as181.c

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#include <math.h>
#include <stdio.h>
#include <grass/cdhc.h>
#include "local_proto.h"


/* Local function prototypes */
static double poly(double c[], int nord, double x);


/*-Algorithm AS 181
 * by J.P. Royston, 1982.
 * Applied Statistics 31(2):176-180
 *
 * Translation to C by James Darrell McCauley, mccauley@ecn.purdue.edu.
 *
 * Calculates Shapiro and Wilk's W statistic and its sig. level
 *
 * Originally used:
 * Auxiliary routines required: ALNORM = algorithm AS 66 and NSCOR2
 * from AS 177.

 * Note: ppnd() from as66 was replaced with ppnd16() from as241.
 */
void wext(double x[], int n, double ssq, double a[], int n2, double eps,
          double *w, double *pw, int *ifault)
{
    double eu3, lamda, ybar, sdy, al, un, ww, y, z;
    int i, j, n3, nc;
    static double wa[3] = { 0.118898, 0.133414, 0.327907 };
    static double wb[4] = { -0.37542, -0.492145, -1.124332, -0.199422 };
    static double wc[4] = { -3.15805, 0.729399, 3.01855, 1.558776 };
    static double wd[6] = { 0.480385, 0.318828, 0.0, -0.0241665, 0.00879701,
        0.002989646
    };
    static double we[6] = { -1.91487, -1.37888, -0.04183209, 0.1066339,
        -0.03513666, -0.01504614
    };
    static double wf[7] = { -3.73538, -1.015807, -0.331885, 0.1773538,
        -0.01638782, -0.03215018, 0.003852646
    };
    static double unl[3] = { -3.8, -3.0, -1.0 };
    static double unh[3] = { 8.6, 5.8, 5.4 };
    static int nc1[3] = { 5, 5, 5 };
    static int nc2[3] = { 3, 4, 5 };
    double c[5];
    int upper = 1;
    static double pi6 = 1.90985932, stqr = 1.04719755;
    static double zero = 0.0, tqr = 0.75, one = 1.0;
    static double onept4 = 1.4, three = 3.0, five = 5.0;
    static double c1[5][3] = {
        {-1.26233, -2.28135, -3.30623},
        {1.87969, 2.26186, 2.76287},
        {0.0649583, 0.0, -0.83484},
        {-0.0475604, 0.0, 1.20857},
        {-0.0139682, -0.00865763, -0.507590}
    };
    static double c2[5][3] = {
        {-0.287696, -1.63638, -5.991908},
        {1.78953, 5.60924, 21.04575},
        {-0.180114, -3.63738, -24.58061},
        {0.0, 1.08439, 13.78661},
        {0.0, 0.0, -2.835295}
    };

    *ifault = 1;

    *pw = one;
    *w = one;

    if (n <= 2)
        return;

    *ifault = 3;
    if (n / 2 != n2)
        return;

    *ifault = 2;
    if (n > 2000)
        return;

    *ifault = 0;
    i = n - 1;

    for (*w = 0.0, j = 0; j < n2; ++j)
        *w += a[j] * (x[i--] - x[j]);

    *w *= *w / ssq;
    if (*w > one) {
        *w = one;

        return;
    }
    else if (n > 6) {           /* Get significance level of W */
        /*
         * N between 7 and 2000 ... Transform W to Y, get mean and sd,
         * standardize and get significance level
         */

        if (n <= 20) {
            al = log((double)n) - three;
            lamda = poly(wa, 3, al);
            ybar = exp(poly(wb, 4, al));
            sdy = exp(poly(wc, 4, al));
        }
        else {
            al = log((double)n) - five;
            lamda = poly(wd, 6, al);
            ybar = exp(poly(we, 6, al));
            sdy = exp(poly(wf, 7, al));
        }

        y = pow(one - *w, lamda);
        z = (y - ybar) / sdy;
        *pw = alnorm(z, upper);

        return;
    }
    else {
        /* Deal with N less than 7 (Exact significance level for N = 3). */
        if (*w >= eps) {
            ww = *w;
            if (*w >= eps) {
                ww = *w;
                if (n == 3) {
                    *pw = pi6 * (atan(sqrt(ww / (one - ww))) - stqr);

                    return;
                }

                un = log((*w - eps) / (one - *w));
                n3 = n - 3;
                if (un >= unl[n3 - 1]) {
                    if (un <= onept4) {
                        nc = nc1[n3 - 1];

                        for (i = 0; i < nc; ++i)
                            c[i] = c1[i][n3 - 1];

                        eu3 = exp(poly(c, nc, un));
                    }
                    else {
                        if (un > unh[n3 - 1])
                            return;

                        nc = nc2[n3 - 1];

                        for (i = 0; i < nc; ++i)
                            c[i] = c2[i][n3 - 1];

