Evaluating FVS-NI Basal Area Increment Model Revisions under Structural Based Prediction

Robert Froese, Ph.D., R.P.F.School of Forest Resources and Environmental ScienceMichigan Technological University, Houghton MI 49931

This presentation has four parts

Introduction

Approach

Relevance

Performance

The issue, the question and the model formulations examined

The methods and the data sets

How does SBP affect model building and model application?

How do data structure, assumptions and the methodology interact?

This presentation has four parts

The issue, the question and the model formulations examined

The methods and the data sets

How does SBP affect model building and model application?

How do data structure, assumptions and the methodology interact?

Introduction

Approach

Relevance

Performance

This presentation has four parts

The issue, the question and the model formulations examined

The methods and the data sets

How does SBP affect model building and model application?

How do data structure, assumptions and the methodology interact?

Introduction

Approach

Relevance

Performance

This presentation has four parts

The issue, the question and the model formulations examined

The methods and the data sets

How does SBP affect model building and model application?

How do data structure, assumptions and the methodology interact?

Introduction

Approach

Relevance

Performance

Competition variables have sampling error that varies in forestry problems

• Why?

– Inventory plot size and density are far from standardized in forestry

– As a stand becomes patchy or older, sampling variance increases

• Sampling errors attenuate regression coefficients towards zero, leading to type II errors in model development

• If sampling variance of predictors is different between fitting and application ordinary least squares (OLS) regression coefficients are not unbiased

Fuller (1987) derived an unbiased estimator for the underlying linear structural model

%β =M̂ xx−1M̂ xy

where

M̂ xy = n−1 Xt'Yt − Σuwtt( )

t

n

∑M̂ xx = n−1 Xt

'Xt − Σuutt( )t

n

∑

Σuw

Σuu

is a vector of covariances between errors in Y and X

is a matrix of error variances and covariances for errors in X

Stage and Wykoff (1998) developed Structural Based Prediction

1. Derive estimators for sampling variance of competition variables

2. Estimate coefficients following Fuller’s (1987) logic

3. Revise coefficients during simulation to take into account the current estimate of sampling variance

β̂ = M̂xx + Σ̂uutt⎡⎣ ⎤⎦−1

M̂xy + Σ̂uwtt⎡⎣ ⎤⎦

This study had two objectives

1. Wykoff (1997) and Froese (2003) tested revisions using OLS and afterwards fit the model using SBP; would they have reached the same conclusions if they tested revisions using SBP?

2. SBP has not been tested on independent data. Does SBP perform in practice according to theory; namely, are predictions made using the DDS model fit using SBP less biased than predictions made with the model fit using OLS?

The Prognosis BAI model is a multiple linear regression on the logarithmic scale

ln DDS( ) =HAB+ LOC +b1 ⋅ln DBH( ) +b2 ⋅LOC : DBH

+b3 ⋅cos ASP( )⋅SL +b4 sin ASP( )⋅SL +b5 ⋅SL +b6 ⋅SL2 +b7 ⋅EL +b8 ⋅EL2

+b9 ⋅CR+b10 ⋅CR

ln DBH +1( )+b11 ⋅

SBADBH

+b12 ⋅HAB :SBA+b13 ⋅(1−P90)⋅PBA

Wykoff 1997

Froese 2003

ln DDS( ) =HAB+ LITH +b1 ⋅ln DBH( ) +b2 ⋅DBH

+b3 ⋅cos ASP( )⋅SL +b4 sin ASP( )⋅SL +b5 ⋅SL +b6 ⋅SL2 +b7 ⋅ANP +b8 ⋅GSP +b9 ⋅GST

+b10 ⋅CR+b11 ⋅CR

ln DBH +1( )+b12 ⋅

SBADBH

+b13 ⋅HAB :SBA+b14 ⋅(1−SPCT )⋅PBA

The approach involves two parts

• evaluating model revisions– Froese 2003 revisions form the basis– Repeat under SBP using FIA data– compare RMSE of prediction residuals under

OLS and SBP

• testing on independent data– use the Froese 2003 model formulation, fit using

FIA data under OLS and SBP– generate predictions for independent testing data– compare bias and RMSE of prediction residuals

under OLS and SBP

Introduction

Approach

Relevance

Performance

The fitting data came from FIA inventories in the Inland Empire

FIA “map” design8,295 trees (20%)

FIA “old” design32,754 trees (80%)

All increment data from increment cores!

