages should be recomputed with a sufficient number of significant digits eliminating these ties. Without this precaution, a weakening of the significance level for the tests statistics has to be expected.

Furthermore, the definition of randomness being independent of the type of the underlying distribution of the elements of the time series and randomness being needed for testing the fit of any particular type of distribution, applying parametric tests for testing randomness is not only begging the question, but involves the risk of biased conclusions.

The alternatives to randomness

Testing stability of F(X_i) against trend

The trend statistic t_n (Mann 1945) is defined as

    t_n = nb (X_i >X_j) for all couples i > j.                                              (7)

With n_i = nb (X_j < X_i) for j < i, we have also

    t_n = S _i n_i.                                                                                        (8)

With n_i = 0, 1,.. , i, having for a random series, var n_i = (i² - 1)/12, we have also with (8)

    E(t_n) = n(n - 1)/4 and var t_n = S _i var n_i = n(n - 1)(2n + 5)/72,       (9)

large or small values of t_n being significant for a grouping of either the largest or of the smallest values towards the end of the series and thus of an increasing or of a decreasing trend.

The statistic t_n has an approximate normal distribution giving acceptable significance levels when n > 10. With the correction for continuity made in increasing |t_n - E(t_n)| by 1/2 (Sneyers, 1975), as to be expected, this accuracy is improved and found to be already valid for n > 4.

 

Testing independence against serial correlation

In this case, replacing j respectively by i+1 in the statistic (6), with (X_n+1 = X₁), the statistic r becomes

    r_s = cov (X_i, X_i+1) / var X_i,                                                             (10)

which, under the hypothesis of randomness, has also 0 mean and variance 1/(n - 1), positive values of r_d being significant for persistence and negative ones, for alternance.