3.3 Multi-temporal runoff-sediment discharge cointegration
relationships
3.3.1 ADF test
First, ADF tests were performed for the raw runoff and sediment
discharge series. The results are summarized in Table 2.
—————————————place
Table 2
here——————————————-
From the table, unit roots exist for X and Y before differentiation at
the 1%, 5%, and 10% significance levels. Thus, the series are
non-stationary. Contrarily, after the first-order differentiation, there
are no unit roots for both series, and they are now stationary series.
Therefore, X and Y are first-order integrated series, i.e.,\(X\sim I(1)\) and \(Y\sim I(1)\).
Subsequently, the multi-temporal runoff and sediment discharge
components were subjected to stationarity tests. The ADF test results
are listed.
—————————————place
Table 3
here——————————————-
—————————————place
Table 4
here——————————————-
According to the above test results, unit roots do not exist for the
undifferentiated series of the IMF1, IMF2, and RES components of the
runoff and sediment discharge at the 1%, 5%, and 10% significance
levels. The series of these components are stationary. Additionally,
relatively satisfactory goodness of fit and higher R2values were noted for the double cumulative curves of the IMF1, IMF2,
and RES components. In these plots, the cumulative runoff and sediment
discharge gave a straight line, and they showed consistent variation. It
is believed that the equations describing the runoff-sediment discharge
relationships of the IMF1, IMF2, and RES components can be determined
using linear regression. Furthermore, structural breaks are noted in the
double cumulative curves of IMF3 and IMF4. The series of these
components are found to be first-order integrated series according to
the ADF tests. Thus, for these components, OLS was employed to estimate
the regression parameters and to determine the cointegration equations.
3.3.2 Multi-temporal
regression equation
Linear regression was used to determine the runoff-sediment discharge
regression equations for IMF1, IMF2, and RES. The corresponding scatter
plots and linear regression equations are illustrated below.
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Figure 6
here——————————————-
From the scatter plots, high goodness of fit of the linear regression
lines are noted for the components of runoff and sediment discharge
measured at Tangnaihai station. The R2 values for the
IMF1, IMF2, and RES components were 0.8105, 0.8357, and 0.9975,
respectively. These indicated that the runoff and sediment discharge
were significantly correlated with these components. At these scales,
linear regression equations can be used to express the runoff-sediment
discharge correlations. Therefore, the Engle–Granger two-step method
was necessary only for the cointegration regression for the raw series
and the IMF3 and IMF4 components. After that, the residual series of
these series were subjected to the ADF tests. The results are listed in
Table 5.
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Table 5
here——————————————-
Table 5 shows that all the residual series were stationary. Thus, it was
concluded that the raw, IMF3, and IMF4 series all show cointegration
relationships.
By employing linear regression and the cointegration theory, regression
equations were obtained for the raw series and the multi-temporal
component series. They are summarized in Table 6.
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Table 6
here——————————————-
In general, higher goodness of fit was noted for the regression model
for the raw series. This indicated that the runoff and sediment
discharge were significantly correlated at the macroscopic scale. Yet,
their relationships at different time scales varied slightly. Among the
series subjected to linear regression, the RES component demonstrated
the highest goodness of fit. This suggested that the runoff and sediment
discharge showed a relatively strong correlation in terms of their
overall variation. Among those subjected to cointegration regression,
the IMF3 component series (indicative of a medium/long-time scale) and
the IMF4 component series (long-time scale) showed lower goodness of
fit. This demonstrated weak runoff-sediment discharge cointegration
relationships at these time scales.
When the cointegration test results were integrated with the double
cumulative curves, the runoff and sediment discharge generally observed
at Tangnaihai station consistently varied with each other
proportionally. However, the relationship variation of the raw runoff
and sediment discharge series was caused by multi-temporal variations.
The weak runoff-sediment discharge relationships on the medium/long and
long-time scales would affect that of the raw series.
