Reconstructability Analysis (RA) is a technique for modeling data that is used to quantify the relationships in categorical data [4,5,6]. It can also be applied to continuous data if these data are discretized (binned). RA is rooted in Shannon’s Information Theory [7], and Ashby’s Constraint Analysis [8], with graphical modeling added by Klir, Krippendorf, and others [9,6]. RA calculates the amount of Information “transmitted” between discrete variables, resulting in a quantification of their relations. RA is a categorical technique that can be used in a way that is analogous to Multiple Linear Regression analysis of continuous data. Reconstructability Analysis is a rich framework for analyzing and quantifying the relationships of parts to wholes, though in this project we use only a small portion of its capabilities.

We use a form of RA in which the data is treated as a directed system, where multiple IVs are used to quantify the entropy-reduction of a single DV. The data are expected to have tabular form, with each row representing an observation, and each column representing a single variable with discrete values. One column is the DV and the other columns are the IVs. (In principle, RA could address multiple DVs, but the OCCAM software used in this research allows only a single DV for each analysis.) This form of RA can determine not only which IVs are the best predictors of the DV (feature selection), but also to make specific predictions of the DV (derived from calculated conditional probabilities of the DV) given the composite IV state. The alternative to a directed system is a neutral system where no IV-DV distinction is made. For neutral systems one is interested in all the relations that exist among all the variables; in effect, each variable is a DV and all other variables are IVs. Directed system analysis is much simpler than neutral system analysis because one selects only one variable as the DV. Another important distinction in RA modeling is between loopless models and models with loops. Loopless models have only one IV-DV relation that predicts the DV, while models with loops have multiple predictive IV-DV relations where these relations are fused together via a maximum entropy formalism to yield a net DV prediction from all of the predicting IVs. Models with loops are usually better predictors but they require iteration and thus longer computation times. In this study we use models with loops, because computation is fast enough to become a non-issue.

To illustrate the idea of a single predictive IV-DV relation, suppose we have three IVs called A, B, and C, and let us call the DV by the name Z. A predictive relation will include Z and some subset of {A, B, C}. For example, the best model with one predictor might have the predictive relation BZ, meaning that B predicts Z better than A and C. The best no-loop model with two IV predictors might have the predictive relation be BCZ, where there is a 3-way interaction effect between B, C, and Z. For models with loops, a two-predictor model might have the predictive relations be BZ:CZ, where B and C predict Z separately, but their predictions are fused via a maximum entropy formalism. These are the types of models you will see in this study. Note that in these models, the order of relations is irrelevant and the order of variables within any relation is also irrelevant, so that the previously mentioned model with loops could equally well have been written CZ:ZB, for example.

In addition to one or more predictive relations, directed systems models have an “IV component,” which is a relation among all and only the IVs; in the current example, ABC. This relation assures that all predictive models are hierarchically nested for statistical considerations; by lumping together all the IVs in a single relation, it abstains from positing any specific relations among different IVs, allowing the focus of the model to be on the predictive relations that include the DV. So, in the example given in the previous paragraph, the IV component would be ABC; the best loopless model would actually be ABC:BCZ and the best with-loops model example would be ABC:BZ:CZ. In this later model, the IV component causes the existence of the loop, which can be seen by replacing the IV component by the BC projection that it encompasses, and writing the model as BC:CZ:ZB. By contrast, if one omitted the IV component, the BZ:CZ model does not have a loop. Models of this sort are known as naïve-Bayes models and are not included in RA as implemented in OCCAM. For a discussion of the relation between RA and these naïve-Bayes models and other Bayesian network models, see Harris and Zwick, 2021 [10].

RA uses hypergraphs as models, information theoretic measures to quantify the goodness of models, and statistical methods to assess confidence in these measures. The set of candidate models defines a Lattice of Structures (hypergraphs) of all possible models for that many variables. Figure 5 shows an example of a four-variable system. The nine models not greyed out are the candidate models if the system is directed; the additional eleven greyed out models would need to be considered if the system were neutral. In these models, a box is a relation; a line (which may be branched) is a variable. The most complex possible model, at the top, is one with a single tetradic relation that fully connects all the variables. The least complex model, at the bottom, is one with a fully independent (disconnected) set of variables, which might be called monadic relations. Intermediate models consist of multiple relations, each of which connects a subset of either three or two variables. Models without loops have relations shown in bold.

A candidate model is assessed by comparing it to a reference model, either the top (the data, the saturated model) or the bottom (the independence model). In this study, the bottom is the reference. Models are assessed based on how well they predict (reduce the Shannon entropy of) the DV using both information theoretic and more generic machine learning metrics. Information captured in the model (1 for the data, 0 for the independence model) and the collinear reduction of DV entropy, when traded off against the added complexity of the candidate model beyond the independence model, helps to select a best model. This model specifies a conditional probability distribution of the DV, given the values of its predicting IVs.

The OCCAM implementation of RA, hosted on the servers at Portland State University, is divided into two primary functions, called “Search” and “Fit.” In Search we

supply a data file in a suitable format and receive as output a set of candidate models with metrics from which we choose a “best” model. The data and the chosen model are supplied to the Fit function, and a set of tables are returned which have every possible combination of states of the IVs and the conditional probabilities of DV states, given the IV states, from which the DV predictions are obtained for these IV states, along with various metrics, including statistical p-values.

The discrete character of the NLCD suggests RA as a potentially valuable analysis technique. NLCD data, for the purposes of this study, define 15 classes of land use types (Figure 1). Since these data are classified in a format readily amenable to RA, we decided to analyze them with OCCAM to see what predictive relations would emerge. For this study, we reclassify the data to 5 classes, which are Water, Developed, & Agriculture (1), Grasses (2), Shrubs (3), Mixed and Deciduous Forest (4), and Evergreen Forest (5). This reduction in the cardinality of the variables make post-processing analysis more manageable, as we will see in the results section. The data are analyzed for five years: 2001, 2006, 2011, 2016, and 2021, referred to respectively as times 1 through 5. NLCD data are reported in a grid of 30m cell size, with an accuracy of around 80% (USGS), having been classified from satellite images.

Our earliest analysis with RA was synchronic rather than diachronic, using only a single year. It showed that most of the information in the data is captured by the neighbors directly North, West, East, and South, thus allowing the diagonal neighbors to be ignored [11]. In the field of Cellular Automata this 4-cell neighborhood is known as the von Neumann Neighborhood (VNN), while the full set of 8 neighbors is the Moore Neighborhood (Figure 6). Hence, we have used VNN for all of the subsequent experiments. To summarize the results from prior work, we found that for the geographic region being examined, the VNN captured 88% of the information in the data, showing also a strong southern influence on the DV (central cell). For example, one finding of this earlier study was that if the southern cell in the VNN was Grasses, regardless of what the north, east, and west cells were, the probability that the DV would also be grasses was 57%. These initial results were promising enough to undertake the multi-year Evergreen Forest study reported in this paper.

The following supporting information can be downloaded at the website of this paper posted on Preprints.org. dmm.sysc.pdx.edu/percy, python notebook with code for extracting and processing data from NLCD for use in OCCAM