Radial basis function network: Difference between revisions

Revision as of 18:35, 16 April 2008

Revision as of 18:46, 16 April 2008

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[[Image:Unnormalized radial basis functions.svg|thumb|350px|right|Figure 2: Two unnormalized radial basis functions in one input dimension. The basis function centers are located at <math> c_1=0.75 </math> and <math> c_2=3.25 </math>.

]]

[[Image:060803 normalized radial basis functions.png|thumb|350px|right|Figure 3: Two normalized radial basis functions in one input dimension. The basis function centers are located at <math> c_1=0.75 </math> and <math> c_2=3.25 </math>.]]

===Normalized===

====Normalized architecture====

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is known as a "normalized radial basis function".

[[Image:060804 3060803 normalized radial basis functions.png|thumb|350px|right|Figure 43: ThreeTwo normalized radial basis functions in one input dimension. The additional basis function hascenters centerare located at <math> c_3c_1=20.75 </math> and <math> c_2=3.25 </math>.]]

====Theoretical motivation for normalization====

There is theoretical justification for this architecture in the case of stochastic data flow. Assume a [[stochastic kernel]] approximation for the joint probability density

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and

:<math> \int \sigma \big ( \left \vert y - e_i \right \vert \big ) \, dy =1</math>.

[[Image:060803060804 normalized3 radialnormalized basis functions.png|thumb|350px|right|Figure 34: TwoThree normalized radial basis functions in one input dimension. The additional basis function centershas are locatedcenter at <math> c_1c_3=02.75 </math> and <math> c_2=3.25 </math>.]]

[[Image:060804 4 normalized basis functions.png|thumb|350px|right|Figure 5: Four normalized radial basis functions in one input dimension. The fourth basis function has center at <math> c_4=0 </math>. Note that the first basis function (dark blue) has become localized.]]