# Comparison of Model-Assisted Endogenous Poststratification Methods for Estimation of Above-Ground Biomass Change in Oregon, USA

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## Abstract

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## 1. Introduction

#### Objective

- A GREG estimator, GREG-EPS, based on the interactions of multiple categorical variables [13].
- An EPS estimator using the method proposed by Breidt and Opsomer [11] that uses a generalized linear model to form strata, GL-EPS.
- An EPS estimator using McConville and Toth’s [14] method based on recursive partitioning trees, TREE-EPS.

## 2. Material and Methods

#### 2.1. Study Area

^{2}in size, and 48.4% is forested land, of which 64% is publicly owned [15]. Oregon has very strong west-to-east environmental gradients. The area west of the Cascade Range is moist and mild, with forest dominated by Douglas-fir, Pseudostuga menziesii (Mirbel) Franco, red alder, Alnus rubra Bong, western hemlock and Tsuga heterophylla (Raf.) Sarg. Areas east of the Cascade Range have a dry, continental climate, and the forests are dominated by ponderosa pine, Pinus ponderosa Lawson & C. Lawson, lodgepole pine, Pinus contorta Douglas ex Loudon, and western juniper, Juniperus occidentalis Hook.

#### 2.2. Auxiliary Information

#### 2.3. Estimation Framework

#### 2.3.1. Population

#### 2.3.2. Target Parameter

#### 2.3.3. PNW-FIA Sampling Design and Sample

#### 2.4. Development of Models for EPS

#### 2.4.1. General Model Selection Considerations

#### 2.4.2. GREG-EPS

- First, arranged all combinations on a tree that was constructed using the variables $CATACDIS{T}_{t}$, $ECO$ and $OWN$. Branches in the first level of the tree were defined using $CATACDIS{T}_{t}$, and branches for the second and third levels were based on $ECO$ and $OWN,$ respectively. Each combination of $ECO$,$OWN$,$CATPCPI$, $CATACDIS{T}_{t}$, $CATPCPI$, $MAXFSE{V}_{t}$, and $\mathrm{\Delta}NLC{D}_{t}$, resulted in a leaf that was placed in its corresponding branch depending on $CATACDIS{T}_{t}$, $ECO$ and $OWN$.
- Leaves with four plots per year or more were set as fixed leaves. The remaining leaves were merged with other leaves (fixed or not) in the same branch. The merging process ran separately in each branch and started with the leaf with a smaller area in the branch. The selected leaf was merged with the closest leaf in the same branch. The Gower distance [21], computed with all categorical variables, was used to determine which leaf was the closest to the selected leaf. This distance was selected because it allows treating differently categorical variables with an implicit ordering (i.e., all categorical variables derived from a continuous one) and categorical variables without an implicit ordering (e.g., $OWN)$. The two leaves were merged into a single leaf, and the expected number of plots per year was recomputed based on the area of the group resulting from the merge. If the expected number of plots per year of the resulting leaf was four or more, or if the small leaf was merged to a fixed leaf, the result was tagged as fixed, and it was not considered as a target for further merging steps.
- Step 2 was repeated until all the resulting leaves in a branch had an expected number of four plots per year or all leaves in one branch were merged into a single leaf.
- When merging all leaves in one branch did not yield an area from which to expect four plots per year, the merging process continued but considered merging groups from branches of the previous level of the tree (i.e., the algorithm continued with branches defined by $CATACDIS{T}_{t}$ and $ECO$ first, and then with branches defined by $CATACDIS{T}_{t}$ only).

#### 2.4.3. GL-EPS

- The maximum value of $\mathrm{\Delta}AG{B}_{it}$ reported by [16] was 8.75 Mg ha
^{−1}year^{−1}. Based on this value, we defined the following five positive intervals $\left(0,\left.2\right],\right.$ $\left(2,\left.4\right],\right.\left(4,\left.6\right]\right.$ $\left(6,\left.8\right],\right.$ and $\left(8,\infty \right)$. - For disturbances causing losses in forest AGB with magnitudes comparable to growth, we used the thresholds used for growth but with negative signs. The last interval $(-\infty ,\left.-8\right]$ accommodates large and negative values of $\mathrm{\Delta}AG{B}_{it}$ occurring after stand-replacing disturbances such as clear cuts.

