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

2. 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.

3. Study Data:

4. 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 =

5. 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)

6. 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

7. 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

8. 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

9. 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

10. 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

11. 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)

12. 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

13. { } 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 =

14. 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 +

15. Fits of Error Type Probabilities0%
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)

16. 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.

17. 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)

18. 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)

19. Parameter Modeling: Fitted Exponential Coefficients Dbh Class20 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 Dbh Dbh-2)

20. 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)

21. 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 *Dbh-1)*(error size)0.59]
adjusted R2 =
Positive fit R2 =

22. 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)

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

24. 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
Normal 2
Normal 3
Normal 4
Normal 5
6.8926

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

26. "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