Taylor Diagram
Interactive Taylor Diagram for evaluating model performance by combining correlation, standard deviation, and RMSE in polar coordinates
The Taylor Diagram, introduced by Karl E. Taylor in 2001, is a polar coordinate plot that simultaneously displays three complementary statistics about how well a model matches reference (observed) data: the Pearson correlation coefficient (r), the ratio of modeled to observed standard deviation (σ_f/σ_r), and the centered Root Mean Square Error (RMSE). All three statistics are geometrically related — given any two, the third is determined by the law of cosines. This allows multiple models to be compared at a glance on a single diagram.
The diagram uses a polar coordinate system where: (1) The radial distance from the origin represents the standard deviation σ of the model field. (2) The azimuthal angle from the horizontal axis encodes the correlation coefficient r via arccos(r). (3) The distance between the model point and the reference point (plotted on the horizontal axis at distance σ_r) equals the centered RMSE, by the law of cosines: RMSE² = σ_f² + σ_r² − 2σ_fσ_r·r. This geometric identity is what makes the diagram possible.
The reference point lies on the horizontal axis at radial distance σ_r. A perfect model would plot at the same point as the reference (σ_f = σ_r, r = 1, RMSE = 0). Models with higher correlation appear closer to the right side (smaller azimuthal angle). Dashed concentric arcs centered on the reference point are RMSE contours — models on the same arc have the same RMSE. Radial dashed circles show standard deviation ratios. A good model has: (1) high correlation (near horizontal), (2) σ_f ≈ σ_r (near reference radial distance), and (3) small RMSE (close to reference point).
The standard deviation measures the amplitude of variability. If σ_model > σ_reference, the model overestimates variability. If σ_model < σ_reference, the model underestimates variability. On the Taylor Diagram, this is the radial distance from the origin. The ratio σ_model/σ_reference is often more informative than absolute values.
The Pearson correlation coefficient r measures pattern similarity between model and observations, ignoring amplitude differences. r = 1 means perfect agreement; r = 0 means no linear relationship; r = -1 means perfect anti-correlation. On the Taylor Diagram, r maps to azimuthal angle via arccos(r). Note that correlation alone is insufficient — perfect pattern (r=1) with wrong amplitude still has errors.
The centered RMSE removes mean bias and measures pattern/amplitude error: RMSE² = (1/N)Σ(f_i - r_i)² after subtracting means. On the Taylor Diagram, RMSE is the Euclidean distance between model and reference points, via: RMSE² = σ_f² + σ_r² − 2σ_fσ_r·r. Concentric arcs centered on the reference point are constant-RMSE contours.
Taylor Diagrams are extensively used in IPCC assessment reports to compare CMIP climate models against observed climatology. Different models (GFDL, HadGEM, MPI-ESM, etc.) are plotted as points, allowing quick identification of which models best reproduce observed patterns of temperature, precipitation, and other fields.
NWP centers use Taylor Diagrams to compare forecast skill across different lead times, model versions, or resolutions. ECMWF routinely produces Taylor Diagrams comparing HRES, ensemble mean, and control forecasts against analyses. By plotting the same model at different lead times (T+24h through T+240h), forecast quality degradation is visualized.
Beyond climate and weather: (1) Remote sensing — validating satellite products against ground truth. (2) Hydrology — comparing rainfall-runoff models. (3) Air quality modeling — evaluating PM2.5 and ozone forecasts. (4) Machine learning — comparing regression models against test data. (5) Signal processing — evaluating reconstructed signals. The Taylor Diagram is one of the most effective single-plot tools for multi-model comparison.