1. Non-Parametric Methods in Forest Models James D. Arney, Ph.D. Forest Biometrics Research Institute FBRI Annual Meeting December 4, 2012

2. Non-Parametric Statistics ► Distribution free methods which do not rely on assumptions that the data are drawn from a given probability distribution. ► Non-parametric statistic can refer to a statistic (a function on a sample) whose interpretation does not depend on the population fitting any parameterized distributions.

3. Why Non-Parametrics ► As non-parametric methods make fewer assumptions, their applicability is much wider than the corresponding parametric methods. In particular, they may be applied in situations where less is known about the application in question. Also, due to the reliance on fewer assumptions, non- parametric methods are more robust.

4. Why Non-Parametrics ► Another justification for the use of non- parametric methods is simplicity. In certain cases, even when the use of parametric methods is justified, non-parametric methods may be easier to use. Due both to this simplicity and to their greater robustness, non-parametric methods are seen by some statisticians as leaving less room for improper use and misunderstanding.

5. Non-Parametric Models ► Non-parametric models differ from parametric models in that the model structure is not specified a priori but is instead determined from data. The term nonparametric is not meant to imply that such models completely lack parameters but that the number and nature of the parameters are flexible and not fixed in advance.

7. Interpolate Observations ► Care is taken to sample across the response surface. ► Original observations (x and y) may occur at random points across this response surface. ► Observations of Y (dependent variable) are interpolated to systematic intervals of X. ► Y i = Sum(Y k /(1 + (X k -X i ) 2 ) a weighted mean

9. Smooth Y Observation Trends ► Least Squares – every observation has an impact on every estimate. ► Binomial Smoothing – only local observations have effect on local estimate. ► Y i = 0.0625*(Y i-2 ) + 0.25*(Y i-1 ) + 0.375*(Y i ) + 0.25*(Y i+1 ) + 0.0625*(Y i+2 ) ► Only Y values at regular intervals are used. ► Resulting trend is loaded to FPS Library.

10. Weights based on Pascal’s Triangle Non-Parametric Smoothing 11 211 4121 81331 1614641 Wgt0.06250.250.3750.250.0625

16. The Data determines the model, not the Investigator.

19. The Data determines the model, not the Investigator.