Prewhitening-Aided Innovative Trend Analysis Method for Trend Detection in Hydrometeorological Time Series

Abstract

Detection of trends in hydrometeorological time series is essential for understanding the complex variability of hydrometeorological data. Although different types of methods have been proposed, accurately identifying trends and their statistical significance is still challenging due to the complex characteristics of hydroclimatic data and the limitations of diverse methods. In this article, we propose a new trend detection approach, namely the prewhitening-aided innovative trend analysis (ITA). This method first corrects the significance test formula of the original ITA method, followed by a prewhitening method to eliminate serial autocorrelation and ensure independence. Results of Monte–Carlo experiments verified the superiority of the prewhitening-aided ITA method to the previous ITA methods. Moreover, serial correlations had significant impacts on the performance of diverse methods. Comparatively, the traditional ITA method kept high Type I errors and tended to overestimate the significance of trends. The four ITA methods, which were improved in previous studies, performed better than the traditional ones but could not overcome the influence of either positive or negative correlation characteristics of time series. The four prewhitening-aided ITA methods performed much better as they could effectively handle serial correlation. Among all the nine methods concerned in this study, the variance correction prewhitening-aided ITA (VCPWITA0) method performed the best. Detection of trends in precipitation data in the Qinghai-Tibet Plateau further verified the superiority of the VCPWITA0 method. The proposed method fully exploited the advantages of both ITA and prewhitening, and thus, it provided a new approach for detecting trends and has the potential for wide use in hydrological and climate sciences.

1. Introduction

Detection of trends in hydrometeorological time series is important for understanding the spatiotemporal variability of hydrometeorological processes at large timescales [1,2,3,4,5,6,7]. Many methods have been developed for trend detection [4,8,9], including the moving average method, Mann–Kendall (MK) test [10,11], Spearman rank correlation analysis [12], least squares method, Sen’s slope estimation [13], and linear regression [14]. They are mainly applicable to detecting the monotonic trend in a time series [15,16]. Generally, each method has its own assumptions and requirements for data, such as distribution form, homogeneity, correlation, and sample length of time series [4]. However, observed hydrometeorological data usually do not meet these assumed conditions and requirements [17], and results of trend detection often have biases and even errors [18]. As a result, accurately detecting trends in hydrometeorological time series remains a challenge.

Nonparametric test methods do not require the normal distribution assumption of time series, and thus, they are more widely used for the detection of trends than parametric test methods [4], but independence is a prerequisite for nonparametric methods in trend detection. However, observed hydrometeorological time series usually indicate obvious autocorrelation characteristics, as in streamflow [3,4,19], temperature [1,2], water quality [20], and precipitation [21]. They violate the assumption of independence and often cause overestimation of trends [22]. To overcome the influence of dependence, Sen [23] proposed a graphical-based nonparametric method for trend detection called Innovation Trend Analysis (ITA); after that, he improved the trend indicators and significance evaluation in ITA [24]. The ITA method was shown to be not restricted by serial dependence and distribution type, and the trends of time series can be visualized graphically. It allows for not only the overall observation of time series but also searches for trends in different data ranges such as low-, median-, and high-value ranges. The ITA method has been extensively used to detect trends of environmental and hydrometeorological variables, and its superiority compared to conventional methods has been verified [5,6,20,21,25,26].

However, it can be found from the literature that the significance test formula of the ITA method still depended on the independence assumption [24], and the mathematical derivation of the sub-series in the method is unreasonable [27,28], which overestimates the trend’s significance. As a result, the argument that the ITA method has stronger capabilities of trend detection than conventional methods is debatable [21,29,30,31,32]. Some studies have tried to improve the ITA method. For instance, Wu et al. [33] applied the bootstrap method to calculate the trend critical value of the ITA test statistic (Bootstrap_ITA), but it cannot be accurately formulated. Wang et al. [27] proposed the variance correction analysis-based ITA method (VCA_ITA). However, it not only involves the selection of models and the estimation of parameters but also requires some assumptions and transformations on the original time series, which include uncertainties. Alashan [34] employed covariance to improve the critical trend formula in ITA (COV_ITA), but its effectiveness needs further verification.

Generally, prewhitening and variance correction are two effective approaches to eliminating serial autocorrelation and to ensuring independence. Comparatively, the former is more widely used. The commonly used prewhitening methods include ordinary prewhitening, trend-free prewhitening, iterative prewhitening, and variance corrected prewhitening [18,22,35,36,37]. These methods have been successfully combined with the MK test to improve the accuracy of trend detection. In recent years, the application of prewhitening methods in nonparametric trend detection has been an active area of research. For instance, Sheoran et al. [38] proposed an improved version of prewhitening for trend analysis in autocorrelated time series and conducted comparative evaluations with existing prewhitening methods. Collaud Coen et al. [39] not only analyzed the effects of various prewhitening methods on the statistical significance and slope in the Mann–Kendall test but also introduced a new method combining three prewhitening approaches, termed 3PW. O’Brien et al. [40] compared widely used prewhitening methods with several newer variants, integrating them with the Mann–Kendall nonparametric trend test to address positive and negative autocorrelation effects. However, there has been a lack of studies combining prewhitening methods with other nonparametric trend tests, particularly the ITA method.

