Copula diagnostics used in the Research Version
The Copula tab in the current Research Version is a bivariate, rank-based dependence diagnostic. It converts the selected paired variables to pseudo-observations in the unit square, draws the U–V scatter plot, and reports dependence and drought-persistence summaries. It is useful for describing association and persistence, but it is not a drought forecast by itself.
User-selected pairing
Step 1: build the paired variables
The user selects a variable pairing in the Copula tab. For the option shown as SPI vs SPI (lag 1; persistence), the paired variables are consecutive SPI values:
\[
X_i = SPI_t, \qquad Y_i = SPI_{t+1}
\]
This pairing is used to examine whether dry or wet SPI conditions tend to persist into the next time step.
Other pairings follow the same structure. For example, a precipitation–SPI pairing uses one variable from precipitation and the other from SPI for the same matched time steps. Records with missing or invalid paired values are excluded before the diagnostic is calculated.
Pseudo-observations
Step 2: convert pairs to U and V values
The Copula tab does not plot the original SPI or precipitation values directly. It first converts each paired variable to a rank-based probability value between 0 and 1:
\[
U_i = \frac{rank(X_i)}{n+1}, \qquad V_i = \frac{rank(Y_i)}{n+1}
\]
Here, \(n\) is the number of valid paired observations. This is why the chart is labeled U–V scatter, and both axes range from 0 to 1.
In the U–V scatter plot, each dot is one valid paired time step. For SPI-related variables, small values near 0 represent the driest or lowest-ranked conditions, while values near 1 represent the wettest or highest-ranked conditions.
Dependence summary
Step 3: calculate correlation and Kendall tau
The summary box reports the correlation between the rank-based U and V values as the Gaussian rho value:
\[
\hat{\rho}=Corr(U,V)
\]
Correlation is bounded by \(-1 \le \hat{\rho} \le 1\). A positive value means the two variables tend to move in the same rank direction. A negative value means they tend to move in opposite rank directions.
Kendall tau is also reported as a nonparametric rank-dependence measure:
\[
\hat{\tau}=\frac{C-D}{\frac{1}{2}n(n-1)}
\]
\(C\) is the number of concordant pairs and \(D\) is the number of discordant pairs.
Values near zero indicate weak dependence. In the SPI lag-1 persistence example, a positive value means low SPI ranks tend to be followed by low SPI ranks and high SPI ranks tend to be followed by high SPI ranks.
Gaussian family
Why tail-dependence values are not shown in the tool
The current Research Version uses a Gaussian copula to summarize overall dependence. In a Gaussian copula, lower-tail and upper-tail dependence are theoretically zero for any non-perfect correlation.
Because these values are not informative in normal DMAP-AI runs, the Research Version does not show lower-tail or upper-tail dependence in the Copula summary. The more useful outputs are Gaussian rho, Kendall tau, the U–V scatter, sample size, and the persistence summaries such as P(drought continues next step).
Sample size
Sample size used
The summary item Sample size used is the number of valid paired observations after the selected pairing is built and missing or invalid values are removed:
\[
n = \#\{(X_i,Y_i): X_i \text{ and } Y_i \text{ are valid and finite}\}
\]
For the lag-1 SPI persistence option, one time step is lost because each pair needs both \(SPI_t\) and \(SPI_{t+1}\).
Persistence / duration
Drought persistence summary for SPI ≤ -1.0
When the selected pairing is SPI vs SPI lag 1, the summary box also reports persistence statistics based on a drought threshold of \(SPI \le -1.0\). Define a drought indicator as:
\[
D_t=
\begin{cases}
1, & SPI_t \le -1.0 \\
0, & SPI_t > -1.0
\end{cases}
\]
The probability that drought continues into the next step is calculated from times that are already in drought:
\[
P(\text{drought continues next step})=
\frac{\#\{t:D_t=1 \text{ and } D_{t+1}=1\}}{\#\{t:D_t=1\}}
\]
The probability of a new drought next step is calculated from times that are not currently in drought:
\[
P(\text{new drought next step})=
\frac{\#\{t:D_t=0 \text{ and } D_{t+1}=1\}}{\#\{t:D_t=0\}}
\]
The approximate mean drought length is the average length of consecutive drought runs:
\[
\bar{L}=\frac{1}{m}\sum_{j=1}^{m} L_j
\]
\(L_j\) is the length of drought run \(j\), and \(m\) is the number of drought runs. If the available record is too limited for this estimate, the tool can report NA.
How to read the Copula tab
How the displayed results should be interpreted
| Displayed item | Meaning in the current Research Version |
|---|
| U–V scatter | Rank-based paired values in the unit square. Each dot is one valid paired time step. |
| Pair type | The selected variable pairing, such as SPI vs SPI lag 1 for persistence analysis. |
| Family: Gaussian | The dependence summary is reported in a Gaussian-copula-style framework. |
| Gaussian rho | Correlation of the rank-based U and V values. |
| Kendall tau | Rank-dependence measure based on concordant and discordant pairs. |
| Persistence / duration | Empirical transition statistics using \(SPI \le -1.0\) as the drought threshold. |
The Copula tab is a descriptive diagnostic. It should be interpreted together with the SPI table, charts, and drought severity tab. It does not replace the main SPI calculation and should not be interpreted as an official drought forecast.
