# Permutation Entropy-Based Analysis of Temperature Complexity Spatial-Temporal Variation and Its Driving Factors in China

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## Abstract

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## 1. Introduction

## 2. Research Materials

#### 2.1. Air Temperature Data

#### 2.2. Explanatory Variables of Spatial Variation of Temperature Fluctuation Complexity (TFC)

## 3. Methods

#### 3.1. A Permutation Entropy (PE)-Based Method to Quantify TFC

- Reconstruction of the phase space: For a time series of daily average temperature x(i) (i = 1, 2, …, n), we reconstructed a m-dimensional space and get: X(i) = [x(i), x(i + 1), …, x(i + (m − 1)l)]. m and l are positive integers. l is set to 1. m is crucial for the reconstruction of the phase space.
- Recoding the reconstructed sequences: Rearrange X(i) in ascending order with [x(I + (j
_{1}− 1)l) ≤ x(I + (j_{2}− 1)l) ≤ … ≤ x(I + (j_{m}− 1)l)]. For each X(i), there is a symbolic sequence (permutation) as A(g) = [j_{1}, j_{2}, …, j_{m}] (g = 1, 2, …, k), where A is a set of symbolic sequences for all X(i). The maximum of possible permutations is m!, k ≤ m!. - Calculation of PE: The probability of each symbol sequence is recorded as [P
_{1}, P_{2}, …, P_{k}]. P_{k}is calculated as the number of occurrences of sequence k divided by total number of sequences. The PE of k symbolic sequences of time series x(i) can be defined as: $PE(m)=-{\displaystyle \sum _{v=1}^{k}{P}_{v}\mathrm{ln}{P}_{v}}$. When P_{v}= 1/m!, PE(m) reached the maximum ln(m!). Finally PE(m) is normalized by ln(m!) and there is a more elegant form 0 ≤ PE(m) ≤ 1.

#### 3.2. A GeoDetector-Based Method to Detect Driving Factors of TFC Spatial Variation

^{2}and N are the variance and sample size of Y respectively. ${\sigma}_{i}^{2}$ and N

_{i}are the variance and sample size of Y in subzone i (X = i) respectively.

#### 3.3. Mann-Kendall Method to Investigate TFC Temporal Variation

- Given PE series {PE
_{i}} i = 1, 2, …, n, where n is 39 for annual PE series and n is 35 for seasonal PE series, S_{k}is the number of PE_{i}exceeding PE_{j}(1 ≤ j ≤ i).$${S}_{K}={\displaystyle \sum _{i=1}^{k}{r}_{i}\text{}(i=1,\text{}2,\text{}\dots ,\text{}n);\text{}{r}_{i}}=\{\begin{array}{l}1\text{}P{E}_{i}P{E}_{j}\\ 0\text{}P{E}_{i}\le P{E}_{j}\end{array}\text{}(j=1,\text{}2,\text{}\dots ,\text{}i)$$ - {UF} is calculated to depict the trend from PE
_{1}to PE_{k}. Under the assumption of random and independence of time series, UF_{k}(k = 1, 2, …, n) quickly converges to the standard normal distribution as n increases (n > 10),$$U{F}_{k}=\frac{{S}_{k}-E({S}_{k})}{\sqrt{Var({S}_{k})}},\text{}E({S}_{k})=\frac{n(n-1)}{4},\text{}Var({S}_{k})=\frac{n(n-1)(2n+5)}{72}$$ - {UB} is calculated to depict the trend from PE
_{k}to PE_{n}. Reverse the sequence {PE_{i}} and repeat the step2 to get UF’_{k}.UB_{k}= −UF’_{k}., where UB_{1}= 0 and k = n, n − 1, …, 1.

_{1}to PE

_{k}when UF

_{k}> 0. UF

_{k}< 0 indicates a decline. Based on the hypothesis test (null hypothesis is {PE

_{i}} is completely random), null hypothesis is rejected if |UF

_{k}| > Z

_{0.05/2}there is significant trend of the PE sequence. |UF

_{k}| < Z

_{0.05/2}is the opposite. From the normal distribution table, Z

_{0.05/2}is 1.96. If the two curves of the {UF} and {UB} have an intersection between the significant lines, there is the mutation of {PE

_{i}} in the year of the intersection point of {UF} and {UB}. That means the intersection point is the beginning of the abrupt change of {PE

_{i}}.

