Last time when I read the paper “A General Regression Neural Network” by Donald Specht, it was exactly 10 years ago when I was in the graduate school. After reading again this week, I decided to code it out with SAS macros and make this excellent idea available for the SAS community.

The prototype of GRNN consists of 2 SAS macros, %grnn_learn() for the training of a GRNN and %grnn_pred() for the prediction with a GRNN. The famous Boston Housing dataset is used to test these two macros with the result compared with the outcome from the R implementation below. In this exercise, it is assumed that the smoothing parameter SIGMA is known and equal to 0.55 in order to simplify the case.

pkgs <- c('MASS', 'doParallel', 'foreach', 'grnn')
lapply(pkgs, require, character.only = T)
registerDoParallel(cores = 8)

data(Boston)
X <- Boston[-14]
st.X <- scale(X)
Y <- Boston[14]
boston <- data.frame(st.X, Y)

pred_grnn <- function(x, nn){
  xlst <- split(x, 1:nrow(x))
  pred <- foreach(i = xlst, .combine = rbind) %dopar% {
    data.frame(pred = guess(nn, as.matrix(i)), i, row.names = NULL)
  }
}

grnn <- smooth(learn(boston, variable.column = ncol(boston)), sigma = 0.55)
pred_grnn <- pred_grnn(boston[, -ncol(boston)], grnn)
head(pred_grnn$pred, n = 10)
# [1] 24.61559 23.22232 32.29610 32.57700 33.29552 26.73482 21.46017 20.96827
# [9] 16.55537 20.25247

The first SAS macro to train a GRNN is %grnn_learn() shown below. The purpose of this macro is store the whole specification of a GRNN in a SAS dataset after the simple 1-pass training with the development data. Please note that motivated by the idea of MongoDB, I use the key-value paired scheme to store the information of a GRNN.

libname data '';

data data.boston;
  infile 'housing.data';
  input x1 - x13 y;
run;

%macro grnn_learn(data = , x = , y = , sigma = , nn_out = );
options mprint mlogic nocenter;
********************************************************;
* THIS MACRO IS TO TRAIN A GENERAL REGRESSION NEURAL   *;
* NETWORK (SPECHT, 1991) AND STORE THE SPECIFICATION   *;
*------------------------------------------------------*;
* INPUT PARAMETERS:                                    *;
*  DATA  : INPUT SAS DATASET                           *;
*  X     : A LIST OF PREDICTORS IN THE NUMERIC FORMAT  *;
*  Y     : A RESPONSE VARIABLE IN THE NUMERIC FORMAT   *;
*  SIGMA : THE SMOOTH PARAMETER FOR GRNN               *;
*  NN_OUT: OUTPUT SAS DATASET CONTAINING THE GRNN      *;
*          SPECIFICATION                               *;
*------------------------------------------------------*;
* AUTHOR:                                              *;
*  [email protected]                                   *;
********************************************************;

data _tmp1;
  set &data (keep = &x &y);
  where &y ~= .;
  array _x_ &x;
  _miss_ = 0;
  do _i_ = 1 to dim(_x_);
    if _x_[_i_] = . then _miss_ = 1; 
  end;
  if _miss_ = 0 then output;
run;

proc summary data = _tmp1;
  output out = _avg_ (drop = _type_ _freq_)
  mean(&x) = ;
run;

proc summary data = _tmp1;
  output out = _std_ (drop = _type_ _freq_)
  std(&x) = ;
run;

proc standard data = _tmp1 mean = 0 std = 1 out = _data_;
  var &x;
run;

data &nn_out (keep = _neuron_ _key_ _value_);
  set _last_ end = eof;
  _neuron_ + 1;
  length _key_ $32;
  array _a_ &y &x;
  do _i_ = 1 to dim(_a_);
    if _i_ = 1 then _key_ = '_Y_';
    else _key_ = upcase(vname(_a_[_i_]));
    _value_ = _a_[_i_];
    output;
  end; 
  if eof then do;
    _neuron_ = 0;
    _key_  = "_SIGMA_";
    _value_  = σ
    output;
    set _avg_;
    array _b_ &x;
    do _i_ = 1 to dim(_b_);
      _neuron_ = -1;
      _key_ = upcase(vname(_b_[_i_]));
      _value_ = _b_[_i_];
      output;
    end;
    set _std_;
    array _c_ &x;
    do _i_ = 1 to dim(_c_);
      _neuron_ = -2;
      _key_ = upcase(vname(_c_[_i_]));
      _value_ = _c_[_i_];
      output;
    end;
  end;
run;

proc datasets library = work;
  delete _: / memtype = data;
run;
quit;

********************************************************;
*              END OF THE MACRO                        *;
********************************************************;
%mend grnn_learn;

