Equitable transform applied to phenology and temperature in a changing climate: Scaling to maintain individuality

Abstract

Abstract We describe an equitable transform that can be used to estimate missing data points, determine systematic patterns in data, observe baseline changes, and detect different amplitudes in replicated sequences. It is applicable to short discrete two‐ or three‐dimensional data sets such as biological life cycles or water content in similar media. The technique is independent of the continuity or ordering of sequences but is distinct from methods such as principal component analysis. It is ideally used to preprocess noisy or incomplete data sets prior to analysis with other well‐established techniques. This transformation maintains systematic differences between individual sequences when the underlying pattern is a separable function in two variables added to another function in one of these variables. The equitable transform partitions the original noisy data into the underlying signal determined from the data and its residuals. Points from one sequence can be transformed to any other sequence without losing any information. Information about one or more sequences can be used to infer others with missing data. A link to a github R package is provided so the transform can easily be run on any two‐dimensional data set. Simulated two‐dimensional data sets are used to demonstrate its utility in recovering missing data and scaling/offsetting in one of the dimensions. We used the transform to determine that winter temperatures at a High Arctic site have warmed by 1.8° ± 0.4°C/decade and summer temperatures by 1.1° ± 0.2°C/decade from 1986 to 2007. Applied to 18 yr of phenology data for the tundra plant Dryas integrifolia at the same site, we determined that the annual cycles of phenology events could be modeled accurately. Phenology, in some circumstances, can be described as offsetting and scaling the rate at which life cycle events occur. We introduce the idea of scaling and shifting the seasonal cycle of a reference plant via the equitable transform in order to approximate the behavior of multiple phenological cycles. Relative phenology rates of Dryas integrifolia were found to have increased over time indicating duration of phenological stages have become shorter in recent years, likely in response to the warming climate.

Keywords

• Arctic plant
• averaging
• Boltzmann transformation
• climate change
• equitable matrix
• fractal