# 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

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**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