NAG Toolbox: nag_correg_quantile_linreg (g02qg)

Purpose

Syntax

Description

References

Parameters

Compulsory Input Parameters

1:     $\mathrm{sorder}$ – int64int32nag_int scalar

Determines the storage order of variates supplied in dat.

Constraint: $\mathbf{sorder}=1$ or $2$.

2:     $\mathrm{intcpt}$ – string (length ≥ 1)

Indicates whether an intercept will be included in the model. The intercept is included by adding a column of ones as the first column in the design matrix, $X$.

$\mathbf{intcpt}=\text{'Y'}$

An intercept will be included in the model.

$\mathbf{intcpt}=\text{'N'}$

An intercept will not be included in the model.

Constraint: $\mathbf{intcpt}=\text{'N'}$ or $\text{'Y'}$.

3:     $\mathrm{weight}$ – string (length ≥ 1)

Indicates if weights are to be used.

$\mathbf{weight}=\text{'W'}$

A weighted regression model is fitted to the data using weights supplied in array wt.

$\mathbf{weight}=\text{'U'}$

An unweighted regression model is fitted to the data and array wt is not referenced.

Constraint: $\mathbf{weight}=\text{'U'}$ or $\text{'W'}$.

4:     $\mathrm{dat}\left(\mathit{lddat},:\right)$ – double array

The first dimension, $\mathit{lddat}$, of the array dat must satisfy

• if $\mathbf{sorder}=1$, $\mathit{lddat}\ge \mathbf{n}$;
• otherwise $\mathit{lddat}\ge \mathbf{m}$.

The second dimension of the array dat must be at least $\mathbf{m}$ if $\mathbf{sorder}=1$ and at least $\mathbf{n}$ if $\mathbf{sorder}=2$.

The $\mathit{i}$th value for the $\mathit{j}$th variate, for $\mathit{i}=1,2,\dots ,\mathbf{n}$ and $\mathit{j}=1,2,\dots ,\mathbf{m}$, must be supplied in

• $\mathbf{dat}\left(i,j\right)$ if $\mathbf{sorder}=1$, and
• $\mathbf{dat}\left(j,i\right)$ if $\mathbf{sorder}=2$.

The design matrix $X$ is constructed from dat, isx and intcpt.

5:     $\mathrm{isx}\left(\mathbf{m}\right)$ – int64int32nag_int array

Indicates which independent variables are to be included in the model.

$\mathbf{isx}\left(j\right)=0$

The $j$th variate, supplied in dat, is not included in the regression model.

$\mathbf{isx}\left(j\right)=1$

The $j$th variate, supplied in dat, is included in the regression model.

Constraints:

• $\mathbf{isx}\left(\mathit{j}\right)=0$ or $1$, for $\mathit{j}=1,2,\dots ,\mathbf{m}$;
• if $\mathbf{intcpt}=\text{'Y'}$, exactly $\mathbf{ip}-1$ values of isx must be set to $1$;
• if $\mathbf{intcpt}=\text{'N'}$, exactly ip values of isx must be set to $1$.

6:     $\mathrm{y}\left(\mathbf{n}\right)$ – double array

$y$, the observations on the dependent variable.

7:     $\mathrm{tau}\left(\mathbf{ntau}\right)$ – double array

The vector of quantiles of interest. A separate model is fitted to each quantile.

Constraint: $\sqrt{\epsilon }<\mathbf{tau}\left(\mathit{j}\right)<1-\sqrt{\epsilon }$ where $\epsilon$ is the machine precision returned by nag_machine_precision (x02aj), for $\mathit{j}=1,2,\dots ,\mathbf{ntau}$.

8:     $\mathrm{b}\left(\mathbf{ip},\mathbf{ntau}\right)$ – double array

If $\mathbf{Calculate\; Initial\; Values}=\mathrm{NO}$, $\mathbf{b}\left(\mathit{i},\mathit{l}\right)$ must hold an initial estimates for ${\hat{\beta }}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,\mathbf{ip}$ and $\mathit{l}=1,2,\dots ,\mathbf{ntau}$. If $\mathbf{Calculate\; Initial\; Values}=\mathrm{YES}$, b need not be set.

