Despite the fact that pixels (i.e. picture elements) are the basic sampling units of maps, we are aware of no software package or tool that allows users to model changes that may occur at such fine spatial resolutions over broad geographic extents.

Curve Fit is an extension to the application ArcMap that allows users to conduct linear or nonlinear regression analysis on the range of values found within input raster data sets (geo-referenced images), independently for each pixel.

Outputs consist of raster surfaces of regression model parameter estimates, standard errors, goodness-of-fit estimates and multimodel inference measures.

Curve fit outputs characterize continuous spatial or temporal change across a series of raster data sets.

Geographic patterns can change through time and/or across space, and these changes can lead to differences in the movement pattern and body condition of organisms, their interactions with each other and their environment, and ultimately lead to population and community-level changes (Turner 1989). When quantifying landscape patterns using remotely sensed data, it is important to recognize that each pixel (i.e. picture element) has a temporal and spatial context. A pixel's temporal context refers to its past and present classification and is often used to model landscape change at the finest possible resolution (e.g. Baker 1989). The spatial context of a pixel depends on the classification of neighbouring pixels, and the size of the area considered as the neighbourhood and forms the basis of morphological spatial pattern analysis (MSPA, Vogt et al. 2007; Riitters et al. 2007)—the reclassification of pixels based on the role they play in their neighbourhood. Despite the fact that pixels are the basic unit of a map and that they have a spatial and temporal context, we are unaware of any software package or tool that can be used to quantify changes that take place at the pixel level with changes in time or spatial scale.

Features of Curve Fit

We developed Curve Fit, an extension to the GIS application ArcMap, to allow users to run regression analysis on a series of raster data sets. A full description of the extension, installation instructions and brief example of its use can be found athttp://www.umesc.usgs.gov/management/dss/curve_fit.html, we briefly summarize it here. The user enters an array of values for an explanatory variable X (e.g. a spatial scale, temporal value, etc.). A raster data set representing the corresponding landscape variable Y (e.g. temperature, biomass, habitat density, diversity, etc.) is paired with each X value entered by the user. Curve Fit then uses either linear or nonlinear regression techniques (depending on user selection) to calculate a unique mathematical model at each pixel of the input raster data sets. If a nonlinear model is selected, the user has the option to constrain parameters and manipulate solution iterations and tolerances. Models supported include:

Linear:Y=aX+b

Polynomial:Y=anXn+an−1Xb−1+…aX+a

Power:Y=aXb

Exponential:Y=ae−bx

Four Parameter Logistic:Y=d+a−d1+(Xc)b

Curve Fit output products include model fit, error, P-value and multimodel inference statistics (AIC or BIC); as well as parameter value estimates, standard errors, P-values, t-statistics and residual errors. Each output product is a raster data set matching the resolution and extent of the input data sets. The user can select output formats of 64-bit double precision or 32-bit single precision. The approach is similar to geographically weighted regression (GWR, Fotheringham, Brunsdon & Charlton 2002), which ArcMap already has an extension for. However, whereas the purpose of GWR is to determine how the coefficients of explanatory variables vary in space, the purpose of the Curve Fit approach is to explore how the prediction itself (e.g. some property of the landscape) changes with a single explanatory variable (e.g. time or spatial scale). Below, we provide an example for how one might use Curve Fit to map multiscale landscape patterns.

Application of Curve Fit

One of the most significant recent advancements in mapping landscape patterns has been the development of spatial-contextual approaches (e.g. Riitters et al. 2000, 2002; Vogt et al. 2007; Wickham et al. 2007; Bar Massada & Radeloff 2010). These approaches define the character of a pixel based on all other pixels found in its neighbourhood. Neighbourhoods are represented using fixed-area windows centred over each pixel. The process is relatively simple: one calculates some index within the window (e.g. habitat density or diversity), returns the value to the centre pixel and then moves the window to the next pixel and the process is repeated. By doing this, a new map is produced that displays scale (i.e. window or neighbourhood)-specific values for each pixel. Because the results depend on the size of the window used in the analysis, results are typically reported for multiple window sizes. For example, we used the ‘variety’ focal statistics tool in ArcMap 10.0 (ESRI 2010) to calculate the number of different aquatic area types in the neighbourhood (i.e. window) of each pixel, for multiple neighbourhood sizes for a portion of the Upper Mississippi River (Fig. 1). We used 10 different neighbourhood sizes (1–100 ha), resulting in 10 new scale-specific maps (3 shown in Fig. 1, Panels B-D).

