Transcript GROWTH & YIELD MODELLING IN NEW ZEALAND

WHY ARE YOU USING THAT
REGRESSION?
Western Mensurationist Meeting
Jim Flewelling
July, 2003
FOCUS
• POPULATIONS
– VARIANCE IN RELATIONSHIPS
• OBJECTIVES
– USE OF REGRESSION
• TECHNIQUES are SECONDARY
TWO WORLDS
• SURVEY SAMPLING
– Fixed Populations
– Objective refers to Population
• REGRESSION ANALYSIS
– Relationships between variables
– Objectives refer to individuals or populations
SURVEY SAMPLING
• Fixed Population.
• Specified probability-sampling processes.
• Estimation of population parameters
– unbiased estimators.
SURVEY SAMPLING
“If we are to infer from sample to population,
the selection process is an integral part of
the inference.”
- Stuart (1984, p. 4)
REGRESSIONS IN SURVEY
SAMPLING
• AUXILIARY INFORMATION (X)
– known for population.
– Increased precision.
• MODEL-ASSISTED ESTIMATORS
COMMON (Särndal et al.,1992)
• MODEL-BASED ESTIMATORS
MODEL-ASSISTED SURVEY
SAMPLING
Ratio of Means Estimator:
YˆR  ( y / x )  X
Asymptotically unbiased,
whether or not y proportional to x.
Could be used to estimate individual y’s.
No claim of unbiasedness here.
MODEL-BASED SURVEY
SAMPLING
• Assumptions from Regression Analysis.
– True model
– E(e|x) = 0
– Errors are independent.
• Random selection avoids a source of bias.
• Inference from regression theory, not the
distribution of samples.
• Theory from Royall (1970).
REGRESSION ANALYSIS
• Least Squares - Legrendre (1805) and Gauss.
• Sir Francis Galton (1877, 1885):
Offspring of seeds “did not tend to resemble their parent
seeds in size, but to be always more mediocre [i.e.,
more average] than they - to be smaller than the
parents, if the parents were large … the mean filial
regression towards mediocrity was directly proportional
to the parental deviation from it.” (quoted from Draper
& Smith)
LEAST SQUARES
REGRESSION
yˆ  y  ˆ  (sy / sx )  ( x  x )
Var
< Var(y)
GEOMETRIC MEAN
REGRESSION
yˆ  y  (sign ˆ )  (sy / sx )  ( x  x)
Preserves Variance
Discussion by Ricker (1984)
HEIGHT-AGE CURVES
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Site Curves (Curtis)
Site Index Prediction Functions
Geometric Mean Regression
Stochastic Differential Equation
Height Growth Models
Percentile Models
Site Curves and SI Prediction
Functions
• Curtis et al. (1974)
• Site Curve - Yield table construction
– H = f(A, SI).
• SI Prediction Function - Site Classification
– SI = f(A, H).
SITE CURVES, SI
PREDICTION, and GMR
SI = H (index age)
HA = H (age A)
3 Lines:
All at mean (HA, SI)
Slope = SI/HA
{ , 1, 1/ }
Straight-line
assumption valid for
bivariate normal.
Stochastic Differential Equation
(Garcia, 1979)
• dH/dt = (b/c)H{(a/H)c -1}
– b is plot-specific, (a, c) are global.
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Integrates to Chapman-Richards.
Add Wiener process error to growth.
Add measurement errors at intervals.
Fit with Maximum Likelihood.
It’s a growth-model; also base-age invariant
site curve.
Height Growth Model
• Family of H-A curves.
• From any one age, predict height difference to next or
previous age.
• Parameters adjusted to minimize errors in predicted
growth. (Bonnor et al, 1995), Flewelling et al (2001).
• Crude, ignores measurement errors, and correlations
between periods. Flexible model form.
• It’s a growth model - attempts to model H-A trajectories
of plots. Base-age invariant.
Percentile Models
• Concept by Pienaar and Clutter (Clutter et
al, 1983).
• Example by Bi (2002).
• Extends to irregular data. (Flewelling, 1982,
unpublished).
• Current econometrics theory, rich history.
Percentile Models
• Pienaar and Clutter:
Percentiles as a labeling device: “useful in
illustrating the fact that index age is not a
fundamental or required concept in the use
of site index to express site quality.”
Percentile Models, Example
• Bi et al ( 2002)
• Temporary plots (age and site assumed
orthogonal).
• H(t) assumed to have normal distribution.
• Q0.75 and Q0.25, fit as functions of t.
– methodology from Koenker and Bassett (1978)
• Mean H(t) fit with weighted regression.
Percentile Model, Irregular data.
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Sectioned tree data, height every year.
