Regresion

ionA Tutorial on the Piecewise Regression Approach Applied to Bedload Transport Data
Sandra E. Ryan Laurie S. Porth

United States Department of Agriculture Forest Service Rocky Mountain Research Station General Technical Report RMRS-GTR-189 May 2007

Ryan, Sandra E.; Porth, Laurie S. 2007. A tutorial on the piecewise regression approach applied to bedload transport data. Gen. Tech. Rep.RMRS-GTR-189. Fort Collins, CO: U.S. Department of Agriculture, Forest Service, Rocky Mountain Research Station. 41 p.

Abstract
This tutorial demonstrates the application of piecewise regression to bedload data to define a shift in phase of transport so that the reader may perform similar analyses on available data. The use of piecewise regression analysis implicitly recognizes differentfunctions fit to bedload data over varying ranges of flow. The transition from primarily low rates of sand transport (Phase I) to higher rates of sand and coarse gravel transport (Phase II) is termed “breakpoint” and is defined as the flow where the fitted functions intersect. The form of the model used here fits linear segments to different ranges of data, though other types of functions may be used.Identifying the transition in phases is one approach used for defining flow regimes that are essential for self-maintenance of alluvial gravel bed channels. First, the statistical theory behind piecewise regression analysis and its procedural approaches are presented. The reader is then guided through an example procedure and the code for generating an analysis in SAS is outlined. The results frompiecewise regression analysis from a number of additional bedload datasets are presented to help the reader understand the range of estimated values and confidence limits on the breakpoint that the analysis provides. The identification and resolution of problems encountered in bedload datasets are also discussed. Finally, recommendations on a minimal number of samples required for the analysis areproposed.

Keywords: Piecewise linear regression, breakpoint, bedload transport

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Authors
Sandra E. Ryan, Research Hydrologist/Geomorphologist U.S. Forest Service Rocky Mountain Research Station 240 West Prospect RoadFort Collins, CO 80526 E-mail: sryanburkett@fs.fed.us Phone: 970-498-1015 Fax: 970-498-1212 Laurie S. Porth, Statistician U.S. Forest Service Rocky Mountain Research Station 240 West Prospect Road Fort Collins, CO 80526 E-mail: lporth@fs.fed.us Phone: 970-498-1206 Fax: 970-498-1212 Statistical code and output shown in boxed text in the document (piecewise regression procedure and bootstrapping),as well as an electronic version of the Little Granite Creek dataset are available on the Stream System Technology Center website under “software” at http://stream.fs.fed.us/publications/software.html.

Contents
Introduction.................................................................................... 1 Data................................................................................................. 2 Statistical Theory........................................................................... 2 Tutorial Examples .......................................................................... 4 Little Granite Creek Example ................................................... 4 Hayden Creek Example ......................................................... 18 Potential Outliers...