9:     $\mathrm{iopts}\left(:\right)$ – int64int32nag_int array

Note: the dimension of this array is dictated by the requirements of associated functions that must have been previously called. This array must be the same array passed as argument iopts in the previous call to nag_correg_optset (g02zk).

Optional parameter array, as initialized by a call to nag_correg_optset (g02zk).

10:   $\mathrm{opts}\left(:\right)$ – double array

Note: the dimension of this array is dictated by the requirements of associated functions that must have been previously called. This array must be the same array passed as argument opts in the previous call to nag_correg_optset (g02zk).

Optional parameter array, as initialized by a call to nag_correg_optset (g02zk).

11:   $\mathrm{state}\left(:\right)$ – int64int32nag_int array

Note: the actual argument supplied must be the array state supplied to the initialization routines nag_rand_init_repeat (g05kf) or nag_rand_init_nonrepeat (g05kg).

If $\mathbf{Interval\; Method}=\mathrm{BOOTSTRAP\; XY}$, state contains information about the selected random number generator. Otherwise state is not referenced.

Optional Input Parameters

1:     $\mathrm{n}$ – int64int32nag_int scalar

Default: the dimension of the array y and the first dimension of the array dat. (An error is raised if these dimensions are not equal.)

The total number of observations in the dataset. If no weights are supplied, or no zero weights are supplied or observations with zero weights are included in the model then $\mathbf{n}=n$. Otherwise $\mathbf{n}=n+$ the number of observations with zero weights.

Constraint: $\mathbf{n}\ge 2$.

2:     $\mathrm{m}$ – int64int32nag_int scalar

Default: the dimension of the array isx and the first dimension of the array dat. (An error is raised if these dimensions are not equal.)

$m$, the total number of variates in the dataset.

Constraint: $\mathbf{m}\ge 0$.

3:     $\mathrm{ip}$ – int64int32nag_int scalar

Default: the first dimension of the array b.

$p$, the number of independent variables in the model, including the intercept, see intcpt, if present.

Constraints:

• $1\le \mathbf{ip}<\mathbf{n}$;
• if $\mathbf{intcpt}=\text{'Y'}$, $1\le \mathbf{ip}\le \mathbf{m}+1$;
• if $\mathbf{intcpt}=\text{'N'}$, $1\le \mathbf{ip}\le \mathbf{m}$.

4:     $\mathrm{wt}\left(:\right)$ – double array

The dimension of the array wt must be at least $\mathbf{n}$ if $\mathbf{weight}=\text{'W'}$

If $\mathbf{weight}=\text{'W'}$, wt must contain the diagonal elements of the weight matrix $W$. Otherwise wt is not referenced.

When

$\mathbf{Drop\; Zero\; Weights}=\mathrm{YES}$

If $\mathbf{wt}\left(i\right)=0.0$, the $i$th observation is not included in the model, in which case the effective number of observations, $n$, is the number of observations with nonzero weights. If $\mathbf{Return\; Residuals}=\mathrm{YES}$, the values of res will be set to zero for observations with zero weights.

$\mathbf{Drop\; Zero\; Weights}=\mathrm{NO}$

All observations are included in the model and the effective number of observations is n, i.e., $n=\mathbf{n}$.

Constraints:

• If $\mathbf{weight}=\text{'W'}$, $\mathbf{wt}\left(\mathit{i}\right)\ge 0.0$, for $\mathit{i}=1,2,\dots ,\mathbf{n}$;
• The effective number of observations $\ge 2$.

5:     $\mathrm{ntau}$ – int64int32nag_int scalar

Default: the dimension of the array tau and the second dimension of the array b. (An error is raised if these dimensions are not equal.)

The number of quantiles of interest.

