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of output neurons to provide. If $outputs is not specified, $outputs
defaults to equal $size. $outputs may not exceed $size. If $outputs
exceeds $size, the <CODE>new()</CODE> constructor will return undef.</P>
<P>The optional parameter, $topology_flag, defaults to 0 when not used. There are
three valid topology flag values:</P>
<P><STRONG>0</STRONG> <EM>default</EM>
My feed-foward style: Each neuron in layer X is connected to one input of every
neuron in layer Y. The best and most proven flag style.</P>
<PRE>
        ^   ^   ^               
        O\  O\ /O       Layer Y
        ^\\/^/\/^
        | //|\/\|
        |/ \|/ \|               
        O   O   O       Layer X
        ^   ^   ^</PRE>
<P>(Sorry about the bad art...I am no ASCII artist! :-)</P>
<P><STRONG>1</STRONG>
In addition to flag 0, each neuron in layer X is connected to every input of 
the neurons ahead of itself in layer X.</P>
<P><STRONG>2</STRONG> <EM>(``L-U Style'')</EM>
No, its not ``Learning-Unit'' style. It gets its name from this: In a 2 layer, 3
neuron network, the connections form a L-U pair, or a W, however you want to look
at it.</P>
<PRE>
        ^   ^   ^
        |   |   |
        O--&gt;O--&gt;O
        ^   ^   ^
        |   |   |
        |   |   |
        O--&gt;O--&gt;O
        ^   ^   ^
        |   |   |</PRE>
<P>As you can see, each neuron is connected to the next one in its layer, as well
as the neuron directly above itself.</P>
<P>Before you can really do anything useful with your new neural network
object, you need to teach it some patterns. See the <A HREF="#item_learn"><CODE>learn()</CODE></A> method, below.</P>
<P></P>
<DT><STRONG><A NAME="item_learn">$net-&gt;learn($input_map_ref, $desired_result_ref [, options ]);</A></STRONG><BR>
<DD>
This will 'teach' a network to associate an new input map with a desired resuly.
It will return a string containg benchmarking information. You can retrieve the
pattern index that the network stored the new input map in after <A HREF="#item_learn"><CODE>learn()</CODE></A> is complete
with the <CODE>pattern()</CODE> method, below.
<P><B>UPDATED:</B> You can now specify strings as inputs and ouputs to learn, and they will be crunched
automatically. Example:</P>
<PRE>
        $net-&gt;learn('corn', 'cob');
        # Before update, you have had to do this:
        # $net-&gt;learn($net-&gt;crunch('corn'), $net-&gt;crunch('cob'));</PRE>
<P>Note, the old method of calling crunch on the values still works just as well.</P>
<P><B>UPDATED:</B> You can now learn inputs with a 0 value. Beware though, it may not <A HREF="#item_learn"><CODE>learn()</CODE></A> a 0 value 
in the input map if you have randomness disabled. See NOTES on using a 0 value with randomness
disabled.</P>
<P>The first two arguments may be array refs (or now, strings), and they may be of different lengths.</P>
<P>Options should be written on hash form. There are three options:
</P>
<PRE>
         inc    =&gt;      $learning_gradient
         max    =&gt;      $maximum_iterations
         error  =&gt;      $maximum_allowable_percentage_of_error</PRE>
<P>$learning_gradient is an optional value used to adjust the weights of the internal
connections. If $learning_gradient is ommitted, it defaults to 0.20.
</P>
<P>
$maximum_iterations is the maximum numbers of iteration the loop should do.
It defaults to 1024.  Set it to 0 if you never want the loop to quit before
the pattern is perfectly learned.</P>
<P>$maximum_allowable_percentage_of_error is the maximum allowable error to have. If 
this is set, then <A HREF="#item_learn"><CODE>learn()</CODE></A> will return when the perecentage difference between the
actual results and desired results falls below $maximum_allowable_percentage_of_error.