The main objective of this study is to improve the accuracy of trend detection by combining the ITA method with prewhitening methods to overcome the influence of serial correlation. In the proposed method, the errors in the mathematical derivation of the original ITA method are first corrected; then, the serial correlation of the original time series is prewhitened to detect trends accurately. For comparison, we combine each of the above four prewhitening methods with the ITA method to verify their efficiencies and further compare the four prewhitening-aided ITA methods to the three improved ITA methods mentioned above (Bootstrap_ITA, VCA_ITA, COV_ITA), as well as the original ITA method and the ITA0 method which only modifies mathematical formula of the original ITA method but without prewhitening. It should be noted that the proposed method can tackle both positive and negative serial correlation, a research gap for trend detection in present studies. These methods were applied to precipitation data in the flood season (July–September) in the Qinghai-Tibet Plateau to verify the superiority of the proposed method.

2. Brief Description of ITA Method

2.1. Innovation Trend Analysis (ITA)

The ITA method can detect diverse forms of trend in time series, including monotonic upward, non-monotonic upward, monotonic downward, and non-monotonic downward trend. The specific idea of ITA is to first divide the original time series into equal parts and arrange them in ascending order, denoted as $\left\{y_{1i}\right\}$ and $\left\{y_{2i}\right\}$ (i = 1, 2…, n/2, where n is the length of time series), and then take $\left\{y_{1i}\right\}$ as the abscissa and $\left\{y_{2i}\right\}$ as the ordinate to draw a scatter plot in the Cartesian coordinate system and further compare it with the trend-free straight line (Figure 1). If the scattered points are distributed above (below) the 45° line, it indicates a monotonic upward (downward) trend. If the scattered points are distributed near the 45° line, it indicates an insignificant trend. The scattered points being farther from the 45° line indicate a more significant trend [23,41].

Regarding the ITA method, the statistic Z_ITA for the significance test of trend is [24]:

$$Z_{ITA}={\displaystyle \frac{S}{{\sigma }_S}}$$

(1)

where S is the slope of the trend, and ${\sigma }_s$ is the standard deviation of S,

$$S={\displaystyle \frac{2}{n}}\left({\overline{y}}_2-{\overline{y}}_1\right)$$

(2)

$${\sigma }_S={\displaystyle \frac{2\sqrt{2}}{n}}\sqrt{{\sigma }_{{\overline{y}}_1}{\sigma }_{{\overline{y}}_2}}\sqrt{1-{\rho }_{{\overline{y}}_1{\overline{y}}_2}}={\displaystyle \frac{2\sqrt{2}}{n\sqrt{n}}}\sigma \sqrt{1-{\rho }_{{\overline{y}}_1{\overline{y}}_2}}$$

(3)

In Equation (2), ${\overline{y}}_1$ is the arithmetic mean of $\left\{y_{1i}\right\}$, and ${\overline{y}}_2$ is the arithmetic mean of $\left\{y_{2i}\right\}$. In Equation (3), $\sigma$ is the standard deviation of the original time series, and ${\rho }_{{\overline{y}}_1{\overline{y}}_2}$ is the correlation coefficient between ${\overline{y}}_1$ and ${\overline{y}}_2$. Z_ITA is tested according to the standard normal distribution. If $\left|Z_{ITA}\right|>Z_{\alpha /2}$, the trend is considered significant at the significance level α. $Z_{ITA}>0$ represents a positive (upward) trend, while $Z_{ITA}<0$ represents a negative (downward) trend.

2.2. Improved ITA Method Based on Covariance (COV_ITA)

In the original ITA method, Equation (3) was obtained by assuming that sub-series $\left\{y_{1i}\right\}$ and $\left\{y_{2i}\right\}$ have the same standard deviation (${\sigma }_{{\overline{y}}_1}={\sigma }_{{\overline{y}}_2}=\sigma /\sqrt{n}$) and $E\left({{\overline{y}}_2}^2\right)=E\left({{\overline{y}}_1}^2\right)$. However, this assumption usually cannot be met for those time series that have skewed and dependent characteristics. If the following statistical formula is used:

$${\sigma }_{{\overline{y}}_1}^2=E\left({{\overline{y}}_1}^2\right)-E^2\left({\overline{y}}_1\right)$$

(4)

$${\sigma }_{{\overline{y}}_2}^2=E\left({{\overline{y}}_2}^2\right)-E^2\left({\overline{y}}_2\right)$$

(5)

$$Cov\left({\overline{y}}_1{\overline{y}}_2\right)=E\left({\overline{y}}_1{\overline{y}}_2\right)-E\left({\overline{y}}_1\right)E\left({\overline{y}}_2\right)$$

(6)

We can obtain the following:

$${\sigma }_s={\displaystyle \frac{2}{n}}\left(Var\right({\overline{y}}_1)+Var({\overline{y}}_2)-2Cov({\overline{y}}_1{\overline{y}}_2){)}^{1/2}$$

(7)

Equation (7) is obtained without making any assumptions, and it does not require the time series to be normally distributed or independent. By substituting ${\sigma }_s$ of Equation (7) into Equation (1), the statistical significance of the trend can be more accurately evaluated [34]. However, its effectiveness needs further verification.

