stats.py

import warnings

import climpred.stats as st
import numpy as np
import numpy.polynomial.polynomial as poly
import scipy
import xarray as xr

from .checks import has_dims, has_missing, is_xarray
from .timeutils import TimeUtilAccessor
from .utils import match_nans


def _check_y_not_independent_variable(y, dim):
    """Checks that `y` is not the independent variable in statistics functions.

    Args:
        y (xr.DataArray or xr.Dataset): Dependent variable from statistics functions.
        dim (str): Dimension statistical function is being applied over.

    Raises:
        ValueError: If `y` is a DataArray and equal to `dim`. This infers that something
            like a time axis is being placed in the dependent variable.

    """
    if isinstance(y, xr.DataArray) and (y.name == dim):
        raise ValueError(
            f'Dependent variable y should not be the same as the dim {dim} being '
            'applied over. Change your y variable to x.'
        )


def _convert_time_and_return_slope_factor(x, dim):
    """Converts `x` to numeric time (if datetime) and returns slope factor.

    The numeric time accounts for differences in length of months, leap years, etc.
    when fitting a regression and also ensures that the numpy functions don't break
    with datetimes.

    Args:
        x (xr.DataArray or xr.Dataset): Independent variable from statistical functions.
        dim (str): Dimension statistical function is being applied over.

    Returns:
        x (xr.DataArray or xr.Dataset): If `x` is a time axis, converts to numeric
            time. Otherwise, return the original `x`.
        slope_factor (float): Factor to multiply slope by if returning regression
            results. This accounts for the fact that datetimes are converted to
            "days since 1990-01-01" numeric time and thus the answer comes out
            in the original units per day (e.g., degC/day). This auto-converts to
            the original temporal frequency (e.g., degC/year) if the calendar
            can be inferred.
    """
    slope_factor = 1.0
    if isinstance(x, xr.DataArray):
        if x.timeutils.is_temporal:
            slope_factor = x.timeutils.slope_factor
            x = x.timeutils.return_numeric_time()
    return x, slope_factor


def _handle_nans(x, y, nan_policy):
    """Modifies `x` and `y` based on `nan_policy`.

    Args:
        x, y (xr.DataArray or ndarrays): Two time series to which statistical function
            is being applied.
        nan_policy (str): One of ['none', 'propagate', 'raise', 'omit', 'drop']. If
            'none' or 'propagate', return unmodified so the nans can be propagated
            through in the functions. If 'raise', raises a warning if there are any
            nans in `x` or `y`. If 'omit' or 'drop', removes values that contain
            a nan in either `x` or `y` and returns resulting `x` and `y`.

    Returns:
        x, y (xr.DataArray or ndarrays): Modified `x` and `y` datasets.

    Raises:
        ValueError: If `nan_policy` is 'raise' and there are nans in either `x` or `y`;
            if `nan_policy` is not one of ['none', 'propagate', 'raise', 'omit',
            'drop']; or if `x` or `y` are larger than 1-dimensional.
    """
    # Only support 1D, since we are doing `~np.isnan()` indexing for 'omit'/'drop'.
    if (x.ndim > 1) or (y.ndim > 1):
        raise ValueError(
            f'x and y must be 1-dimensional. Got {x.ndim} for x and {y.ndim} for y.'
        )

    if nan_policy in ['none', 'propagate']:
        return x, y
    elif nan_policy == 'raise':
        if has_missing(x) or has_missing(y):
            raise ValueError(
                "Input data contains NaNs. Consider changing `nan_policy` to 'none' "
                "or 'omit'. Or get rid of those NaNs somehow."
            )
        else:
            return x, y
    elif nan_policy in ['omit', 'drop']:
        if has_missing(x) or has_missing(y):
            x_mod, y_mod = match_nans(x, y)
            # The above function pairwise-matches nans. Now we remove them so that we
            # can compute the statistic without the nans.
            x_mod = x_mod[np.isfinite(x_mod)]
            y_mod = y_mod[np.isfinite(y_mod)]
            return x_mod, y_mod
        else:
            return x, y
    else:
        raise ValueError(
            f"{nan_policy} not one of ['none', 'propagate', 'raise', 'omit', 'drop']"
        )


def _polyfit(x, y, order, nan_policy):
    """Helper function for performing ``np.poly.polyfit`` which is used in both
    ``polyfit`` and ``rm_poly``.

