Multiplicative regression via constrained least squares

This paper considers multiplicative models for predicting a response variable as a product of predictor variables. In the ideal case of known model parameters, the minimum mean squared error predictor is derived and its performance is shown to be fundamentally limited by the magnitude of the multiplicative error component. For estimating model parameters from data, the methods of logarithmically-transformed ordinary least squares (OLS) and nonlinear least squares (NLS) are discussed. We then propose a constrained least squares (CLS) regression method that combines the NLS objective function with a constraint based on the OLS solution. In experiments on log-normal and gamma-distributed data, CLS yields significant improvements in mean squared prediction error by avoiding large errors in parameter estimates and better accommodating model mismatch. We also compare the performances of the regression methods using real-world health care usage data.