2.3. Improved ITA Method Based on Bootstrap Method (Bootstrap_ITA)

Wu et al. [33] employed the bootstrap method to improve the ITA method by introducing a new indicator. Assuming that the original time series is $\left\{x_k,k=1,2,\dots n\right\}$, it is bisected and arranged in an ascending order to obtain two component series with the same length, denoted as $\left\{y_{1i}\right\}$ and $\left\{y_{2i}\right\}$ (i = 1, 2…, n/2), respectively. The new indicator P is calculated as:

$$P={\displaystyle \frac{1}{n}}\sum _{i=1}^n{P}_i={\displaystyle \frac{1}{n}}\sum _{i=1}^n{\displaystyle \frac{10(y_{2i}-y_{1i})}{{\overline{y}}_1}}$$

(8)

This method detects the upward or downward trend based on the indicator P and uses the bootstrap method to determine its statistical significance. The specific process can be simply described as follows:

(1)

Divide the time series data $\left\{x_k,k=1,2,\dots n\right\}$ into two sub-series with equal size, and arrange them in an ascending order to calculate P in Equation (8);

(2)

Randomly select n sample data (some of which may be sampled multiple times) from $\left\{x_k,k=1,2,\dots n\right\}$ to form a new sample time series. After resampling M times, arrange the obtained new indicators in ascending order as: $P_1,P_2,\dots P_M$;

(3)

Given significance level α, calculate $L=M\times \alpha /2$,$H=M\times (1-\alpha /2)$, and then the confidence interval is $\left[P_L,P_H\right]$. If the indicator P falls in the confidence interval, the trend is not significant at the significance level α; otherwise, it is considered significant; besides, p > 0 represents an upward trend, while p < 0 represents a downward trend.

The bootstrap method is effective for doing significance tests in the presence of unknown distribution patterns, and Minh et al. [42] pointed out that M = 1000 can ensure the reliability of results. However, this method cannot be accurately formulated (see the above step (2)).

2.4. ITA Method Based on Variance Correction Analysis (VCA_ITA)

Wang et al. [27] pointed out that the original ITA method was based on the basic assumption of series independence. However, when facing the dependence characteristics of time series, the trend test in the original ITA method will distort the meaning of the trend due to the variance expansion of the slope estimator. Thus, it is necessary to re-derive the slope variance expression. They improved the ITA method based on variance correction analysis using the aggregated variance method. The cross-correlation between two mean estimations (${\overline{y}}_1$ and ${\overline{y}}_2$) can be expressed as:

$${\rho }_{{\overline{y}}_1{\overline{y}}_2}={\displaystyle \frac{E\left({\overline{y}}_1{\overline{y}}_2\right)-E\left({\overline{y}}_1\right)E\left({\overline{y}}_2\right)}{{\sigma }_{{\overline{y}}_1}{\sigma }_{{\overline{y}}_2}}}$$

(9)

Considering that $E\left({\overline{y}}_1\right)=E\left({\overline{y}}_2\right)$ and ${\sigma }_{{\overline{y}}_1}={\sigma }_{{\overline{y}}_2}$, the initial expression of slope variance in the original ITA method can be changed as:

$$V(S)={\displaystyle \frac{8}{n^2}}\left[{\sigma }_{{\overline{y}}_1}^2+E^2\left({\overline{y}}_1\right)-E\left({\overline{y}}_1{\overline{y}}_2\right)\right]$$

(10)

After further derivation, the general expression for the variance of the trend slope in the persistence structure (series independence or dependence) is obtained as follows:

$$V(S)={\displaystyle \frac{8}{n^2}}\left[{\sigma }_{{\overline{y}}_1}^2-{\displaystyle \frac{4}{n^2}}{\sigma }^2·R\right]$$

(11)

Among them, R represents the sum of autocorrelation coefficients.

For the lag-1 auto-correlation (i.e., AR (1)) model, which has the correlation coefficient of ρ, the final form of the slope variance is as follows:

$$V(S)={\displaystyle \frac{16}{n^3}}{\sigma }^2\left[{\displaystyle \frac{1+\rho }{1-\rho }}-{\displaystyle \frac{2}{n}}·R\right]$$

(12)

Furthermore, the standard deviation of its slope can be expressed as

$${\sigma }_s=\sqrt{V\left(S\right)}$$

(13)

By substituting ${\sigma }_s$ of Equation (13) into Equation (1), the statistical significance of the trend can be evaluated. However, this method not only involves the selection of models and the estimation of parameters (see Equation (12)) but also requires some assumptions and transformations on the original time series, which include uncertainties. More detailed information on the VCA_ITA method can be found in Wang et al. [27].