7,932 trees (44%) from control plots10,659 trees (56%) from treated plots

The testing data came from the USFS Region 1 Permanent Plot Program

• Installed in managed stands, mostly pre-commercial thinning• Control plots were left untreated• Geographically restricted to National Forests

– Coeur d’Alene, Flathead, Kanisku, Kootenai, Lolo and St. Joe

• Diameter increment from successive re-measurements, not cores

Evaluating model revisions

similar results for all species:

– when change in precision due to revisions in the model formulation is assessed the outcome of revisions is more favourable under SBP

– when change in precision due to the model framework is assessed, SBP always results in a degradation in model performance, but the degradation is less for every species under the revised DDS model formulation developed in Chapter 4

– SBP increased RMSE by 1.0 - 4.8% for the Wykoff 1997 version, but only 0.7 – 3.6% for the Froese 2003 version

Introduction

Approach

Relevance

Performance

SBP usually reduced bias as expected when applied to independent data

SpeciesOLS Bias SBP Bias PRMSE (bias corrected)

Abs. Rel. % Abs. Rel. % OLS SBP Diff. %

ABGR -0.02 -0.5 -0.11 -2.4 0.630 0.641 1.7

ABLA -0.08 -1.8 -0.08 -1.7 0.604 0.579 -4.3

HARD -0.45 -11.5 -0.44 -11.2 0.662 0.643 -3.0

HIEL 0.03 0.8 0.02 0.5 0.459 0.417 -10.1

LAOC -0.20 -4.8 -0.16 -3.9 0.610 0.597 -2.2

PICO -0.16 -3.7 -0.16 -3.8 0.620 0.588 -5.4

PIEN -0.26 -6.0 -0.22 -5.0 0.616 0.595 -3.5

PIPO -0.49 -10.5 -0.52 -11.3 0.678 0.667 -1.6

PSME -0.24 -5.5 -0.24 -5.5 0.586 0.583 -0.5

THPL 0.09 2.1 0.05 1.2 0.677 0.695 2.6

Precision was improved more consistently with SBP

SpeciesOLS Bias SBP Bias PRMSE (bias corrected)

Abs. Rel. % Abs. Rel. % OLS SBP Diff. %

ABGR -0.02 -0.5 -0.11 -2.4 0.630 0.641 1.7

ABLA -0.08 -1.8 -0.08 -1.7 0.604 0.579 -4.3

HARD -0.45 -11.5 -0.44 -11.2 0.662 0.643 -3.0

HIEL 0.03 0.8 0.02 0.5 0.459 0.417 -10.1

LAOC -0.20 -4.8 -0.16 -3.9 0.610 0.597 -2.2

PICO -0.16 -3.7 -0.16 -3.8 0.620 0.588 -5.4

PIEN -0.26 -6.0 -0.22 -5.0 0.616 0.595 -3.5

PIPO -0.49 -10.5 -0.52 -11.3 0.678 0.667 -1.6

PSME -0.24 -5.5 -0.24 -5.5 0.586 0.583 -0.5

THPL 0.09 2.1 0.05 1.2 0.677 0.695 2.6

Predictions are similar in magnitude under each method, with exceptions

Trends in residuals across stand basal area were slightly improved with SBP

Results for Pseudotsuga menziesii

Trends in residuals across PBAL were also slightly improved with SBP

Results for Pseudotsuga menziesii

SBP effects may be overwhelmed by poor model performance on these data

Results for Pseudotsuga menziesii

The effect of SBP is confounded with other issues in the test and test data

• The test data are different in more ways than sampling design

• SBP would be enhanced by methodological revisions– Poisson model– Estimation algorithm

Introduction

Approach

Relevance

Performance

SBP produces stable results despite complexity and confounding influences

• model testing very encouraging– bias reduced for all species

but those that have other problems

– precision actually improved for most species

• at minimum, these results suggest model users need not fear spurious results using the DDS model implemented with SBP

Summary

Model revision decisions are insensitive to regression methodology

SBP increases RMSE but decreases PRMSE

SBP reduces bias in most situations as expected

Methodological revisions are desirable

1−P90( )⋅PBA⇒

1−SPCT( )⋅PBA

Larix occidentalis

RMSE +1.7%PRMSE −2.2%

Larix occidentalis

OLS BIAS -4.8%

SBP BIAS -3.9%