NAG C Library Function Document nag_robust_m_regsn_param_var (g02hfc)

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1 g02 Correlaton and Regresson Analyss g02hfc 1 Purpose NAG C Lbrary Functon Document nag_robust_m_regsn_param_var (g02hfc) nag_robust_m_regsn_param_var (g02hfc) calculates an estmate of the asymptotc varance-covarance matrx for the bounded nfluence regresson estmates (M-estmates). It s ntended for use wth nag_robust_m_regsn_user_fn (g02hdc). 2 Specfcaton vod nag_robust_m_regsn_param_var (Nag_OrderType order, double (*ps)(double t, Nag_Comm *comm), double (*psp)(double t, Nag_Comm *comm), Nag_RegType regtype, Nag_CovMatrxEst covmat_est, double sgma, Integer n, Integer m, const double x[], Integer pdx, const double rs[], const double wgt[], double cov[], Integer pdc, double comm_arr[], Nag_Comm *comm, NagError *fal) 3 Descrpton For a descrpton of bounded nfluence regresson see nag_robust_m_regsn_user_fn (g02hdc). Let be the regresson parameters and let C be the asymptotc varance-covarance matrx of ^. Then for Huber type regresson where 2 ¼ 1 þ m n C ¼ f H ðx T XÞ 1 ^ 2 ; f H ¼ 1 n m P n 2 ¼1 ðr =^Þ P 1 n 0 r ^ 2 2 P 1 n 0 n ¼1 ðr =^Þ 1 P n 0 2 n ¼1 ðr =^Þ ; 2 1 n P n ¼1 0 r ^ see Huber (1981) and Marazz (1987b). For Mallows and Schweppe type regressons, C s of the form where S 1 ¼ 1 n XT DX and S 2 ¼ 1 n XT PX. 2 ^ S 1 1 S n 2 S 1 1 ; D s a dagonal matrx such that the th element approxmates Eð 0 ðr =ðw ÞÞÞ n the Schweppe case and Eð 0 ðr =Þw Þ n the Mallows case. P s a dagonal matrx such that the th element approxmates Eð 2 ðr =ðw ÞÞw 2 Þ n the Schweppe case and Eð 2 ðr =Þw 2 Þ n the Mallows case. Two approxmatons are avalable n nag_robust_m_regsn_param_var (g02hfc): g02hfc.1

2 g02hfc NAG C Lbrary Manual 1. Average over the r Schweppe Mallows D ¼ 1 P n 0 r j n j¼1 ^w w D ¼ 1 P n 0 r n j¼1 j^ P ¼ 1 P n 2 r j n j¼1 ^w w 2 P ¼ 1 P n 2 r n j¼1 j^ 2. Replace expected value by observed Schweppe Mallows D ¼ 0 r w ^w D ¼ 0 r w ^ P ¼ 2 r w 2 P ^w ¼ 2 r w 2 ^ See Hampel et al. (1986) and Marazz (1987b). In all cases ^ s a robust estmate of. nag_robust_m_regsn_param_var (g02hfc) s based on routnes n ROBETH; see Marazz (1987b). 4 References Hampel F R, Ronchett E M, Rousseeuw P J and Stahel W A (1986) Robust Statstcs. The Approach Based on Influence Functons Wley Huber P J (1981) Robust Statstcs Wley Marazz A (1987b) Subroutnes for robust and bounded nfluence regresson n ROBETH Cah. Rech. Doc. IUMSP, No. 3 ROB 2 Insttut Unverstare de Médecne Socale et Préventve, Lausanne 5 Parameters 1: order Nag_OrderType Input On entry: the order parameter specfes the two-dmensonal storage scheme beng used,.e., rowmajor orderng or column-major orderng. C language defned storage s specfed by order ¼ Nag RowMajor. See Secton of the Essental Introducton for a more detaled explanaton of the use of ths parameter. Constrant: order ¼ Nag RowMajor or Nag ColMajor. 2: ps Functon ps must return the value of the functon for a gven value of ts argument. Its specfcaton s: w w 2 double ps (double t, Nag_Comm *comm) 1: t double Input On entry: the argument for whch ps must be evaluated. 2: comm NAG_Comm * Input/Output The NAG communcaton parameter (see the Essental Introducton). 3: psp Functon psp must return the value of 0 ðtþ ¼ d dt Its specfcaton s: ðtþ for a gven value of ts argument. g02hfc.2

