Stand Age Class to Size Class Crosswalk by Forest Type Group in Minnesota, USA

Abstract

Many forestry models require the input of stand size class information, a variable with multiple definitions across forest inventories. This work describes a crosswalk between stand age class and stand size class for several forest type groups in Minnesota to facilitate consistency and availability of the latter information. Refinements to the crosswalk include ratio adjustments that redistribute the number of plots (or hectares) from one estimated size class to another, based on known crosswalk error rates. The results showed that 61.9% of all plots were correctly classified, and 95.5% were within one size class. Correct classifications for individual forest type/size class combinations ranged from 16.7–79.2%. Applying the crosswalk to a validation dataset produced percent errors from −46.4% to 49.4%, but with the ratio adjustment, errors dropped to −20.1% to 14.4%. These results suggest that the crosswalk and ratio adjustments provide a coarse, yet reasonable, substitute to using more complex stand size classification methodologies, particularly when forecasting future stand conditions.

1. Introduction

Stand size class plays a crucial role in many forest and ecological contexts, including when scheduling harvest, prescribing silvicultural treatments, and evaluating wildlife habitat. Descriptive and predictive models of forests may include stand size class information as a response or explanatory variable (e.g., [1,2]). A size class represents the predominant tree size or stage of tree development within a forest stand or landscape (e.g., seedling/sapling class). The U.S. Forest Service, Forest Inventory and Analysis (FIA) program measures forest resources across the entire country and its territories. This program defines three size classes based on the stocking majority of large diameter trees (≥27.94 cm for hardwoods; ≥22.86 cm for softwoods), medium diameter trees (≥12.7 cm and less than large trees), and small diameter trees (<12.7 cm), with diameters generally measured at diameter-breast-height (dbh; 1.37 m above ground) [3]. In addition, most state natural resource agencies (e.g., Minnesota Department of Natural Resources (MNDNR)) record size class during their inventories and use this variable in management decisions [4].

For other operational forest inventories in Minnesota, many do not use size classification methodologies as complex as those used by FIA or MNDNR, which require detailed information and algorithms (see [4,5]). This complexity hinders the addition of such detail in existing and future inventories for other ownerships in the state (e.g., county lands, private landowners, tribal lands). Perhaps more critically, applied modeling and forecasting efforts benefit from straightforward models that increment through time (e.g., [1]). Linking the stand age with other forest attributes (e.g., stand size class) facilitates temporal stand and habitat projections by enabling iterative updates to these variables of interest. Further, the use of stand age alone avoids the complexities of incrementing additional predictor variables before estimating the response. Therefore, this research sought to (1) create an age class to size class crosswalk that allowed for rapid stand size class imputation using stand age class information; (2) provide ratio adjustments that re-assign hectares from one estimated size class to another, based on known crosswalk error rates; and (3) validate the crosswalk and the ratio adjustments using additional observations and quantify the error reduction (or addition) from using the ratio adjustments.

2. Materials and Methods

Per plot, the FIA program uses four 0.0169 ha subplots to measure stand and larger tree (≥12.7 cm dbh) attributes and four 0.0013 ha microplots to measure smaller tree (<12.7 cm dbh) attributes. If a plot encompasses a major shift in certain attributes (e.g., ownership, forest type, stand size class), the plot is subdivided into conditions based on the differences. Annually, FIA measures 20% of the plots in Minnesota, finishing a complete measurement every five years. Training data for this study came from FIA measurement periods from 1999–2013 and comprised 9263 plots [6]. This dataset includes three consecutive statewide inventories (1999–2003, 2004–2008, and 2009–2013). Further, the data was pooled to increase sample sizes [7]. Only sampled plots with one FIA condition and adequate stocking (≥10%) were included to minimize variability and to have an assigned forest type, respectively. For a full description of the FIA database, see [3,8].