## 4. Results and Analysis

#### 4.1. Spatial Variation of Annual TFC and Its Driving Factors Analysis

#### 4.2. Spatial Variation of Seasonal TFC and Its Driving Factors Analysis

#### 4.3. Temporal Variation of Annual TFC and Seasonal TFC

_{k}and UB

_{k}had one distinct intersection between the significant lines (Z

_{0.05/2}= +1.96) in 1986. It is indicated that there is the mutation of TFC intensity in 1986. In other words, the abrupt change of annual TFC started in the year of 1986. Since 1986, UF

_{k}had been below 0. And UF was below −1.96 in 2011. This indicates that the annual TFC decreased substantially from 1979–2011. Meanwhile, the UB in 2011 was over 0, but less than 1.96. It means that there was an insignificant increase of annual TFC from 2011–2017.

_{k}kept on less than −1.96 after 1986 in the Cluster 2–4 and their TFC showed the downward trend. The mutation process of Cluster 1 is quite complex and its trend is not clear. UF and UB have more than one intersection between the significant lines (Figure S4a). There was the most evident downward trend of annual TFC from 1979–2017 in Cluster 2, with the weakest TFC. UF

_{k}in 2000–2011 had been lower than −1.96 and the annual TFC declined strikingly in Cluster 2 from 1979–2011. In general, the trend and mutation results of regional average TFC are representative.

_{k}and UB

_{k}have an intersection between the significant lines in these three seasons. There is a significant abrupt change of TFC series and their starting year of the abrupt change was basically 1986–1992. In summer, the abrupt change of summer TFC started in 1986 (Figure 11b). There was a downward trend from 1983–2001 (with summer TFC from 0.892–0.877). Then summer TFC rose to 0.887 from 2001–2017 (Figure S5b). In autumn, the abrupt change of TFC series started in 1989 (Figure 11c). Autumn TFC decreased markedly from 0.881–0.867 in 1983–2011. Then it ascended to 0.873 after 2011 (Figure S5c). In winter, the abrupt change of TFC series started in 1992 (Figure 11d). Winter TFC experienced an apparent decline in 1983–2012, dropping from 0.873–0.867, and increased to 0.875 after 2012 (Figure S5d).