3. Prewhitening-Aided ITA Method

The ITA method is well-known for its more powerful ability to detect trends, especially for analyzing hidden changing trends that cannot be detected by traditional tests and methods. However, as described above, there are some unreasonable mathematical derivations in the trend recognition program and statistical significance test in the original ITA method [24]. The original ITA method is based especially on the assumption of data independence. To address the two key issues, we re-derived the significance test formula and corrected the ITA method. We further combined the prewhitening methods with the corrected ITA method to improve the detection of trends in hydroclimatic time series. In the improved method, prewhitening is used to remove serial correlation and ensure serial independence.

3.1. Correction of Formula in ITA

When using the ITA method for trend recognition, the significance assessment of the trend is a crucial step. It requires calculating the standard deviation of the trend slope, which involves Equation (3). However, Equation (3) of the original ITA method is incorrect and has two errors. First, for independent and identically distributed data, ${\sigma }_{{\overline{y}}_1}={\sigma }_{{\overline{y}}_2}=\sqrt{2}\sigma /\sqrt{n}$ is the correct standard deviation of the average estimator of sub-series with a length of n/2 [43]. However, the original ITA method took the form of ${\sigma }_{{\overline{y}}_1}$ and ${\sigma }_{{\overline{y}}_2}$ as $\sigma /\sqrt{n}$, resulting in an underestimation of the slope of the standard deviation value ${\sigma }_S$ and an overestimation of the trend’s significance. Second, for independent data, the cross-correlation between any two sub-series should be zero. Thus, there is theoretically no need to calculate the cross-correlation ${\rho }_{{\overline{y}}_1{\overline{y}}_2}$ in Equation (3); otherwise, it will lead to overestimation of the cross-correlation values. After correcting the above two key issues, the expression for the variance in the slope estimator is corrected as:

$$V(S)={\displaystyle \frac{8}{n^2}}{\sigma }_{{\overline{y}}_1}{\sigma }_{{\overline{y}}_2}(1-{\rho }_{{\overline{y}}_1{\overline{y}}_2})={\displaystyle \frac{8}{n^2}}{\displaystyle \frac{{2\sigma }^2}{n}}(1-{\rho }_{{\overline{y}}_1{\overline{y}}_2})$$

(14)

$$V(S)={\displaystyle \frac{16}{n^3}}{\sigma }^2$$

(15)

Therefore, the first key point of the prewhitening-aided ITA method is to correct Equation (3) as:

$${\sigma }_s=\sqrt{V(S)}={\displaystyle \frac{4}{n\sqrt{n}}}\sigma$$

(16)

By substituting ${\sigma }_s$ of Equation (16) into Equation (1), the statistical significance of the trend can be more accurately evaluated.

3.2. Combination of Corrected ITA Method and Prewhitening Methods

Observed hydrometeorological data usually have correlation (i.e., dependence) characteristics, which can affect the accuracy of trend recognition. Prewhitening methods can effectively remove correlation in a time series, which is necessary for improving the performance of the ITA method. Thus, it is believed that combining the ITA method corrected above with the prewhitening method can identify trends more accurately and evaluate their statistical significance. It is the main idea of the prewhitening-aided ITA method proposed in this study, and its specific steps for detecting trends are explained as follows:

(1)

Use the Theil Sen Median method (TSA) [13] to calculate the linear trend slope of the original time series X_t = (X₁, X₂, … X_n):

$$b=Median({\displaystyle \frac{X_j-X_i}{j-i}}),\forall j>i$$

(17)

where b is the calculated linear trend slope, “Median” is the median function, and X_i is the i-th observation value.

(2)

Choose an appropriate prewhitening method to handle the original time series X_t.

(3)

Follow the steps of the ITA method to first divide the prewhitened time series into equal parts and arrange them in ascending order. Then, take the first and second halves as the abscissa and ordinate, respectively, to draw a scatter plot in the Cartesian coordinate system (Figure 1). If the scattered points are distributed above (below) the 45° line, it indicates a monotonic upward (downward) trend. If the scattered points are distributed near the 45° line, it indicates an insignificant trend.

(4)

Use Equation (16) to calculate the corrected standard deviation of the trend slope in the prewhitened time series.

(5)

Conduct trend significance assessment. Combine Equations (16) and (1) to obtain the value of Z_ITA. If $\left|Z_{ITA}\right|>Z_{\alpha /2}$, it is considered that the trend is significant at the significance level α, and $Z_{ITA}>0$ represents a positive (upward) trend, and $Z_{ITA}<0$ represents a negative (downward) trend. In this study, the significance level of α = 0.05 was used.

The commonly used prewhitening methods include Ordinary Prewhitening (PW) [35], Trend-Free Prewhitening (TFPW) [18], Wang and Swail Prewhitening (WSPW) [36,37], and Variance Correction Prewhitening (VCPW) [22]. We used Monte–Carlo experiments to identify the most effective prewhitening method and couple it with the corrected ITA method to develop the most effective prewhitening-aided ITA method for the detection of trends in hydrometeorological time series. A detailed introduction to each prewhitening method can be found in Appendix A.

4. Monte–Carlo Experiments

4.1. Design of Monte–Carlo Experiments

We designed Monte–Carlo (MC) experiments to demonstrate the superiority of the proposed prewhitening-aided ITA method and to compare the performances of the four prewhitening methods. The specific MC experiment schemes and parameter settings are shown in

Table 1. Design of the Monte–Carlo experiment schemes and parameter settings.

