• 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.

1.Statistical properties of random rank series

If the elements x_i of a series of size n are replaced by their ranks X_i, this new series is a permutation of the series of whole numbers X_i = 1, 2,.. , n, and testing randomness comes down to applying the mathematical properties of these numbers.

Having for i = 1, 2,.. , n the summations S and means E

S i = n(n + 1)/2 -> E(i) = (n + 1)/2

S i² = n(n + 1)(2n + 1)/6 -> E(i²) = (n + 1)(2n + 1)/6 = (2n² + 3n +1)/6

S i³ = [n(n + 1)/2]² -> E(i³) = n[(n + 1)/2)]². (1)

we have thus

var X_i = E(i²) - [E(i)]² = [(n + 1)/2][(2n + 1)/3 - (n + 1)/2] = (n² - 1)/12. (2)

Moreover, for X_i ¹ X_j, the number of pairs (X_i, X_j) being n_ij = n(n - 1)/2,

and having (X_i - X_j)² = (X_j - X_i)², (3)

we have also

E(X_i - X_j) = 0 and var (X_i - X_j) = E(X_i - X_j)²,

and the n_ij values of (X_i - X_j) are the n(n - 1) / 2 values of the sets of numbers (1, 2,.. , i) for i = 1, 2,.. , (n - 1).

It follows that

Var (X_i - X_j) = S _1,(n-1) i E(i²)/[n(n-1)/2)]

= S _1,(n-1) [(2i³ + 3i² + i)/6]/[n(n-1)/2)]

= [2E(i³⁾) + 3E(i²) + E(i)]/3n

= n(n + 1)/6

 

2. Testing rank randomness in a time series.

2.1 The trend test

In the test statistic t_n = S _1,n n_i, n_i is for X_i the number of inequalities X_j < X_i for j < i.

For each n_i, the equally possible values being 0, 1,.. , (i - 1), we have

var n_i = (i² - 1)/12.

Moreover, the values n_i being independent, we have

var t_n = S _1,n var n_i = {[n(n + 1)(2n + 1)/6] - n }/12

= n[(n + 1)(2n + 1) - 6]/72

= n(2n² + 3n - 5)/72

= n(n - 1)(2n + 5)/ 72.

 

2.2 The serial correlation test

For the series X_i, with X_i, X_j where i, j = 1, 2,.. , n and X_i ¹ X_j, we have

cov (X_i, X_j) = [S _1,n (2X_i² - (X_i - X_j)²]/2n = var X_i - [S _1,n(X_i - X_j)²]/2n

Moreover, cov (X_i, X_j) = 0, implying var X_i = [var (X_i - X_j)]/2,

var cov (X_i, X_j) = S _1,n [var (X_i - X_j)²]/2n.

With var (X_i - X_j)² = var [X_i(X_i - X_j) + X_j (X_i - X_j)]

we have finally

var cov (X_i, X_j) = [var X_i. var (X_i - X_j)]/2n

It follows that for the serial correlation

r = r (X_i, X_j) = [cov (X_i, X_j)]/ var X_i,

we have

E[(r) = 0 and

var r = [var cov (X_i, X_j)]/[var X_i]² = 1/(n - 1).