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

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## Abstract

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## 1. Introduction

_{1}: Regional temperature anomalies in Europe show particular time-scales with higher or lower sample entropy than expected under the assumption of white noise, i.e., there are signals in the time-series that are only apparent at specific time-scales.

_{2}: The sample entropy of European temperature anomalies shows significant differences at particular time-scales when comparing 1851–1960 with 1961–2014, i.e., the temporal scaling properties of the temperature data have changed at specific temporal scales of observation.

## 2. Methods

_{i}), θ is the set of all values x

_{i}that X can assume with p(x

_{i}) > 0, and E is the expectation operator. A special case of p log(p) = 0 is defined if p = 0.

_{1}, …, x

_{i}, …, x

_{N}], we first construct consecutive coarse-grained time-series, [yτ

^{(}

^{τ)}], for different values of the temporal scale factor, τ. First, we divide the original times series into non-overlapping windows of length τ, trimming any residual < τ elements at the end of the time-series. Second, we average all data points inside each window. The resulting coarse-grained time-series elements are calculated as:

^{(}

^{τ)}) is identical to the original time-series. For τ > 1, each coarse-grained time-series contains $\left(\frac{N}{\tau}\right)$ elements, i.e., the length of the original time-series divided by the scale factor (omitting the remainder of the time-series with <τ elements). In the last step, we estimate the sample entropy for each coarse-grained time-series (y

^{(}

^{τ)}) and plot it against the scale factor τ.

_{m}(i) of length m, we count the number of vectors in the time-series that are similar to it, i.e., that lie within a certain small distance that is proportional to the tolerance parameter r, scaled by the standard deviation of the original time-series. For a time-series of length N, the average number of such matching state vectors for all states u

_{m}(i), i = 1, …, N − m+1, divided by the total number of vectors of that length in the time-series, is denoted by U

^{m}(r). Repeating this calculation for the vectors u

_{m + 1}(i) leads to the average number of matching state vectors U

^{m}

^{+ 1}(r) of length m + 1. The sample entropy S

_{E}[8] is then defined as:

**Figure 1.**Area for which CRUTEM4v surface temperature anomaly data were extracted and analysed for each 5° × 5° grid box using multi-scale entropy (MSE). Country outlines data source: GISCO—Eurostat (European Commission). Administrative boundaries: EuroGeographics

^{©}, UN-FAO, Turkstat.

## 3. Results

**Figure 2.**Multi-scale sample entropy of: (

**a**) an artificial uniform white noise time-series; and (

**b**) a sine function.

**Figure 3.**Time-series from 1851–1960 of CRUTEM4v variance-adjusted surface temperature anomaly data for grid box [8|35] centred on 52.5° latitude and −7.5° longitude, with m = 3 and r = 0.3. Scale factor is shown in months. (

**a**) original time-series; (

**b**) multi-scale sample entropy for the grid box (95% confidence intervals in green) against scale factor; (

**c**) sample variance for different scale factors; (

**d**) Shannon information theoretic entropy for different scale factors.

**Figure 4.**Time-series from 1961–2014 of CRUTEM4v variance-adjusted surface temperature anomaly data for grid box [8|35] centred on 52.5° latitude and −7.5° longitude, with m = 3 and r = 0.3. Scale factor is shown in months. (

**a**) original time-series; (

**b**) multi-scale sample entropy for the grid box (95% confidence intervals in green) against scale factor; (

**c**) sample variance for different scale factors; (

**d**) Shannon information theoretic entropy for different scale factors. For the MSE in (b), missing data points represent scales at which no template matches were found.

**Figure 5.**Sample entropy, variance and Shannon entropy for the time-series of CRUTEM4v variance-adjusted surface temperature anomaly data for grid box [8|35] with m = 3 and r = 0.3. Black = 1851–1960; Red = 1961–2014. (

**a**) multi-scale sample entropy for the grid box against scale factor; (

**b**) sample variance for different scale factors; (

**c**) Shannon information theoretic entropy for different scale factors (months).

**Figure 6.**Multi-scale sample entropy of the CRUTEM4v variance-adjusted surface temperature anomaly data over Central Europe for m = 3 and r = 0.3. Scale factor is shown in months. Black = 1850 to 1960. Red = 1961 to 2014. Bold lines show the mean of all grid boxes and thin lines show the upper and lower bounds of the 95% confidence interval based on the t-distribution.

_{1}that regional temperature anomalies in Europe show particular time-scales with higher sample entropy than expected under the assumption of white noise. This only holds for the more recent data from 1961 to 2014, where signals in the time-series are apparent at specific time-scales where spikes in sample entropy are detected.