3 g02 Correlaton and Regresson Analyss g02hfc double psp (double t, Nag_Comm *comm) 1: t double Input On entry: the argument for whch psp must be evaluated. 2: comm NAG_Comm * Input/Output The NAG communcaton parameter (see the Essental Introducton). 4: regtype Nag_RegType Input On entry: the type of regresson for whch the asymptotc varance-covarance matrx s to be calculated. If regtype ¼ Nag HuberReg, Huber type regresson. If regtype ¼ Nag MallowsReg, Mallows type regresson. If regtype ¼ Nag SchweppeReg, Schweppe type regresson. 5: covmat_est Nag_CovMatrxEst Input On entry: fregtype 6¼ Nag HuberReg, covmat_est must specfy the approxmaton to be used. If covmat est ¼ Nag CovMatAve, averagng over resduals. If covmat est ¼ Nag CovMatObs, replacng expected by observed. If regtype ¼ Nag HuberReg, covmat_est s not referenced. 6: sgma double Input On entry: the value of ^, as gven by nag_robust_m_regsn_user_fn (g02hdc). Constrant: sgma > 0. 7: n Integer Input On entry: the number, n, of observatons. Constrant: n > 1. 8: m Integer Input On entry: the number, m, of ndependent varables. Constrant: 1 m < n. 9: x½dmš const double Input Note: the dmenson, dm, of the array x must be at least maxð1; pdx mþ when order ¼ Nag ColMajor and at least maxð1; pdx nþ when order ¼ Nag RowMajor. Where Xð; jþ appears n ths document, t refers to the array element f order ¼ Nag ColMajor, x½ðj 1Þpdx þ 1Š; f order ¼ Nag RowMajor, x½ð 1Þpdx þ j 1Š. On entry: the values of the X matrx,.e., the ndependent varables. Xð; jþ must contan the jth element of X, for ¼ 1; 2;...;n, j ¼ 1; 2;...;m. 10: pdx Integer Input On entry: the strde separatng matrx row or column elements (dependng on the value of order) n the array x. g02hfc.3

4 g02hfc NAG C Lbrary Manual Constrants: f order ¼ Nag ColMajor, pdx n; f order ¼ Nag RowMajor, pdx m. 11: rs½nš const double Input On entry: the resduals from the bounded nfluence regresson. These are gven by nag_robust_m_regsn_user_fn (g02hdc). 12: wgt½nš const double Input On entry: f regtype 6¼ Nag HuberReg, wgt must contan the vector of weghts used by the bounded nfluence regresson. These should be used wth nag_robust_m_regsn_user_fn (g02hdc). If regtype ¼ Nag HuberReg, wgt s not referenced. Constrant: fregtype 6¼ Nag HuberReg, wgt½š 0:0 for ¼ 0; 1;... 13: cov½dmš double Output Note: the dmenson, dm, of the array c must be at least pdc m. If order ¼ Nag ColMajor, the ð; jþth element of the matrx C s stored n cov½ðj 1Þpdc þ 1Š and f order ¼ Nag RowMajor, the ð; jþth element of the matrx C s stored n cov½ð 1Þpdc þ j 1Š. On ext: the estmate of the varance-covarance matrx. 14: pdc Integer Input On entry: the strde separatng matrx row or column elements (dependng on the value of order) n the array cov. Constrant: pdc m. 15: comm_arr½dmš double Output Note: the dmenson, dm, of the array comm_arr must be at least m ðnþmþ1þþ2n. On ext: f regtype 6¼ Nag HuberReg, comm arr½ 1Š, for ¼ 1; 2;...;n, wll contan the dagonal elements of the matrx D and comm arr½ 1Š, for ¼ n þ 1;nþ 2;...; 2n, wll contan the dagonal elements of matrx P. 16: comm NAG_Comm * Input/Output The NAG communcaton parameter (see the Essental Introducton). 17: fal NagError * Input/Output The NAG error parameter (see the Essental Introducton). 6 Error Indcators and Warnngs NE_INT On entry, n = hvalue. Constrant: n > 1. On entry, pdx ¼hvalue. Constrant: pdx > 0. On entry, pdc ¼hvalue. Constrant: pdc > 0. On entry, m ¼hvalue. Constrant: m 1. g02hfc.4

5 g02 Correlaton and Regresson Analyss g02hfc NE_INT_2 On entry, m = hvalue, n = hvalue. Constrant: 1 m < n. On entry, pdx ¼hvalue, n ¼hvalue. Constrant: pdx n. On entry, pdx ¼hvalue, m ¼hvalue. Constrant: pdx m. On entry, pdc ¼hvalue, m ¼hvalue. Constrant: pdc m. On entry, m ¼hvalue, pdc ¼hvalue. Constrant: pdc m. On entry, n m: n ¼hvalue, m ¼hvalue. NE_ENUM_INT On entry, regtype ¼hvalue, wgt ¼hvalue. Constrant: f regtype 6¼ Nag HuberReg, wgt½š 0:0 for ¼ 0;...;. NE_CORRECTION_FACTOR Correcton factor = 0 (Huber type regresson). NE_POS_DEF X T X matrx not postve defnte. NE_REAL On entry, sgma ¼hvalue. Constrant: sgma 0. NE_REAL_ARRAY_ELEM_CONS On entry, an element of wgt < 0. NE_SINGULAR S1 matrx s sngular or almost sngular. NE_ALLOC_FAIL Memory allocaton faled. NE_BAD_PARAM On entry, parameter hvalue had an llegal value. NE_INTERNAL_ERROR An nternal error has occurred n ths functon. Check the functon call and any array szes. If the call s correct then please consult NAG for assstance. 7 Accuracy In general, the accuracy of the varance-covarance matrx wll depend prmarly on the accuracy of the results from nag_robust_m_regsn_user_fn (g02hdc). g02hfc.5

