Trend Identification (Mean)

Many factors can produce nonstationarity in annual maximum series (AMS) data, including changes in climate, land use/cover, and water management. This vignette demonstrates how to use the ffaframework to check for evidence of nonstationarity in the mean of a time series.

List of Tests

Mean Trend Tests

Function	Purpose
eda_mk_test	Tests for a monotonic trend (Mann-Kendall test).
eda_bbmk_test	Tests for a monotonic trend under serial correlation (block-bootstrap MK test).

Stationarity Tests

Function	Purpose
eda_spearman_test	Tests for serial correlation (Spearman test).
eda_kpss_test	Tests for a stochastic trend (KPSS test).
eda_pp_test	Tests for a deterministic trend (Phillips-Perron test).

Variability Trend Tests

Function	Purpose
MW-MK Test	Tests for a trend in the variability (moving-window MK test)
eda_white_test	Tests for time-dependence in the variability (White test).

Trend Estimation (Mean & Variability)

Function	Purpose
eda_sens_trend	Estimates slope and intercept of a linear trend (Sen’s slope estimator).
eda_runs_test	Evaluates residuals’ structure under linear model assumptions (Wald-Wolfowitz runs test).

Case Study

This vignette will explore data from the Bow River at Banff (05BB001) hydrological monitoring station. The remoteness of this station means that trends annual maxima are caused by changes in climate as opposed to changes in land use or cover. Data for this station is provided as Application_3.1.csv in the ffaframework package.

library(ffaframework)

csv_path <- system.file("extdata", "Application_3.1.csv", package = "ffaframework")
df <- read.csv(csv_path, comment.char = "#")
df <- subset(df, !is.na(max)) # Remove missing values

head(df)
#>    year max
#> 10 1909 314
#> 11 1910 230
#> 12 1911 264
#> 13 1912 174
#> 14 1913 232
#> 15 1914 214

plot_ams_data(df$max, df$year, title = "Bow River at Banff (05BB001)")

Assessing Trends in the Mean

Mann-Kendall (MK) Test

The MK test is a non-parametric test used to detect monotonic trends in a time series. Under the null hypothesis, there is no trend.

Pass a vector of annual maximum series (AMS) data to eda_mk_test() to perform the MK test.

mk_test <- eda_mk_test(df$max)

print(mk_test$p_value)
#> [1] 0.006773098

Conclusion: At a p-value of \(0.0067\), we reject the null hypothesis. There is statistical evidence of a trend in the mean.

Spearman Test

The Spearman test is used to check for serial correlation, which can cause the MK test to identify a spurious trend. The smallest lag at which the serial correlation is not statistically significant is known as the “least lag”.

spearman_test <- eda_spearman_test(df$max)

print(spearman_test$least_lag)
#> [1] 1

plot_spearman_test(spearman_test)

Conclusion: A least lag of 1 provides no evidence of serial correlation.

Further Tests

If the Spearman test identified serial correlation, we would use the block-bootstrap Mann-Kendall test (with eda_bbmk_test) to check for a monotonic trend under serial correlation. The BBMK test uses a reshuffling procedure to remove serial correlation from the data.

The Phillips-Perron (eda_pp_test) and KPSS (eda_kpss_test) tests are used to check for a unit root (or stochastic trend) in the data. A unit root is an extreme type of serial correlation that creates unpredictable variation in the data. Since we did not find evidence of serial correlation in the Spearman test, it is not necessary to apply these tests. The Phillips-Perron and KPSS tests have opposite hypotheses: the Phillips-Perron assumes the presence of a unit root as the null hypothesis, while the KPSS test does not.

Sen’s Trend Estimator

While the Mann-Kendall identified a significant monotonic trend, it did not estimate its slope or intercept. We can estimate the monotonic trend using Sen’s trend estimator, which uses a non-parametric approach that is robust to outliers.

eda_sens_trend() takes two arguments: the annual maximum series and the corresponding vector of years. Then, plot_sens_trend() can be used to generate a plot of the results. It requires the AMS data and corresponding vector of years. It also takes an optional trend in the mean (mean_trend), or a trend in the variability (variability_trend).

mean_trend <- eda_sens_trend(df$max, df$year)

plot_sens_trend(df$max, df$year, mean_trend = mean_trend)

Note: The covariate is computed using the formula \((\text{years} - 1900) / 100\).

Runs Test

The previous statistical tests have assumed the nonstationarity is linear. The Wald-Wolfowitz runs test assess the feasibility of the linearity assumption by checking the residuals for randomness. If the residuals are random (the null hypothesis), there is evidence that the underlying trend is linear. The eda_runs_test() function takes the output of eda_sens_trend() as an argument.

runs_test <- eda_runs_test(mean_trend)

print(runs_test$p_value)
#> [1] 0.4392721

plot_runs_test(runs_test, "Runs Test (AMS Mean)")

Conclusion: At a p-value of \(0.682\), there is evidence that a linear trend is suitable for the data.

Conclusion

The Mann-Kendall test identified evidence of nonstationarity in the mean of the AMS. The Spearman test did not find evidence of serial correlation, validating the results of the Mann-Kendall test. The runs test found that a linear model is suitable for the nonstationarity. Flood frequency analysis of this dataset requires a time-dependent probability model.

Recommendation: Use NS-FFA.