Toward a Characterization of Measurement Error Sean Canavan David Hann Oregon State University

Recall: Measurement error enters into forestry in many different ways and forms The errors can have very negative effects on model parameters, model estimates, and the variances of model parameters and model estimates. Correction techniques do exist for countering the effects of measurement errors in many situations, but typically require knowing something about the form of the errors. People have generally made the assumption that the errors are Normal in distribution.

Study Data:

Study Data: Dbh: n = 2175 < 0 : 529, = 0 : 368, > 0 : 1278 < 0 : 529, = 0 : 368, > 0 : 1278 0.8” – 72.1” Species: DF, TF, PP, SP, IC Ht: n = 1238 < 0 : 722, = 0 : 30, > 0 : 486 8.4’ – 231.7’ Positive fit R2 = 0.9486

The Normal Assumption: It is often assumed that measurement errors follow a Normal distribution - (Nester 1981, Garcia 1984, Smith 1986, Päivinen & Yli-Kojola 1989, Gertner 1991, McRoberts et al. 1994, Kozak 1998, Kangas 1998, Kangas & Kangas 1999, Phillips et al. 2000, Williams & Schreuder 2000) Bias assumption: μ = 0 Variance assumption: homogeneous (σ2 constant) heterogeneous (σ2 not constant)

Normal(0,1) PDF 0.45 0.4 0.35 0.3 0.25 f(x) 0.2 0.15 0.1 0.05 -5 -4 -3 -2 -1 1 2 3 4 5 x

The Normal Assumption: It is often assumed that measurement errors follow a Normal distribution - (Nester 1981, Garcia 1984, Smith 1986, Päivinen & Yli-Kojola 1989, Gertner 1991, McRoberts et al. 1994, Kozak 1998, Kangas 1998, Kangas & Kangas 1999, Phillips et al. 2000, Williams & Schreuder 2000) Bias assumption: μ = 0 Variance assumption: homogeneous (σ2 constant) heterogeneous (σ2 not constant) What happens when there are many correct measurements? example: Dbh measured to a tenth of an inch

Measurement Error Value Cumulative Probability 25% Correct 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 -1.5 -1 -0.5 1 1.5 nsig = 115 50% nsig = 50 75% nsig = 12 100% nsig = 6 Measurement Error Value Cumulative Probability

Error Distribution Modeling: First Approach: PDF modeling Second Approach: CDF modeling Part 1: Modeling Error Type Probabilities Part 2: Modeling the Positive and Negative error portions of the curve

Normal(0,1) CDF 1 0.9 0.8 0.7 0.6 F(x) = P(X < x) 0.5 0.4 0.3 0.2 0.1 -5 -4 -3 -2 -1 1 2 3 4 5 x

Empirical Dbh Error CDF Surface 1.00 0.75 Cumulative Probability 0.50 0.25 0.0 10.0 20.0 Dbh (inches) 30.0 40.0 1.8 1.2 0.6 0.0 -0.6 Error (inches)

Error Distribution Modeling: First Approach: PDF modeling Second Approach: CDF modeling Part 1: Modeling Error Type Probabilities Part 2: Modeling the Positive and Negative error portions of the curve

{ } Fitted CDF Equation: P(X = x) = Pr(ε < 0)*Negative Error CDF ε < 0 Pr(ε < 0) + Pr(ε = 0) ε = 0 Pr(ε < 0) + Pr(ε = 0) + Pr(ε > 0)*Positive Error CDF ε > 0 P(X = x) = 0.00 0.20 0.40 0.60 0.80 1.00 -1.5 -1 -0.5 0.5 1 1.5 Error Size Cumulative Probabiility } Pr(ε = 0) Pr(ε < 0) Pr(ε > 0) Positive fit R2 = 0.9486

Part 1: Error Type Probability Modeling Multinomial Regression in S-Plus GLM with a Poisson link function Overdispersion/Quasilikelihood Counts by 1-inch Dbh Classes / 5-ft. & 10-ft. Ht Classes Candidate predictors: Dbh, Dbh½, Dbh2, Dbh-1 Ht, Ht½, Ht2, Ht-1 Probability model forms: ) ( 2 1 i Dbh f e + ) ( 2 1 Dbh f e +

