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Artificial Neuron Models 人工神经元模型.doc
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Artificial Neuron Models 人工神经元模型.doc
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- 1 -
Artificial Neuron Models
Computational neurobiologists have constructed very elaborate computer models of
neurons in order to run detailed simulations of particular circuits in the brain. As
Computer Scientists, we are more interested in the general properties of neural networks,
independent of how they are actually "implemented" in the brain. This means that we can
use much simpler, abstract "neurons", which (hopefully) capture the essence of neural
computation even if they leave out much of the details of how biological neurons work.
People have implemented model neurons in hardware as electronic circuits, often
integrated on VLSI chips. Remember though that computers run much faster than brains -
we can therefore run fairly large networks of simple model neurons as software
simulations in reasonable time. This has obvious advantages over having to use special
"neural" computer hardware.
A Simple Artificial Neuron
Our basic computational element (model neuron) is often called a node or unit. It
receives input from some other units, or perhaps from an external source. Each input has
an associated weight w, which can be modified so as to model synaptic learning. The unit
computes some function f of the weighted sum of its inputs:
�
�
j
jiji
ywfy )(
Its output, in turn, can serve as input to other units.
� The weighted sum
�
j
jij
yw
is called the net input to unit i, often written net
i
.
� Note that w
ij
refers to the weight from unit j to unit i (not the other way around).
� The function f is the unit's activation function. In the simplest case, f is the
identity function, and the unit's output is just its net input. This is called a linear
unit.
- 2 -
Linear Regression
Fitting a Model to Data
Consider the data below (for more complete auto data, see data description, raw data, and
maple plots):
(Fig. 1)
Each dot in the figure provides information about the weight (x-axis, units: U.S. pounds)
and fuel consumption (y-axis, units: miles per gallon) for one of 74 cars (data from
1979). Clearly weight and fuel consumption are linked, so that, in general, heavier cars
use more fuel.
Now suppose we are given the weight of a 75th car, and asked to predict how much fuel
it will use, based on the above data. Such questions can be answered by using a model - a
short mathematical description - of the data (see also optical illusions). The simplest
useful model here is of the form
y = w
1
x + w
0
(1)
This is a linear model: in an xy-plot, equation 1 describes a straight line with slope w
1
and intercept w
0
with the y-axis, as shown in Fig. 2. (Note that we have rescaled the
coordinate axes - this does not change the problem in any fundamental way.)
How do we choose the two parameters w
0
and w
1
of our model? Clearly, any straight line
drawn somehow through the data could be used as a predictor, but some lines will do a
better job than others. The line in Fig. 2 is certainly not a good model: for most cars, it
will predict too much fuel consumption for a given weight.
- 3 -
(Fig. 2)
The Loss Function
In order to make precise what we mean by being a "good predictor", we define a loss
(also called objective or error) function E over the model parameters. A popular choice
for E is the sum-squared error:
(2)
In words, it is the sum over all points i in our data set of the squared difference between
the target value t
i
(here: actual fuel consumption) and the model's prediction y
i
,
calculated from the input value x
i
(here: weight of the car) by equation 1. For a linear
model, the sum-sqaured error is a quadratic function of the model parameters. Figure 3
shows E for a range of values of w
0
and w
1
. Figure 4 shows the same functions as a
contour plot.
(Fig. 3) (Fig. 4)
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