The evaluation of model performance is a widely discussed and yet extremely controversial topic in hydrology, atmospheric science, and environmental studies. Generally, it is not super straightforward to quantify the accuracy of tools that simulate complex systems. One of the reasons is that these systems usually have various sub-systems. For example, a complex river system will simulate regulated streamflow, water allocation, dam operations, etc. This can imply that there are multiple goodness-of-fit objectives that cannot be fully optimized simultaneously. In this situation, there will always be tradeoffs in the accuracy of the system representation.
The other reason is that these complex systems are often state-dependent, and their behavior non-linearly changes as the system evolves. Finally, there are variables in the system that have several properties, and it is usually impossible to improve all of their properties at the same time (Gupta et al., 1998). Take streamflow as an example. In calculating model accuracy, we can focus on the daily fluctuation of streamflow or look into the changes weekly, monthly, and annually. We can also concentrate on the seasonality of streamflow, extreme low and high cases, and the persistence of these extreme events. Therefore, our results are not fully defensible if we only focus on one performance metric and make inferences that ignore many other components of the system. In this blog post, I am going to explain a visualization technique that allows us to draw three or more metrics of the system on the same plot. The plot is called a Taylor Diagram. The Taylor Diagram is not really a solution to the problem that I explained, but it does provide a systematic and mathematically sound way of demonstrating a few popular goodness-of-fit measures. The original version of the Taylor Diagram includes three performance metrics: i) standard deviation of simulated vs. observed; ii) correlation coefficient between observed and simulated; and iii) centered root mean square error (CRMSE). However, there are other versions of the Taylor Diagram that include more than these three metrics (e.g., here). The Taylor Diagram was developed by Karl E. Taylor (original paper) and has been frequently used by meteorologists and atmospheric scientists. In this blog post, I focus on streamflow.
Underlying Mathematical Relationships
As mentioned above, the Taylor Diagram draws the following three statistical metrics on a single plot: standard deviation, CRMSE, and correlation. The equation below relates these three:
In this equation:
This relationship can be easily derived using the definition of CRMSE, which is:
In this equation:
We can expand this equation to the following:
The first two components of the equation are standard deviations of observed and simulated time series. To understand the third component of the equation, we need to recall the mathematical definition of correlation.
If we multiply both sides of the above equation by standard deviation of observed and simulated, we see that the third components of the equations 3 and equation 4 are actually the same. The Taylor Diagram uses polar coordinates to visualize all of these components. Readers can refer to the original paper to find more discussion and the trigonometric proofs behind this form of plot.
In this blog post, I am presenting a simple R package that can be used to create Taylor Diagrams. In this example, I use the same streamflow datasets that I explained in the previous blog post. First, you need to download the time series from GitHub and use the following commands to read these two time series into R.
Observed_Arrow<-read.table("~/Taylor Diagram/Arrow_observed.txt", header = T) Simulated_Arrow<-read.table("~/Taylor Diagram/Arrow_simulated.txt", header = T)
In this post, I use the “openair” R library, which was originally developed for atmospheric chemistry data analysis. In addition to the Taylor Diagram, the package provides many helpful visualization options that can be accessed from here. Note that I was able to find at least one more R package that can be used to create Taylor Diagrams (i.e., plotrix ). There are also Python and MATLAB libraries that can be used to create Taylor Diagrams.
You can use the following to install the “openair” package and activate the library.
The following command can be used to create a Taylor Diagram.
combined_dataset<-cbind(Observed_Arrow[ 18263:length(Observed_Arrow[,1]),], Arrow_sim=Simulated_Arrow[1:10592,4]) TaylorDiagram(combined_dataset, obs = "ARROW_obs", mod = "Arrow_sim")
Interpretation of the Taylor Diagram
The Taylor Diagram indicates the baseline observed point where correlation is 1 and CRMSE is 0 (purple color). If the simulation point is close to the observed point, it means that they are similar in terms of standard deviation, their correlation is high, and their CRMSE is close to zero. There is also a black dashed line that represents the standard deviation of the observed time series. If the red dot is above the line, it means that the simulated data set has a higher variation. The other helpful information that we gain from the Taylor Diagram is the correlation between observed and simulated values. Higher correlation shows a higher level of agreement between observed and simulated data. The correlation goes down as a point moves toward higher sectors in the graph. Centered root mean square error also shows the quality of the simulation process; however, it puts more weight on outliers. The golden contour lines on the polar plot show the value of CRMSE. Basically, on this figure, the red point has a higher standard deviation, the correlation is around 90%, and the CRMSE is close to 20,000.
The package also allows us to look at the performance of the model during different months or years, and we can compare different simulation scenarios with the observed data. Here, I use the function to draw a point for each month. The “group” argument can be used to do that.
TaylorDiagram(combined_dataset, obs = "ARROW_obs", mod = "Arrow_sim", group = "Month")
The function also provides a simple way to break down the data into different plots for other attributes. For example, I created four panels for four different annual periods:
TaylorDiagram(combined_dataset, obs = "ARROW_obs", mod = "Arrow_sim", group = "Month", type = "Year")