Performs the Noether test for cyclical trend.
#include <imsls.h>
float *imsls_f_noether_cyclical_trend (int n_observations, float x[], ..., 0)
The type double function is imsls_d_noether_cyclical_trend.
int
n_observations (Input)
Number of observations in x. n_observations must
be greater than or equal to 3.
float x[]
(Input)
Array of length n_observations
containing the data in chronological order.
Array, p, of length 3 containing the probabilities of stat[1] or more, stat[2] or more, or stat[3] or more monotonic sequences.
If stat[0] is less than 1, p[0] is set to NaN (not a number).
#include <imsls.h>
float *imsls_f_noether_cyclical_trend ((int n_observations, float x[],
IMSLS_FUZZ, float fuzz,
IMSLS_STAT, int **stat,
IMSLS_STAT_USER, int stat[], IMSLS_N_MISSING, int *n_missing,
IMSLS_RETURN_USER, float p[],
0)
IMSLS_FUZZ,
float
fuzz (Input)
Nonnegative constant used to determine ties in computing
ranks in the combined samples. A tie is declared when two observations in the
combined sample are within fuzz of each other.
Default value for fuzz is
0.0.
IMSLS_STAT,
int
**stat (Output)
Address of a pointer to an internally
allocated array of length 6 containing the following statistics:
|
Row |
Statistics |
|
stat[0] |
The number of consecutive sequences of length three used to detect cyclical trend when tying middle elements are eliminated from the sequence, and the next consecutive observation is used. |
|
stat[1] |
The number of monotonic sequences of length three in the set defined by stat[0]. |
|
stat[2] |
The number of nonmonotonic sequences where tied threesomes are counted as nonmonotonic. |
|
stat[3] |
The number of monotonic sequences where tied threesomes are counted as monotonic. |
|
stat[4] |
The number of middle observations eliminated because they were tied in forming the stat[0] sequences. |
|
stat[5] |
The number of tied sequences found in forming the stat[2] and stat[3] sequences. A sequence is called a tied sequence if the middle element is tied with either of the two other elements. |
IMSLS_STAT_USER,
int
stat[] (Output)
Storage for array stat is provided by
the user.
See IMSLS_STAT.
IMSLS_N_MISSING,
int
*n_missing (Output)
Number of missing values in X.
IMSLS_RETURN_USER,
float p[]
(Input)
User allocated array of length 3 containing the return
values.
The count variables, stat[1], stat[2], and stat[3], differ in the manner in which ties are handled. A tie can occur in a set (of size three) only if the middle element is tied with either of the two ending elements. Tied ending elements are not considered. In stat[1], tied middle observations are eliminated, and a new set of size 3 is obtained by using the next observation in the sample. In stat[2], the original set of size three is used, and tied middle observations are counted as nonmonotonic. In stat[3], tied middle observations are counted as monotonic.
The probabilities of occurrence of the counts are obtained from the binomial distribution with p = 1/3, where p is the probability that a random sample of size three from a continuous distribution is monotonic. The binomial sample size is, of course, the number of sequences of size three found (adjusted for ties).
Hypothesis test:
H0 : q =
Pr(Xi > Xi−1 > Xi−2) + Pr(Xi < Xi
−1 < Xi−2 ) £ 1/3
H1 : q > 1/3
Reject if p[0]
(or p[1]
or p[2]
depending on the method used for handling ties) is less than the significance
level of the test.
Assumption: The observations are independent and are from a continuous distribution.
A test for cyclical trend in a sequence of 1000 randomly generated observations is performed. Because of the sample used, there are no ties and all three test statistics yield the same result.
#include <imsls.h>
#include <stdio.h>
void main()
{
float *pvalue=NULL;
int nobs = 1000, nmiss, *stat = NULL;
float *x = NULL;
imsls_random_seed_set(123457);
x = imsls_f_random_uniform(nobs, 0);
pvalue = imsls_f_noether_cyclical_trend(nobs, x,
IMSLS_STAT, &stat,
IMSLS_N_MISSING, &nmiss,
0);
imsls_f_write_matrix("P", 0, 2, pvalue, 0);
imsls_i_write_matrix("STAT", 0, 5, stat, 0);
printf("\n n missing = %d\n", nmiss);
}
P
0
1 2
0.6979
0.6979 0.6979
STAT
0
1 2 3
4 5
333 107 107
107 0 0
n missing = 0
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