Constraint: $\mathbf{ntau}\ge 1$.

Output Parameters

1:     $\mathrm{df}$ – double scalar

The degrees of freedom given by $n-k$, where $n$ is the effective number of observations and $k$ is the rank of the cross-product matrix $X^{\mathrm{T}}X$.

2:     $\mathrm{b}\left(\mathbf{ip},\mathbf{ntau}\right)$ – double array

$\mathbf{b}\left(\mathit{i},\mathit{l}\right)$, for $\mathit{i}=1,2,\dots ,\mathbf{ip}$, contains the estimates of the parameters of the regression model, $\hat{\beta }$, estimated for $\tau =\mathbf{tau}\left(\mathit{l}\right)$.

If $\mathbf{intcpt}=\text{'Y'}$, $\mathbf{b}\left(1,l\right)$ will contain the estimate corresponding to the intercept and $\mathbf{b}\left(i+1,l\right)$ will contain the coefficient of the $j$th variate contained in dat, where $\mathbf{isx}\left(j\right)$ is the $i$th nonzero value in the array isx.

If $\mathbf{intcpt}=\text{'N'}$, $\mathbf{b}\left(i,l\right)$ will contain the coefficient of the $j$th variate contained in dat, where $\mathbf{isx}\left(j\right)$ is the $i$th nonzero value in the array isx.

3:     $\mathrm{bl}\left(\mathbf{ip},:\right)$ – double array

The second dimension of the array bl will be $\mathbf{ntau}$ if $\mathbf{Interval\; Method}\ne \mathrm{NONE}$.

If $\mathbf{Interval\; Method}\ne \mathrm{NONE}$, $\mathbf{bl}\left(i,l\right)$ contains the lower limit of an $\left(100\times \alpha \right)\%$ confidence interval for $\mathbf{b}\left(\mathit{i},\mathit{l}\right)$, for $\mathit{i}=1,2,\dots ,\mathbf{ip}$ and $\mathit{l}=1,2,\dots ,\mathbf{ntau}$.

If $\mathbf{Interval\; Method}=\mathrm{NONE}$, bl is not referenced.

The method used for calculating the interval is controlled by the optional parameters Interval Method and Bootstrap Interval Method. The size of the interval, $\alpha$, is controlled by the optional parameter Significance Level.

4:     $\mathrm{bu}\left(\mathbf{ip},:\right)$ – double array

The second dimension of the array bu will be $\mathbf{ntau}$ if $\mathbf{Interval\; Method}\ne \mathrm{NONE}$.

If $\mathbf{Interval\; Method}\ne \mathrm{NONE}$, $\mathbf{bu}\left(i,l\right)$ contains the upper limit of an $\left(100\times \alpha \right)\%$ confidence interval for $\mathbf{b}\left(\mathit{i},\mathit{l}\right)$, for $\mathit{i}=1,2,\dots ,\mathbf{ip}$ and $\mathit{l}=1,2,\dots ,\mathbf{ntau}$.

If $\mathbf{Interval\; Method}=\mathrm{NONE}$, bu is not referenced.

The method used for calculating the interval is controlled by the optional parameters Interval Method and Bootstrap Interval Method. The size of the interval, $\alpha$ is controlled by the optional parameter Significance Level.

5:     $\mathrm{ch}\left(\mathbf{ip},\mathbf{ip},:\right)$ – double array

The last dimension of the array ch will be $\mathbf{ntau}$ if $\mathbf{Interval\; Method}\ne \mathrm{NONE}$ and $\mathbf{Matrix\; Returned}=\mathrm{COVARIANCE}$ and at least $\mathbf{ntau}+1$ if $\mathbf{Interval\; Method}\ne \mathrm{NONE}$, $\mathrm{IID}$ or $\mathrm{BOOTSTRAP\; XY}$ and $\mathbf{Matrix\; Returned}=\mathrm{H\; INVERSE}$

Depending on the supplied optional parameters, ch will either not be referenced, hold an estimate of the upper triangular part of the covariance matrix, $\Sigma$, or an estimate of the upper triangular parts of $n{J}_n$ and $n^{-1}{H}_n^{-1}$.

