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"stepwise" integrating a 2D array

• To: mathgroup at smc.vnet.net
• Subject: [mg91158] "stepwise" integrating a 2D array
• From: Claus <clausenator at gmail.com>
• Date: Fri, 8 Aug 2008 07:12:16 -0400 (EDT)
• Organization: Comp.Center (RUS), U of Stuttgart, FRG

Hi,
I have an array, consisting of n tuples (x,y,z).
The first entry is the point at the bottom left (thinking in a 
coordinate system, smallest (x,y) pair).
Now I want to do something like an "stepwise"/empirical integral over (x,y).

Schematically, if I had a 2x2 array (only the z-values),like

a1  a2
b1  b2

I would want an array like this:

a1+b1    b1+b2+a1+a2
b1       b1+b2

In the code below, "xyzVals" is the array that I want to integrate, 
containing the (x,y,z) tuples.

I could do a bunch of for loops, but I am trying to not use do loops in 
mathematica whenever possible.
Is it possible to do it without?
I tried some of the examples from
http://www.verbeia.com/mathematica/tips/HTMLLinks/Tricks_A-K_3.html
and I also thought about an array with sub-arrays,
but couldn't get it for me to work.

Thanks for any pointers!
Claus


In[13]:= X = RandomReal[{0, 1}, 500];
Y = RandomReal[{0, 1}, 500];
BinWidth = 0.2;
DensVals =
  Flatten[BinCounts[
    Transpose[{X, Y}], {0, 1, BinWidth}, {0, 1, BinWidth}]]

Out[16]= {16, 16, 17, 26, 23, 23, 16, 18, 25, 17, 13, 26, 26, 21, 19, \
15, 32, 20, 21, 15, 11, 26, 16, 19, 23}

In[17]:= xyzVals = Table[{i/(1/BinWidth) + (BinWidth/2),
    j/(1/BinWidth) + (BinWidth/2),
    DensVals[[(1/BinWidth) i + j + 1]]}
   , {i, 0, (1/BinWidth) - 1}
   , {j, 0, (1/BinWidth) - 1}]

Out[17]= {{{0.1, 0.1, 16}, {0.1, 0.3, 16}, {0.1, 0.5, 17}, {0.1, 0.7,
    26}, {0.1, 0.9, 23}}, {{0.3, 0.1, 23}, {0.3, 0.3, 16}, {0.3, 0.5,
    18}, {0.3, 0.7, 25}, {0.3, 0.9, 17}}, {{0.5, 0.1, 13}, {0.5, 0.3,
    26}, {0.5, 0.5, 26}, {0.5, 0.7, 21}, {0.5, 0.9, 19}}, {{0.7, 0.1,
    15}, {0.7, 0.3, 32}, {0.7, 0.5, 20}, {0.7, 0.7, 21}, {0.7, 0.9,
    15}}, {{0.9, 0.1, 11}, {0.9, 0.3, 26}, {0.9, 0.5, 16}, {0.9, 0.7,
    19}, {0.9, 0.9, 23}}}