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POU-Matrix for Rectangular Maps
 

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    <p>I have read about the U-Matrix in many places including this site. The best explanation of a U-Matrix is found <a href="https://stackoverflow.com/questions/13631673/u-make-a-u-matrix">here</a> in this site with the explanation why there is so little correct information (the original <a href="http://www.uni-marburg.de/fb12/datenbionik/pdf/pubs/1990/UltschSiemon90" rel="nofollow noreferrer">paper</a> is not at all useful) about how the U-Matrix is properly calculated.</p> <p>The answer to the above question completely explains the concept for a hexagonal map. But the logic of calculating the U-matrix in the answer to the linked question do not hold when the map is rectangular. </p> <p>For example consider 3 x 3 rectangular lattice as shown below.</p> <p><img src="https://i.stack.imgur.com/iq1nV.png" alt="The Rectangular Map"></p> <p>Using the above lattice I can calculate the U-Matrix as shown below.</p> <p><img src="https://i.stack.imgur.com/pVDQq.png" alt="The U-Matrix"></p> <p>The yellow colored squares are the distances between the blue colored squares. I'm certain about the yellow colored squares. I'm also certain about the blue colored squares since we only need to take the average or the median of its surrounding.</p> <p>So my question is : <strong>How to calculate the red squares?</strong></p> <p>I found a few sources including the ones mentioned in the previous question I have cited above. The best explanations I got for a rectangular U-Matrix is the following</p> <p><a href="http://www.lana.lt/journal/43/NA16409.pdf" rel="nofollow noreferrer">Description 1</a> -> In this paper the authors does not completely explain how to calculate the red squares. Just explains the average of the surrounding needs to be taken. Which is not clear and in my opinion not proper (see below)</p> <p><a href="http://www.escet.urjc.es/~mrubio/papers/ManuelRubioKes2002.pdf" rel="nofollow noreferrer">Description 2</a> -> In this paper the authors have clearly stated how to calculate the red squares but the logic they have presented seem flawed.</p> <p><strong>My explanation why the above might not be proper</strong></p> <p>If one takes the average of its surrounding to calculate the red squares as mentioned by description 1 the calculation of blue squares would be directly affected. For example consider calculating the value of blue square number 1 in U-Matrix. If we are to take the average of its surrounding we need distances (1,2) , (1,4) and (1,5). If we fill the corresponding red square with (1,5) the calculation for the blue square 4 is wrong since we did not calculate (2,4) and the same red square should be the place to have it. So the equation of dividing the addition of (1,5) and (2,4) by 2*(1.414...) will not work since there is a component which does not belong to the average. In the case of blue square 1 the distance portion of (2,4) do not belong there.</p> <p>I programmed using the description in the second paper and the U-Matrix generated for a simple data set is not satisfactory. Whilst the average of distances around a given node performs better than the U-Matrix for the same data set as given below. (The images are U-Matrix followed by the average)</p> <p><img src="https://i.stack.imgur.com/GLxjm.jpg" alt="U-Matrix"> <img src="https://i.stack.imgur.com/ramv4.jpg" alt="enter image description here"></p>
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