Could really use some help with the notation in this example

[CraigH]

http://img59.imageshack.us/img59/6414/puag.png

The picture above is used to help understand artificial neural networks, however for now this doesn't matter as I'm only concerned with the maths.

A number is input at the start of each both horizontal lines at the left hand side. The numbers travel along the lines from left to right and at each node (circle) the numbers at that node are summed together and sent out the other side.

Each line also has an associated "weight", this weight is multiplied with the number traveling along the line before it is sent to the attached node. For example if the numbers (2,2,2) are inputted, and all of the lines have a weight of 2, then the final output will be 192.

This is all I understand so far. The rest I have guessed using the information given in the picture above.

I'm guessing that i is a variable that ranges from 1 to 3, j is a variable that ranges from 1 to 2, and k is a variable that is always 1.

so in the formula

x_{j}=\sum_{i}w_{ij}*I_{i}

x_{j} is the value of the 'j'th hidden node

and

I_{i} is the value of the 'i'th input

e.g I_{1} will be the 1st input (going into the top horizontal line)

w_{ij} will be the "weight" of the line before the node, however I do not understand what to do when there are two subscripts next to it.

So for example the value of the 2nd hidden node will be

x_{2}=\displaystyle\sum\limits_{i=1}^3 w_{i2}*I_{i}

I just do not understand how you evaluate the sum when you have 2 subscripts under a variable as is the case with w_{ij}

can someone please explain?

Or have I completely misunderstood the whole network and I'm getting everything wrong?

Thanks!

 

[Mark44]

As I interpret this, i indicates which input layer (1, 2, or 3), and j indicates which hidden layer (1 or 2).

$$\sum_{i = 1}^3 w_{i2} * I_i $$
means w₁₂ * I₁ + w₂₂ * I₂ + w₃₂ * I₃

 

[CraigH]

means w₁₂ * I₁ + w₂₂ * I₂ + w₃₂ * I₃

But what does w₁₂ mean?

say that the 1st input line has a weight of 3 and the second hidden layer line has a weight of 4 would w₁₂ mean 3*4 ?

so I₁ would be multiplied by 12?

 

[Mark44]

But what does w₁₂ mean?

say that the 1st input line has a weight of 3 and the second hidden layer line has a weight of 4 would w₁₂ mean 3*4 ?

so I₁ would be multiplied by 12?

I don't think so. There are six lines that join the three input nodes to the two hidden nodes. The weights are associated with the lines, not the nodes. That's my take, anyway, but I could be wrong.

 

[mfb]

The layers don't have weights, the individual connections have. w₁₂ is the weight how input i=1 influences the hidden node j=2.

 

[CraigH]

Thanks for your answers mark44/mfb. I'm still not fully sure with this question but i'll come back to it in the morning when I'm less tired and have another go at it. Thanks!

 

[CraigH]

e. w₁₂ is the weight how input i=1 influences the hidden node j=2.

ah so w₁₂ is the weight of the line between i=1 and j=2. This seems obvious now. Thank you!