Non-Parametric Non-Parametric MethodsMethods

ininForest ModelsForest Models

James D. Arney, Ph.D.James D. Arney, Ph.D.Forest Biometrics Research Forest Biometrics Research

InstituteInstituteFBRI Annual MeetingFBRI Annual MeetingDecember 4, 2012December 4, 2012

Non-Parametric StatisticsNon-Parametric Statistics

►Distribution freeDistribution free methods which do not methods which do not rely on assumptions that the data are rely on assumptions that the data are drawn from a given probability drawn from a given probability distribution.distribution.

►Non-parametric statisticNon-parametric statistic can refer to a can refer to a statistic (a function on a sample) statistic (a function on a sample) whose interpretation does not depend whose interpretation does not depend on the population fitting any on the population fitting any parameterized distributions.parameterized distributions.

Why Non-ParametricsWhy Non-Parametrics

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

Why Non-ParametricsWhy Non-Parametrics

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

Non-Parametric ModelsNon-Parametric Models

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

Observed Tree Diameter Growth by Density Level

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Interpolate ObservationsInterpolate Observations

►Care is taken to Care is taken to sample across the response sample across the response surfacesurface..

►Original observations (x and y) may occur at Original observations (x and y) may occur at random pointsrandom points across this response surface. across this response surface.

►Observations of Y (dependent variable) are Observations of Y (dependent variable) are interpolated to interpolated to systematic intervalssystematic intervals of X. of X.

►YYii = Sum(Y = Sum(Ykk/(1 + (X/(1 + (Xkk-X-Xii))2 2 ) a weighted mean) a weighted mean

Observed and Interpolated Tree Diameter Growthat Systematic intervals of Density

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ClassY

Smooth Y Observation Smooth Y Observation TrendsTrends

►Least SquaresLeast Squares – every observation has an – every observation has an impact on every estimate.impact on every estimate.

►Binomial SmoothingBinomial Smoothing – only local observations – only local observations have effect on local estimate.have effect on local estimate.

►YYii = 0.0625*(Y = 0.0625*(Yi-2i-2) + 0.25*(Y) + 0.25*(Yi-1i-1) + 0.375*(Y) + 0.375*(Yii) + ) + 0.25*(Y0.25*(Yi+1i+1) + 0.0625*(Y) + 0.0625*(Yi+2i+2))

►Only Y values at regular intervals are used.Only Y values at regular intervals are used.►Resulting trend is loaded to FPS Library.Resulting trend is loaded to FPS Library.

Weights based on Pascal’s Weights based on Pascal’s TriangleTriangle

Non-Parametric SmoothingNon-Parametric Smoothing

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88 11 33 33 11

1616 11 44 66 44 11

WgtWgt 0.0620.06255

0.250.25 0.3750.375 0.250.25 0.0620.06255

Observed and Smoothed Tree Diameter GrowthAcross the Range of the Density Response Surface

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Observed Tree Diameter Growth by Density Levelwith Least Squares Regression estimation

y = -0.0024x + 1.9634R2 = 0.9226

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Linear (Obs#)

Observed Tree Diameter Growth by Density Levelwith Least Squares Regression estimation

y = -0.8987Ln(x) + 6.1756R2 = 0.9725

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Log. (Obs#)

Observed Tree Diameter Growth by Density Levelwith Least Squares Regression estimation

y = 3E-06x2 - 0.005x + 2.4326R2 = 0.9868

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Poly. (Obs#)

Observed Tree Diameter Growth by Density Levelwith Least Squares Regression estimation

y = 3.8499e-0.0043x

R2 = 0.9438

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Expon. (Obs#)

Observed and Smoothed Tree Diameter GrowthAcross the Range of the Density Response Surface

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The Data determines the model, not the Investigator.

Flewelling Parametric Taper ModelComparison to 5,656 Felled-Tree Non-Parametric

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Walters & Hann Parametric Taper ModelComparison to 5,656 Felled-Tree Non-Parametric

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Smoothed Non-Parametric Taper ProfilesAcross the Range of Observed Taper Dimensions

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