%grnn_learn(data = data.boston, x = x1 - x13, y = y, sigma = 0.55, nn_out = data.grnn);

proc print data = data.grnn (obs = 10) noobs;
run;
/* SAS PRINTOUT OF GRNN DATA:
_neuron_    _key_     _value_
    1        _Y_      24.0000
    1        X1       -0.4194
    1        X2        0.2845
    1        X3       -1.2866
    1        X4       -0.2723
    1        X5       -0.1441
    1        X6        0.4133
    1        X7       -0.1199
    1        X8        0.1401
    1        X9       -0.9819
*/

After the training of a GRNN, the macro %grnn_pred() would be used to generate predicted values from a test dataset with all predictors. As shown in the print-out, first 10 predicted values are identical to those generated with R.

libname data '';

%macro grnn_pred(data = , x = , id = NA, nn_in = , out = grnn_pred);
options mprint mlogic nocenter;
********************************************************;
* THIS MACRO IS TO GENERATE PREDICTED VALUES BASED ON  *;
* THE SPECIFICATION OF GRNN CREATED BY THE %GRNN_LEARN *;
* MACRO                                                *;
*------------------------------------------------------*;
* INPUT PARAMETERS:                                    *;
*  DATA : INPUT SAS DATASET                            *;
*  X    : A LIST OF PREDICTORS IN THE NUMERIC FORMAT   *;
*  ID   : AN ID VARIABLE (OPTIONAL)                    *;
*  NN_IN: INPUT SAS DATASET CONTAINING THE GRNN        *;
*         SPECIFICATION GENERATED FROM %GRNN_LEARN     *;
*  OUT  : OUTPUT SAS DATASET WITH GRNN PREDICTIONS     *;
*------------------------------------------------------*;
* AUTHOR:                                              *;
*  [email protected]                                   *;
********************************************************;

data _data_;
  set &data;
  array _x_ &x;
  _miss_ = 0;
  do _i_ = 1 to dim(_x_);
    if _x_[_i_] = . then _miss_ = 1;
  end;
  if _miss_ = 0 then output;
run;

data _data_;
  set _last_ (drop = _miss_);
  %if &id = NA %then %do;
  _id_ + 1;
  %end;
  %else %do;
  _id_ = &id;
  %end;
run;

proc sort data = _last_ sortsize = max nodupkey;
  by _id_;
run;

data _data_ (keep = _id_ _key_ _value_);
  set _last_;
  array _x_ &x;
  length _key_ $32;
  do _i_ = 1 to dim(_x_);
    _key_ = upcase(vname(_x_[_i_]));
    _value_ = _x_[_i_];
    output;
  end;
run;

proc sql noprint;
select _value_ ** 2 into :s2 from &nn_in where _neuron_ = 0;

create table
  _last_ as 
select
  a._id_,
  a._key_,
  (a._value_ - b._value_) / c._value_ as _value_
from
  _last_ as a,
  &nn_in as b,
  &nn_in as c
where
  compress(a._key_, ' ') = compress(b._key_, ' ') and
  compress(a._key_, ' ') = compress(c._key_, ' ') and
  b._neuron_ = -1                                 and
  c._neuron_ = -2;

create table
  _last_ as
select
  a._id_,
  b._neuron_,
  sum((a._value_ - b._value_) ** 2) as d2,
  mean(c._value_)                   as y,
  exp(-(calculated d2) / (2 * &s2)) as exp
from
  _last_  as a,
  &nn_in as b,
  &nn_in as c
where
  compress(a._key_, ' ') = compress(b._key_, ' ') and
  b._neuron_ = c._neuron_                         and
  b._neuron_ > 0                                  and
  c._key_ = '_Y_'
group by
  a._id_, b._neuron_;

create table
  _last_ as
select
  a._id_,
  sum(a.y * a.exp / b.sum_exp) as _pred_
from
  _last_ as a inner join (select _id_, sum(exp) as sum_exp from _last_ group by _id_) as b
on
  a._id_ = b._id_
group by
  a._id_;
quit;

proc sort data = _last_ out = &out sortsize = max;
  by _id_;
run;

********************************************************;
*              END OF THE MACRO                        *;
********************************************************;
%mend grnn_pred;

%grnn_pred(data = data.boston, x = x1 - x13, nn_in = data.grnn);

proc print data = grnn_pred (obs = 10) noobs;
run;
/* SAS PRINTOUT:
_id_     _pred_
  1     24.6156
  2     23.2223
  3     32.2961
  4     32.5770
  5     33.2955
  6     26.7348
  7     21.4602
  8     20.9683
  9     16.5554
 10     20.2525
*/

After the development of these two macros, I also compare predictive performances between GRNN and OLS regression. It turns out that GRNN consistently outperforms OLS regression even with a wide range of SIGMA values. With a reasonable choice of SIGMA value, even a GRNN developed with 10% of the whole Boston Housing dataset is able to generalize well and yield a R^2 > 0.8 based upon the rest 90% data.