# An introduction of Taylor diagram

## What is Taylor diagram?

Taylor diagram is invented by Karl E. Taylor for the comparison between model and observational data (Taylor, 2001). Essentially, it includes *three metrics and their visualisation in a polar coordinate system*.

### Theoretical basis

- Pearson’s coefficient
- Standard Deviation (SD)
- centred RMSE (cRMSE)

These three metrics are related using a cosine rule: $$ c^2 = a^2 + b^2 + 2ab \cos(\theta)\ $$

$$ \text{cRMSE}^2 = \sigma_{\text{model}}^2 + \sigma_{\text{data}}^2 - 2 \sigma_{\text{model}} \sigma_{\text{data}} \cdot \rho $$

As definition, $\rho = \cos(\theta)$ and $\theta = \arccos(\rho)$

### Visualisation in a polar coordinate system

When we get the angle ($\theta$) and the radius ($\sigma$), we can plot a point in a polar coordinate system. In a more complex figure, we add the reference information such as its SD. Then the cRMSE that measures the model-data distance can be calculated.

### Interpretation of Taylor diagram

Clearly, cRMSE is the metric to focus on. A model with smaller cRMSE has better performance.

## References

Taylor, K.E. (2001). Summarizing multiple aspects of model performance in a single diagram. J. Geophys. Res. 106: 7183–7192.