Case Study
This vignette will explore data from the Chilliwack River at
Chilliwack Lake (08MH016) 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 CAN-08MH016.csv in the
ffaframework package.
library(ffaframework)
df <- data_local("CAN-08MH016.csv")
head(df)
#> year max
#> 1 1922 62.9
#> 2 1923 74.5
#> 3 1924 79.0
#> 4 1925 35.1
#> 5 1926 62.3
#> 6 1927 65.7
plot_ams_data(df$max, df$year, title = "Chilliwack River at Chilliwack Lake (08MH016)")
Assessing Trends in the Variance
MWMK Test
The MWMK test is used to detect trends in the variability of a time
series. First, a moving-window algorithm is used to estimate the
variability of the AMS data. Then, the Mann-Kendall test is applied to
the series of standard deviations to check for a trend.
The data_mw_variability estimates the moving-window
standard deviations and the eda_mk_test function performs
the Mann-Kendall test.
mw <- data_mw_variability(df$max, df$year)
mwmk_test <- eda_mk_test(mw$std)
print(mwmk_test$p_value)
#> [1] 0.001463998
Conclusion: At a p-value of \(0.0015\), we reject the null
hypothesis. There is evidence of a linear trend in the variability in
the data.
White Test
The White test checks for heteroskedasticity, or general
time-dependence in the variability. The null hypothesis is
homoskedasticity, or constant variability in the data.
white_test <- eda_white_test(df$max, df$year)
print(white_test$p_value)
#> [1] 0.1175955
Conclusion: At a p-value of \(0.1176\), we fail to reject the
null hypothesis. There is no statistical evidence of
heteroskedasticity.
Sen’s Trend Estimator
While the previous tests provide evidence for a monotonic trend in
the variability, they do not estimate the slope or intercept of this
trend. We can estimate the monotonic trend using Sen’s trend estimator,
which uses a nonparametric approach that is robust to outliers.
eda_sens_trend() takes two arguments: the data (either
an annual maximum series or vector of standard deviations) and the
corresponding vector of years. Alternatively,
plot_ams_data() can be used to run Sen’s trend estimator on
the data and/or variability series and then plot the results. It takes
the optional arguments plot_mean and
plot_variability in addition to the data and
years.
plot_ams_data(
df$max,
df$year,
plot_mean = "Constant",
plot_variability = "Trend",
title = "Sen's Trend Estimator (AMS Variability)"
)
Note: The covariate is computed using the formula
\((\text{years} - 1900) / 100\).
Runs Test
Sen’s trend estimator assumed that the nonstationarity is linear. The
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 residuals from
eda_sens_trend() as the first argument and the observation
years as the second argument.
sens_trend <- eda_sens_trend(mw$std, mw$year)
runs_test <- eda_runs_test(sens_trend$residuals, mw$year)
print(runs_test$p_value)
#> [1] 0.3311375
plot_runs_test(runs_test, title = "Runs Test (AMS Variability)")
Conclusion: At a p-value of \(0.3311\), we fail to reject the
null hypothesis. There is evidence that the residuals are random and a
linear trend is suitable for the data.
Conclusion
The MWMK and White tests find evidence of nonstationarity in the
variability of the AMS. The runs test confirms 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.