If $\mathbf{Interval\; Method}=\mathrm{NONE}$ or $\mathbf{Matrix\; Returned}=\mathrm{NONE}$, ch is not referenced.

If $\mathbf{Interval\; Method}=\mathrm{BOOTSTRAP\; XY}$ or $\mathrm{IID}$ and $\mathbf{Matrix\; Returned}=\mathrm{H\; INVERSE}$, ch is not referenced.

Otherwise, for $i,j=1,2,\dots ,\mathbf{ip},j\ge i$ and $l=1,2,\dots ,\mathbf{ntau}$:

• If $\mathbf{Matrix\; Returned}=\mathrm{COVARIANCE}$, $\mathbf{ch}\left(i,j,l\right)$ holds an estimate of the covariance between $\mathbf{b}\left(i,l\right)$ and $\mathbf{b}\left(j,l\right)$.
• If $\mathbf{Matrix\; Returned}=\mathrm{H\; INVERSE}$, $\mathbf{ch}\left(i,j,1\right)$ holds an estimate of the $\left(i,j\right)$th element of $n{J}_n$ and $\mathbf{ch}\left(i,j,l+1\right)$ holds an estimate of the $\left(i,j\right)$th element of $n^{-1}{H}_n^{-1}$, for $\tau =\mathbf{tau}\left(l\right)$.

The method used for calculating $\Sigma$ and $H_n^{-1}$ is controlled by the optional parameter Interval Method.

6:     $\mathrm{res}\left(\mathbf{n},:\right)$ – double array

The second dimension of the array res will be $\mathbf{ntau}$ if $\mathbf{Return\; Residuals}=\mathrm{YES}$.

If $\mathbf{Return\; Residuals}=\mathrm{YES}$, $\mathbf{res}\left(\mathit{i},\mathit{l}\right)$ holds the (weighted) residuals, $r_{\mathit{i}}$, for $\tau =\mathbf{tau}\left(\mathit{l}\right)$, for $\mathit{i}=1,2,\dots ,\mathbf{n}$ and $\mathit{l}=1,2,\dots ,\mathbf{ntau}$.

If $\mathbf{weight}=\text{'W'}$ and $\mathbf{Drop\; Zero\; Weights}=\mathrm{YES}$, the value of res will be set to zero for observations with zero weights.

If $\mathbf{Return\; Residuals}=\mathrm{NO}$, res is not referenced.

7:     $\mathrm{state}\left(:\right)$ – int64int32nag_int array

8:     $\mathrm{info}\left(\mathbf{ntau}\right)$ – int64int32nag_int array

$\mathbf{info}\left(i\right)$ holds additional information concerning the model fitting and confidence limit calculations when $\tau =\mathbf{tau}\left(i\right)$.

Code	Warning
$0$	Model fitted and confidence limits (if requested) calculated successfully
$1$	The function did not converge. The returned values are based on the estimate at the last iteration. Try increasing Iteration Limit whilst calculating the parameter estimates or relaxing the definition of convergence by increasing Tolerance.
$2$	A singular matrix was encountered during the optimization. The model was not fitted for this value of $\tau$.
$4$	Some truncation occurred whilst calculating the confidence limits for this value of $\tau$. See Algorithmic Details for details. The returned upper and lower limits may be narrower than specified.
$8$	The function did not converge whilst calculating the confidence limits. The returned limits are based on the estimate at the last iteration. Try increasing Iteration Limit.
$16$	Confidence limits for this value of $\tau$ could not be calculated. The returned upper and lower limits are set to a large positive and large negative value respectively as defined by the optional parameter Big.

It is possible for multiple warnings to be applicable to a single model. In these cases the value returned in info is the sum of the corresponding individual nonzero warning codes.

