# Multi-Scale Entropy Analysis as a Method for Time-Series Analysis of Climate Data

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Centre for Landscape and Climate Research, University of Leicester, University Road, Leicester LE1 7RH, UK

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Centre for Ecology and Hydrology, Maclean Building, Benson Lane, Crowmarsh Gifford, Wallingford, Oxfordshire, OX10 8BB, UK

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Department of Civil and Transport Engineering, Norwegian University of Science and Technology, Høgskoleringen 7A, 7491 Trondheim, Norway

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School of Environmental Sciences, University of East Anglia, Climatic Research Unit, Norwich Research Park, Norwich NR4 7TJ, UK

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Center of Excellence for Climate Change Research/Department of Meteorology, King Abdulaziz University, Jeddah 21589, Saudi Arabia

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Author to whom correspondence should be addressed.

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These authors contributed equally to this work.

Received: 18 November 2014 / Revised: 14 February 2015 / Accepted: 28 February 2015 / Published: 6 March 2015

Evidence is mounting that the temporal dynamics of the climate system are changing at the same time as the average global temperature is increasing due to multiple climate forcings. A large number of extreme weather events such as prolonged cold spells, heatwaves, droughts and floods have been recorded around the world in the past 10 years. Such changes in the temporal scaling behaviour of climate time-series data can be difficult to detect. While there are easy and direct ways of analysing climate data by calculating the means and variances for different levels of temporal aggregation, these methods can miss more subtle changes in their dynamics. This paper describes multi-scale entropy (MSE) analysis as a tool to study climate time-series data and to identify temporal scales of variability and their change over time in climate time-series. MSE estimates the sample entropy of the time-series after coarse-graining at different temporal scales. An application of MSE to Central European, variance-adjusted, mean monthly air temperature anomalies (CRUTEM4v) is provided. The results show that the temporal scales of the current climate (1960–2014) are different from the long-term average (1850–1960). For temporal scale factors longer than 12 months, the sample entropy increased markedly compared to the long-term record. Such an increase can be explained by systems theory with greater complexity in the regional temperature data. From 1961 the patterns of monthly air temperatures are less regular at time-scales greater than 12 months than in the earlier time period. This finding suggests that, at these inter-annual time scales, the temperature variability has become less predictable than in the past. It is possible that climate system feedbacks are expressed in altered temporal scales of the European temperature time-series data. A comparison with the variance and Shannon entropy shows that MSE analysis can provide additional information on the statistical properties of climate time-series data that can go undetected using traditional methods.