#### 2.4.4. TREE-EPS

#### 2.5. Estimators of $\Delta AGB$ and Variance Estimators

#### 2.5.1. Approximation to Sampling Design Weights and Point and Variance Estimators

#### 2.5.2. Point and Variance Estimators

#### 2.5.3. Comparison to Current PNW-FIA Estimators and Horvitz-Thompson Estimators

## 3. Results

#### 3.1. EPS and PS Assisting Models and Summaries

^{2}, for the assisting models for EPS methods ranged between 35.93% to 40.19%, while R

^{2}for FIA-PS was 20.45%. Root mean squared errors, RMSE, for the assisting models ranged from 3.66 Mg ha

^{−1}year

^{−1}to 3.19 Mg ha

^{−1}year

^{−1}for EPS methods, while RMSE for FIA-PS was 4.19 Mg ha

^{−1}year

^{−1}. The proportion of plots with observed and predicted values of $\mathrm{\Delta}AGB$ with the same sign (i.e., positive changes predicted as positive and negative changes predicted as negative changes) ranged from 84.25% to 84.94% for EPS methods. For FIA-PS, this percentage was 74.13%. These results indicate that the EPS variants analyzed in this study can improve the precision of the current estimates of $\mathrm{\Delta}AGB$. GL-EPS was the method that experienced the largest improvements in terms of Adj-R

^{2}when $\mathrm{\Delta}CM{S}_{t}$ was added to the stack of auxiliary variables, Adj-R

^{2}was 36.38% for GL-EPS and 40.19% for GL-EPS CMS. Adding $\mathrm{\Delta}CM{S}_{t}$ to the pool of auxiliary variables had a minor effect on GREG-EPS (i.e., Adj-R

^{2}was 37.15% for GREG-EPS and 38.38% for GREG-EPS-CMS). The model for TREE-EPS had an Adj-R

^{2}of 36.25% and explained slightly more variance than the model for TREE-EPS-CMS, with an Adj-R

^{2}of 35.93%.

#### 3.2. Estimates of Changes for the State for Specific 10-Year Periods

^{−1}year

^{−1}to 0.88 Mg ha

^{−1}year

^{−1}depending on the 10-year period and estimator. Differences between estimated totals for different periods tended to be of larger magnitude than differences between methods for a given period (Figure 4).

#### 3.3. Estimators of Running Means

^{−1}year

^{−1}and 0.65 Mg ha

^{−1}year

^{−1}, and confidence intervals for the estimates provided by any method contained the estimates provided by the other methods (Figure 7). For the running means of eight 10-year periods, adding $\mathrm{\Delta}CM{S}_{t}$ to the pool of auxiliary variables consistently resulted in values of $\mathrm{\Delta}{\hat{V}}_{M}$ larger than those obtained by EPS models not using this auxiliary variable. Differences between GL-EPS and GL-EPS-CMS were largest at 6.7%, i.e., 38.3% vs. 45.0%, and differences between GREG-EPS and GREG-EPS-CMS and TREE-EPS and TREE-EPS-CMS were of small magnitude < 1%, i.e., 36.1% vs. 36.23% and 33.95 vs. 34.2%, respectively.

## 4. Discussion

#### 4.1. Similarities between Model-Assisted Estimators

^{−1}year

^{−1}and 0.65 Mg ha

^{−1}year

^{−1}. These results indicate a consistent increase in forest AGB in the study area but do not inform about the type of changes that are occurring. The accumulation of forest AGB can result from many processes of a very different nature and with very different ecological implications. For example, both: (1) increases in the amount of land sustaining mature forest structures with larger trees and more fire resilient structures, and (2) increases in the amount of young and dense forest structures resulting from fire suppression efforts contribute to total forest AGB accumulation. The ecological implications of each type of increase mentioned above are very different and could be evaluated by a more detailed analysis of the field-measured plot data. However, PS and EPS allow constructing additive tables for non-overlapping categories in a straightforward manner. Thus, these methods can be used to derive estimates of change by typologies.

