# 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

1. Pearson’s coefficient
2. Standard Deviation (SD)
3. 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.