A NON-SAMPLING BASED METHOD FOR PROPAGATING PARAMETER UNCERTAINTY INTO STATE-TRANSITION MODELS

Purpose: Probabilistic sensitivity analysis (PSA) has been the standard for accounting the effect of parameter uncertainty on model outcomes.  The mean and variance of model outcomes can be calculated after numerous Monte Carlo samplings of model parameters and running the model for each sampled set.  However PSA becomes computationally expensive when the sample space is enormous.  We introduce the concept of Polynomial Chaos Expansion (PCE), a non-sampling-based method for modeling parameter uncertainty.

Method: PCE method projects random variables or processes with finite second-order moment onto another vector space in which the quantity of interest can be represented as a series expansion of orthogonal polynomials with respect to an orthogonalizing measure (Equation 1).  The probability density functions commonly used for model parameters in PSA, i.e. uniform, normal, beta, and gamma distributions, correspond to the orthogonalizing measures of Legendre, Hermite, Jacobi, and Laguerre polynomials, respectively.  The mean of a model outcome can be calculated without random sampling and is given by the 0-th coefficient in the series expansion.  These coefficients are calculated as the inner product of the model and the parameter with the polynomial functions with respect to the measure.  PCE essentially converts the problem of random sampling to integration over probability space.  As an example of how PCE works, we applied PCE and PSA methods to account for the uncertainty in mortality rate in a model with healthy and death states.  Normal distribution was used to characterize the parameter uncertainty for both methods.  The PCE method expanded the model and the mortality rate in terms of Hermite polynomials with normal distribution as the measure.  The mean survival probabilities (MSP) were calculated and compared between methods.

Result: The 0-th coefficient of the series expansion (MSP), is given by the solution to the following equation:

 

The PCE method with degree of 2 and one-million quadrature points for integrating the above equation was faster than PSA with 1000 Monte Carlo samples in generating the MSP.  Variations in mortality rate generated by PSA and PCE methods produced 10-year MSPs of 0.73 and 0.72, respectively.  The variance estimates are 0.0053 and 0.0052 for PSA and PCE, respectively.

Conclusion: PCE may serve as a computationally efficient alternative to PSA method for accounting for parameter uncertainty in state-transition models.