Universality in Multiparameter Multiparameter Fitting Sloppy Models Fitting Sloppy Models Casey Kevin S Brown Chris Myers VeitElser PietBrouwer Cell Dynamics Fitting Exponentials Polynomials ID: 171593
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Universality in Universality in Multiparameter Multiparameter Fitting: Sloppy Models Fitting: Sloppy Models Casey, Kevin S. Brown, Chris Myers, VeitElser, PietBrouwer Cell Dynamics Fitting Exponentials, Polynomials Fits good: measured bad Fit Ensemble:Interpolation Ensemble: Extrapolation Fitting Decaying Exponentials Fitting Decaying Exponentials eAeAeAty),,(= Classic Ill-Posed Inverse Problem Given Geiger counter measurements from a radioactive pile, can we recover the identity of the elements and/or predict future radioactivity? Good fits with bad decay rates! yy))(()( 3532125 6 Parameter Fit Sloppy Systems Biology Sloppy Systems Biology Fits good: Where is Sloppiness From? Where is Sloppiness From? Fitting Polynomials to Data Fitting Monomials to Data y = anxn Functional Forms Same Hessian Hij= 1/(i+j+1) Hilbert matrix: famous Orthogonal Polynomials y = bnLn(x) Functional Forms Distinct Eigen Parameters Hessian Hij= ij Sloppiness arises when bare parameters skew in eigenbasis Small Determinant!|H| =n Why are they Sloppy? Why are they Sloppy? The VandermondeEnsemble Assumptions: i.Parameters are nearly degenerate:ii.Residuals iii.Cost is sum of squares of residuals: ddεεεεεε#%##21211112/)1()()det(∝−=NNjijiεεε kjikVAkrr AVAVJJHTT== VandermondeMatrix Random instance i/ V ? Random model A? Also, equal spacingsin logs, level repulsion