    Args:
        x, y (ndarrays): Independent and dependent variables used in the polynomial
            fit.
        order (int): Order of polynomial fit to perform.
        nan_policy (str, optional): Policy to use when handling nans. Defaults to
            "none".

            * 'none', 'propagate': If a NaN exists anywhere on the given dimension,
                return nans for that whole dimension.
            * 'raise': If a NaN exists at all in the datasets, raise an error.
            * 'omit', 'drop': If a NaN exists in `x` or `y`, drop that index and
                compute the slope without it.

    Returns:
        fit (ndarray, xarray object): If ``nan_policy`` is 'none' or 'propagate' and
            a nan exists in the time series, returns all nans. Otherwise, returns the
            polynomial fit.
    """
    x_mod, y_mod = _handle_nans(x, y, nan_policy)
    # This catches cases where a given grid cell is full of nans, like in land masking.
    if (nan_policy in ['omit', 'drop']) and (x_mod.size == 0):
        return np.full(len(x), np.nan)
    # This catches cases where there is missing values in the independent axis, which
    # breaks polyfit.
    elif (nan_policy in ['none', 'propagate']) and (has_missing(x_mod)):
        return np.full(len(x), np.nan)
    else:
        # fit to data without nans, return applied to original independent axis.
        coefs = poly.polyfit(x_mod, y_mod, order)
        return poly.polyval(x, coefs)


def _warn_if_not_converted_to_original_time_units(x):
    """Administers warning if the independent variable is in datetimes and the
    calendar frequency could not be inferred.

    Args:
        x (xr.DataArray or xr.Dataset): Independent variable for statistical functions.
    """
    if isinstance(x, xr.DataArray):
        if x.timeutils.is_temporal:
            if x.timeutils.freq is None:
                warnings.warn(
                    'Datetime frequency not detected. Slope and std. errors will be '
                    'in original units per day (e.g., degC per day). Multiply by '
                    'e.g., 365.25 to convert to original units per year.'
                )


@is_xarray(0)
def ACF(ds, dim='time', nlags=None):
    """Compute the ACF of a time series to a specific lag.

    Args:
      ds (xarray object): dataset/dataarray containing the time series.
      dim (str): dimension to apply ACF over.
      nlags (optional int): number of lags to compute ACF over. If None,
                            compute for length of `dim` on `ds`.

    Returns:
      Dataset or DataArray with ACF results.

    Notes:
      This is preferred over ACF functions from MATLAB/scipy, since it doesn't
      use FFT methods.
    """
    # Drop variables that don't have requested dimension, so this can be
    # applied over the full dataset.
    if isinstance(ds, xr.Dataset):
        dropVars = [i for i in ds if dim not in ds[i].dims]
        ds = ds.drop(dropVars)

    # Loop through every step in `dim`
    if nlags is None:
        nlags = ds[dim].size

    acf = []
    # The 2 factor accounts for fact that time series reduces in size for
    # each lag.
    for i in range(nlags - 2):
        res = autocorr(ds, lag=i, dim=dim)
        acf.append(res)
    acf = xr.concat(acf, dim=dim)
    return acf


@is_xarray(0)
def autocorr(ds, lag=1, dim='time', return_p=False):
    """Calculated lagged correlation of a xr.Dataset.

    Args:
        ds (xarray object): Dataset to compute autocorrelation with.
        lag (int, optional): Lag to compute autocorrelation at. Defaults to 1.
        dim (str, optional): Dimension to compute autocorrelation over.
            Default to 'time'.
        return_p (bool, optional): If True, return just the correlation coefficient.
            If False, return both the correlation coefficient and p value.