## 5. Discussion

## 6. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Li, J.; Chou, J. Global analysis theory of climate system and its applications. Sci. Bull.
**2003**, 48, 1034–1039. [Google Scholar] [CrossRef] - Adachi, S.A.; Nishizawa, S.; Yoshida, R.; Yamaura, T.; Ando, K.; Yashiro, H.; Kajikawa, Y.; Tomita, H. Contributions of changes in climatology and perturbation and the resulting nonlinearity to regional climate change. Nat. Commun.
**2017**, 8, 2224. [Google Scholar] [CrossRef] [PubMed] - Atta-ur-Rahman; Dawood, M. Spatio-statistical analysis of temperature fluctuation using Mann-Kendall and Sen’s slope approach. Clim. Dyn.
**2017**, 48, 783–797. [Google Scholar] [CrossRef] - Zheng, Z. Characteristics of long-term climate change in Beijing with Detrended Fluctuation Analysis. Chin. J. Geophys.
**2007**, 50, 1084–1088. [Google Scholar] [CrossRef] - Kerr, R.A. Climate change. Confronting the bogeyman of the climate system. Science
**2005**, 310, 432–433. [Google Scholar] [CrossRef] - Hidalgo, H.G.; Alfaro, E.J.; Quesada-Montano, B. Observed (1970–1999) climate variability in Central America using a high-resolution meteorological dataset with implication to climate change studies. Clim. Chang.
**2017**, 141, 1–16. [Google Scholar] [CrossRef] - Rolland, C. Spatial and seasonal variations of air temperature lapse rates in Alpine regions. J. Clim.
**2003**, 16, 1032–1046. [Google Scholar] [CrossRef] - Cavanaugh, N.R.; Shen, S.S.P. Northern hemisphere climatology and trends of statistical moments documented from GHCN-daily surface air temperature station data from 1950 to 2010. J. Clim.
**2014**, 27, 5396–5410. [Google Scholar] [CrossRef] - Anwer, M. Nature of centennial global climate change from observational records. Am. J. Clim. Chang.
**2015**, 4, 337–354. [Google Scholar] [CrossRef] - Zunino, L.; Kulp, C.W. Detecting nonlinearity in short and noisy time series using the permutation entropy. Phys. Lett. A
**2017**, 381, 3627–3635. [Google Scholar] [CrossRef] - Zhenlin, Y.; Hanna, E.; Callaghan, T.V. Modelling surface-air-temperature variation over complex terrain around Abisko, Swedish Lapland: Uncertainties of measurements and models at different scales. Geogr. Ann. Ser. A Phys. Geogr.
**2011**, 93, 89–112. [Google Scholar] - Li, S.C.; Liu, F.Y.; Zhao, Z.Q. Climate complexity and spatial variation in China. Clim. Environ. Res.
**2008**, 13, 31–38. [Google Scholar] - Cheng, C.X.; Shi, P.J.; Song, C.Q.; Gao, J.B. Geographic big-data: A new opportunity for geography complexity study. Acta Geogr. Sin.
**2018**, 73, 1397–1406. [Google Scholar] - Song, C.Q.; Cheng, C.X.; Shi, P.J. Geography complexity: New connotations of geography in the new era. Acta Geogr. Sin.
**2018**, 73, 1204–1213. [Google Scholar] - Zhang, T.; Shen, S.; Cheng, C.X.; Song, C.Q.; Ye, S.J. Long-range correlation analysis of soil temperature and moisture on A’rou Hillsides, Babao River Basin. J. Geophys. Res. Atmos.
**2018**, 123, 12606–12620. [Google Scholar] [CrossRef] - Gao, P.; Zhang, H.; Li, Z. A hierarchy-based solution to calculate the configurational entropy of landscape gradients. Landsc. Ecol.
**2017**. [Google Scholar] [CrossRef] - Gao, P.; Liu, Z.; Liu, G.; Zhao, H.; Xie, X. Unified metrics for characterizing the fractal nature of geographic features. Ann. Am. Assoc. Geogr.
**2017**, 107, 1315–1331. [Google Scholar] [CrossRef] - Li, S.C.; Zhao, Q.F.; Wu, S.H.; Dai, E.F. Measurement of climate complexity using sample entropy. Int. J. Clim.