Trend Slope	Data Type	Data Length	Lag-1 Auto-Correlation	Trend Slope	Data Type	Data Length	Lag-1 Auto-Correlation	Trend Slope	Data Type	Data Length	Lag-1 Auto-Correlation
b = 0	Series 1	60	0	b = 0.001	Series 11	60	0	b = 0.002	Series 21	60	0
	Series 2	120	0		Series 12	120	0		Series 22	120	0
	Series 3	60	0.4		Series 13	60	0.4		Series 23	60	0.4
	Series 4	60	0.8		Series 14	60	0.8		Series 24	60	0.8
	Series 5	120	0.4		Series 15	120	0.4		Series 25	120	0.4
	Series 6	120	0.8		Series 16	120	0.8		Series 26	120	0.8
	Series 7	60	−0.4		Series 17	60	−0.4		Series 27	60	−0.4
	Series 8	60	−0.8		Series 18	60	−0.8		Series 28	60	−0.8
	Series 9	120	−0.4		Series 19	120	−0.4		Series 29	120	−0.4
	Series 10	120	−0.8		Series 20	120	−0.8		Series 30	120	−0.8

. A total of three influencing factors were considered; to be specific, the trend slope b was set as 0, 0.001, and 0.002, respectively; the length was set as n = 60 and 120 to represent short and long time series, respectively. The correlation coefficients ($\rho$) were set as three types: uncorrelated, positively correlated, and negatively correlated, with 0.4 representing weak correlation and 0.8 representing strong correlation. The synthetic time series generated by each MC experimental scheme was repeated 2000 times with the mean value ($\mu$) of 1 and the coefficient of variation ($C_v$) of 0.2.

The synthetic time series was generated by using the AR (1) model:

$$X_i=\mu +\rho (X_{i-1}-\mu )+{\epsilon }_i+bi with i=1, 2\dots , n$$

(18)

where $\mu$ is the average value of the time series $X_i$, $\rho$ is its lag-1 autocorrelation coefficient, ${\epsilon }_i$ is a white noise component with mean ${\mu }_{\epsilon }$ = 0 and variance ${\sigma }_{\epsilon }^2={\sigma }^2(1-{\rho }^2)$, and ${\sigma }^2$ is the variance of $X_i$. If ${\xi }_i$ is a random variable following the normal distribution with mean ${\mu }_{\xi }$ = 0 and variance ${\sigma }_{\xi }^2=1$, Equation (18) can be rewritten as:

$$X_i=\mu +\rho (X_{i-1}-\mu )+\sigma \sqrt{1-{\rho }^2}{\xi }_i+bi$$

(19)

In the MC experiments, we use four prewhitening-aided ITA methods and three improved ITA methods (Bootstrap_ITA, VCA_ITA, COV_ITA), as well as the original ITA method and the ITA0 method (i.e., only correct the mathematical formula of the original ITA method but without prewhitening), to detect the trend in each synthetic time series, and further compare the performances of the nine methods. To be specific, we used two indicators to evaluate the results of each method: Type I error rate and the power of trend detection. Type I error rate (denoted as R_rej) is the probability that an insignificant trend is misjudged as a significant trend, and it was calculated as [18]:

$$R_{rej}=N_{rej}/N$$

(20)

where N = 2000 and N_rej is the number of time series with their insignificant trends being overestimated as significant trends.

The power of trend detection is the probability that a significant trend can be truly detected, and it can be calculated as:

$$R_{pow}=N_{pow}/N$$

(21)

where N = 2000 and N_pow is the number of time series with their significant trends being accurately identified.

4.2. Effect of Formula Correction in ITA

The Type I error rates of the original ITA method, three improved ITA methods, and the ITA0 method were compared and the results are shown in Figure 2. When analyzing these synthetic time series without significant correlation characteristics, except for the original ITA method, all the other four methods gave similar low Type I error rates, implying their better performances. Besides, changes in data length had little influence on the results.

The results of Type I error rates for analyzing synthetic time series with significant positive and negative correlation characteristics are depicted in Figure 3. When positive lag-1 autocorrelation was present, the ITA0, COV_ITA, and Bootstrap_ITA methods invariably produced higher Type I error rates than the assigned significance level of 5%. The VCA_ITA method consistently provided the lowest Type I error rates, while the original ITA method gave the highest Type I error rates. As expected, except for the original ITA method and VCA_ITA method, the other three methods gave higher Type I error rates along with the increase in positive correlation characteristics (i.e., lag-1 autocorrelation coefficient). The length of time series also had a weak impact on the results. Despite hydrometeorological data usually exhibiting positive serial dependence, several studies also detected negative values of the lag-1 autocorrelation coefficient [18,40]. When negative lag-1 autocorrelation was present, the ITA0, COV_ITA, Bootstrap_ITA, and VCA_ITA methods maintained low Type I error rates that were kept within the assigned significance level of 5%, which was not the case for the original ITA method.