#### 4.2. Model Selection

#### 4.3. Differences between Model-Assisted Estimators

#### 4.4. Other Considerations of Practical Importance

#### 4.4.1. Estimation of Variance and Estimation of Change for Periods Not Matching the PNW-FIA Panel Frequency

#### 4.4.2. Auxiliary Variables

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**Location of the study area, outlined in black, and Estimation Units, EU, defined by the US Forest Service Forest Inventory and Analysis program in Oregon. OL, Other lands, NF, National Forest, lands not designated as wilderness areas (intensified sample), and WL, National Forest lands designated as wilderness areas.

**Figure 2.**Observed versus predicted plots for the models used to derive EPS estimators and the PS model of PNW-FIA. Predicted values were obtained using the models fitted to the sample of the eight 10-year periods used in the model selection. The adjusted coefficient of determination (Adj-R2), root mean squared error (RMSE), (+,+) & (−,−) indicate the percentage of ground plots where the observed value had the same sign as the predicted value (i.e., positive changes predicted as positive changes and vice versa), (−,+) & (+,−), percentage of ground observations with positive changes of AGB but negative predicted values of change, and vice versa.

**Figure 3.**Maps depicting the change in above-ground biomass (AGB) were created using different EPS estimators for the period 2004–2014. Stratum colors were assigned based on the estimated stratum means. Red colors correspond to negative changes in stratum means (i.e., losses in AGB). In addition, green colors correspond to positive changes in stratum means (i.e., gains in AGB).

**Figure 4.**Estimated change and estimated 95% confidence intervals were computed as ${\widehat{\overline{\mathrm{\Delta}AGB}}}_{t}\pm 1.96\sqrt{{\widehat{V}}_{M}\left({\widehat{\overline{\mathrm{\Delta}AGB}}}_{t}\right)}$ for the periods 2001–2011 to 2008–2018 for all methods under analysis, including the Horvitz-Thompson estimator and the post-stratification estimators currently used by PNW-FIA.

**Figure 5.**Estimated increases in performance, $\mathrm{\Delta}{\hat{V}}_{M}$, with respect to the Horvitz Thompson, HT, of FIA-PS and EPS estimators. $\mathrm{\Delta}{\hat{V}}_{M}$ is 1 minus the ratio of the variance of the FIA-PS or EPS estimator to the variance of the HT estimator.

**Figure 6.**Estimated change for all periods considered in the study (

**left**). Estimated improvements with respect to the Horvitz-Thompson estimator (

**middle**) and estimated improvements of endogenous poststratification methods with respect to the poststratification currently used (

**right**). Note that single 10-year periods appear in the diagonal.

**Figure 7.**Estimated change and estimated 95% confidence intervals were computed as ${\widehat{\overline{\mathrm{\Delta}AGB}}}_{2001,2018}\pm 1.96\sqrt{{\widehat{V}}_{M}\left({\widehat{\overline{\mathrm{\Delta}AGB}}}_{2001,2018}\right)}$ for the running mean for the periods 2001–2011 to 2008–2018 for all methods under analysis, including the Horvitz-Thompson estimator and the post-stratification estimators currently used by PNW-FIA.

**Table 1.**Auxiliary information database summary. Categorical variables (Ca), Continuous variables (Co). Categorical variables derived from continuous ones are marked with an asterisk (i.e., *Ca).

