The Effect of Record L.ength on a Nonlinear Regression Model for Weekly Stream Temperatures

Abstract

A four parameter, logistic stream temperature model using weekly air temperature as the predictor of weekly stream temperature was fitted by least squares regression to records varying in length from 12 to 32 years. The records were from four streams in Minnesota and three streams in Oklahoma. The purpose of the study was to test if stream temperature models formulated from 3-year samples were representative of stream temperature models developed from the seven, full-length records. This test was done because the model had previously been applied to 3-year records from 585 streams and associated weather stations in the US (Mohseni et aI., 1997). Each full-length record was divided into 3-year samples containing up to 156 weekly air temperature and stream temperature data. The logistic stream temperature model was then fitted to the 3~year samples, as well as the full~length records. The models formulated from the full~length records were assumed to represent the "true" weekly air temperature/stream temperature relationships or "population" relationships. F-tests were used to determine whether statistical similarity between the 3~year sample and the full~length models existed. The results showed that approximately 33% of the 3-year sample relationships were not statistically similar to their respective population models. Further analysis of the 3-year sample and population regression parameters revealed notable discrepancies, especially for the parameter representing upper bound stream temperature. Twenty-six of thirty-one 3-year samples produced estimates of this parameter less than their respective popUlation model. In addition, the 3-year sample estimates of upper bound stream temperature demonstrated a large variance. Nonlinear, least squares parameter estimates were found to be inherently biased. The bias of nonlinear regression parameters is reduced with increasing sample length. Three-year weekly air temperature and stream temperature records can not exhibit the natural variance found in longer records. Records of more than 3-year duration are therefore necessary for the consistent representation of long-term weekly air temperature/stream temperature relationships.