**2006**, 26, 2131–2139. [Google Scholar] - Jiang, L.; Li, N.N.; Fu, Z.T.; Zhang, J.P. Long-range correlation behaviors for the 0-cm average ground surface temperature and average air temperature over China. Appl. Clim.
**2015**, 119, 25–31. [Google Scholar] [CrossRef] - Cao, Y.H.; Tung, W.W.; Gao, J.B.; Protopopescu, V.A.; Hively, L.M. Detecting dynamical changes in time series using the permutation entropy. Phys. Rev. E
**2004**, 70, 046217. [Google Scholar] [CrossRef][Green Version] - Zheng, X.X.; Zhou, G.W.; Li, D.D.; Zhou, R.C.; Ren, H.H. Application of variational mode decomposition and permutation entropy for rolling bearing fault diagnosis. Int. J. Acoust. Vib.
**2019**, 24, 303–311. [Google Scholar] [CrossRef] - Li, X.L.; Cui, S.Y.; Voss, L.J. Using permutation entropy to measure the electroencephalographic effects of sevoflurane. Anesthesiology
**2008**, 109, 448–456. [Google Scholar] [CrossRef] [PubMed] - Stosic, T.; Telesca, L.; Ferreira, D.V.D.; Stosic, B. Investigating anthropically induced effects in streamflow dynamics by using permutation entropy and statistical complexity analysis: A case study. J. Hydrol.
**2016**, 540, 1136–1145. [Google Scholar] [CrossRef] - Zunino, L.; Zanin, M.; Tabak, B.M.; Perez, D.G.; Rosso, O.A. Forbidden patterns, permutation entropy and stock market inefficiency. Phys. A
**2009**, 388, 2854–2864. [Google Scholar] [CrossRef] - Yang, Y.L.; Zhou, M.N.; Niu, Y.; Li, C.G.; Cao, R.; Wang, B.; Yan, P.F.; Ma, Y.; Xiang, J. Epileptic seizure prediction based on permutation entropy. Front. Comput. Neurosci.
**2018**, 12, 55. [Google Scholar] [CrossRef] [PubMed] - Li, X.L.; Ouyang, G.X.; Liang, Z.H. Complexity measure of motor current signals for tool flute breakage detection in end milling. Int. J. Mach. Tools Manuf.
**2008**, 48, 371–379. [Google Scholar] [CrossRef] - Zheng, Y.F.; Yin, L.F.; Wu, R.J. Investigation of the spatial and temporal distribution of extreme heat in mainland of china with detrended fluctuation and permutation entropy. J. Trop. Meteorol.
**2012**, 28, 251–257. [Google Scholar] - Hou, W.; Feng, G.L.; Dong, W.J.; Li, J.P. A technique for distinguishing dynamical species in the temperature time series of north China. Acta Phys. Sin.
**2006**, 55, 2663–2668. [Google Scholar] - Hao, C.Y.; Wu, S.H.; Li, S.C. Measurement of climate complexity using permutation entropy. Geogr. Res.
**2007**, 26, 46–52. [Google Scholar] - Feddema, J.J. The importance of land-cover change in simulating future climates. Science
**2005**, 310, 1674–1678. [Google Scholar] [CrossRef] - Jiang, T.; Liu, X.; Wu, L. Method for mapping rice fields in complex landscape areas based on pre-trained convolutional neural network from HJ-1 A/B data. ISPRS Int. J. Geo-Inf.
**2018**, 7, 418. [Google Scholar] [CrossRef] - Ma, J.; Xie, S.P. Regional patterns of sea surface temperature change: A source of uncertainty in future projections of precipitation and atmospheric circulation. J. Clim.
**2013**, 26, 2482–2501. [Google Scholar] [CrossRef] - Giorgi, F.; Francisco, R. Uncertainties in regional climate change prediction: A regional analysis of ensemble simulations with the HADCM2 coupled AOGCM. Clim. Dyn.
**2000**, 16, 169–182. [Google Scholar] [CrossRef] - Tereshchenko, I.E.; Filonov, A.E. Air temperature fluctuations in Guadalajara, Mexico, from 1926 to 1994 in relation to urban growth. Int. J. Clim.
**2001**, 21, 483–494. [Google Scholar] [CrossRef] - Vincze, M.; Borcia, I.D.; Harlander, U. Temperature fluctuations in a changing climate: An ensemble-based experimental approach. Sci. Rep.
