# noqa: D205,D400
"""
Ensemble Robustness metrics.
============================
Robustness metrics are used to estimate the confidence of the climate change signal
of an ensemble. This submodule is inspired by and tries to follow the guidelines of
the IPCC, more specifically the 12th chapter of the Working Group 1's contribution to
the AR5 [AR5WG1C12]_ (see box 12.1).
References
----------
.. [AR5WG1C12] https://www.ipcc.ch/site/assets/uploads/2018/02/WG1AR5_Chapter12_FINAL.pdf
"""
from typing import Tuple, Union
import numpy as np
import scipy.stats as spstats
import xarray as xr
from xclim.core.formatting import update_history
[docs]def change_significance(
fut: Union[xr.DataArray, xr.Dataset],
ref: Union[xr.DataArray, xr.Dataset] = None,
test: str = "ttest",
**kwargs,
) -> Tuple[Union[xr.DataArray, xr.Dataset], Union[xr.DataArray, xr.Dataset]]:
"""Robustness statistics qualifying how the members of an ensemble agree on the existence of change and on its sign.
Parameters
----------
fut : Union[xr.DataArray, xr.Dataset]
Future period values along 'realization' and 'time' (..., nr, nt1)
or if `ref` is None, Delta values along `realization` (..., nr).
ref : Union[xr.DataArray, xr.Dataset], optional
Reference period values along realization' and 'time' (..., nt2, nr).
The size of the 'time' axis does not need to match the one of `fut`.
But their 'realization' axes must be identical.
If `None` (default), values of `fut` are assumed to be deltas instead of
a distribution across the future period.
`fut` and `ref` must be of the same type (Dataset or DataArray). If they are
Dataset, they must have the same variables (name and coords).
test : {'ttest', 'welch-ttest', 'threshold', None}
Name of the statistical test used to determine if there was significant change. See notes.
kwargs
Other arguments specific to the statistical test.
For 'ttest' and 'welch-ttest':
p_change : float (default : 0.05)
p-value threshold for rejecting the hypothesis of no significant change.
For 'threshold': (Only one of those must be given.)
abs_thresh : float (no default)
Threshold for the (absolute) change to be considered significative.
rel_thresh : float (no default, in [0, 1])
Threshold for the relative change (in reference to ref) to be significative.
Only valid if `ref` is given.
Returns
-------
change_frac
The fraction of members that show significant change [0, 1].
Passing `test=None` yields change_frac = 1 everywhere. Same type as `fut`.
pos_frac
The fraction of members showing significant change that show a positive change ]0, 1].
Null values are returned where no members show significant change.
The table below shows the coefficient needed to retrieve the number of members
that have the indicated characteristics, by multiplying it to the total
number of members (`fut.realization.size`).
+-----------------+------------------------------+------------------------+
| | Significant change | Non-significant change |
+-----------------+------------------------------+------------------------+
| Any direction | change_frac | 1 - change_frac |
+-----------------+------------------------------+------------------------+
| Positive change | pos_frac * change_frac | N.A. |
+-----------------+------------------------------+ |
| Negative change | (1 - pos_frac) * change_frac | |
+-----------------+------------------------------+------------------------+
Notes
-----
Available statistical tests are :
'ttest' :
Single sample T-test. Same test as used by [tebaldi2011]_. The future
values are compared against the reference mean (over 'time'). Change is qualified
as 'significant' when the test's p-value is below the user-provided `p_change`
value.
'welch-ttest' :
Two-sided T-test, without assuming equal population variance. Same
significance criterion as 'ttest'.
'threshold' :
Change is considered significative if the absolute delta exceeds a given
threshold (absolute or relative).
None :
Significant change is not tested and, thus, members showing no change are
included in the `sign_frac` output.
References
----------
.. [tebaldi2011] Tebaldi C., Arblaster, J.M. and Knutti, R. (2011) Mapping model agreement on future climate projections. GRL. doi:10.1029/2011GL049863
Example
-------
This example computes the mean temperature in an ensemble and compares two time
periods, qualifying significant change through a single sample T-test.