3.3.3 Variable
structure cointegration model
Because structural breaks were observed for the IMF3 and IMF4
runoff-sediment discharge relationships, the corresponding cointegration
relationships were weaker. Hence, variable structure cointegration
models were constructed to determine reasonable models by considering
structural breaks. The most prominent break was noted for the IMF3
component in 2005 and that for the IMF4 component was observed in 2001.
In model construction, these points were treated as structural breaks
and virtual variables \(D_{1t}\) and \(D_{2t}\) were introduced.
For the IMF3 component, a virtual variable was introduced as follows:
\(D_{1t}=\left\{\par
\begin{matrix}0,t\leq 2005\\
1,t>2005\\
\end{matrix}\right.\ \) (15)
Three models corresponding to different scenarios (section 2.3.2) were
examined to determine the most reasonable model. The models considering
three different variation scenarios are summarized in Table 7.
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Table 7
here——————————————-
The R2 values of the models corresponding to scenarios
1, 2, and 3 were 0.606516, 0.788878, and 0.800332, respectively. These
values were significantly higher than that of the original cointegration
model (0.410943). The highest goodness of fit was achieved by model 3,
therefore, it is believed that this model can more accurately depict the
IMF3 runoff-sediment discharge relationship.
The virtual variable introduced into the model for the IMF4 component is
given as follows:
\(D_{2t}=\left\{\par
\begin{matrix}0,t\leq 2001\\
1,t>2001\\
\end{matrix}\right.\ \) (16)
Table 8 shows the models considering the three scenarios:
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Table 8
here——————————————-
Under the three scenarios, the models give R2 values
of 0.792911, 0.807911, and 0.841282, respectively. They were much higher
than that of the original cointegration model for the IMF4 component
(0.342336). More specifically, model 3 demonstrated the highest goodness
of fit. Therefore, it could more accurately describe the runoff-sediment
discharge relationship variation for IMF4 component.
3.3.4 Model
accuracy
With the help of the runoff-sediment discharge regression models
established for the components, the runoff during the research period
was simulated. The results are illustrated in Figure 7. The
cointegration model constructed using the raw series was called an
original model. The model built based on the multi-temporal components
was named composite model 1. Lastly, composite model 2 refers to the
variable structure cointegration model constructed using the
multi-temporal components and considering structural breaks.
—————————————place
Figure 7
here——————————————-
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Figure 8
here——————————————-
—————————————place
Figure 9
here——————————————-
Figure 7 reveals that, in general, the models satisfactorily simulated
the raw runoff. In particular, near the end of the research period, the
composite model 2 showed higher accuracy. Its results showed the same
variation trend as that of the raw series. From Figure 8, the average
relative errors by the original model, the composite model 1, and the
composite model 2 were 11.43%, 10.76%, and 7.82%, respectively.
According to the Standard for Hydrological Information and Hydrological
forecasting (GBT 22482-2008), an error tolerance of 20% is appropriate
in hydrological forecasting. Hence, the relative errors of all the
models fall within the acceptable range. In particular, the greatest
average relative error resulted from the original model, whereas the
smallest was generated by composite model 2 considering structural
breaks. Figure 9 illustrates that this model yields relative errors less
than 10% for 41 years out of the 54 years simulated. The relative
errors by this model were 10%–20% for 9 years and they were greater
than 30% for the remaining 4 years. Hence, most of the relative errors
by this model were low, while few were high. Therefore, the resulting
average relative error by composite model 2 was comparatively small.
From Figure 8, at the beginning of the research period, the three models
give the same relative error. However, the errors in the original model
and the composite model 1 increased after 2000. The errors in 1997,
2002, and 2009 were the greatest. Furthermore, the composite model 2
provided more satisfactory runoff simulation results and smaller
relative errors for these years. Because this model considered the local
variation characteristics and structural breaks of the multi-temporal
components, its overall simulation accuracy was higher. The model could
more accurately reflect the long-term equilibrium and short-term
fluctuating relationships between the runoff and sediment discharge in
the source region of the Yellow River.