If you do not include 'error', or $maximum_allowable_percentage_of_error is set to -1,
then <A HREF="#item_learn"><CODE>learn()</CODE></A> will not return until it gets an exact match for the desired result OR it
reaches $maximum_iterations.</P>
<P></P>
<DT><STRONG><A NAME="item_learn_set">$net-&gt;learn_set(\@set, [ options ]);</A></STRONG><BR>
<DD>
<B>UPDATED:</B> Inputs and outputs in the dataset can now be strings. See information on auto-crunching
in <A HREF="#item_learn"><CODE>learn()</CODE></A>
<P>This takes the same options as <A HREF="#item_learn"><CODE>learn()</CODE></A> and allows you to specify a set to learn, rather
than individual patterns. A dataset is an array refrence with at least two elements in the
array, each element being another array refrence (or now, a scalar string). For each pattern to
learn, you must specify an input array ref, and an ouput array ref as the next element. Example:
</P>
<PRE>
        my @set = (
                # inputs        outputs
                [ 1,2,3,4 ],  [ 1,3,5,6 ],
                [ 0,2,5,6 ],  [ 0,2,1,2 ]
        );</PRE>
<P>See the paragraph on measuring forgetfulness, below. There are 
two learn_set()-specific option tags available:</P>
<PRE>
        flag     =&gt;  $flag
        pattern  =&gt;  $row</PRE>
<P>If ``flag'' is set to some TRUE value, as in ``flag =&gt; 1'' in the hash of options, or if the option ``flag''
is not set, then it will return a percentage represting the amount of forgetfullness. Otherwise,
<A HREF="#item_learn_set"><CODE>learn_set()</CODE></A> will return an integer specifying the amount of forgetfulness when all the patterns 
are learned.</P>
<P>If ``pattern'' is set, then <A HREF="#item_learn_set"><CODE>learn_set()</CODE></A> will use that pattern in the data set to measure forgetfulness by.
If ``pattern'' is omitted, it defaults to the first pattern in the set. Example:</P>
<PRE>
        my @set = (
                [ 0,1,0,1 ],  [ 0 ],
                [ 0,0,1,0 ],  [ 1 ],
                [ 1,1,0,1 ],  [ 2 ],  #  &lt;---
                [ 0,1,1,0 ],  [ 3 ]
        );
</PRE>
<P>
If you wish to measure forgetfulness as indicated by the line with the arrow, then you would
pass 2 as the &quot;pattern&quot; option, as in &quot;pattern =&gt; 2&quot;.</P>
<P>Now why the heck would anyone want to measure forgetfulness, you ask? Maybe you wonder how I 
even measure that. Well, it is not a vital value that you have to know. I just put in a 
``forgetfulness measure'' one day because I thought it would be neat to know.</P>
<P>How the module measures forgetfulness is this: First, it learns all the patterns in the set provided,
then it will run the very first pattern (or whatever pattern is specified by the ``row'' option)
in the set after it has finished learning. It will compare the <A HREF="#item_run"><CODE>run()</CODE></A> output with the desired output
as specified in the dataset. In a perfect world, the two should match exactly. What we measure is
how much that they don't match, thus the amount of forgetfulness the network has.</P>
<P>NOTE: In version 0.77 percentages were disabled because of a bug. Percentages are now enabled.</P>
<P>Example (from examples/ex_dow.pl):</P>
<PRE>
        # Data from 1989 (as far as I know..this is taken from example data on BrainMaker)
        my @data = ( 
                #       Mo  CPI  CPI-1 CPI-3    Oil  Oil-1 Oil-3    Dow   Dow-1 Dow-3   Dow Ave (output)
                [       1,      229, 220,  146,         20.0, 21.9, 19.5,       2645, 2652, 2597],      [       2647  ],
                [       2,      235, 226,  155,         19.8, 20.0, 18.3,       2633, 2645, 2585],      [       2637  ],
                [       3,      244, 235,  164,         19.6, 19.8, 18.1,       2627, 2633, 2579],      [       2630  ],
                [       4,      261, 244,  181,         19.6, 19.6, 18.1,       2611, 2627, 2563],      [       2620  ],
                [       5,      276, 261,  196,         19.5, 19.6, 18.0,       2630, 2611, 2582],      [       2638  ],
                [       6,      287, 276,  207,         19.5, 19.5, 18.0,       2637, 2630, 2589],      [       2635  ],
                [       7,      296, 287,  212,         19.3, 19.5, 17.8,       2640, 2637, 2592],      [       2641  ]                 
        );