9:     $\mathrm{ifail}$ – int64int32nag_int scalar

$\mathbf{ifail}=\mathbf{0}$ unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

   $\mathbf{ifail}=11$

Constraint: $\mathbf{sorder}=1$ or $2$.

   $\mathbf{ifail}=21$

On entry, $\mathbf{intcpt}=\_$ was an illegal value.

   $\mathbf{ifail}=31$

On entry, weight had an illegal value.

   $\mathbf{ifail}=41$

Constraint: $\mathbf{n}\ge 2$.

   $\mathbf{ifail}=51$

Constraint: $\mathbf{m}\ge 0$.

   $\mathbf{ifail}=71$

Constraint: $\mathit{lddat}\ge \mathbf{n}$.

   $\mathbf{ifail}=72$

Constraint: $\mathit{lddat}\ge \mathbf{m}$.

   $\mathbf{ifail}=81$

Constraint: $\mathbf{isx}\left(i\right)=0$ or $1$ for all $i$.

   $\mathbf{ifail}=91$

Constraint: $1\le \mathbf{ip}<\mathbf{n}$.

   $\mathbf{ifail}=92$

On entry, ip is not consistent with isx or intcpt.

   $\mathbf{ifail}=111$

Constraint: $\mathbf{wt}\left(i\right)\ge 0.0$ for all $i$.

   $\mathbf{ifail}=112$

Constraint: $\text{effective number of observations}\ge \_$.

   $\mathbf{ifail}=121$

Constraint: $\mathbf{ntau}\ge 1$.

   $\mathbf{ifail}=131$

On entry is invalid.

   $\mathbf{ifail}=201$

On entry, either the option arrays have not been initialized or they have been corrupted.

   $\mathbf{ifail}=221$

On entry, state vector has been corrupted or not initialized.

   $\mathbf{ifail}=231$

A potential problem occurred whilst fitting the model(s).
Additional information has been returned in info.

   $\mathbf{ifail}=-99$

An unexpected error has been triggered by this routine. Please contact NAG.

   $\mathbf{ifail}=-399$

Your licence key may have expired or may not have been installed correctly.

   $\mathbf{ifail}=-999$

Dynamic memory allocation failed.

Accuracy

Further Comments

Example

Open in the MATLAB editor: g02qg_example

function g02qg_example


fprintf('g02qg example results\n\n');

