1. © 2003 By Default! A Free sample background from www.powerpointbackgrounds.com Slide 1 Isotonic regression and normalisation of environmental quality data Anders Grimvall, Mohamed Hussian, and Oleg Burdakov

2. © 2003 By Default! A Free sample background from www.powerpointbackgrounds.com Slide 2 Motivation The response of environmental systems to different types of physical forcing is often monotonic and non-linear The response of environmental systems to different types of physical forcing is often monotonic and non-linear Temporal trends in environmental quality can emerge more clearly if the collected data are normalised by removing natural fluctuations Temporal trends in environmental quality can emerge more clearly if the collected data are normalised by removing natural fluctuations

3. © 2003 By Default! A Free sample background from www.powerpointbackgrounds.com Slide 3 Tot-P concentrations vs water discharge April mean values from Brunsbüttel on the Elbe River in Germany, 1985-2000

4. © 2003 By Default! A Free sample background from www.powerpointbackgrounds.com Slide 4 Concentrations of tropospheric ozone vs humidity Average monthly values from Ähtäri in central Finland

5. © 2003 By Default! A Free sample background from www.powerpointbackgrounds.com Slide 5 Isotonic regression Let Let denote n observations of p explanatory variables and one response variable. Minimise Minimise under the constraints

6. © 2003 By Default! A Free sample background from www.powerpointbackgrounds.com Slide 6 The Pool-Adjacent-Violators (PAV) algorithm The Pool-Adjacent-Violators (PAV) algorithm (Ayer, 1955; Barlow et al., 1972; Hanson et al., 1973) (Ayer, 1955; Barlow et al., 1972; Hanson et al., 1973) Simple averaging techniques Simple averaging techniques (e.g. Mukerjee & Stern, 1994) Widely used algorithms for isotonic regression

7. © 2003 By Default! A Free sample background from www.powerpointbackgrounds.com Slide 7 The PAV algorithm in one explanatory variable

8. © 2003 By Default! A Free sample background from www.powerpointbackgrounds.com Slide 8 A PAV algorithm for monotonic regression in several explanatory variables A PAV-algorithm for monotonic response in two- or multi- factor experiments was developed by Dykstra & Robertson in 1984. A PAV-algorithm for monotonic response in two- or multi- factor experiments was developed by Dykstra & Robertson in 1984. Limitations of this algorithm: Limitations of this algorithm: –Very inefficient or unusable for typical multiple regression data where at least one of the explanatory variables is continuous –Unclear how seasonality can be handled

9. © 2003 By Default! A Free sample background from www.powerpointbackgrounds.com Slide 9 A simple averaging technique

10. © 2003 By Default! A Free sample background from www.powerpointbackgrounds.com Slide 10 Simple averaging in two explanatory variables Amount of Tot-N carried by the Elbe at Brunsbüttel February values 1985-2000 Tot-N transport (kton/month) Fitted values Year Water discharge (10 9 m 3 /month)

11. © 2003 By Default! A Free sample background from www.powerpointbackgrounds.com Slide 11 Handling seasonality

12. © 2003 By Default! A Free sample background from www.powerpointbackgrounds.com Slide 12 Estimation of monotone trend in the presence of seasonal variation using simple averaging Monthly mean concentrations of total nitrogen at Brunsbüttel on the Elbe River Year Month Fitted Tot-N concentration (mg/l)

13. © 2003 By Default! A Free sample background from www.powerpointbackgrounds.com Slide 13 Improved simple averaging techniques For, the estimators are non-decreasing in x. The value of minimizing is given by

14. © 2003 By Default! A Free sample background from www.powerpointbackgrounds.com Slide 14 Averaging techniques: Pro’s and con’s Can handle several independent variables and seasonal variation Can handle several independent variables and seasonal variation Sensitive to outliers Sensitive to outliers Do not give optimal (mean square) solutions Do not give optimal (mean square) solutions

15. © 2003 By Default! A Free sample background from www.powerpointbackgrounds.com Slide 15 Monte-Carlo adjustment of a monotonic response function Employ an updating formula Employ an updating formula where is a vector defining the direction (shape) of the adjustment, and h is a scaling factor. Pick the direction at random and select the scaling factor so that the mean square error is minimised without violating the monotonic constraints. Pick the direction at random and select the scaling factor so that the mean square error is minimised without violating the monotonic constraints.