_{2}that the sample entropy of European temperature anomalies shows significant differences at particular time-scales when comparing 1851–1960 with 1961–2014. We infer that the temporal scaling properties of the temperature data have changed at the specific temporal scales of observation shown by significant deviations in Figure 6.The regional scale analysis in Figure 6 is based on the assumption that regional aggregation of the estimation of the sample entropy at different time-scales is more robust than a detailed grid-box level analysis because it is based on a larger number of meteorological stations. However, the visualization of the spatial patterns of the changes in temporal scaling behaviour may lead to further insights. The challenge is to visualize a complex dataset as a 2D map. For each grid box the MSE analysis results in a multi-scale plot like the one shown in Figure 3. Consideration needs to be given to the metrics of interest before mapping the results from this analysis. We can examine the difference between the sample entropy at a particular timescale between 1850–1960 and 1961–2014, for example. The selection of a timescale at which to visualize the differences is not straightforward. Here, the timescale with the largest absolute difference in sample entropy was chosen. In the map of Central Europe in Figure 7a, the colour scale represents the magnitude of the largest change in sample entropy (increase or decrease at any scale factor). It shows that Southern France, Eastern Spain and Corsica have experienced a decrease in sample entropy between 1850–1960 and 1961–2014. Portugal, the Bretagne, Great Britain, Ireland, Northern Germany and Eastern Europe showed a significant increase in sample entropy at particular timescales. At which timescales do these changes in sample entropy occur? Figure 7b shows a map where the colours represent the temporal scale factor at which the largest absolute difference in sample entropy occurs. In Figure 7b it is apparent that there is no consistent pattern of the scale factors at which the largest change in sample entropy occurred, but many areas show scale factors between 20 and 40.

**Figure 7.**Map of Central Europe indicating the grid boxes where the CRUTEM4v variance-adjusted surface temperature anomaly data show a statistically significant (p < 0.05) change in multi-scale sample entropy. Non-significant values are set to zero. (

**a**) Magnitude of the largest change in sample entropy detected across all scale factors. Negative values show a decrease from 1850–1960 to 1961–2014. (