>>> from xclim import ensembles
>>> ens = ensembles.create_ensemble(temperature_datasets)
>>> tgmean = xclim.atmos.tg_mean(tas=ens.tas, freq='YS')
>>> fut = tgmean.sel(time=slice('2020', '2050'))
>>> ref = tgmean.sel(time=slice('1990', '2020'))
>>> chng_f, pos_f = ensembles.change_significance(fut, ref, test='ttest')
If the deltas were already computed beforehand, the 'threshold' test can still
be used, here with a 2 K threshold.
>>> delta = fut.mean('time') - ref.mean('time')
>>> chng_f, pos_f = ensembles.change_significance(delta, test='threshold', abs_thresh=2)
"""
test_params = {
"ttest": ["p_change"],
"welch-ttest": ["p_change"],
"threshold": ["abs_thresh", "rel_thresh"],
}
changed = None
if ref is None:
delta = fut
n_valid_real = delta.notnull().sum("realization")
if test not in ["threshold", None]:
raise ValueError(
"When deltas are given (ref=None), 'test' must be one of ['threshold', None]"
)
else:
delta = fut.mean("time") - ref.mean("time")
n_valid_real = fut.notnull().all("time").sum("realization")
if test == "ttest":
p_change = kwargs.setdefault("p_change", 0.05)
# Test hypothesis of no significant change
pvals = xr.apply_ufunc(
lambda f, r: spstats.ttest_1samp(f, r, axis=-1, nan_policy="omit")[1],
fut,
ref.mean("time"),
input_core_dims=[["realization", "time"], ["realization"]],
output_core_dims=[["realization"]],
vectorize=True,
dask="parallelized",
output_dtypes=[float],
)
# When p < p_change, the hypothesis of no significant change is rejected.
changed = pvals < p_change
elif test == "welch-ttest":
p_change = kwargs.setdefault("p_change", 0.05)
# Test hypothesis of no significant change
# equal_var=False -> Welch's T-test
pvals = xr.apply_ufunc(
lambda f, r: spstats.ttest_ind(
f, r, axis=-1, equal_var=False, nan_policy="omit"
)[1],
fut,
ref,
input_core_dims=[["realization", "time"], ["realization", "time"]],
output_core_dims=[["realization"]],
exclude_dims={"time"},
vectorize=True,
dask="parallelized",
output_dtypes=[float],
)
# When p < p_change, the hypothesis of no significant change is rejected.
changed = pvals < p_change
elif test == "threshold":
if "abs_thresh" in kwargs and "rel_thresh" not in kwargs:
changed = abs(delta) > kwargs["abs_thresh"]
elif "rel_thresh" in kwargs and "abs_thresh" not in kwargs and ref is not None:
changed = abs(delta / ref.mean("time")) > kwargs["rel_thresh"]
else:
raise ValueError("Invalid argument combination for test='threshold'.")
elif test is not None:
raise ValueError(
f"Statistical test {test} must be one of {', '.join(test_params.keys())}."
)
if test is not None:
delta_chng = delta.where(changed)
change_frac = changed.sum("realization") / n_valid_real
else:
delta_chng = delta
change_frac = xr.ones_like(delta.isel(realization=0))
# Test that models agree on the sign of the change
# This returns NaN (cause 0 / 0) where no model show significant change.
pos_frac = (delta_chng > 0).sum("realization") / (change_frac * n_valid_real)
# Metadata
kwargs_str = ", ".join(
[f"{k}: {v}" for k, v in kwargs.items() if k in test_params[test]]
)
test_str = (
f"Significant change was tested with test {test} with parameters {kwargs_str}."