Figure 4 provides the power of trend detection for the original ITA method, the three improved ITA methods, and the ITA0 method when analyzing synthetic time series without significant correlation characteristics. Compared to the original ITA method, which has high trend detection power, all four other methods similarly provided lower trend detection power for both weak and strong trends. However, by considering the results in Figure 2 together, it should be noticed that the contradiction between the high Type I error rates and high power of trend detection for the original ITA method implies its defects of overestimating trends’ statistical significance, as reported widely in previous studies [21,28]. Comparatively, it was thought that the results from the other four methods were more reasonable.

Figure 5 illustrates the results of analyzing synthetic time series with significant positive and negative correlation characteristics. As anticipated, in the presence of positive ρ₁, the powers of all these methods become higher along with the increase in data length and trends’ slopes. A similar result occurred in the presence of negative ρ₁. However, the ITA0, COV_ITA, and Bootstrap_ITA methods had weak power in the presence of largely negative ρ₁ for short data length and low slope of trend. The VCA_ITA method had a low power for high positive ρ₁ values.

In summary, the original ITA method performed the worst among all the five methods, no matter for analyzing time series with or without serial correlations. Comparatively, in the absence of series correlation, the other four methods gave similar low Type I error rates and high power of trend detection, which implied the effectiveness of our formula correction (i.e., Equation (16)) in the original ITA method, as explained in Section 3.1. However, when the analyzed time series had significant serial correlations, all these methods (also including the ITA0 method) performed poorly, which implied the necessity of further prewhitening of the original time series for accurately detecting trends, which was the motivation for the prewhitening-aided ITA method proposed in this study.

We further compared the ITA0 method with the four prewhitening-aided ITA methods. The trend significance test results for those synthetic time series without serial correlations, shown in Figure 6, indicate that the five methods provide similar low Type I error rates. It was understandable because as these synthetic time series had no significant correlation characteristics, there was no big difference in doing prewhitening or not for trend detection. Besides, results in Figure 7 show that when positive serial correlation was present, the ITA0 and TFPWITA0 methods consistently maintained much higher Type I error rates than the significance level of 5%, particularly in the presence of higher ρ1 values. Comparatively, the PWITA0, WSPWITA0, and VCPWITA0 methods provided much lower Type I error rates in all these MC experiment schemes. Besides, it was noted that changes in data length had little influence on the results.

Figure 8 shows that all five methods have similar trend detection power for analyzing those synthetic time series that do not exhibit significant correlation characteristics. Besides, Figure 9 illustrates the power of the five methods for analyzing the synthetic time series with significant positive and negative correlation characteristics. It indicates that in the presence of positive ρ₁, the PWITA method has the lowest power among the five methods. On the contrary, the ITA0 method has the lowest power for negative values of ρ₁. Comparatively, the other three methods have high trend detection power in all these MC experiment schemes. In summary, the power of the ITA0 method is significantly affected by serial correlation, while the prewhitening-aided ITA methods are more powerful, indicating the necessity of prewhitening before detecting trends.

Although many studies focused on the important criterion of Type I error rates being close to a significance level of concern [37,44], both low Type I error rate and high power of trend detection should be considered together for evaluating the performances of methods for trend detection and choosing the best one [40]. Based on the above results of the two indicators, the advantages and disadvantages of the nine methods compared in this study are summarized in

Table 2. Advantages and disadvantages of the nine ITA-related methods considered in this study.

Method	How It Works	Advantages/Disadvantages
Original ITA	-Directly handle data without pretreatment	-High Type I error rate -High power of trend detection
COV_ITA	-Directly handle data without pretreatment -Represent the trend slope variance using mathematical equations in covariance form	-High Type I error rate -High power of trend detection
Bootstrap_ITA	-Directly handle data without pretreatment -Use the bootstrap method to determine the significance interval of the trend	-High Type I error rate -High power of trend detection
VCA_ITA	-Directly handle data without pretreatment -Correct the variance of the trend slope	-Low Type I error rate -Low power of trend detection
ITA0	-Correct the mathematical formula -Directly handle data without prewhitening	-High Type I error rate -High power of trend detection
PW_ITA0	-Correct the mathematical formula -Handle data by first removing the correlation using the PW method	-Low Type I error rate -Low power of trend detection
TFPW_ITA0	-Correct the mathematical formula -Handle data by first removing the correlation using the TFPW method	-High Type I error rate -High power of trend detection
WSPW_ITA0	-Correct the mathematical formula -Handle data by first removing the correlation using the WSPW method	-Low Type I error rate -High power of trend detection
VCPW_ITA0	-Correct the mathematical formula -Handle data by first removing the correlation using the VCPW method	-Low Type I error rate -High power of trend detection

. To be specific, the Type I error rate of the original ITA method was very high in many cases, far exceeding the specified significance level of 5%, which indicates its weakness and the importance of formula correction conducted in Section 3.1. The Type I error rates of the COV_ITA, Bootstrap_ITA, ITA0, and TFPWITA0 methods were especially high when dealing with time series with positive correlations. The VCA_ITA and PWITA0 methods had weak power of trend detection when handling time series with positive correlations. Serial negative correlation seriously affected the power of the COV_ITA, Bootstrap_ITA, and ITA0 methods. All these results clearly reflected the big influence of serial correlation and advocated the essential step of prewhitening for trend detection. The WSPWITA0 and VCPWITA0 methods were much less affected by serial correlation than the other seven methods, as they maintained a relatively low Type I error rate and high power of trend detection. Further comparison showed that the VCPWITA0 method was superior to the WSPWITA0 method in most cases. Thus, the former was recommended in this study for the detection of trends in hydrometeorological time series.