Fits of Error Type Probabilities 0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100% 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 Dbh (inches) Probability P(e > 0) P(e = 0) P(e < 0)

Part 2: Modeling Positive and Negative CDFs Negative Errors: CDFs by Dbh class Step 1: Exponential fits model form: exp(β*Error Size) actually fit: 1 – exp(β*Error Size) Step 2: Parameter Modeling βi = f(Dbh) Step 3: Combined Equation Fit 1 – exp(f(Dbh)*Error Size) So we want a function that looks like an exponential, but changes as Dbh changes. Or, we want to fit exponential functions by Dbh and look at how the parameter changes with Dbh.

Cumulative Probability 21.5” Class 2.5” Class 0.2 0.4 0.6 0.8 1 -1 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 21.5” Class 2.5” Class 45.0” Class Error Size (inches)

Part 2: Modeling Positive and Negative CDFs Negative Errors: CDFs by Dbh class Step 1: Exponential fits model form: exp(β*Error Size) actually fit: 1 – exp(β*Error Size) Step 2: Parameter Modeling βi = f(Dbh) Step 3: Combined Equation Fit 1 – exp(f(Dbh)*Error Size)

Parameter Modeling: Fitted Exponential Coefficients Dbh Class 20 40 60 80 100 120 5 10 15 25 30 35 45 50 Dbh Class Fitted Exponential Coefficients 1000 1200 10.04exp(-0.03Dbh + 1.77Dbh-1 + 0.59Dbh-2)

Part 2: Modeling Positive and Negative CDFs Negative Errors: CDFs by Dbh class Step 1: Exponential fits model form: exp(β*Error Size) actually fit: 1 – exp(β*Error Size) Step 2: Parameter Modeling βi = f(Dbh) Step 3: Combined Equation Fit 1 – exp(f(Dbh)*Error Size)

Combined equation fit: Variable power on error size: 1 – exp[b0*exp(b1Dbh + b2Dbh-1 + b3Dbh-2)*(error size)c1] Resulting CDF equation: exp[10.04*exp(-0.03*Dbh + 1.77*Dbh-1)*(error size)0.59] adjusted R2 = 0.8664 Positive fit R2 = 0.9486

Fitted Dbh Error CDF Surface 1.00 0.75 Cumulative Probability 0.50 0.25 0.00 8 16 Dbh (inches) 24 32 40 2.00 1.00 0.00 -1.00 Error (inches)

Alternative Surfaces (Dbh): Normal 1: Unbiased, homogeneous Normal: μ = 0.0, σ = 0.2237 Normal 2: Constant bias, homogeneous Normal: μ = 0.0901, σ = 0.2237 Normal 3: Non-constant bias, homogeneous Normal: μ = 0.003983*Dbh + 0.000121*Dbh2, σ = 0.2237 Normal 4: Unbiased, heterogeneous Normal: μ = 0.0, σ = σD*exp[0.1145*Dbh] Normal 5: Non-constant bias, heterogeneous Normal: μ = μ = 0.003983*Dbh + 0.000121*Dbh2, σ = σD*exp[0.1145*Dbh]

Sum of Squared Differences Comparison of Surface Fits: Sum of Squared Differences Distribution Dbh (n=2175) Ht (n=1238) Ours 4.7893 7.9583 Normal 1 41.1679 26.3670 Normal 2 61.5424 19.2941 Normal 3 39.1445 18.9771 Normal 4 20.3516 17.3235 Normal 5 19.4147 6.8926

Conclusions: Case of many correct measurements Case of few correct measurements Drawing random samples Species differences Changing precision levels: Dbh: 0.1”  1.0”  368  1087 out of 2175 Ht: 0.1’  1.0’  30  274 out of 1238

"Sampling gets you to the final answer, if you do it often enough. Measuring everything correctly gets you to the correct answer. Don't get those mixed up." Olde Statistical Sayings Inventory and Cruising Newsletter Issue No. 32, October 1995