Universality in Universality in MultiparameterMultiparameterFitting: Sloppy ModelsFitting: Sloppy Models

James P. Sethna, Josh Waterfall, Ryan Gutenkunst, Fergal Casey, Kevin S. Brown, Chris Myers, Veit Elser, Piet Brouwer

Cell Dynamics

Fitting Exponentials, Polynomials

Fits good: measured bad

FitEnsemble:InterpolationEnsemble: Extrapolation

Fitting Decaying ExponentialsFitting Decaying Exponentials

ttt eAeAeAty 321 321),,(γγγ −−− ++=γA

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!

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P, S, I3532 125

6 Parameter Fit

Fitting ExponentialsFitting Exponentials

Sloppy direction

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χ2 contours

eigen

param

s

Best Fit

Hessian ∂2C/∂θ2 at Best FitSloppy Directions ⇔Small Eigenvalues

Horizontal scale shrunk by 1000 times!

Aspect ratio = Human hairFits Good

Measuring Parameters Bad

bare params

Is Sloppiness Universal?Is Sloppiness Universal?

Syst

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NOT SLOPPY

Random Matrix Theory Connection:Explain Eigenvalue

Distributions

Sloppy Systems:

• Enormous range of eigenvalues

• Roughly equal density in log

• Observed in broad range of systems

Sloppy Systems BiologySloppy Systems Biology

Fits good: measured

bad

Where is Sloppiness From?Where is Sloppiness From?Fitting Polynomials to Data

Fitting Monomials to Datay = ∑an xn

Functional Forms SameHessian Hij = 1/(i+j+1)Hilbert matrix: famous

Orthogonal Polynomialsy = ∑bn Ln(x)

Functional Forms DistinctEigen ParametersHessian Hij = δij

Sloppiness arises when bare parameters skew in eigenbasis Small Determinant!|H| = ∏ λn

Why are they Sloppy?Why are they Sloppy?The Vandermonde Ensemble

Assumptions:i. Parameters are nearly degenerate: θι = θ0 + εiii. Residuals symmetric in parameters: mi = ∑j εjiiii.Cost is sum of squares of residuals: C(θ) = ∑rk2({mi})

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Vandermonde Matrixd=N-1 Random instance εi / V ?

Random model A?Also, equal spacings in logs, level repulsion

ReferencesReferences“The sloppy model universality class and the Vandermonde matrix”, Joshua J. Waterfall, Fergal P. Casey, Ryan N. Gutenkunst, Kevin S. Brown, Christopher R. Myers, Piet W. Brouwer, Veit Elser, and James P. Sethna, http://arxiv.org/abs/cond-mat/0605387.“Sloppy systems biology: tight predictions with loose parameters”, Ryan N. Gutenkunst, Joshua J. Waterfall, Fergal P. Casey, Kevin S. Brown, Christopher R. Myers & James P. Sethna (submitted).“The Statistical Mechanics of Complex Signaling Networks: Nerve Growth Factor Signaling”, Kevin S. Brown, Colin C. Hill, Guillermo A. Calero, Christopher R. Myers, Kelvin H. Lee, James P. Sethna, and Richard A. Cerione, Physical Biology 1, 184-195 (2004) . “Statistical Mechanics Approaches to Models with Many Poorly Known Parameters”, Kevin S. Brown and James P. Sethna, Phys. Rev. E 68, 021904 (2003). “Bayesian Ensemble Approach to Error Estimation of InteratomicPotentials”, Søren L. Frederiksen, Karsten W. Jacobsen, Kevin S. Brown, and James P. Sethna, Phys. Rev. Letters 93, 165501 (2004).“Bayesian Error Estimation in Density Functional Theory”, J. J. Mortensen, K. Kaasbjerg, S. L. Frederiksen, J. K. Norskov, James P. Sethna, K. W. Jacobsen, Phys. Rev. Letters 95, 216401 (2005).

Universality in Universality in MultiparameterMultiparameterFitting: Sloppy ModelsFitting: Sloppy Models

James P. Sethna, Josh Waterfall, Ryan Gutenkunst, Fergal Casey, Kevin S. Brown, Chris Myers, Veit Elser, Piet Brouwer

Cell Dynamics

Fitting Exponentials, Polynomials

Fits good: measured bad

Universality in Multiparameter Fitting: Sloppy ModelsFitting Decaying ExponentialsFitting ExponentialsIs Sloppiness Universal?Sloppy Systems BiologyWhere is Sloppiness From?Why are they Sloppy?ReferencesUniversality in Multiparameter Fitting: Sloppy Models