5. Case Study

We used precipitation data in the Qinghai-Tibet Plateau (QTP) to further validate the accuracy and effectiveness of our proposed method for trend detection. The hydroclimatic conditions in QTP are complex and variable [45,46]. Accurately revealing the evolution characteristics of hydroclimatic variables in this region is the necessary premise for understanding the spatiotemporal variability of hydroclimatic processes and further assessing the influence of climate change and its adaption. Here, 36 meteorological stations (see Figure 10) in QTP and its surrounding areas were selected to detect trends of precipitation during flood season (July–September) from 1959 to 2019. The data were obtained from the China Meteorological Data Sharing Service System (https://www.cma.gov.cn/).

The shift in the average value of the data time series is referred to as inhomogeneity for data values [47]. Homogeneous data sets are required to correctly interpret the trends of time series. Before checking the trend of the hydroclimatic time series, the change point test should be performed. The most widely used statistical methods for evaluating heterogeneity and identifying abrupt changes in time series include the Pettitt test, F-test, T-test, and Rank Sum test, among others [48,49,50]. To ensure the reliability of the results, this study used three methods, namely the T-test, Rank Sum test, and Pettitt test, to perform homogeneity tests on the precipitation series at 36 measured stations of Qinghai-Tibet Plateau in the case study. In addition, the critical values of several homogeneity tests depend on the confidence level and sample size, as shown in

Table 3. Critical values of the three homogeneity tests.

Data Size	TT	RST	PET
61	2.001	1.960	289.750

. To maintain consistency with Monte–Carlo experiments, a confidence level of 95% is taken.

Table 4. The results of the homogeneity test in precipitation time series in flood season (July–September) at 36 stations in Qinghai-Tibet Plateau.

Station Number	TT	RST	PET	Homogeneity	Station Number	TT	RST	PET	Homogeneity
1	−1.904	2.290	289	Y	19	1.816	1.402	170	Y
2	−1.651	1.979	246	Y	20	4.019	3.455	476	N
3	−1.352	1.591	189	Y	21	2.034	1.931	266	Y
4	−1.528	1.401	192	Y	22	2.105	2.018	266	N
5	−3.879	3.378	415	N	23	1.152	1.106	144	Y
6	−2.059	1.909	228	Y	24	−1.672	1.895	261	Y
7	−2.524	2.373	327	N	25	−1.392	1.481	184	Y
8	−1.173	1.172	139	Y	26	−1.088	1.463	180	Y
9	−4.052	3.332	451	N	27	−1.895	1.920	251	Y
10	−3.948	3.332	398	N	28	1.781	1.429	196	Y
11	−2.035	2.252	279	N	29	−2.013	2.135	278	N
12	−1.405	1.736	240	Y	30	1.600	1.698	234	Y
13	1.789	1.461	228	Y	31	−1.478	1.945	267	Y
14	−2.647	2.412	313	N	32	−3.085	2.611	358	N
15	1.566	1.370	190	Y	33	2.197	2.805	289	N
16	1.807	1.545	212	Y	34	−2.414	2.581	340	N
17	−3.002	3.165	378	N	35	−1.549	1.454	184	Y
18	1.681	1.730	237	Y	36	−1.207	1.486	206	Y

Note: Bold values indicate that the value has exceeded the critical value. “Y” represents yes, “N” represents no.

lists the calculated values of the statistics (as well as homogeneity results) obtained by each method. If the absolute value of the statistic calculated from the relevant test exceeds the critical values listed in Table 3, it indicates that the method determines that the precipitation series is not homogeneous. If all three or any two tests meet the homogeneity criteria, the precipitation data are considered a homogeneous series. Otherwise, it needs to be corrected or discarded [47,51]. As detailed in Table 4, a comprehensive evaluation of the precipitation data by three homogeneity tests demonstrated that they did not exceed the critical values in high amounts, with nine stations having abrupt changes and four stations possibly having abrupt changes. We have made corrections to these thirteen stations and will continue to use them in our research.