Younger ages: full data set.
Older ages: reduced data set.
Establish tree percentiles at young age.
Reassign censored percentiles older ages.
Compute (and model) means and standard
deviations from heights and percentiles.
Percentile models, econometrics
• Koenker (2000):
• wonderful discussion of least squares,
alternative methods, and statistical history.
• Minimization of summed absolute errors
dates from 1760’s.
Height-Age Curves. Questions
• Should height growth models be the same
as constant percentile curves?
• Are regressions from one age to another
wanted?
• Is there any use for an index age other than
as a label?
POPULATIONS
WHICH PROJECTION IS WANTED?
TREE GROWTH MODELS
•  DBH
• Mortality fractions.
• What ensures that the variance of projected
stand table is correct?
– Need variance models as constraints?
– Different fitting techniques?
– Good luck and occasional checking?
RIGHT INDEPENDENT
VARIABLES?
Regional H-DBH
Curves.
Biased by Age or
position in stand.
Alternative:
local curves,
another variable.
Bayesian Regression
• Neglected in Forestry?
• Empirical Bayes used in volume equations
(Green and Strawderman, 1985).
• Taper and volume equations by forest
district (McTague, Stansfield and Lan,
1992).
• Other opportunities?
Bayesian Opportunity
• Fit y = a0 + a1x1 + a2x2 + a3x3 + …..
– Often by species or other category.
– Coefficients tested and omitted if nonsignificant.
– Or, selected coefficients fit in common for all
species.
– Bayesian regression or other methods better?
OTHER REGRESSION
TECHNIQUES
• ML with better error characterization.
• Mixed models.
• Systems: Seemingly unrelated regression,
2SLS, 3SLS ……..
• Generally are more efficient, better
estimates of parameter variance, possibly
avoid some biases. Necessary?
• Imputation?
SUMMARY
• What does population look like?
• What should be described?
• What techniques allow that?
REFERENCES
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Bi, H., A.D. Kozek and I.S. Ferguson. 2002. Quantile-based site index curves: a brief
introductory note. Proc of IUFRO Symposium on Statistics and Technology in Forestry,
Sept 8-12, 2002 Blacksburg. [ May be a related 2003 paper in J of Agr, Biological, and
Environmental Statistics.]
Bonnor, G.M., R. J. DeJong, P. Boudewyn and J. Flewelling. 1995. A guide to the STIM
growth model. Nat. Res. Canada. Info Rpt X-353.
Clutter, J.L., J.C. Fortson, L.V. Pienaar, G.H. Brister and R.L. Bailey. 1983. Timber
management: a quantitative approach. Krieger Publ., Malamar, FL. 333 p.
Curtis, R.O., D.J. Demars, F.R. Herman 1974. Which dependent variable is site index height - age relationships? For. Sci. 20: 74-87
Draper, N. R. and H. Smith. 1998. Applied Regression Analysis. Wiley. New York. 706
p.
Flewelling, J. 1982. Dominant height trends for plantations of loblolly pine at the
Mississippi/Alabama region of Weyerhaeuser Company. Research Rpt 050-3415/3.
Weyerhaeuser Forestry Research, Hot Springs. (unpublished)
Flewelling, J., R. Collier, B. Gonyea, D. Marshall and E. Turnblom. 2001. Height-age
curves for planted stands of Douglas fir, with adjustments for density. SMC Working
Paper No. 1, Univ. of WA, Seattle.
REFERENCES
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Garcia, O., 1979. A stochastic Differential Equation Model for height growth of forest
stands. Biometrics 39: 1059-1072.
Green, E. and W.E. Strawderman. 1985. The use of Bayes/Empirical Bayes Estimation
in Individual Tree Volume Equation Development. For. Sci. 31: 975-990.
Koenker, R. 2000. Galton, Edgeworth, Frisch, and prospects for quantile regression in
econometrics. J of Econometrics 95: 347-374.
Koenker, R.W. and G.W. Basset. 1978. Regression Quantiles. Econometrica 50, 43-61.
McTague, J.P., W.F. Stansfield, Z. Lan. 1992. Southwestern ponderosa pine, Douglas fir
and white fir volume and taper functions. Report to USFS. Northern Arizona University.
Ricker, W.E. Computation and uses of central trend lines. Can. J. Zool. 62:1897-1905
Royall, R.M. 1970. On finite population sampling theory under certain linear regression
models. Biometrika 57: 377-387.
Särndal, C., B. Swensson, J. Wretman . 1992. Model assisted survey sampling. SpringerVerlag, New York. 694 p.
Stuart, A. 1984. The ideas of sampling. Macmillan, New York. 91 p.
COMMENTS?
QUESTIONS?