1% Simple illustration of constructing tree-based models within Aleph
    2% To run do the following:
    3%       a. Load Aleph
    4%       b. read_all(wedge).
    5%       c. induce_tree.
    6
    7% Model trees are constructed by specifying a predicate that
    8% will be used for model construction for examples in a leaf.
    9% The user has to provide a definition for this predicate that
   10% is able to: (a) construct the model; and (b) predict using
   11% the model constructed. The trick used is the same as that
   12% for lazy evaluation.
   13
   14% Learning a model tree
   15% The function to be learnt is:
   16%             y = f(x) =  x + 1 (x =< 0)
   17%                      = -x + 1 ( x > 0)
   18% That is:
   19%
   20%                 |
   21%                /|\
   22%               / | \
   23%              /  |  \
   24%      --------------------------
   25%                 0
   26
   27% what Aleph actually learns with the data given is:
   28%             y = f(x) =  x + 1 (x =< -0.5)
   29%                      = -x + 1 ( x > -0.5)
   30% adding more examples rectifies this: see wedge.f

   35:-use_module(library(aleph)).   36:- if(current_predicate(use_rendering/1)).   37:- use_rendering(prolog).   38:- endif.   39:- aleph.   40
   41%%%%%%%%%%%%%%%%%%%%%%%%%%%%
   42% specify tree type
   43
   44:- aleph_set(tree_type,model).   45:- aleph_set(evalfn,mse).   46:- aleph_set(minpos,2).       % minimum examples in leaf for splitting
   47:- aleph_set(mingain,0.01).	% toy example needs this to be low
   48:- aleph_set(dependent,2).	% second argument of f/2 is to predicted
   49:- aleph_set(verbosity,10).   50%:- aleph_set(mingain,-1e10).
   51
   52% specify predicate definition to use for model construction
   53:- model(predict/3).   54
   55%%%%%%%%%%%%%%%%%%%%%%%%%%%%
   56% Mode declarations
   57
   58:- modeh(1,f(+x,-y)).   59:- modeb(1,lteq(+x,#threshold)).
   60:- modeb(1,predict(+x,-y,#params)).
   61
   62:- determination(f/2,lteq/2).   63:- determination(f/2,predict/3).   64
   65:-begin_bg.   66%%%%%%%%%%%%%%%%%%%%%%%%%%%%
   67% Type definitions
   68
   69threshold(-0.5).
   70threshold(0.0).
   71threshold(0.5).
   72
   73params([_Slope,_Constant,_Sd]).
   74
   75list([_|_]).
   76
   77%%%%%%%%%%%%%%%%%%%%%%%%%%%%
   78% Background
   79
   80lteq(X,Y):-
   81        var(Y), !,
   82        X = Y.
   83lteq(X,Y):-
   84        number(X), number(Y),
   85        X =< Y.
   86
   87% definition for model construction (parameter estimation)
   88predict(X,Y,[M,C,Sd]):-
   89	list(X),
   90	list(Y), !,
   91	l_regress1(Y,X,M,C,Sd).
   92% definition for prediction
   93predict(X,Y,[M,C,_]):-
   94	number(X), var(Y), !,
   95	Y is M*X + C.
   96% definition for model checking
   97predict(X,Y,[M,C,Sd]):-
   98	number(Y), number(X), !,
   99	Y1 is M*X + C,
  100	Diff is Y - Y1,
  101	abs_val(Diff,ADiff),
  102	ADiff =< 3*Sd.
  103
  104% very simple univariate linear regression
  105l_regress1([YVals|_],[XVals|_],M,C,0.0):-
  106        YVals = [Y1,Y2|_],
  107        XVals = [X1,X2|_],
  108        M is (Y2-Y1)/(X2-X1),
  109        C is Y1 - M*X1.
  110
  111%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  112% Constraints
  113
  114% remove redundant checks for =<
  115% prune((_:-Body)):-
  116% 	has_literal(lteq(X,Y),Body,Left),
  117% 	has_literal(lteq(X1,Y1),Left,_),
  118% 	X == X1,
  119% 	Y1 =< Y.
  120% 
  121% has_literal(L,(L,L1),L1).
  122% has_literal(L,(_,L1),Left):-
  123% 	!,
  124% 	has_literal(L,L1,Left).
  125% has_literal(L,L,true).
  126
  127%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  128% Utilities
  129
  130abs_val(X,Y):- X < 0, !, Y is -X.
  131abs_val(X,X):- X >= 0.
  132
  133:-end_bg.  134:-begin_in_pos.  135
  136f(-1.0,0.0).
  137f(-0.5,0.5).
  138 f(-0.25,0.75).	% adding this results in the correct theory
  139f(0.0,1.0).
  140f(0.5,0.5).
  141f(1.0,0.0).
  142
  143:-end_in_pos.  144:-begin_in_neg.  145:-end_in_neg.