# Significance Testing¶

Significance testing is important for assessing whether a given initialized prediction system is skillful. Some questions that significance testing can answer are:

• Is the correlation coefficient of a lead time series significantly different from zero?
• What is the probability that the retrospective forecast is more valuable than a historical simulation?
• Are correlation coefficients statistically significant despite temporal and spatial autocorrelation?

All of these questions deal with statistical significance. See below on how to use `climpred` to address these questions. Please also have a look at the significance testing example.

## p value for temporal correlations¶

For the correlation metrics, like `_pearson_r()` and `_spearman_r()`, `climpred` also hosts the associated p-value, like `_pearson_r_p_value()`, that this correlation is significantly different from zero. `_pearson_r_eff_p_value()` also incorporates the reduced degrees of freedom due to temporal autocorrelation. See example.

## Bootstrapping with replacement¶

Testing statistical significance through bootstrapping is commonly used in the field of climate prediction [could add some example citations here]. Bootstrapping relies on resampling the underlying data with replacement for a large number of iterations, as proposed by the decadal prediction framework of Goddard et al. 2013 [Goddard2013]. This means that the initialized ensemble is resampled with replacement along a dimension (`init` or `member`) and then that resampled ensemble is verified against the observations. This leads to a distribution of initialized skill. Further, a reference forecast uses the resampled initialized ensemble, e.g. `compute_persistence()`, which creates a reference skill distribution. Lastly, an uninitialized skill distribution is created from the underlying historical members or the control simulation.

The probability or p value is the fraction of these resampled initialized metrics beaten by the uninitialized or resampled reference metrics calculated from their respective distributions. Confidence intervals using these distributions are also calculated.

This behavior is incorporated into `climpred` by the base function `bootstrap_compute()`, which is wrapped by `bootstrap_hindcast()` and `bootstrap_perfect_model()` for the respective prediction simulation type. See example

## Field significance¶

Please use `esmtools.testing.multipletests()` to control the false discovery rate (FDR) in geospatial data from the above obtained p-values [Wilks2016]. See the FDR example.