**2017**, 7, 254. [Google Scholar] [CrossRef] - Bai, L.; Jiang, L.; Yang, D.Y.; Liu, Y.B. Quantifying the spatial heterogeneity influences of natural and socioeconomic factors and their interactions on air pollution using the geographical detector method: A case study of the Yangtze River Economic Belt, China. J. Clean. Prod.
**2019**, 232, 692–704. [Google Scholar] [CrossRef] - Luo, W.; Jasiewicz, J.; Stepinski, T.; Wang, J.F.; Xu, C.D.; Cang, X.Z. Spatial association between dissection density and environmental factors over the entire conterminous United States. Geophys. Res. Lett.
**2016**, 43, 692–700. [Google Scholar] [CrossRef] - Shi, S.Q.; Han, Y.; Yu, W.B.; Cao, Y.Q.; Cai, W.M.; Yang, P.; Wu, W.B.; Yu, Q.Y. Spatio-temporal differences and factors influencing intensive cropland use in the Huang-Huai-Hai Plain. J. Geogr. Sci.
**2018**, 28, 1626–1640. [Google Scholar] [CrossRef][Green Version] - Chen, Y.; Hu, Q.; Yang, Y.M.; Qian, W.H. Anomaly based analysis of extreme heat waves in Eastern China during 1981–2013. Int. J. Clim.
**2017**, 37, 509–523. [Google Scholar] [CrossRef] - Gao, L.; Schulz, K.; Chen, X.W.; Lin, G.F. Analysis of extreme temperatures in China based on ERA-interim reanalysis data. South North Water Transf. Water Sci. Technol.
**2014**, 12, 75–78. [Google Scholar] - Gao, L.; Hao, L. Verification of ERA-Interim reanalysis data over China. J. Subtrop. Resour. Environ.
**2014**, 9, 75–81. [Google Scholar] - Song, Z.Y.; Qin, W.H.; Wei, Y.J. Evaluation of surface air temperature in three reanalysis datasets on islands adjacent to Zhejiang. Adv. Mar. Sci.
**2018**, 36, 499–511. [Google Scholar] - Bandt, C.; Pompe, B. Permutation entropy: A natural complexity measure for time series. Phys. Rev. Lett.
**2002**, 88, 174102. [Google Scholar] [CrossRef] [PubMed] - Riedl, M.; Muller, A.; Wessel, N. Practical considerations of permutation entropy—A tutorial review. Eur. Phys. J. Spec. Top.
**2013**, 222, 249–262. [Google Scholar] [CrossRef] - Li, X.L.; Ouyang, G.X.; Richards, D.A. Predictability analysis of absence seizures with permutation entropy. Epilepsy Res.
**2007**, 77, 70–74. [Google Scholar] [CrossRef] - Wang, J.F.; Zhang, T.L.; Fu, B.J. A measure of spatial stratified heterogeneity. Ecol. Indic.
**2016**, 67, 250–256. [Google Scholar] [CrossRef] - Wang, J.F.; Xu, C.D. Geodetector: Principle and prospective. Acta Geogr. Sin.
**2017**, 72, 116–134. [Google Scholar] - Hirsch, R.M.; Slack, J.R. A Nonparametric trend test for seasonal data with serial dependence. Water Resour. Res.
**1984**, 20, 727–732. [Google Scholar] [CrossRef] - Liu, Q.; Yang, Z.F.; Cui, B.S.; Sun, T. Temporal trends of hydro-climatic variables and runoff response to climatic variability and vegetation changes in the Yiluo River basin, China. Hydrol. Process.
**2010**, 23, 3030–3039. [Google Scholar] [CrossRef] - Xu, Z.X.; Gong, T.L.; Li, J.Y. Decadal trend of climate in the Tibetan Plateau-regional temperature and precipitation. Hydrol. Process.
**2008**, 22, 3056–3065. [Google Scholar] [CrossRef] - Deniz, A.; Toros, H.; Incecik, S. Spatial variations of climate indices in Turkey. Int. J. Clim.
**2015**, 31, 394–403. [Google Scholar] [CrossRef] - Kapala, A.; Machel, H.; Flohn, H. Behaviour of the centres of action above the Atlantic since 1881. Part II: Associations with regional climate anomalies. Int. J. Clim.