sorder = int64(1);
c1 = 'y';
weight = 'u';
dat = [ 420.1577;  541.4117;  901.1575;  639.0802;  750.8756;  945.7989;
        829.3979;  979.1648; 1309.8789; 1492.3987;  502.8390;  616.7168;
        790.9225;  555.8786;  713.4412;  838.7561;  535.0766;  596.4408;
        924.5619;  487.7583;  692.6397;  997.8770;  506.9995;  654.1587;
        933.9193;  433.6813;  587.5962;  896.4746;  454.4782;  584.9989;
        800.7990;  502.4369;  713.5197;  906.0006;  880.5969;  796.8289;
        854.8791; 1167.3716;  523.8000;  670.7792;  377.0584;  851.5430;
       1121.0937;  625.5179;  805.5377;  558.5812;  884.4005; 1257.4989;
       2051.1789; 1466.3330;  730.0989; 2432.3910;  940.9218; 1177.8547;
       1222.5939; 1519.5811;  687.6638;  953.1192;  953.1192;  953.1192;
        939.0418; 1283.4025; 1511.5789; 1342.5821;  511.7980;  689.7988;
       1532.3074; 1056.0808;  387.3195;  387.3195;  410.9987;  499.7510;
        832.7554;  614.9986;  887.4658; 1595.1611; 1807.9520;  541.2006;
       1057.6767;  800.7990; 1245.6964; 1201.0002;  634.4002;  956.2315;
       1148.6010; 1768.8236; 2822.5330;  922.3548; 2293.1920;  627.4726;
        889.9809; 1162.2000; 1197.0794;  530.7972; 1142.1526; 1088.0039;
        484.6612; 1536.0201;  678.8974;  671.8802;  690.4683;  860.6948;
        873.3095;  894.4598; 1148.6470;  926.8762;  839.0414;  829.4974;
       1264.0043; 1937.9771;  698.8317;  920.4199; 1897.5711;  891.6824;
        889.6784; 1221.4818;  544.5991; 1031.4491; 1462.9497;  830.4353;
        975.0415; 1337.9983;  867.6427;  725.7459;  989.0056; 1525.0005;
        672.1960;  923.3977;  472.3215;  590.7601;  831.7983; 1139.4945;
        507.5169;  576.1972;  696.5991;  650.8180;  949.5802;  497.1193;
        570.1674;  724.7306;  408.3399;  638.6713; 1225.7890;  715.3701;
        800.4708;  975.5974; 1613.7565;  608.5019;  958.6634;  835.9426;
       1024.8177; 1006.4353;  726.0000;  494.4174;  776.5958;  415.4407;
        581.3599;  643.3571; 2551.6615; 1795.3226; 1165.7734;  815.6212;
       1264.2066; 1095.4056;  447.4479; 1178.9742;  975.8023; 1017.8522;
        423.8798;  558.7767;  943.2487; 1348.3002; 2340.6174;  587.1792;
       1540.9741; 1115.8481; 1044.6843; 1389.7929; 2497.7860; 1585.3809;
       1862.0438; 2008.8546;  697.3099;  571.2517;  598.3465;  461.0977;
        977.1107;  883.9849;  718.3594;  543.8971; 1587.3480; 4957.8130;
        969.6838;  419.9980;  561.9990;  689.5988; 1398.5203;  820.8168;
        875.1716; 1392.4499; 1256.3174; 1362.8590; 1999.2552; 1209.4730;
       1125.0356; 1827.4010; 1014.1540;  880.3944;  873.7375;  951.4432;
        473.0022;  601.0030;  713.9979;  829.2984;  959.7953; 1212.9613;
        958.8743; 1129.4431; 1943.0419;  539.6388;  463.5990;  562.6400;
        736.7584; 1415.4461; 2208.7897;  636.0009;  759.4010; 1078.8382;
        748.6413;  987.6417;  788.0961; 1020.0225; 1230.9235;  440.5174;
        743.0772];
y   = [ 255.8394;  310.9587;  485.6800;  402.9974;  495.5608;  633.7978;
        630.7566;  700.4409;  830.9586;  815.3602;  338.0014;  412.3613;
        520.0006;  452.4015;  512.7201;  658.8395;  392.5995;  443.5586;
        640.1164;  333.8394;  466.9583;  543.3969;  317.7198;  424.3209;
        518.9617;  338.0014;  419.6412;  476.3200;  386.3602;  423.2783;
        503.3572;  354.6389;  497.3182;  588.5195;  654.5971;  550.7274;
        528.3770;  640.4813;  401.3204;  435.9990;  276.5606;  588.3488;
        664.1978;  444.8602;  462.8995;  377.7792;  553.1504;  810.8962;
       1067.9541; 1049.8788;  522.7012; 1424.8047;  517.9196;  830.9586;
        925.5795; 1162.0024;  383.4580;  621.1173;  621.1173;  621.1173;
        548.6002;  745.2353;  837.8005;  795.3402;  418.5976;  508.7974;
        883.2780;  742.5276;  242.3202;  242.3202;  266.0010;  408.4992;
        614.7588;  385.3184;  515.6200; 1138.1620;  993.9630;  299.1993;
        750.3202;  572.0807;  907.3969;  811.5776;  427.7975;  649.9985;
        860.6002; 1143.4211; 2032.6792;  590.6183; 1570.3911;  483.4800;
        600.4804;  696.2021;  774.7962;  390.5984;  612.5619;  708.7622;
        296.9192; 1071.4627;  496.5976;  503.3974;  357.6411;  430.3376;
        624.6990;  582.5413;  580.2215;  543.8807;  588.6372;  627.9999;
        712.1012;  968.3949;  482.5816;  593.1694; 1033.5658;  693.6795;
        693.6795;  761.2791;  361.3981;  628.4522;  771.4486;  757.1187;
        821.5970; 1022.3202;  679.4407;  538.7491;  679.9981;  977.0033;
        561.2015;  728.3997;  372.3186;  361.5210;  620.8006;  819.9964;
        360.8780;  395.7608;  442.0001;  404.0384;  670.7993;  297.5702;
        353.4882;  383.9376;  284.8008;  431.1000;  801.3518;  448.4513;
        577.9111;  570.5210;  865.3205;  444.5578;  680.4198;  576.2779;
        708.4787;  734.2356;  433.0010;  327.4188;  485.5198;  305.4390;
        468.0008;  459.8177;  863.9199;  831.4407;  534.7610;  392.0502;
        934.9752;  813.3081;  263.7100;  769.0838;  630.5863;  645.9874;
        319.5584;  348.4518;  614.5068;  662.0096; 1504.3708;  406.2180;
        692.1689;  588.1371;  511.2609;  700.5600; 1301.1451;  879.0660;
        912.8851; 1509.7812;  484.0605;  399.6703;  444.1001;  248.8101;
        527.8014;  500.6313;  436.8107;  374.7990;  726.3921; 1827.2000;
        523.4911;  334.9998;  473.2009;  581.2029;  929.7540;  591.1974;
        637.5483;  674.9509;  776.7589;  959.5170; 1250.9643;  737.8201;
        810.6772;  983.0009;  708.8968;  633.1200;  631.7982;  608.6419;
        300.9999;  377.9984;  397.0015;  588.5195;  681.7616;  807.3603;
        696.8011;  811.1962; 1305.7201;  442.0001;  353.6013;  468.0008;
        526.7573;  890.2390; 1318.8033;  331.0005;  416.4015;  596.8406;
        429.0399;  619.6408;  400.7990;  775.0209;  772.7611;  306.5191;
        522.6019];

