next up previous contents
Next: Local models Up: Modeling Previous: Modeling

Lookup models

The elastic net formed by the SOM in the input space can be interpreted as an implicit lookup model of the phenomena that produced the training data. The lookup model can be used e.g. for sensitivity analysis [12]. Let tex2html_wrap_inline3414 be the BMU of vector tex2html_wrap_inline3156 . When a vector component i is varied by tex2html_wrap_inline3420 :


where tex2html_wrap_inline3272 is a unit vector along the ith axis, the BMU changes to another map unit tex2html_wrap_inline3426 . Calculating tex2html_wrap_inline3428 gives the response of the model to the change. The quantization error tex2html_wrap_inline3430 can be used to estimate the reliability of the result.

If the idea of elastic net is taken a bit further the precision of the model can be increased. Let i be the BMU of a sample vector tex2html_wrap_inline3156 . From the neighbors tex2html_wrap_inline3194 of unit i, find the next best-matching unit j, and finally from units tex2html_wrap_inline3442 find the unit k closest to the sample vector. The three map units are adjacent and together define a plane in the input space. By calculating the projection tex2html_wrap_inline3446 of tex2html_wrap_inline3156 on that plane, a closer approximation of tex2html_wrap_inline3156 than the best-matching reference vector tex2html_wrap_inline3162 is obtained:


where tex2html_wrap_inline3454 is the unit-length normal vector of the plane. To ensure continuity of the projection, the tex2html_wrap_inline3446 must be constrained to be within the triangle tex2html_wrap_inline3458 .

Juha Vesanto
Tue May 27 12:40:37 EET DST 1997