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Could really use some help with the notation in this example

  1. Sep 24, 2013 #1
    http://img59.imageshack.us/img59/6414/puag.png [Broken]

    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

    [itex]x_{j}=\sum_{i}w_{ij}*I_{i}[/itex]

    [itex]x_{j}[/itex] is the value of the 'j'th hidden node

    and

    [itex]I_{i}[/itex] is the value of the 'i'th input

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

    [itex]w_{ij}[/itex] 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

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

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

    can someone please explain?

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

    Thanks!
     
    Last edited by a moderator: May 6, 2017
  2. jcsd
  3. Sep 24, 2013 #2

    Mark44

    Staff: Mentor

    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
     
  4. Sep 24, 2013 #3
    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?
     
  5. Sep 24, 2013 #4

    Mark44

    Staff: Mentor

    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.
     
  6. Sep 24, 2013 #5

    mfb

    User Avatar
    2016 Award

    Staff: Mentor

    The layers don't have weights, the individual connections have. w12 is the weight how input i=1 influences the hidden node j=2.
     
  7. Sep 24, 2013 #6
    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!
     
  8. Sep 26, 2013 #7
    ah so w12 is the weight of the line between i=1 and j=2. This seems obvious now. Thank you!
     
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