The use of Curve Fit begins by entering the array of window sizes (X) corresponding with each map of aquatic area richness (Y). The model form is then selected; based on prior experience (De Jager & Rohweder 2011), we chose to use the power function Y = aX^{b}, where a is the estimated aquatic area richness at the unit window size and b is the rate of increase in aquatic area richness with increasing window size. Milne (1997) showed that the rate of increase in richness with area (b) is related to a fractal dimension, with faster rates of increase corresponding to more complex geometric shapes. Curve Fit outputs selected and shown in Fig. 1 include b (Panel E) and the residual sums of squares (RSS, Panel F) for each pixel.

We plotted the data for two contrasting locations (Fig. 1, Panel G) to illustrate differences in spatial scaling. The location identified by the square exhibited a faster rate of increase in richness with increasing window size as compared to the location identified by the triangle. To the degree that species display strong affinities for specific aquatic habitats, we predict that the area identified by the square would harbour greater biodiversity. Future studies could use the maps in Fig. 1 to establish sampling designs that would test this prediction.

Small pockets of high RSS can be seen in Fig. 1 (Panel F), and they generally occur in areas where b approaches 0·5. These particular locations have relatively high levels of aquatic area diversity, but the power function does a poor job of characterizing the relationship between richness and window size. It is possible that linear or sigmoidal functions could provide a better fit to the data in those locations. For this reason, Curve Fit provides the user with the ability to fit multiple models to the data and conduct contests among competing models using information theoretics (AIC, Akaike 1973) to rank and select the most parsimonious model at the pixel level, thereby displaying a continuous surface of best model types.

Discussion

We suggest that the information found in multiple, spatially or temporally specific maps can be condensed to a smaller subset of maps that display parameter estimates and statistics characterizing continuous changes with increasing scale. Such approaches are at the core of landscape allometry (Mandelbrot 1983; Milne et al. 1992) and have a history of use in landscape assessment (Turner et al. 1989; Milne 1997) and modelling landscape change (Baker 1989). However, there is currently no software package or tool that we are aware of that allows users to conduct such analyses over large geographic areas at the finest spatial resolution. In this paper, we have highlighted the use of Curve Fit for assessment of multiscale spatial patterns, but it can be used for any process that includes multiple raster data sets that span some common explanatory variable. We suggest that this approach could be very powerful for examining rates of change in landscape patterns over time, which may vary spatially.

Curve Fit receives premade raster data sets, allowing for estimation of continuous change in any landscape metric. For example, one could use moving-window algorithms to map measures of spatial autocorrelation (e.g. semi-variance, Moran's I) at different scales and use Curve Fit to model changes in spatial autocorrelation with increasing scale (i.e. lag distance). Although it should be pointed out that Curve Fit does not currently support traditional variogram models (e.g. spherical, exponential, gaussian).

The fact that Curve fit receives premade raster data sets also means that results will reflect any limitations found within those data sets. For example, pixels located close to the boundary of maps can be biased when using moving windows if the window is large enough to encompass unmapped space (areas outside of the boundary). Although methods have been proposed to handle this bias (McGarigal, Cushman & Ene 2012), any artefacts found in the raster data sets will subsequently influence the Curve Fit output. Furthermore, as with any statistical analysis, attention should be paid to the relationship between the number of raster data sets (i.e. data points) and the number of model parameters used in the fitting process.

Acknowledgements

Tim Fox (tfox@usgs.gov) developed Curve Fit using the programming language VB. Net. It runs on Windows 7 and is freely available online (http://www.umesc.usgs.gov/management/dss/curve_fit.html). Tim Fox and Nathan De Jager conceived the idea for Curve Fit while conducting landscape modelling and assessment studies funded by the Long Term Resources Monitoring Component of the Upper Mississippi River Restoration-Environmental Management Program, a partnership among four federal agencies (USACE, USEPA, USFWS, USGS) and five state natural resource agencies (Illinois, Iowa, Minnesota, Missouri, Wisconsin). The support of this partnership is greatly appreciated. Use of trade, product or firm names does not imply endorsement by the US Government.