Type | Source | Variables, Acronym | Pre-Processing | Variable Type | Temporal |
---|---|---|---|---|---|

Proxies for potential forest AGB productivity | 1 Arc second Shuttle Radar Topography Mission, SRTM, Google earth engine | Elevation, $ELEV$ | Resampling bilinear interpolation | Co | Static |

Elevation categories, $CATELEV$ | Division in 3 elevation categories of equal area | *Ca | |||

Slope, $SLP$ | Computed from $ELEV$ | Co | |||

Heat load index, $HTL$ | Computed from $ELEV$ | Co | |||

800 m resolution PRISM 30-year normals & Sun hours from SRTM | Paterson climate productivity index, $PCPI$ | Resampling bilinear interpolation.Solar radiation ArcGIS tool | Co | ||

Categories Paterson climate productivity index, $CATPCPI$ | Division in 3 categories of equal area | *Ca | |||

US Forest Service | Cleland’s level 3 ecoregions, $ECO$ | Rasterization | Ca | ||

Ownership | Bureau of land management, BLM | Ownership, $OWN$ | Rasterization & reclassification | Ca | |

Proxies for disturbance | Monitoring trends in burn severity, MTBS. | Fire severity, $MAXFSE{V}_{t}$ | Maximum fire severity for 10-year periods. Resampling nearest neighbors | Ca | Dynamic |

Landscape Change Monitoring System, LCMS. | Disturbances, $ACDIS{T}_{t}$ | Computation of accumulated disturbances for 10-year periods | Co | ||

MTBS- LCMS | Disturbance-categories, $CATACDIS{T}_{t}$ | Reclassification of $MAXFSE{V}_{t}$ and thresholds for $ACDIS{T}_{t}$ | *Ca | ||

MRLC National Land Cover Database, NLCD. | Land cover change, $\mathrm{\Delta}NLC{D}_{t}$ | Resampling nearest-neighbor. Reclassification and computation of change | Ca | ||

Change in multi-year CMS AGB map | Fekety and Hudak, (2019) & Hudak et al., (2020) [8,9] | Independent prediction of $\mathrm{\Delta}AGB$ derived from Fekety and Hudak, (2019) [8] predictions of AGB for multiple years, $\mathrm{\Delta}CM{S}_{t}$ | Resampling with bilinear interpolation. Computation of differences in predicted forest AGB between years. | Co | |

Categories of $\mathrm{\Delta}AGB$ change derived from independent predictions of AGB for multiple years, $CAT\mathrm{\Delta}CM{S}_{t}$ | Reclassification of $\mathrm{\Delta}CM{S}_{t}$ based on intervals defined from values reported by [16] | *Ca |

**Table 2.**The total number of FIA plots used in model fitting was reported by time period and Estimation Units (EU). The number of excluded plots, which were classified as water in all NLCD maps, are also reported. Estimation units were: National Forest lands not designated as wilderness areas (NF), National Forest lands designated as wilderness areas (WL), and other lands (OL).

Period | Total Number of Plots | Number of Plots by EU | Excluded Plots | ||
---|---|---|---|---|---|

NF | OL | WL | |||

2001–2011 | 1310 | 675 | 592 | 28 | 15 |

2002–2012 | 1412 | 681 | 682 | 29 | 20 |

2003–2013 | 1402 | 687 | 656 | 29 | 30 |

2004–2014 | 1418 | 703 | 671 | 29 | 15 |

2005–2015 | 1420 | 704 | 662 | 33 | 21 |

2006–2016 | 1348 | 680 | 623 | 22 | 23 |

2007–2017 | 1331 | 650 | 645 | 18 | 18 |

2008–2018 | 1340 | 674 | 616 | 25 | 25 |

**Table 3.**Total number of strata, number of sampled strata, and proportion of area sampled by 10-year periods and for all 10-year periods combined.

2001–2011 | 2002–2012 | 2003–2013 | 2004–2014 | 2005–2015 | 2006–2016 | 2007–2017 | 2008–2018 | All Periods | ||
---|---|---|---|---|---|---|---|---|---|---|

GREG-EPS | Total # of strata | 84 | ||||||||

Sampled strata | 84 | 84 | 83 | 84 | 83 | 84 | 84 | 84 | 84 | |

% area sampled | 100.00 | 100.00 | 99.89 | 100.00 | 99.80 | 100.00 | 100.00 | 100.00 | 100.00 | |

