Recognizing van Deusen’s mixed estimator for annual forest inventory as a linear mixed model

Edgar C.B., Westfall J.A.

An analysis of United States national forest inventory observations in the Laurentian Mixed Forest reveals a marked increase in forest disturbance between 1999 and 2015. The Laurentian Mixed Forest Province ecological subregion spans the northern sections of Michigan, Minnesota, and Wisconsin and includes forest area of between 16.7 and 17.5 million hectares depending on the year. Forest disturbance ranges from a low of 0.13 million hectares (0.8% of forest area) in 2000 to a high of 2.09 million hectares (11.9% of forest area) in 2014. The year 2015 is notable for being the first year since 2000 where forest disturbance declines, albeit modestly (11.4% of forest area). The marked increase is attributable to disturbances occurring continuously over time between remeasurement. Disturbances with the highest annual averages are insect damage to trees, disease damage to trees, and deer/ungulate at 291 thousand, 189 thousand, and 126 thousand hectares per year, respectively. Disturbances occurring in a specific year, what we term discrete disturbances, show no discernible trend during the period. The most extensive discrete disturbances are wind in 1999, 2011, and 2012 at 108 thousand, 62 thousand, and 61 thousand hectares, respectively. Standard estimates from national forest inventory lack specificity as to the actual year of the disturbance. The estimates reported here are actual annual estimates of disturbance that apply estimation methods accounting for the retrospective nature of the disturbance observation. The timing (year) and location (ecological section) of the two most extensive wind events coincide with historical records.

Dettmann G.T., Radtke P.J., Coulston J.W., Green P.C., Wilson B.T., Moisen G.G.

Small area estimation is a growing area of research for making inferences over geographic, demographic, or temporal domains smaller than those in which a particular survey data set was originally intended to be used. We aimed to review a body of literature to summarize the breadth and depth of small area estimation and related estimation strategies in forest inventory and management to-date, as well as the current state of terminology, methods, concerns, data sources, research findings, challenges, and opportunities for future work relevant to forestry and forest inventory research. Estimation methodologies explored include direct, indirect, and composite estimation within design-based and model-based inference bases. A variety of estimation methods in forestry have been applied to extensive multi-resource inventory systems like national forest inventories to increase the precision of estimates on small domains or subsets of the overall populations of interest. To avoid instability and large variances associated with small sample sizes when working with small area domains, forest inventory data are often supplemented with information from auxiliary sources, especially from remote sensing platforms and other geospatial, map-based products. Results from many studies show gains in precision compared to direct estimates based only on field inventory data. Gains in precision have been demonstrated in both project-level applications and national forest inventory systems. Potential gains are possible over varying geographic and temporal scales, with the degree of success in reducing variance also dependent on the types of auxiliary information, scale, strength of model relationships, and methodological alternatives, leaving considerable opportunity for future research and growth in small area applications for forest inventory.

Hou Z., Domke G.M., Russell M.B., Coulston J.W., Nelson M.D., Xu Q., McRoberts R.E.

• We propose a data assimilation procedure for updating estimates with USFS FIA data. • This procedure incorporates the design-based and model-based inferences. • Updated estimates are comparable with estimates requiring 5+ years pooled FIA data. • This procedure is 100% compatible with the FIA database that is publicly available. • This procedure is unbiased and efficient, suitable for official reporting instruments. The United Nations Framework Convention on Climate Change requires annual estimates for forestry and ecological indicators to monitor the change in forest resources, the sustainability of forest management, and the emission and sink of forest carbon. It is particularly important to update estimates of forestland area in a timely fashion and at flexible geographical scales, not only for its value in monitoring biological diversity at the ecosystem scale, but also because of its close association with other indicators such as forest biomass and carbon. However, in the US, the Forest Survey Handbook advises that the sampling error should not exceed 3% per 404686 ha (one million acres) of forestland area, a demanding standard barely met by pooling the Forest Inventory and Analysis (FIA) panel data measured in an inventory cycle of 5–10 years. Consequently, this study aims to propose and illustrate an updating procedure using data assimilation that integrates a design-based estimator with a model-based mixed estimator for updating annual estimates at two population levels, the state- and county-levels. The three states in the USA, Minnesota (MN), Georgia (GA) and California (CA), representing the Northern, the Southern and the Pacific Northwest FIA programs, constitute the study areas. FIA data collected were based on a 5-year inventory cycle for MN (2006–2010) and GA (2005–2009), and a 10-year cycle for CA (2001–2010). The total number of sample plots was 17764 for MN, 6323 for GA, and 16740 for CA. Distinguishing features attribute to this procedure include: (1) unbiasedness: the integration of design-based estimates into the mixed estimator introduces a favorable property – unbiasedness, which could be the property national forest inventories concern the most; (2) efficiency: considerable improvements in estimation precision greater than 55%, achieving sampling errors as small as those relying on using 5–10 years pooled FIA data; (3) time: compared with the temporal trends reflected by design-based estimates, the updated trends were of much smoother trend lines and narrower confidence intervals that would better depict temporal changes for a population at flexible spatial scales; (4) space: this procedure is scale-invariant, meaning its efficiency is not affected by an inventory employing either a large- or small-area estimation, which was demonstrated at the two population levels; and (5) generalizability: this procedure is unbiased and efficient, 100% compatible with the FIA database which is readily available to the public, and thus suitable for various official reporting instruments.

Grafström A., Matei A.

When sampling from a continuous population (or distribution), we often want a rather small sample due to some cost attached to processing the sample or to collecting information in the field. Moreover, a probability sample that allows for design‐based statistical inference is often desired. Given these requirements, we want to reduce the sampling variance of the Horvitz–Thompson estimator as much as possible. To achieve this, we introduce different approaches to using the local pivotal method for selecting well‐spread samples from multidimensional continuous populations. The results of a simulation study clearly indicate that we succeed in selecting spatially balanced samples and improve the efficiency of the Horvitz–Thompson estimator.

Rao J.N., Molina I.

Van Deusen P.C.

Three estimators of current status and trend are compared for an annual interpenetrating panel design. The five-panel annual inventory design is simulated over a 10-year period with flat, increasing, and quadratic growth trends. The simulated comparisons show that the mixed estimator performs well relative to the 5-year moving average in terms of bias and mean squared error in all cases. The one-panel mean can have less bias than the moving average when there is a trend, but it is more variable. The moving average tends to lag evolving trends, which can result in very large bias.

Van Deusen P.C.

Flexible models are proposed for assessing trend in annual forest survey data with increasing levels of generality. The models can be fit to the data using a mixed estimator and compared using the Akaike or Schwarz information criterion. The models also provide smoothed estimates of annual means.

Eilers P.H., Marx B.D.

B-splines are attractive for nonparametric modelling, but choosing the optimal number and positions of knots is a complex task. Equidistant knots can be used, but their small and discrete number allows only limited control over smoothness and fit. We propose to use a relatively large number of knots and a difference penalty on coefficients of adjacent B-splines. We show connections to the familiar spline penalty on the integral of the squared second derivative. A short overview of $B$-splines, of their construction and of penalized likelihood is presented. We discuss properties of penalized B-splines and propose various criteria for the choice of an optimal penalty parameter. Nonparametric logistic regression, density estimation and scatterplot smoothing are used as examples. Some details of the computations are presented.

Searle S.R., Casella G., McCulloch C.E.