# Meaningful Trend in Climate Time Series: A Discussion Based On Linear and Smoothing Techniques for Drought Analysis in Taiwan

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## Abstract

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## 1. Introduction

## 2. Study Region and Data

## 3. Methodology

#### 3.1. Standardized Precipitation Index (SPI)

#### 3.2. Smoothing Technique: Regularized Minimal-Energy Tensor-Product Spline (RMTS)

#### 3.3. Trend Detection Based On Linear and Smoothing Techniques

#### 3.3.1. Linear Regression

#### 3.3.2. Locally Weighted Least Squares Regression

#### 3.3.3. Test for Trend Using First Derivatives

## 4. Results and Discussion

#### 4.1. Overall Trend in Taiwan’s Drought from 1960 to 2019

#### 4.2. Trends in the Earlier and Later 30 Years

#### 4.3. Discussion of Meaningful Trends

## 5. Conclusions and Recommendations

- Trend detection using LR showed great differences from that using the smoothing techniques, and LR seemed to be less robust as it falsely identified too many grids with significant trends and non-Gaussian residuals. LR trend lines were not found meaningful in many occasions of our case examining the SPI series in Taiwan since the data did not present much linearity.
- When all the methods reached a consensus in the patterns of detected trends with significance, intuitively we could have more confidence in such detected trends. By calculating pattern correlations as the quantification metric of pattern similarity between detected trends, we found that the recent drying trend at the shorter time scales over eastern Taiwan in 1990–2019 should be the most trustworthy.
- Regardless of the methods, detected trends in the entire period (1960–2019), the earlier 30 years (1960–1989), or the later 30 years (1990–2019) were all different. While the general wetting trend was identified over a great portion of Taiwan’s territory in the past 60 years, some migrations of drying or wetting trends actually took place in different time intervals.

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Spatial patterns of trends in the SPI series from 1960 to 2019 detected using linear regression; (

**a**–

**d**) the SPI at 3-, 6-, 9-, and 12-month periods. Grids with significant trends are dotted.

**Figure 3.**The same as Figure 2, but for detected trends using first derivatives of regularized minimal-energy tensor-product B-splines (RMTB).

**Figure 4.**The same as Figure 2, but for detected trends using locally estimated scatterplot smoothing (LOESS).

**Figure 5.**SPI12 time series along with the best fitting linear trend lines and RMTB curves at six CWB stations located over the (

**a**–

**c**) western, (

**d**,

**e**) mountainous, and (

**f**) southern regions of Taiwan.

**Figure 6.**The same as Figure 5, but for four CWB stations over the eastern side of Taiwan.

**Figure 7.**Spatial patterns of trends in the SPI series in the earlier period (1960–1989) detected using linear regression; (

**a**–

**d**) the SPI at 3-, 6-, 9-, and 12-month periods.

**Figure 8.**The same as Figure 7, but for detected trends using first derivatives of RMTB.

**Figure 9.**The same as Figure 7, but for detected trends using LOESS.

**Figure 10.**Spatial patterns of trends in the SPI series in the later period (1990–2019) detected using linear regression; (

**a**–

**d**) the SPI at 3-, 6-, 9-, and 12-month periods.

**Figure 11.**The same as Figure 10, but for detected trends using first derivatives of RMTB.

**Figure 12.**The same as Figure 10, but for detected trends using LOESS.

**Table 1.**Pattern correlations between detected trend in the SPI using one of the three methods and that using another method over three different time periods. The first, second, and third values in each parenthesis indicate correlations derived from the paired methods: linear regression (LR) vs. first derivative; LR vs. locally estimated scatterplot smoothing (LOESS), and first derivative vs. LOESS, respectively. Correlation values greater than 0.35 are in bold.

Time Period | SPI3 | SPI6 | SPI9 | SPI12 |
---|---|---|---|---|

1960–2019 | (0.03, 0.02, 0.67) | (0.04, 0.05, 0.74) | (0.09, 0.10, 0.66) | (0.19, 0.22, 0.68) |

1960–1989 | (0.04, 0.06, 0.79) | (0.03, 0.22, 0.44) | (0.26, 0.23, 0.37) | (0.01, 0.19, 0.08) |

1990–2019 | (0.32, 0.36, 0.91) | (0.40, 0.38, 0.86) | (0.49, 0.52, 0.71) | (0.28, 0.40, 0.13) |

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**MDPI and ACS Style**

Huang, S.-H.; Mahmud, K.; Chen, C.-J.
Meaningful Trend in Climate Time Series: A Discussion Based On Linear and Smoothing Techniques for Drought Analysis in Taiwan. *Atmosphere* **2022**, *13*, 444.
https://doi.org/10.3390/atmos13030444

**AMA Style**

Huang S-H, Mahmud K, Chen C-J.
Meaningful Trend in Climate Time Series: A Discussion Based On Linear and Smoothing Techniques for Drought Analysis in Taiwan. *Atmosphere*. 2022; 13(3):444.
https://doi.org/10.3390/atmos13030444

**Chicago/Turabian Style**

Huang, Shih-Han, Khalid Mahmud, and Chia-Jeng Chen.
2022. "Meaningful Trend in Climate Time Series: A Discussion Based On Linear and Smoothing Techniques for Drought Analysis in Taiwan" *Atmosphere* 13, no. 3: 444.
https://doi.org/10.3390/atmos13030444