6 g02hfc NAG C Lbrary Manual 8 Further Comments Ths routne s only for stuatons n whch X has full column rank. Care has to be taken n the choce of the functon snce f 0 ðtþ ¼0 for too wde a range then ether the value of f H wll not exst or too many values of D wll be zero and t wll not be possble to calculate C. 9 Example The asymptotc varance-covarance matrx s calculated for a Schweppe type regresson. The values of X, ^ and the resduals and weghts are read n. The averagng over resduals approxmaton s used. 9.1 Program Text /* nag_robust_m_regsn_param_var (g02hfc) Example Program. * * Copyrght 2002 Numercal Algorthms Group. * * Mark 7, */ #nclude <math.h> #nclude <stdo.h> #nclude <nag.h> #nclude <nag_stdlb.h> #nclude <nagg02.h> statc double ps(double t, Nag_Comm *comm); statc double psp(double t, Nag_Comm *comm); nt man(vod) /* Scalars */ double sgma; Integer ext_status,, c, x, j, k, m, n; Integer pdc, pdx; NagError fal; Nag_OrderType order; Nag_Comm comm; /* Arrays */ double *cov=0, *rs=0, *wgt=0, *comm_arr=0, *x=0; #fdef NAG_COLUMN_MAJOR #defne COV(I,J) cov[(j-1)*pdc + I - 1] #defne X(I,J) x[(j-1)*pdx + I - 1] order = Nag_ColMajor; #else #defne COV(I,J) cov[(i-1)*pdc + J - 1] #defne X(I,J) x[(i-1)*pdx + J - 1] order = Nag_RowMajor; #endf ext_status = 0; INIT_FAIL(fal); Vprntf("g02hfc Example Program Results\n"); /* Skp headng n data fle */ Vscanf("%*[^\n] "); /* Read n the dmensons of X */ Vscanf("%ld%ld%*[^\n] ", &n, &m); /* Allocate memory */ f (!(cov = NAG_ALLOC(m * m, double))!(rs = NAG_ALLOC(n, double))!(wgt = NAG_ALLOC(n, double))!(comm_arr = NAG_ALLOC(m*(n+m+1)+2*n, double)) g02hfc.6

7 g02 Correlaton and Regresson Analyss g02hfc!(x = NAG_ALLOC(n * m, double)) ) Vprntf("Allocaton falure\n"); ext_status = -1; goto END; #fdef NAG_COLUMN_MAJOR pdc = m; pdx = n; #else pdc = m; pdx = m; #endf Vprntf("\n"); /* Read n the X matrx */ for ( = 1; <= n; ++) for (j = 1; j <= m; ++j) Vscanf("%lf", &X(,j)); Vscanf("%*[^\n] "); /* Read n sgma */ Vscanf("%lf%*[^\n] ", &sgma); /* Read n weghts and resduals */ for ( = 1; <= n; ++) Vscanf("%lf%lf%*[^\n] ", &wgt[ - 1], &rs[ - 1]); /* Set other parameter values */ x = 5; c = 3; /* Set parameters for Schweppe type regresson */ g02hfc(order, ps, psp, Nag_SchweppeReg, Nag_CovMatAve, sgma, n, m, x, pdx, rs, wgt, cov, pdc, comm_arr, &comm, &fal); f (fal.code!= NE_NOERROR) Vprntf("Error from g02hfc.\n%s\n", fal.message); ext_status = 1; goto END; Vprntf("Covarance matrx\n"); for (j = 1; j <= m; ++j) for (k = 1; k <= m; ++k) Vprntf("%10.4f%s", COV(j,k), k%6 == 0 k == m?"\n":" "); END: f (cov) NAG_FREE(cov); f (rs) NAG_FREE(rs); f (wgt) NAG_FREE(wgt); f (comm_arr) NAG_FREE(comm_arr); f (x) NAG_FREE(x); return ext_status; statc double ps(double t, Nag_Comm *comm) g02hfc.7

8 g02hfc NAG C Lbrary Manual double ret_val; f (t <= -1.5) ret_val = -1.5; else f (fabs(t) < 1.5) ret_val = t; else ret_val = 1.5; return ret_val; statc double psp(double t, Nag_Comm *comm) double ret_val; ret_val = 0.0; f (fabs(t) < 1.5) ret_val = 1.0; return ret_val; 9.2 Program Data g02hfc Example Program Data 5 3 : N M : X1 X2 X : End of X1 X2 and X3 values : SIGMA : Weghts and resduals, WGT and RS : End of weghts and resduals 9.3 Program Results g02hfc Example Program Results Covarance matrx g02hfc.8 (last)