**2015**, 18, 23–36. [Google Scholar] [CrossRef] - Bujalsky, L.; Jirka, V.; Zemek, F.; Frouz, J. Relationships between the normalised difference vegetation index and temperature fluctuations in post-mining sites. Int. J. Min. Reclam. Environ.
**2018**, 32, 254–263. [Google Scholar] [CrossRef] - He, J.L.; Zhao, W.; Li, A.N.; Wen, F.P.; Yu, D.J. The impact of the terrain effect on land surface temperature variation based on Landsat-8 observations in mountainous areas. Int. J. Remote Sens.
**2019**, 40, 1808–1827. [Google Scholar] [CrossRef] - Alo, C.A.; Wang, G. Role of dynamic vegetation in regional climate predictions over western Africa. Clim. Dyn.
**2010**, 35, 907–922. [Google Scholar] [CrossRef] - Wu, L.Y.; Zuo, H.C.; Feng, J.M.; Chen, B.L.; Dong, L.X. Numerical simulation of the impact of land use and green vegetation fraction changes on regional climate in China. J. Lanzhou Univ. Nat. Sci.
**2018**, 54, 54–64. [Google Scholar] - Zhao, L.; Yang, Q.; An, S.Z. Numerical simulation analysis on the impact of change in rangeland vegetation type on climate in the Tianshan Mountains using a Regional Climate Model. Acta Pratacult. Sin.
**2014**, 23, 51–61. [Google Scholar] - Wang, H. Temperature sensitivity of plant phenology in temperate and subtropical regions of China from 1850 to 2009. Int. J. Clim.
**2015**, 35, 913–922. [Google Scholar] [CrossRef] - Glover, J.; Mcculloch, J. The empirical relation between solar radiation and hours of bright sunshine in the high-altitude tropics. Q. J. R. Meteorol. Soc.
**1958**, 84, 172–175. [Google Scholar] [CrossRef] - Haslinger, K.; Anders, I.; Hofstatter, M. Regional climate modelling over complex terrain: An evaluation study of COSMO-CLM hindcast model runs for the Greater Alpine Region. Clim. Dyn.
**2013**, 40, 511–529. [Google Scholar] [CrossRef] - Zhu, J.W.; Zeng, X.D. Influences of the seasonal growth of vegetation on surface energy budgets over middle to high latitudes. Int. J. Clim.
**2017**, 37, 4251–4260. [Google Scholar] [CrossRef] - Qiao, D.; Wang, N. Relationship between winter snow cover dynamics, climate and spring grassland vegetation phenology in inner Mongolia, China. ISPRS Int. J. Geo-Inf.
**2019**, 8, 42. [Google Scholar] [CrossRef] - Wang, X.; Sun, Z.Y.; Zhou, A.G. Alpine cold vegetation response to climate change in the western Nyainqentanglha range in 1972–2009. Sci. World J.
**2014**, 2014, 514736. [Google Scholar] [CrossRef] [PubMed] - Ren, X.; Qian, Y. A coupled regional air-sea model, its performance and climate drift in simulation of the East Asian summer monsoon in 1998. Int. J. Clim.
**2005**, 25, 679–692. [Google Scholar] [CrossRef] - Qian, Y.F.; Zheng, Y.Q.; Zhang, Y.; Miao, M.Q. Responses of China’s summer monsoon climate to snow anomaly over the Tibetan Plateau. Int. J. Clim.
**2003**, 23, 593–613. [Google Scholar] [CrossRef] - Oxoli, D.; Ronchetti, G.; Minghini, M.; Molinari, M.E.; Lotfian, M.; Sona, G.; Brovelli, M.A. Measuring urban land cover influence on air temperature through multiple geo-data-the case of Milan, Italy. ISPRS Int. J. Geo-Inf.
**2018**, 7, 421. [Google Scholar] [CrossRef] - Wang, T.; Sun, J.G.; Han, H.; Yan, C.Z. The relative role of climate change and human activities in the desertification process in Yulin region of northwest China. Environ. Monit. Assess.
**2012**, 184, 7165–7173. [Google Scholar] [CrossRef] - Balsamo, G.; Albergel, C.; Beljaars, A.; Boussetta, S.; Brun, E.; Cloke, H.; Dee, D.; Dutra, E.; Muñoz-Sabater, J.; Pappenberger, F.; et al. ERA-Interim/Land: A global land surface reanalysis data set. HESS
**2015**, 19, 389–407. [Google Scholar] [CrossRef]