The magnitudes of these 36 precipitation time series were obviously different. For a better comparison, we uniformly calculated their dimensionless trends (i.e., b/σ). Considering the influences of series correlation on trend detection, we also calculated the lag-1 autocorrelation coefficient of each precipitation time series. The results in Figure 10 indicated both positive and negative correlations among them as the reason for considering negative correlations in the above MC experiments. To investigate the significance of higher-order correlation, we also calculated the first- to fourth-order autocorrelation coefficients for all 36 precipitation series and generated corresponding boxplots for analysis (Figure 11). The results demonstrate that only some values of the first-order autocorrelation coefficients exceed the statistical significance thresholds. The second-, third-, and fourth-order coefficients all remain within the confidence limits, indicating non-significant correlations. This observation validates the rationality of focusing solely on first-order autocorrelation in our study. We used the above nine methods to detect the trend of precipitation in the region and compared their performances. Figure 12 visually displays the spatial distribution of the precipitation time series with significant trends detected by each method. To be specific, when using the original ITA method, precipitation data at 31 stations (86.11% of the total 36 stations) were detected with significant trend components. However, the VCPWITA0 method only detected significant trends at two stations. The Bootstrap_ITA, VCA_ITA, and WSPWITA0 methods detected significant trends at four stations. The COV_ITA, ITA0, PWITA0, and TFPWITA0 methods detected significant trends at three stations.

Figure 12 presented that the trend detection results of the original ITA method differed from those of other methods, as the former may overestimate the significance of trends in precipitation at many stations. It is noted that we corrected the mathematical formula (Equation (16)) in the original ITA method and denoted it as the ITA0 method, by which significant trends at only three stations were identified. The results of the ITA0 method were similar to those of the other seven methods, which verified the accuracy of Equation (16). In addition, results among the eight methods, except the original ITA method, were similar due to weak correlations of these precipitation time series, as shown in Figure 10.

Regarding the VCPWITA0 method, the trend detection result by the optimal method selected in the MC experiments had differences compared to other methods. To demonstrate the accuracy of the significance test results of the trend by the VCPWITA0 method, we calculated the signal-to-noise ratio (SNR) of the precipitation time series for all 36 stations (see Figure 13). The SNR index used in this study is defined and calculated as the ratio between the variance of the trend component and that of other components in the original precipitation time series [52,53]. Theoretically, a larger SNR value reflects a higher ratio of trend components in the original time series, implying more significance of the trend. Among all precipitation data, the SNR values of the precipitation data at two stations (No. 5 and 9, see Figure 12) were 0.1957 and 0.1706, respectively, which were much higher than those at the other stations. As expected, all nine methods detected that the two trends were statistically significant. However, the results at other stations did not follow the rules. We took the No. 20 station as an example to explain it. At the station, the SNR value of its precipitation time series was 0.0563, which only ranked 8th among the 36 stations; its trend was detected as insignificant by the VCPWITA0 method but significant by the other seven methods. Unexpectedly, for the precipitation time series with SNR values ranked 3rd–7th, although these SNR values were much higher than that at the No. 20 station, their trends were detected as insignificant by all methods. Thus, the trend at the No. 20 station may be overestimated by the seven methods. Through further analysis, the precipitation time series had the strongest positive correlation of 0.338 and a linear trend slope of −0.013. Thus, it was deduced that its significant serial correlation caused the overestimation of the trend’s significance by the seven methods. Comparatively, the VCPWITA0 method overcame the influence of serial correlation; thus, its results should be more reliable. Overall, the superiority of the VCPWITA0 method had been verified, which was consistent with the conclusion obtained from the MC experiments.

6. Conclusions

Trend detection is important for hydrometeorological time series analysis. However, the results of trend detection are usually influenced by many factors, such as distribution form, homogeneity, correlation, and length of hydrometeorological time series. The ITA method, as a new method for trend testing in recent years, has become increasingly used far beyond the field of hydrology due to its unique advantages. However, there are unreasonable mathematical derivations in the trend recognition program and statistical significance tests in the original ITA method. The original ITA method is based especially on the assumption of data independence. To address the two key issues, in this study, we corrected the significance test formula of the ITA method and further combined the prewhitening methods with the corrected ITA method to improve the detection of trends in hydroclimatic time series. Based on the main idea, we proposed the new prewhitening-aided ITA method.

The results of both MC experiments and observed hydrometeorological data verified the effectiveness of our formula correction (i.e., Equation (16)) on the original ITA method and the prewhitening practice for trend detection. Results indicated that the performances of the original ITA, COV_ITA, Bootstrap_ITA, VCA_ITA, and ITA0 methods were significantly affected by serial correlation, while the prewhitening-aided ITA methods were more powerful and effective. Specifically, the Type I error rates of the COV_ITA, Bootstrap_ITA, ITA0, and TFPWITA0 methods were especially high when dealing with time series with positive correlations. The VCA_ITA and PWITA0 methods had weak power of trend detection when handling time series with positive correlations. Serial negative correlation seriously affected the power of the COV_ITA, Bootstrap_ITA, and ITA0 methods. The WSPWITA0 and VCPWITA0 methods were much less affected by serial correlation than the other seven methods, as they maintained a relatively low Type I error rate and high power of trend detection. Comparatively, the VCPWITA0 method performed the best, and it is recommended for detecting trends in hydrometeorological time series.

In summary, this study explored trend recognition by coupling the ITA method with prewhitening. Considering that the characteristics of observed hydrometeorological time series are complex and time-varying, in addition to serial correlation and data length, the impacts of other factors also need to be investigated for accurate trend recognition. Besides, the applicability of our proposed prewhitening-aided ITA method also needs further verification by being applied to more regions and basins.