Use of difference-based methods to explore statistical and mathematical model discrepancy in inverse problems

Abstract

Normalized differences of several adjacent observations, referred to as pseudo-measurement errors in this paper, are used in so-called difference-based estimation methods as building blocks for the variance estimate of measurement errors. Numerical results demonstrate that pseudo-measurement errors can be used to serve the role of measurement errors. Based on this information, we propose the use of pseudo-measurement errors to determine an appropriate statistical model and then to subsequently investigate whether there is a mathematical model misspecification or error. We also propose to use the information provided by pseudo-measurement errors to quantify uncertainty in parameter estimation by bootstrapping methods. A number of numerical examples are given to illustrate the effectiveness of these proposed methods.

The Effect of Statistical Error Model Formulation on the Fit and Selection of Mathematical Models of Tumor Growth for Small Sample Sizes

“…We note that a misspecified statistical error model can lead to an incorrect estimation of the parameters as well as their uncertainly bounds. With the assumption on the errors E j (i.e., i.i.d, mean zero, and constant variance) in the previous section, there are generally two ways of selecting the correct statistical error model, namely, using residual plots or applying second-order differencing techniques directly to the data [4,5]. We briefly discuss these methods below.…”

“…In addition, if the correct mathe- matical model is not used, residual plots could give inaccurate information, as these plots also depend on the solution of the mathematical model as well as on the chosen γ value. A method that relies only on the data itself for identifying the correct observational statistical error model is a second-order differencing technique and is described in detail in [4]. This method is found to be more accurate and efficient than using residual plots [4] as well as not requiring prior solution of inverse problems.…”

“…A method that relies only on the data itself for identifying the correct observational statistical error model is a second-order differencing technique and is described in detail in [4]. This method is found to be more accurate and efficient than using residual plots [4] as well as not requiring prior solution of inverse problems. We attempted to use this second order differencing technique directly with several of our larger data sets, namely the HPV tumors.…”

Parameter Estimation for an Allometric Food Web Model

“…Initial treatment of the inverse problem in [2] suggested that the ATN model cannot capture some aspects of aphid population dynamics. Therefore, we first use difference-based pseudo-measurement errors, as described in [7], to test our statistical model without tacitly assuming a mathematical model. We compute estimates of the measurement error, f γ (t j ;θ)ǫ j , from the observed N j 1 .…”

Modelling populations ofLygus hesperuson cotton fields in the San Joaquin Valley of California: the importance of statistical and mathematical model choice

“…We assume an error model in the form of (8), but a more generalized form of error model for GLS methods can be found in [3,4,14,16,32]. GLS estimatesθ θ θ n and estimated weights {ω j (θ θ θ)} n j=1 are found using a standard iterative method [2,4,14,16,32] as given below .…”