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From: saswss@hotellng.unx.sas.com (Warren Sarle)
Subject: Re: Input Variable Contribution
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Date: Mon, 1 Jul 1996 21:59:04 GMT
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In biocomp@biocomp.seanet.com (Carl M. Cook) wrote: 
> Thus for each record in a data set, hold all inputs at the values in that 
> record then step up and down each variable by a user specified percentage
> and measure the effect.
> ...
>We take this dither analysis one step further in ExamiNeur, looking at the 
>signed up and down side effects too.  We will also be taking it further yet 

In article <4r6rl8$hv0@sjx-ixn5.ix.netcom.com>, acrocad@ix.netcom.com
(Chuck Bass) replied:
|> This is all fine and good for linear systems.  IE the effect
|> of each input variable can be assumed independent and the 
|> output equals sum of the contributions of the linear terms.
|> However, in a NN can we assume linearity?  

No, we can't assume linearity. Note that Carl said that the finite
difference derivative is computed "for each record in a data set".
This gives you a distribution of values which can be summarized in
various ways. In a linear model, the derivative for each input is
constant, so you would get the same value (the regression
coefficient) for every record.

Here is an extract from a post I made on 9 May 95 regarding
sensitivity/importance of inputs:

In linear regression, each input has a single weight associated with it,
and this simplicity facilitates interpretation of the model. However,
the regression weights do not necessarily measure the importance of the
inputs, even if the inputs are standardized. There has been considerable
argument in the literature about measures of importance of inputs in
linear regression. One of the better expository articles is:

  Darlington, R.B. (1968), "Multiple Regression in Psychological Research
  and Practice," Psychological Bulletin, 69, 161-182.

A linear regression weight tells you how much the output changes for a
unit change in the input. In linear regression, this change does not
depend on the value of the input before it is changed.  In a
nonlinear-in-the-inputs model, the change in the output _does_ depend on
the value the input is changed from, but you can still compute an
_average_ change in the output for a unit change in an input, where the
average can be taken over the training data or over some other set of
cases that you want to make predictions for. You can also estimate
standard errors and confidence intervals for this average change--see:

   Baxt, W.G. and White, H. (1995) "Bootstrapping confidence intervals
   for clinical input variable effects in a network trained to identify
   the presence of acute myocardial infarction", Neural Computation,
   7, 624-638.

In addition to the average change in the output per unit change in an
input, I would recommend estimating the average _squared_ change in the
output per unit change in an input, in case both large positive and
large negative changes occur.

The other most important measure of importance of an input is the
proportional change in the generalization error when the input is
omitted from the model. In a linear regression model, this is the
squared semipartial correlation of the input with the output,
controlling for the other inputs. (Unfortunately, many linear regression
programs report this quantity in an unnormalized form as a "sum of
squares" due to the input; if you divide this "sum of squares" by the
"corrected total sum of squares", you get the more interpretable form of
the measure as a proportion.) Even in linear regression, this measure of
importance is not always monotonically related to the regression
weights.  In neural nets, you can estimate the proportional change in
the generalization error when the input is omitted from the model by
retraining the network with that input omitted.

-- 

Warren S. Sarle       SAS Institute Inc.   The opinions expressed here
saswss@unx.sas.com    SAS Campus Drive     are mine and not necessarily
(919) 677-8000        Cary, NC 27513, USA  those of SAS Institute.