                        un = log(un);   /*alog */
                        eu3 = exp(exp(poly(c, nc, un)));
                    }
                    ww = (eu3 + tqr) / (one + eu3);
                    *pw = pi6 * (atan(sqrt(ww / (one - ww))) - stqr);

                    return;
                }
            }
        }
        *pw = zero;

        return;
    }

    return;
}


/*
 * Algorithm AS 181.1   Appl. Statist.  (1982) Vol. 31, No. 2
 * 
 * Obtain array A of weights for calculating W
 */
void wcoef(double a[], int n, int n2, double *eps, int *ifault)
{
    static double c4[2] = { 0.6869, 0.1678 };
    static double c5[2] = { 0.6647, 0.2412 };
    static double c6[3] = { 0.6431, 0.2806, 0.0875 };
    static double rsqrt2 = 0.70710678;
    double a1star, a1sq, sastar, an;
    int j;

    *ifault = 1;
    if (n <= 2)
        return;

    *ifault = 3;
    if (n / 2 != n2)
        return;

    *ifault = 2;
    if (n > 2000)
        return;

    *ifault = 0;
    if (n > 6) {
        /* Calculate rankits using approximate function nscor2().  (AS177) */
        nscor2(a, n, n2, ifault);

        for (sastar = 0.0, j = 1; j < n2; ++j)
            sastar += a[j] * a[j];

        sastar *= 8.0;

        an = n;
        if (n <= 20)
            an--;
        a1sq = exp(log(6.0 * an + 7.0) - log(6.0 * an + 13.0)
                   + 0.5 * (1.0 + (an - 2.0) * log(an + 1.0) - (an - 1.0)
                            * log(an + 2.0)));
        a1star = sastar / (1.0 / a1sq - 2.0);
        sastar = sqrt(sastar + 2.0 * a1star);
        a[0] = sqrt(a1star) / sastar;

        for (j = 1; j < n2; ++j)
            a[j] = 2.0 * a[j] / sastar;
    }
    else {
        /* Use exact values for weights */

        a[0] = rsqrt2;
        if (n != 3) {
            if (n - 3 == 3)
                for (j = 0; j < 3; ++j)
                    a[j] = c6[j];
            else if (n - 3 == 2)
                for (j = 0; j < 2; ++j)
                    a[j] = c5[j];
            else
                for (j = 0; j < 2; ++j)
                    a[j] = c4[j];

      /*-
            goto (40,50,60), n3
         40 do 45 j = 1,2
         45 a(j) = c4(j)
            goto 70
         50 do 55 j = 1,2
         55 a(j) = c5(j)
            goto 70
         60 do 65 j = 1,3
         65 a(j) = c6(j)
      */
        }
    }

    /* Calculate the minimum possible value of W */
    *eps = a[0] * a[0] / (1.0 - 1.0 / (double)n);

    return;
}


/*
 * Algorithm AS 181.2   Appl. Statist.  (1982) Vol. 31, No. 2
 * 
 * Calculates the algebraic polynomial of order nored-1 with array of
 * coefficients c.  Zero order coefficient is c(1)
 */
static double poly(double c[], int nord, double x)
{
    double p;
    int n2, i, j;

    if (nord == 1)
        return c[0];

    p = x * c[nord - 1];

    if (nord != 2) {
        n2 = nord - 2;
        j = n2;

        for (i = 0; i < n2; ++i)
            p = (p + c[j--]) * x;
    }

    return c[0] + p;
}


/*
 * AS R63 Appl. Statist. (1986) Vol. 35, No.2
 * 
 * A remark on AS 181
 * 
 * Calculates Sheppard corrected version of W test.
 * 
 * Auxiliary functions required: ALNORM = algorithm AS 66, and PPND =
 * algorithm AS 111 (or PPND7 from AS 241).
 */
void wgp(double x[], int n, double ssq, double gp, double h, double a[],
         int n2, double eps, double w, double u, double p, int *ifault)
{
    double zbar, zsd, an1, hh;

    zbar = 0.0;
    zsd = 1.0;
    *ifault = 1;

    if (n < 7)
        return;

    if (gp > 0.0) {             /* No correction applied if gp=0. */
        an1 = (double)(n - 1);
        /* correct ssq and find standardized grouping interval (h) */
        ssq = ssq - an1 * gp * gp / 12.0;
        h = gp / sqrt(ssq / an1);
        *ifault = 4;

        if (h > 1.5)
            return;
    }
    wext(x, n, ssq, a, n2, eps, &w, &p, ifault);

    if (*ifault != 0)
        return;

    if (!(p > 0.0 && p < 1.0)) {
        u = 5.0 - 10.0 * p;

        return;
    }

    if (gp > 0.0) {
        /* correct u for grouping interval (n<=100 and n>100 separately) */
        hh = sqrt(h);
        if (n <= 100) {
            zbar = -h * (1.07457 + hh * (-2.8185 + hh * 1.8898));
            zsd = 1.0 + h * (0.50933 + hh * (-0.98305 + hh * 0.7408));
        }
        else {
            zbar = -h * (0.96436 + hh * (-2.1300 + hh * 1.3196));
            zsd = 1.0 + h * (0.2579 + h * 0.15225);
        }
    }

    /* ppnd is AS 111 (Beasley and Springer, 1977) */
    u = (-ppnd16(p) - zbar) / zsd;

    /* alnorm is AS 66 (Hill, 1973) */
    p = alnorm(u, 1);

    return;
}