GREG-EPS-CMS | Total # of strata | 97 | ||||||||

Sampled strata | 95 | 97 | 97 | 97 | 97 | 97 | 94 | 95 | 97 | |

% area sampled | 99.47 | 100.00 | 100.00 | 100.00 | 100.00 | 100.00 | 99.07 | 99.51 | 100.00 | |

GL-EPS | Total # of strata | 30 | ||||||||

Sampled strata | 21 | 20 | 19 | 21 | 20 | 18 | 18 | 20 | 27 | |

% area sampled | 99.75 | 99.87 | 99.80 | 99.85 | 99.74 | 99.83 | 99.56 | 99.78 | 100.00 | |

GL-EPS-CMS | Total # of strata | 30 | ||||||||

Sampled strata | 27 | 23 | 24 | 24 | 23 | 21 | 23 | 25 | 30 | |

% area sampled | 99.65 | 99.30 | 99.60 | 99.76 | 99.60 | 99.77 | 99.53 | 99.71 | 100.00 | |

TREE-EPS | Total # of strata | 44 | ||||||||

Sampled strata | 44 | 44 | 44 | 44 | 44 | 44 | 44 | 44 | 44 | |

% area sampled | 100.00 | 100.00 | 100.00 | 100.00 | 100.00 | 100.00 | 100.00 | 100.00 | 100.00 | |

TREE-EPS-CMS | Total # of strata | 34 | ||||||||

Sampled strata | 34 | 34 | 34 | 34 | 34 | 34 | 34 | 34 | 34 | |

% area sampled | 100.00 | 100.00 | 100.00 | 100.00 | 100.00 | 100.00 | 100.00 | 100.00 | 100.00 | |

FIA-PS | Total # of strata | 191 | ||||||||

Sampled strata | 167 | 173 | 175 | 177 | 171 | 166 | 163 | 167 | 191 | |

% area sampled | 96.79 | 97.68 | 97.64 | 97.43 | 97.50 | 96.61 | 95.40 | 96.92 | 100.00 |

Method | $\mathbf{Median}\mathbf{of}\mathrm{\Delta}{\hat{\mathit{V}}}_{\mathit{M}}$ for 10-Year Periods | $\mathbf{Mean}\mathbf{of}\mathrm{\Delta}{\hat{\mathit{V}}}_{\mathit{M}}$ for 10-Year Periods |
---|---|---|

GREG-EPS | 37.95% | 36.25% |

GREG-EPS-CMS | 40.13% | 35.91% |

GL-EPS | 48.02% | 38.36% |

GL-EPS-CMS | 47.99% | 42.27% |

TREE-EPS | 41.05% | 31.66% |

TREE-EPS-CMS | 40.27% | 30.16% |

FIA-PS | 28.24% | 20.68% |

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## Share and Cite

**MDPI and ACS Style**

Mauro, F.; Monleon, V.J.; Gray, A.N.; Kuegler, O.; Temesgen, H.; Hudak, A.T.; Fekety, P.A.; Yang, Z.
Comparison of Model-Assisted Endogenous Poststratification Methods for Estimation of Above-Ground Biomass Change in Oregon, USA. *Remote Sens.* **2022**, *14*, 6024.
https://doi.org/10.3390/rs14236024

**AMA Style**

Mauro F, Monleon VJ, Gray AN, Kuegler O, Temesgen H, Hudak AT, Fekety PA, Yang Z.
Comparison of Model-Assisted Endogenous Poststratification Methods for Estimation of Above-Ground Biomass Change in Oregon, USA. *Remote Sensing*. 2022; 14(23):6024.
https://doi.org/10.3390/rs14236024

**Chicago/Turabian Style**

Mauro, Francisco, Vicente J. Monleon, Andrew N. Gray, Olaf Kuegler, Hailemariam Temesgen, Andrew T. Hudak, Patrick A. Fekety, and Zhiqiang Yang.
2022. "Comparison of Model-Assisted Endogenous Poststratification Methods for Estimation of Above-Ground Biomass Change in Oregon, USA" *Remote Sensing* 14, no. 23: 6024.
https://doi.org/10.3390/rs14236024