Subsequently, an age class to size class map was explored for several forest types by identifying the age class associated with a change in size class. However, small sample sizes prevented determining a reliable map for several forest types. Thus, we focused on four broad forest type groupings to provide sufficient sample sizes: upland conifer, lowland conifer, northern hardwoods, and aspen-birch. These groupings represent distinct habitats and find common reference in Minnesota (e.g., [9]). We then summarized plot counts by forest type group (hereafter referred to as forest types) and five-year age classes and determined the age class where the majority of plots transitioned from one size class to another for each forest type. The selection of the transition points was straightforward, except for a few cases that required discretion to ensure a reasonable choice (e.g., lowland conifer poletimber to sawtimber). Finally, the crosswalk was applied to the training data to compute the overall error rates by forest type and the error rates for individual forest type/size class combinations.

With only three size classes, the crosswalk will assign a plot to either the correct size class or one of two incorrect alternatives. However, the distributions of plots by size class may overlap considerably, leading to high percentages of incorrect assignments. Therefore, the proportions of correct and incorrect assignments were computed within each level of estimated size class. These values comprise a ratio adjustment to the map that redistributes hectares (or the number of plots) from one estimated size class to another, based on the observed crosswalk error rates (Equation (1)). As such, we describe it as the adjusted crosswalk.

$$h_i^{*}=A_i{h}_i$$

(1)

where $h_i^{*}$ = three element vector of adjusted hectares (or number of plots) by size class, $A_i$ = 3 × 3 matrix of ratio adjustments for redistributing estimated size classes, $h_i$ = three element vector of unadjusted hectares (or number of plots) by size class, and i = an index for forest type. Note that this adjusted crosswalk only applies to applications using aggregate hectares by forest type and size class. When assigning a size class to an individual plot or stand, the unadjusted crosswalk is appropriate and will be subject to the uncorrected error rates.

The validation of the original and adjusted crosswalk involved using them to estimate size classes for the latest complete FIA inventory (2014–2018) (see [10] for a similar approach). The same filters were applied to the validation data as to the training data, resulting in 2913 plots. Both crosswalk estimates were compared to observed classes via percent errors (see Equation (2)). In addition, percent differences (Equation (3)) between estimates using the original and adjusted crosswalk enabled the quantification of the gains (or losses) from using the adjustment.

$$e=\left(Obs-Est\right)/Obs$$

(2)

$$d=\left|e_{cw}|-|e_{acw}\right|$$

(3)

where e = percent error, Obs = observed plot counts (or total hectares) in a forest type/size class combination, Est = estimated plot counts (similar to Obs), d = percent difference, e_cw = percent error using the original crosswalk, and e_acw = percent error using the ratio adjustment. All analyses were conducted using the R statistical program [11].

3. Results and Discussion

Figure 1 shows the distribution of FIA plots by age class and size class for each forest type group. Where the size class distributions intersect represents the age thresholds between size classes, and

Table 1. Stand age class (years) to stand size class crosswalk. The table lists the age ranges associated with each size class for four forest type groups in Minnesota based on Forest Inventory and Analysis (FIA) records spanning 1999–2013.

Forest Type	Size Class
	Seedling/Sapling	Poletimber	Sawtimber
Upland Conifer	0–20	21–55	>55
Lowland Conifer	0–80	81–140	>140
Northern Hardwoods	0–35	36–60	>60
Aspen-Birch	0–30	31–65	>65

gives these age classes associated with each size class. Regardless of the forest type, the pattern remains constant: as stands age, they increase in size class. However, for most forest types, the size class distributions overlap considerably (particularly the poletimber and sawtimber classes), suggesting a moderate to weak relationship between age class and size class.

Reapplying the unadjusted crosswalk from Table 1 to the training data showed that 61.9% of all plots were assigned the correct size class, and 95.5% were within one size class (

Table 2. Age class to size class crosswalk percent accuracy and error for each forest type group. Also included are the error rates across all groups and the sample sizes. Errors are reported as proportions of the observed total number of plots in the associated category. Note that the FIA size classes were coded as 1—sawtimber, 2—poletimber, and 3—seedling/sapling. Thus, negative and positive numbers represent estimates smaller and larger than the actual size class, respectively. Data came from FIA plot records spanning 1999–2013.