**b**) Corresponding scale factor at which the largest absolute difference in sample entropy occurred (months).

## 4. Discussion

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

- Coumou, D.; Rahmstorf, S. A decade of weather extremes. Nat. Clim. Chang.
**2012**, 2, 491–496. [Google Scholar] - Glądalski, M.; Bańbura, M.; Kaliński, A.; Markowski, M.; Skwarska, J.; Wawrzyniak, J.; Zieliński, P.; Bańbura, J. Extreme weather event in spring 2013 delayed breeding time of Great Tit and Blue Tit. Int. J. Biometeorol.
**2014**, 58, 2169–2173. [Google Scholar] [CrossRef] [PubMed] - Robeson, S.M.; Willmott, C.J.; Jones, P.D. Trends in hemispheric warm and cold anomalies. Geophys. Res. Lett.
**2014**, 41, 9065–9071. [Google Scholar] [CrossRef] - Mika, J. Changes in weather and climate extremes: Phenomenology and empirical approaches. Clim. Chang.
**2013**, 121, 15–26. [Google Scholar] [CrossRef] - Song, X.; Zhang, Z.; Chen, Y.; Wang, P.; Xiang, M.; Shi, P.; Tao, F. Spatiotemporal changes of global extreme temperature events (ETEs) since 1981 and the meteorological causes. Nat. Hazards
**2014**, 70, 975–994. [Google Scholar] [CrossRef] - Dutilleul, P.R.L. Spatio-Temporal Heterogeneity; Cambridge University Press: Cambridge, UK, 2011. [Google Scholar]
- Rossi, A.; Massei, N.; Laignel, B. A synthesis of the time-scale variability of commonly used climate indices using continuous wavelet transform. Glob. Planet. Chang.
**2011**, 78, 1–13. [Google Scholar] [CrossRef] - Richman, J.S.; Moorman, J.R. Physiological time-series analysis using approximate entropy and sample entropy. Am. J. Physiol. Heart Circ. Physiol.
**2000**, 278, H2039–H2049. [Google Scholar] [PubMed] - Li, Z.; Zhang, Y.-K. Multi-scale entropy analysis of Mississippi River flow. Stoch. Environ. Res. Risk Assess.
**2008**, 22, 507–512. [Google Scholar] [CrossRef] - Bar-Yam, Y. Multiscale complexity/entropy. Adv. Complex Syst.
**2004**, 7, 47–63. [Google Scholar] [CrossRef] - Costa, M.; Goldberger, A.; Peng, C.-K. Multiscale entropy analysis of biological signals. Phys. Rev. E
**2005**. [Google Scholar] [CrossRef] - Li, X.-J.; Hu, T.-S.; Guo, X.-N.; Zeng, X. Chaos analysis of runoff time series at different timescales. J. Hydraul. Eng.
**2013**, 44, 515–520. [Google Scholar] - Zhou, Y.; Zhang, Q.; Li, K.; Chen, X. Hydrological effects of water reservoirs on hydrological processes in the East River (China) basin: Complexity evaluations based on the multi-scale entropy analysis. Hydrol. Process.
**2012**, 26, 3253–3262. [Google Scholar] [CrossRef] - Shannon, C.E. A mathematical theory of communication. Bell Syst. Tech. J.
**1948**, 27, 379–423. [Google Scholar] [CrossRef] - Grassberger, P.; Procaccia, I. Characterization of strange attractors. Phys. Rev. Lett.
**1983**, 50, 346. [Google Scholar] [CrossRef] - Pincus, S.M. Approximate entropy as a measure of system complexity. Proc. Natl. Acad. Sci. USA
**1991**, 88, 2297–2301. [Google Scholar] [CrossRef] [PubMed] - Kantz, H.; Schreiber, T. Nonlinear Time Series Analysis; Cambridge University Press: Cambridge, UK, 2004. [Google Scholar]
- R Core Team. R: A Language and Environment for Statistical Computing; R Foundation for Statistical Computing: Vienna, Austria, 2014. [Google Scholar]
- Brohan, P.; Kennedy, J.J.; Harris, I.; Tett, S.F.B.; Jones, P.D. Uncertainty estimates in regional and global observed temperature changes: A new data set from 1850. J. Geophys. Res.
**2006**. [Google Scholar] [CrossRef] - Jones, P.D.; Lister, D.H.; Osborn, T.J.; Harpham, C.; Salmon, M.; Morice, C.P. Hemispheric and large-scale land-surface air temperature variations: An extensive revision and an update to 2010. J. Geophys. Res. D: Atmos.
**2012**. [Google Scholar] [CrossRef] - Jones, P.; Osborn, T.; Briffa, K.; Folland, C.; Horton, E.; Alexander, L.; Parker, D.; Rayner, N. Adjusting for sampling density in grid box land and ocean surface temperature time series. J. Geophys. Res. D: Atmos.
**2001**, 106, 3371–3380. [Google Scholar] [CrossRef] - Meinshausen, M.; Smith, S.J.; Calvin, K.; Daniel, J.S.; Kainuma, M.L.T.; Lamarque, J.; Matsumoto, K.; Montzka, S.A.; Raper, S.C.B.; Riahi, K.; et al. The RCP greenhouse gas concentrations and their extensions from 1765 to 2300. Clim. Chang.
**2011**, 109, 213–241. [Google Scholar] [CrossRef] - Coumou, D.; Robinson, A. Historic and future increase in the global land area affected by monthly heat extremes. Environ. Res. Lett.
**2013**, 8, 034018. [Google Scholar] [CrossRef] - Schär, C.; Vidale, P.L.; Lüthi, D.; Frei, C.; Häberli, C.; Liniger, M.A.; Appenzeller, C. The role of increasing temperature variability in European summer heatwaves. Nature
**2004**, 427, 332–336. [Google Scholar] [CrossRef] [PubMed] - Zolina, O.; Simmer, C.; Belyaev, K.; Gulev, S.K.; Koltermann, P. Changes in the duration of European wet and dry spells during the last 60 years. J. Clim.
**2013**, 26, 2022–2047. [Google Scholar] [CrossRef] - Heinrich, G.; Gobiet, A. The future of dry and wet spells in Europe: A comprehensive study based on the ENSEMBLES regional climate models. Int. J. Climatol.
**2012**, 32, 1951–1970. [Google Scholar] [CrossRef] - Ding, Q.; Wang, B. Circumglobal teleconnection in the Northern Hemisphere summer. J. Clim.
**2005**, 18, 3483–3505. [Google Scholar] [CrossRef] - Francis, J.A.; Vavrus, S.J. Evidence linking Arctic amplification to extreme weather in mid-latitudes. Geophys. Res. Lett.
**2012**. [Google Scholar] [CrossRef]

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**MDPI and ACS Style**

Balzter, H.; Tate, N.J.; Kaduk, J.; Harper, D.; Page, S.; Morrison, R.; Muskulus, M.; Jones, P.
Multi-Scale Entropy Analysis as a Method for Time-Series Analysis of Climate Data. *Climate* **2015**, *3*, 227-240.
https://doi.org/10.3390/cli3010227

**AMA Style**

Balzter H, Tate NJ, Kaduk J, Harper D, Page S, Morrison R, Muskulus M, Jones P.
Multi-Scale Entropy Analysis as a Method for Time-Series Analysis of Climate Data. *Climate*. 2015; 3(1):227-240.
https://doi.org/10.3390/cli3010227

**Chicago/Turabian Style**

Balzter, Heiko, Nicholas J. Tate, Jörg Kaduk, David Harper, Susan Page, Ross Morrison, Michael Muskulus, and Phil Jones.
2015. "Multi-Scale Entropy Analysis as a Method for Time-Series Analysis of Climate Data" *Climate* 3, no. 1: 227-240.
https://doi.org/10.3390/cli3010227