isx = [int64(1)];
tau = [0.10; 0.25; 0.50; 0.75; 0.90];
state = zeros(1, 1, 'int64');
ip = 2;
b = zeros(2, 5);
iopts = zeros(100, 1, 'int64');
opts = zeros(100, 1);

% Initialize the optional argument array
[iopts, opts, ifail] = g02zk( ...
                              'Initialize = g02qg', iopts, opts);

% Set optional arguments
[iopts, opts, ifail] = g02zk( ...
                              'Return Residuals = Yes', iopts, opts);
[iopts, opts, ifail] = g02zk( ...
                              'Matrix Returned = Covariance', iopts, opts);
[iopts, opts, ifail] = g02zk( ...
                              'Interval Method = IID', iopts, opts);

% Call the model fitting routine
[df, b, bl, bu, ch, res, state, info, ifail] = ...
  g02qg( ...
         sorder, c1, weight, dat, isx, y, tau, b, iopts, opts, state);

% Display the parameter estimates
% plot setup
t = '\tau';
fig1 = figure;
hold on;
plot(dat,y,'+r');
tt{1} = 'data';
% loop over tau
for l=1:numel(tau)
  fprintf('\nQuantile: %6.3f\n\n', tau(l));
  fprintf('        Lower   Parameter   Upper\n');
  fprintf('        Limit   Estimate    Limit\n');
  for j=1:2
    fprintf('%3d   %7.3f   %7.3f   %7.3f\n', j, bl(j,l), b(j,l), bu(j,l));
  end
  fprintf('\nCovariance matrix\n');
  for i=1:ip
    fprintf('%10.3e ', ch(1:i, i, l));
    fprintf('\n');
  end
  fprintf('\n');
  plot([0 (2000-b(1,l))/b(2,l)],[b(1,l) 2000]);
  tt{l+1} = sprintf('%s = %4.2f',t,tau(l));
end
% plot labels
legend(tt,'Location','SouthEast')
xlabel('Household income');
ylabel('Household food expenditure');
title({'Quantile Regression', ...
       ' Study of Household Expenditure on Food', ...
       'Engels 1857'});
axis([0 5000 0 2000]);
hold off;
if (numel(res) > 0)
  fprintf('First 10 Residuals\n');
  fprintf('                              Quantile\n');
  fprintf('Obs.   %6.3f     %6.3f     %6.3f     %6.3f     %6.3f\n', tau);
  for i=1:10
    fprintf(' %3d %10.5f %10.5f %10.5f %10.5f %10.5f\n', i, res(i, 1:5));
  end
else
  fprintf('Residuals not returned\n');
end