**Figure 1.**(

**a**) Spatial distribution of China’s vegetation zoning with eight subzones. (1) Subtropical evergreen broad-leaved forest area, (2) Cold-temperate coniferous forest area, (3) Warm-temperate deciduous broad-leaved forest area, (4) Temperate grassland area, (5) Temperate desert area, (6) Temperate coniferous and deciduous broad-leaved mixed forest area, (7) Tropical monsoon rainforest and rainforest area and (8) Alpine vegetation area of Qinghai-Tibet Plateau. (

**b**) Spatial distribution of China’s climatic zoning with nine subzones: (1) Middle Temperature Zone, (2) Warm Temperate Zone, (3) Cold Temperate Zone, (4) North Subtropical Zone, (5) Central Subtropical Zone, (6) South Subtropical Zone, (7) Middle Tropical Zone, (8) Marginal Tropical Zone and (9) Plateau Climatic Zone. (

**c**) Spatial distribution of latitude zoning with seven subzones: (1) 18.75–23.75°N, (2) 24–28.5°N, (3) 29.25–34.5°N, (4) 35.25–39.75°N, (5) 40.5–45°N, (6) 45.75–51°N and (7) 51.75–53.25°N. (

**d**) Spatial distribution of China’s altitude zoning with seven subzones: (1) −263–565, (2) 566–1220, (3) 1221–2045, (4) 2046–3085, (5) 3086–4050, (6) 4051–4845, (7) 4846–8535 meter. (

**e**) Spatial distribution of China’s terrain zoning with seven subzones: (1) Plains, (2) Platforms, (3) Hills, (4) Small relief mountains, (5) Medium relief mountains, (6) Large relief mountains and (7) Extreme relief mountains. (

**f**) The name of each province is labelled in Figure 1f.

**Figure 3.**Spatial distribution of the annual temperature fluctuation complexity (TFC) (the average of annual permutation entropy (PE) from 1979–2017).

**Figure 4.**Driving factor explaining ability of spatial variation annual TFC. The inner ring corresponds to the explaining ability of each single factor, and the outer ring corresponds to the explaining ability of the interaction between two factors. A–B represents the interaction between A and B.

**Figure 5.**Spatial distribution of seasonal fluctuation complexity of air temperature (the average of seasonal PE from 1983–2017). (

**a**) Spatial distribution of spring TFC in China. (

**b**) Spatial distribution of summer TFC in China. (

**c**) Spatial distribution of autumn TFC in China. (

**d**) Spatial distribution of winter TFC in China.

**Figure 7.**Driving factor explaining ability on spatial variation of seasonal TFC. The inner rings correspond to the explaining ability of each single factor, and the outer rings correspond to the explaining ability of the interaction between two factors. A–B represents the interaction between A and B. (

**a**) Driving factor explaining ability on spatial variation of spring TFC. (

**b**) Driving factor explaining ability on spatial variation of summer TFC. (

**c**) Driving factor explaining ability on spatial variation of autumn TFC. (

**d**) Driving factor explaining ability on spatial variation of winter TFC.

**Figure 8.**(

**a**) Annual mean series of annual temperature fluctuation complexity over 1979–2017. (

**b**) Temporal trend and mutation of annual temperature fluctuation complexity over 1979–2017. The series of annual PE here is the average of annual PE series at all locations in China, and annual PE series at each location is calculated in Section 3.1. Two black dash lines in Figure 8a correspond to two special years, the mutation starting year 1986 and the lowest PE year 2011 respectively. The black point in Figure 8b marks the intersection of UF

_{k}and UB

_{k}between the significant lines, which is noted as the mutation starting year.

**Figure 10.**Spatial average time series of four annual TFC clusters over 1979–2017. The black dash line in 1986 marked the mutation of TFC intensity. The black dash line in 2011 marked the lowest TFC intensity.

**Figure 11.**Temporal trend and mutation of seasonal temperature fluctuation complexity over 1983–2017. The series of seasonal PE here is the average of seasonal PE series at all locations in China. The seasonal PE series at each location is calculated in Section 3.1. The points marked in Figure 9b–d are the intersection of UF

_{k}and UB

_{k}between the significant lines, which are noted as the mutation starting years. (

**a**) Temporal trend and mutation of spring TFC over 1983–2017. (

**b**) Temporal trend and mutation of summer TFC over 1983–2017. (

**c**) Temporal trend and mutation of autumn TFC over 1983–2017. (

**d**) Temporal trend and mutation of winter TFC over 1983–2017.

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**MDPI and ACS Style**

Zhang, T.; Cheng, C.; Gao, P. Permutation Entropy-Based Analysis of Temperature Complexity Spatial-Temporal Variation and Its Driving Factors in China. *Entropy* **2019**, *21*, 1001.
https://doi.org/10.3390/e21101001

**AMA Style**

Zhang T, Cheng C, Gao P. Permutation Entropy-Based Analysis of Temperature Complexity Spatial-Temporal Variation and Its Driving Factors in China. *Entropy*. 2019; 21(10):1001.
https://doi.org/10.3390/e21101001

**Chicago/Turabian Style**

Zhang, Ting, Changxiu Cheng, and Peichao Gao. 2019. "Permutation Entropy-Based Analysis of Temperature Complexity Spatial-Temporal Variation and Its Driving Factors in China" *Entropy* 21, no. 10: 1001.
https://doi.org/10.3390/e21101001