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Leveraging the power of local spatial autocorrelation in geophysical interpolative clustering

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Abstract

Nowadays ubiquitous sensor stations are deployed worldwide, in order to measure several geophysical variables (e.g. temperature, humidity, light) for a growing number of ecological and industrial processes. Although these variables are, in general, measured over large zones and long (potentially unbounded) periods of time, stations cannot cover any space location. On the other hand, due to their huge volume, data produced cannot be entirely recorded for future analysis. In this scenario, summarization, i.e. the computation of aggregates of data, can be used to reduce the amount of produced data stored on the disk, while interpolation, i.e. the estimation of unknown data in each location of interest, can be used to supplement station records. We illustrate a novel data mining solution, named interpolative clustering, that has the merit of addressing both these tasks in time-evolving, multivariate geophysical applications. It yields a time-evolving clustering model, in order to summarize geophysical data and computes a weighted linear combination of cluster prototypes, in order to predict data. Clustering is done by accounting for the local presence of the spatial autocorrelation property in the geophysical data. Weights of the linear combination are defined, in order to reflect the inverse distance of the unseen data to each cluster geometry. The cluster geometry is represented through shape-dependent sampling of geographic coordinates of clustered stations. Experiments performed with several data collections investigate the trade-off between the summarization capability and predictive accuracy of the presented interpolative clustering algorithm.

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Notes

  1. The predictive clustering framework is originally defined in Blockeel et al. (1998), in order to combine clustering problems and classification/regression problems. The predictive inference is performed by distinguishing between target variables and explanatory variables. Target variables are considered when evaluating similarity between training data such that training examples with similar target values are grouped in the same cluster, while training examples with dissimilar target values are grouped in separate clusters. Explanatory variables are used to generate a symbolic description of the clusters. Although the algorithm presented in Blockeel et al. (1998) can be, in principle, run by considering the same set of variables for both explanatory and target roles, this case is not investigated in the original study.

  2. Inverse distance weighting is a common interpolation algorithm. It has several advantages that endorse its widespread use in geostatistics (Li and Revesz 2002; Karydas et al. 2009; Li et al. 2011): simplicity of implementation; lack of tunable parameters; ability to interpolate scattered data and work on any grid without suffering from multicollinearity.

  3. We can extend this representation of a sensor network by considering a multi-dimensional representation of space. In the multi-dimensional case, multiple variables will be used to identify the location of a station. These multiple variables will be taken into account when computing the distance between sensors.

  4. In the on-line learning phase, missing observations of a variable are interpolated in the data snapshot by using the inverse distance weighted sum of nearby known data in the row.

  5. The spherical law of cosines is used, in order to approximate the geographical distance between the geographic coordinates (e.g. latitude and longitude) of two sensors.

  6. The time cost of computing the local indicators of the spatial autocorrelation property can be made subquadratic by using a spatial data structure, in order to maintain, for each sensor in the network, the sphere of its neighbours. The structure will be updated only when a new sensor is either switched-on or switched-off in the network.

  7. The quadtree decomposition of a cluster divides recursively a cluster quadrant into four subquadrants until final quadrants are determined. As we plan to compute \(Np^{\%}\) final quadrants, the number of levels of this quadtree decomposition is about \(\log _4(Np^{\%})\).

  8. http://www.di.uniba.it/~appice/software/ICT_TICT/index.htm.

  9. We compute RRMSE, in order to scale the error of a target variable with the domain size of the variable.

  10. We note that the local indicators, which are computed by accounting for the pairwise comparison between neighbor stations, can be precomputed before building the tree. Thus, only the variance reduction of local indicators is evaluated over each node. On the other hand, MoranVar computes the global indicator of the spatial autocorrelation over each node. This requires the computation of the pairwise comparison between the neighbor stations that fall in the present node. As neighbors may change in number throughout the tree, the global measure has to be recomputed at each node.

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Acknowledgments

We would like to acknowledge the support of the European Commission through the project MAESTRA—Learning from Massive, Incompletely annotated, and Structured Data (Grant number ICT-2013-612944). The authors thank Saso Dzeroski for providing SIGMEA data and generating IRS and SPOT data, Enric Melé and Joaquima Messeguer for providing FOIXA data, unknown reviewers for their useful suggestions to improve this paper and Lynn Rudd for her help in reading the manuscript. FOIXA and SIGMEA data are collected in the SIGMEA project. SPOT and IRS data are obtained from EEA Corine Image 2006.

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  1. Dipartimento di Informatica, Università degli Studi di Bari “Aldo Moro”, Via Orabona 4, 70125 , Bari, Italy

    Annalisa Appice & Donato Malerba

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  1. Annalisa Appice
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  2. Donato Malerba
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Correspondence to Annalisa Appice.

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Responsible editors: Hendrik Blockeel, Kristian Kersting, Siegfried Nijssen and Filip Železný.

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Appice, A., Malerba, D. Leveraging the power of local spatial autocorrelation in geophysical interpolative clustering. Data Min Knowl Disc 28, 1266–1313 (2014). https://doi.org/10.1007/s10618-014-0372-z

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