Forest Type	Error (Observed-Fitted)					Sample Size
	−2	−1	0	1	2
Upland Conifer	0.006	0.134	0.607	0.180	0.072	777
Lowland Conifer	0.039	0.215	0.552	0.182	0.012	2308
Northern Hardwoods	0.016	0.127	0.566	0.258	0.033	2362
Aspen-Birch	0.010	0.128	0.695	0.145	0.022	3816
All	0.018	0.150	0.619	0.186	0.027	9263

). However, correct classifications varied within the forest type group. The aspen-birch group had the highest accuracy and lowland conifer the lowest (Table 2). This result corresponds directly to the extent of size class distribution overlap within a forest type (Figure 1). In addition, classification errors showed no strong trend toward larger or smaller size classes. Early attempts to increase the proportion of correct category assignments (including alternative approaches such as regression analysis and mixed-effects modeling) showed marginal improvement while reducing utility.

Table 3. Age class to size class crosswalk accuracy and error rates for each forest type/size class combination, along with plot counts for each actual/estimated size class combination. Percentages reported as proportions of the observed total number of plots in a size class (i.e., conditional proportions based on rows). Size classes were coded as 1—sawtimber, 2—poletimber, and 3—seedling/sapling. Data came from FIA plot records spanning 1999–2013.

Forest Type	Observed Size Class	Estimated Size Class
		1		2		3
Upland Conifer	1	0.674	(190)	0.309	(87)	0.018	(5)
	2	0.223	(59)	0.713	(189)	0.064	(17)
	3	0.243	(56)	0.352	(81)	0.404	(93)
Lowland Conifer	1	0.167	(41)	0.472	(116)	0.362	(89)
	2	0.023	(17)	0.466	(347)	0.511	(380)
	3	0.021	(28)	0.306	(403)	0.673	(887)
Northern Hardwoods	1	0.771	(890)	0.196	(226)	0.033	(38)
	2	0.593	(538)	0.325	(295)	0.083	(75)
	3	0.260	(78)	0.237	(71)	0.503	(151)
Aspen-Birch	1	0.525	(403)	0.426	(327)	0.049	(38)
	2	0.211	(319)	0.683	(1034)	0.106	(161)
	3	0.055	(84)	0.153	(235)	0.792	(1215)

breaks down the error by individual forest type/size class combinations. Within a forest type group, the percentages represent the proportion of observed size classes within each estimated size class, and the diagonal elements give the percentage of plots correctly classified. This table shows that the individual error rates had considerably more variation than the overall error rates in Table 2, ranging from 16.7% to 79.2% correct assignments per forest type/size class combination. Correctly assigning the sawtimber size class to lowland conifer stands proved particularly challenging. These stands develop slowly and may often only produce small to moderate sized trees, thus limiting the number of sawtimber stands for defining a clear size class threshold (as evidenced by the flat sawtimber curve in Figure 1). Northern hardwood poletimber stands were also difficult to classify, as the bulk of these stands were older than the age class intersection between the poletimber and sawtimber distributions. Historically, these forests received low priority management, leading to the majority of stands aging into larger size classes. Upland conifer and aspen-birch provided relatively clean breaks between size class distributions, except that most sawtimber aspen-birch stands were younger than the poletimber/sawtimber intersection. The often intensive management and short rotations of aspen-birch limits the number of sawtimber aspen stands and prevents a clearer threshold.

In order to compensate for the large classification errors, ratio adjustments were computed that would redistribute a portion of plots (or hectares) from one estimated size class to the other size classes, based on misclassification rates (i.e., the adjusted crosswalk).

Table 4. Ratio adjustments for each forest type/estimated size class combination. Values in the table are reported as proportions of the estimated total number of plots in a size class (i.e., conditional proportions based on columns). Each proportion represents how plots or hectares within an estimated size class should either remain in the estimated size class (diagonal elements within each forest type group) or be redistributed to the other classes based on known classification error rates (off-diagonal elements). Size classes were coded as 1—sawtimber, 2—poletimber, and 3—seedling/sapling. Data came from FIA plot records spanning 1999–2013.