)
das = {"fut": fut} if ref is None else {"fut": fut, "ref": ref}
pos_frac.attrs.update(
description="Fraction of members showing significant change that agree on a positive change. "
+ test_str,
units="",
test=str(test),
history=update_history(
f"pos_frac from change_significance(fut=fut, ref=ref, test={test}, {kwargs_str})",
**das,
),
)
change_frac.attrs.update(
description="Fraction of members showing significant change. " + test_str,
units="",
test=str(test),
history=update_history(
f"change_frac from change_significance(fut=fut, ref=ref, test={test}, {kwargs_str})",
**das,
),
)
return change_frac, pos_frac
[docs]def robustness_coefficient(
fut: Union[xr.DataArray, xr.Dataset], ref: Union[xr.DataArray, xr.Dataset]
) -> Union[xr.DataArray, xr.Dataset]:
"""Robustness coefficient quantifying the robustness of a climate change signal in an ensemble.
Taken from Knutti and Sedlacek (2013).
The robustness metric is defined as R = 1 − A1 / A2 , where A1 is defined
as the integral of the squared area between two cumulative density functions
characterizing the individual model projections and the multi-model mean
projection and A2 is the integral of the squared area between two cumulative
density functions characterizing the multi-model mean projection and the historical
climate. (Description taken from [knutti2013]_)
A value of R equal to one implies perfect model agreement. Higher model spread or
smaller signal decreases the value of R.
Parameters
----------
fut : Union[xr.DataArray, xr.Dataset]
Future ensemble values along 'realization' and 'time' (nr, nt). Can be a dataset,
in which case the coeffcient is computed on each variables.
ref : Union[xr.DataArray, xr.Dataset]
Reference period values along 'time' (nt). Same type as `fut`.
Returns
-------
R
The robustness coeffcient, ]-inf, 1], float. Same type as `fut` or `ref`.
References
----------
.. [knutti2013] Knutti, R. and Sedláček, J. (2013) Robustness and uncertainties in the new CMIP5 climate model projections. Nat. Clim. Change. doi:10.1038/nclimate1716
"""
def _knutti_sedlacek(reference, future):
def diff_cdf_sq_area_int(x1, x2):
"""Exact integral of the squared area between the non-parametric CDFs of 2 vectors."""
# Non-parametric CDF on points x1 and x2
# i.e. y1(x) is the proportion of x1 <= x
y1 = (np.arange(x1.size) + 1) / x1.size
y2 = (np.arange(x2.size) + 1) / x2.size
x2_in_1 = np.searchsorted(x1, x2, side="right") # Where to insert x2 in x1
x1_in_2 = np.searchsorted(x2, x1, side="right") # Where to insert x1 in x2
# Merge to get all "discontinuities" of the CDF difference
# y1 with repeated value (to the right) where x2 is inserted
# Same for y2. 0s are prepended where needed.
x = np.insert(x1, x2_in_1, x2)
y1_f = np.insert(y1, x2_in_1, np.r_[0, y1][x2_in_1])
y2_f = np.insert(y2, x1_in_2, np.r_[0, y2][x1_in_2])
# Discrete integral of the squared difference (distance) between the two CDFs.
return np.sum(np.diff(x) * (y1_f - y2_f)[:-1] ** 2)
# Get sorted vectors
v_fut = np.sort(future.flatten()) # "cumulative" models distribution
v_favg = np.sort(future.mean(axis=-1)) # Multi-model mean
v_ref = np.sort(reference) # Historical values
A1 = diff_cdf_sq_area_int(v_fut, v_favg) # noqa
A2 = diff_cdf_sq_area_int(v_ref, v_favg) # noqa
return 1 - A1 / A2
R = xr.apply_ufunc( # noqa
_knutti_sedlacek,
ref,
fut,
input_core_dims=[["time"], ["realization", "time"]],
exclude_dims={"time"},
vectorize=True,
dask="parallelized",
output_dtypes=[float],
)
R.attrs.update(
name="R",
long_name="Ensemble robustness coefficient",
description="Ensemble robustness coefficient as defined by Knutti and Sedláček (2013).",
reference="Knutti, R. and Sedláček, J. (2013) Robustness and uncertainties in the new CMIP5 climate model projections. Nat. Clim. Change.",
units="",
history=update_history("knutti_sedlacek(fut, ref)", ref=ref, fut=fut),
)
return R