Taylor diagrams are mathematical diagrams designed to graphically indicate which of several approximate representations (or models) of a system, process, or phenomenon is most realistic. This diagram, invented by Karl E. Taylor in 1994 (published in 2001) facilitates the comparative assessment of different models. It is used to quantify the degree of correspondence between the modeled and observed behavior in terms of three statistics: the Pearson correlation coefficient, the root-mean-square error (RMSE) error, and the standard deviation. Although Taylor diagrams have primarily been used to evaluate models designed to study climate and other aspects of Earth's environment, they can be used for purposes unrelated to environmental science (e.g., to quantify and visually display how well fusion energy models represent reality). Taylor diagrams can be constructed with a number of different open source and commercial software packages, including: GrADS, IDL, MATLAB, NCL, Python, R, and CDAT.
Sample diagram
The sample Taylor diagram shown in Figure 1 provides a summary of the relative skill with which several global climate models simulate the spatial pattern of annual mean precipitation. Eight models, each represented by a different letter on the diagram, are compared, and the distance between each model and the point labeled "observed" is a measure of how realistically each model reproduces observations. For each model, 3 statistics are plotted:
blue contours: the Pearson correlation coefficient (gauging similarity in pattern between the simulated and observed fields) is related to the azimuthal angle; green contours: the centered RMS error in the simulated field is proportional to the distance from the point on the x-axis identified as "observed"; black contours: and the standard deviation of the simulated pattern is proportional to the radial distance from the origin. It is evident from this diagram, for example, that for Model F the correlation coefficient is about 0.65, the RMS error is about 2.6 mm/day and the standard deviation is about 3.3 mm/day. Model F's standard deviation is clearly greater than the standard deviation of the observed field (indicated by the dashed contour at radial distance 2.9 mm/day). The relative merits of various models can be inferred from Figure 1. Simulated patterns that agree well with observations will lie nearest the point marked "observed" on the x-axis. These models have relatively high correlation and low RMS errors. Models lying on the dashed arc have the correct standard deviation (which indicates that the pattern variations are of the right amplitude). In Figure 1 it can be seen that models A and C generally agree best with observations, each with about the same RMS error. Model A, however, has a slightly higher correlation with observations and has the same standard deviation as the observed, whereas model C has too little spatial variability (with a standard deviation of 2.3 mm/day compared to the observed value of 2.9 mm/day). Of the poorer performing models, model E has a low pattern correlation, while model D has variations that are much larger than observed, in both cases resulting in a relatively large (~3 mm/day) centered RMS error in the precipitation fields. Although models D and B have about the same correlation with observations, model B simulates the amplitude of the variations (i.e., the standard deviation) much better than model D, resulting in a smaller RMS error.
Theoretical basis Taylor diagrams display statistics useful for assessing the similarity of a variable simulated by a model (more generally, the "test" field) to its observed counterpart (more generally, the "reference" field). Mathematically, the three statistics displayed on a Taylor diagram are related by the error propagation formula (which can be derived directly from the definition of the statistics appearing in it):
E ′ 2 = σ Δ 2 = σ r 2 + σ t 2 − 2 σ r σ t ρ {\displaystyle E^{\prime 2}=\sigma _{\Delta }^{2}=\sigma _{r}^{2}+\sigma _{t}^{2}-2\sigma _{r}\sigma _{t}\rho } , where ρ is the correlation coefficient between the test and reference fields, E′ is the centered RMS difference between the fields (with any difference in the means first removed), and σ r {\displaystyle \sigma _{r}} and σ t {\displaystyle \sigma _{t}} are the standard deviations of the reference and test fields, respectively. The law of cosines,
c 2 = a 2 + b 2 − 2 a b cos ϕ , {\displaystyle c^{2}=a^{2}+b^{2}-2ab\cos \phi ,}
(where a, b, and c are the length of the sides of the triangle, and ϕ {\displaystyle \phi } is the angle between sides a and b) provides the key to forming the geometrical relationship between the four quantities that underlie the Taylor diagram (shown in Figure 2).
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