Forest Type	Observed Size Class	Estimated Size Class
		1	2	3
Upland Conifer	1	0.623	0.244	0.043
	2	0.193	0.529	0.148
	3	0.184	0.227	0.809
Lowland Conifer	1	0.477	0.134	0.066
	2	0.198	0.401	0.280
	3	0.326	0.465	0.654
Northern Hardwoods	1	0.591	0.382	0.144
	2	0.357	0.498	0.284
	3	0.052	0.120	0.572
Aspen-Birch	1	0.500	0.205	0.027
	2	0.396	0.648	0.114
	3	0.104	0.147	0.859

shows the ratio adjustments for each forest type/size class combination, with the percentages representing the proportions of an estimated size class within the corresponding observed size classes. For example, for those hectares of upland conifer with an estimated sawtimber size class, 62.3% of the hectares will remain in the sawtimber size class, 19.3% will be assigned the poletimber size class, and 18.4% will be given the seedling/sapling size class. The same interpretation holds for the other forest type/estimated size class combinations.

These ratios suggest that for each level of the estimated size class, the majority of plots were correctly classified (except the poletimber class for lowland conifers). Still, the large variability across ratios (e.g., 0.401 to 0.859 for hectares that remain in the estimated size class) results from the considerable overlap observed in Figure 1 and re-enforces the need for using the adjustments when applicable. Note that although similar, Table 3 and Table 4 provide distinct error rates, with those in the former computed relative to all plots within a true size class and those in the latter computed relative to all plots within a predicted size class.

Table 5. Validation statistics (percent error and percent difference) and sample sizes for the age class to size class crosswalk by forest type across all FIA plot records spanning 2014–2018. Sample sizes are those for observed size classes, which are coded as 1—sawtimber, 2—poletimber, and 3—seedling/sapling. Percent errors are given for using the crosswalk alone and with the ratio adjustments. Percent differences quantify the relative change in percent accuracy from including the adjustments, with positive values indicating an improvement.

Forest Type	Observed Size Class	% Error No Adj.	% Error Ratio Adj.	% Diff.	Sample Size
Upland Conifer	1	0.055	0.123	−0.068	109
	2	−0.464	−0.072	0.392	84
	3	0.493	−0.109	0.383	67
Lowland Conifer	1	0.494	−0.051	0.444	87
	2	−0.394	−0.101	0.293	231
	3	0.103	0.060	0.044	464
Northern Hardwoods	1	−0.274	0.095	0.179	419
	2	0.475	−0.106	0.369	255
	3	−0.075	−0.161	−0.086	80
Aspen-Birch	1	−0.318	−0.201	0.116	173
	2	0.162	0.144	0.018	469
	3	−0.044	−0.069	−0.025	475

provides the validation statistics when applying the crosswalk with and without the ratio adjustments to the 2014–2018 FIA data. The original crosswalk produced overall percent errors ranging from −46.4% to 49.4% and behaved the poorest for the conifers. Although northern hardwoods had mixed results, the crosswalk did provide more reasonable estimates for aspen-birch. Adding the ratio adjustment (using Equation (1)) improved results considerably, with overall percent errors ranging from −20.1% to 14.4%. Table 5 also provides the percent difference between using the original crosswalk and using the ratio adjustment. The overall differences ranged from −8.6% to 44.4%, with three forest types (upland conifer, lowland conifer, and northern hardwoods) showing a mostly substantial improvement in accuracy after applying the ratios. For aspen-birch, the positive effect was less pronounced, but still with an overall improvement. Three combinations (sawtimber upland conifer, seedling/sapling northern hardwoods, and seedling/sapling aspen-birch) showed slight decreases in accuracy when using the ratio adjustments, likely due to sampling variation. Still, the majority of large, positive differences suggest that using the ratio adjustments should improve results over the unadjusted crosswalk.

4. Conclusions

Ultimately, the age class to size class crosswalk represents a straightforward method for imputing stand size class information. The unadjusted classification errors ranged from −46.4% to 49.4%, but including the ratio adjustments lowered the errors to −20.1% to 14.4%, indicating that the adjusted crosswalk should be used whenever possible. Natural resource practitioners should recognize the coarseness of the map and its adjustment. Still, the crosswalk provides a reasonable substitute for complex stand size classification methodologies, particularly when needing to efficiently forecast or simulate future forest stand conditions.