Development of a Multifunction Best Fit Computer Program to Model Sediment Transport Data in Rivers
T.V. Hromadka II, A.N. Johnson, and M. Phillips
Introduction
Analysis of sediment load in large rivers is an important problem in the management of the river hydraulics as well as for environmental concerns. Increases in sediment load can result in sediment deposition in the river uniformly or at high concentrations at particular locations. Such sediment deposition can interfere with river flow carrying capacity and also may positive- ly or negatively impact environmental opportunities such as fish and wildlife populations. The usual approach for analysis of sediment load and changes therein is to perform a statistical analy- sis of the sediment load data collected over time at monitoring stations.
Several computer models and pro- grams are currently available that oper- ate upon such sediment load data and prepare the usual fitting of interpolation functions for subsequent use in statisti- cal inferences. The USGS LOADEST program for estimating constituent loads in streams and rivers is one of these available programs. The main objective of this work is to develop a library of “best fit” functions that can be used to fit to the supplied data and com- pute the standard measures of fit error of least squares residual, maximum departure, and average departure. It is contemplated to including polynomials, trigonometric, exponential and power fitting functions. The fitting errors for the three tested measures will then be ranked.
In this current Note, we describe the ongoing research effort to develop an interpolation computer program based upon Mathematica. In the documenta- tion, eleven regression models are tabu- lated and are available in the LOADEST program to be fitted to the data. In the current study, more than 100 regression models are available, with more mod-
els being incorporated. The expanded Mathematica computer program is being directly developed from the program presented in Hromadka et al, 2007.
The typical problem in statistical analysis of data is to consider fitting (regression) functions that minimize some pre-defined measure of goodness- of-fit, such as the well-known least squares residual minimization tech- nique, among others. Generally, a selec- tion of basis functions are chosen such as polynomials or other basis functions, and then all of the basis functions are minimized together as a group accord- ing to the selected measure. In this case, we will develop a Mathematica program that accomplishes the task of fitting prescribed functions to data, but we will build the program to investigate combinations of the basis functions as an alternative to fitting the entire family of basis functions simultaneously. Because the procedure is automated, we have the advantage of doing such an effort without a significant investment of time or effort. The Mathematica program is expandable to include other families of basis functions or other measures of fit. The program measures the goodness of each set of basis functions used and then lists the measures achieved for each attempt.
The use of a set of basis functions to form a linear combination with coef- ficients to be determined by minimizing some best fit measure function to data according to some measure of fit is a pro- cess that is well-known and so will not be repeated here (Gallant 1987, among many others.) A set of basis functions are selected, such as a set of monomials, and then this entire set of basis functions is fitted to the target set of data by a technique such as a weighted Gramm- Schmidt approach or other approach. The computer program Mathematica is particularly well-suited to this task
and provides automated determination of the best fit function to the prescribed data set. The program can be extended by simply adding additional bases to the list of functions being used as fitting functions.
The Mathematica Program
The Mathematica program under development is currently based upon use of the Mathematica FindFit operation. The various combinations of the basis functions are built from the selected basis function family up to the dimen- sion programmed into the notebook. The dimension of this family can be extended by adding additional basis functions. For each combination of these basis func- tions, a new trial function is defined, and the target data set is fitted using the Mathematica FindFit operation and the resulting measure of fit is computed. Once all the combinations of basis func- tions are examined, they are ranked according to goodness of fit by sorting the resulting error residual measures.
Families of basis functions of typi- cal interest include trigonometric basis functions such as: a Sin (b x + c) + d which results in several variations just with these four parameters. For example, a Sin(x), Sin(b x), Sin(x+c), Sin(x)+d, a Sin(b x) and so forth. With a competing goal in fitting functions to data being the possibility that simpler fitting functions may be better models of the underpin- nings of these data, it is instructive to examine all combinations of basis func- tions from a prescribed family.
The Mathematica notebook is eas- ily expandable. The proposed extended program software for loading into a web- site is currently under construction for publication on the web.
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