g02qg example results


Quantile:  0.100

        Lower   Parameter   Upper
        Limit   Estimate    Limit
  1    74.946   110.142   145.337
  2     0.370     0.402     0.433

Covariance matrix
 3.191e+02 
-2.541e-01  2.587e-04 


Quantile:  0.250

        Lower   Parameter   Upper
        Limit   Estimate    Limit
  1    64.232    95.483   126.735
  2     0.446     0.474     0.502

Covariance matrix
 2.516e+02 
-2.004e-01  2.039e-04 


Quantile:  0.500

        Lower   Parameter   Upper
        Limit   Estimate    Limit
  1    55.399    81.482   107.566
  2     0.537     0.560     0.584

Covariance matrix
 1.753e+02 
-1.396e-01  1.421e-04 


Quantile:  0.750

        Lower   Parameter   Upper
        Limit   Estimate    Limit
  1    41.372    62.396    83.421
  2     0.625     0.644     0.663

Covariance matrix
 1.139e+02 
-9.068e-02  9.230e-05 


Quantile:  0.900

        Lower   Parameter   Upper
        Limit   Estimate    Limit
  1    26.829    67.351   107.873
  2     0.650     0.686     0.723

Covariance matrix
 4.230e+02 
-3.369e-01  3.429e-04 

First 10 Residuals
                              Quantile
Obs.    0.100      0.250      0.500      0.750      0.900
   1  -23.10718  -38.84219  -61.00711  -77.14462  -99.86551
   2  -16.70358  -41.20981  -73.81193 -100.11463 -127.96277
   3   13.48419  -37.04518 -100.61322 -157.07478 -200.13481
   4   36.09526    4.52393  -36.48522  -70.97584 -102.95390
   5   83.74310   44.08476   -6.54743  -50.41028  -87.11562
   6  143.66660   89.90799   22.49734  -37.70668  -82.65437
   7  187.39134  142.05288   84.66171   34.21603   -5.80963
   8  196.90443  140.73220   70.44951    7.44831  -38.91027
   9  194.55254  114.45726   15.70761  -75.01861 -135.36147
  10  105.62394   12.32563 -102.13482 -208.16238 -276.22311

Algorithmic Details

Interior Point Algorithm

Central path

Affine scaling step

Nonlinearity Adjustment

Update and convergence

Additional information

Calculation of Covariance Matrix

Optional Parameters

• Band Width Alpha
• Band Width Method
• Big
• Bootstrap Interval Method
• Bootstrap Iterations
• Bootstrap Monitoring
• Calculate Initial Values
• Defaults
• Drop Zero Weights
• Epsilon
• Interval Method
• Iteration Limit
• Matrix Returned
• Monitoring
• QR Tolerance
• Return Residuals
• Sigma
• Significance Level
• Tolerance
• Unit Number

Description of the Optional Parameters

• the keywords, where the minimum abbreviation of each keyword is underlined (if no characters of an optional qualifier are underlined, the qualifier may be omitted);
• a parameter value, where the letters $a$, $i$ and $r$ denote options that take character, integer and real values respectively;
• the default value, where the symbol $\epsilon$ is a generic notation for machine precision (see nag_machine_precision (x02aj)).

Description of Monitoring Information