    Returns:
        r (xarray object): Pearson correlation coefficient.
        p (xarray object): P value, if ``return_p`` is True.
    """
    return st.autocorr(ds, lag=lag, dim=dim, return_p=return_p)


@is_xarray(0)
def corr(x, y, dim='time', lead=0, return_p=False):
    """Computes the Pearson product-moment coefficient of linear correlation.

    This version calculates the effective degrees of freedom, accounting
    for autocorrelation within each time series that could fluff the
    significance of the correlation.

    Args:
        x, y (xarray.DataArray): Time series being correlated.
        dim (str, optional): Dimension to calculate correlation over. Defaults to
            'time'.
        lead (int, optional): If lead > 0, ``x`` leads ``y`` by that many time steps.
            If lead < 0, ``x`` lags ``y`` by that many time steps. Defaults to 0.
        return_p (bool, optional). If True, return both ``r`` and ``p``. Otherwise,
            just return ``r``. Defaults to False.

    Returns:
        r (xarray object): Pearson correlation coefficient.
        p (xarray object): P value, if ``return_p`` is True.

    References:
        * Wilks, Daniel S. Statistical methods in the atmospheric sciences.
          Vol. 100. Academic press, 2011.
        * Lovenduski, Nicole S., and Nicolas Gruber. "Impact of the Southern
          Annular Mode on Southern Ocean circulation and biology." Geophysical
          Research Letters 32.11 (2005).
        * Brady, R. X., Lovenduski, N. S., Alexander, M. A., Jacox, M., and
          Gruber, N.: On the role of climate modes in modulating the air-sea CO2
          fluxes in Eastern Boundary Upwelling Systems, Biogeosciences Discuss.,
          https://doi.org/10.5194/bg-2018-415, 2019.

    """
    # Broadcasts a time series to the same coordinates/size as the grid. If they
    # are both grids, this function does nothing and isn't expensive.
    x, y = xr.broadcast(x, y)

    # Negative lead should have y lead x.
    if lead < 0:
        lead = np.abs(lead)
        return st.corr(y, x, dim=dim, lag=lead, return_p=return_p)
    else:
        return st.corr(x, y, dim=dim, lag=lead, return_p=return_p)


@is_xarray([0, 1])
def linear_slope(x, y, dim='time', nan_policy='none'):
    """Returns the linear slope with y regressed onto x.

    .. note::

        This function will try to infer the time freqency of sampling if ``x`` is in
        datetime units. The final slope will be returned in the original units per
        that frequency (e.g. SST per year). If the frequency cannot be inferred
        (e.g. because the sampling is irregular), it will return in the original
        units per day (e.g. SST per day).

    Args:
        x (xarray object): Independent variable (predictor) for linear regression.
        y (xarray object): Dependent variable (predictand) for linear regression.
        dim (str, optional): Dimension to apply linear regression over.
            Defaults to "time".
        nan_policy (str, optional): Policy to use when handling nans. Defaults to
            "none".

            * 'none', 'propagate': If a NaN exists anywhere on the given dimension,
                return nans for that whole dimension.
            * 'raise': If a NaN exists at all in the datasets, raise an error.
            * 'omit', 'drop': If a NaN exists in `x` or `y`, drop that index and
                compute the slope without it.

    Returns:
        xarray object: Slopes computed through a least-squares linear regression.
    """
    has_dims(x, dim, 'predictor (x)')
    has_dims(y, dim, 'predictand (y)')
    _check_y_not_independent_variable(y, dim)
    x, slope_factor = _convert_time_and_return_slope_factor(x, dim)

    def _linear_slope(x, y, nan_policy):
        x, y = _handle_nans(x, y, nan_policy)
        # This catches cases where a given grid cell is full of nans, like in
        # land masking.
        if (nan_policy in ['omit', 'drop']) and (x.size == 0):
            return np.asarray([np.nan])
        # This catches cases where there is missing values in the independent axis,
        # which breaks polyfit.
        elif (nan_policy in ['none', 'propagate']) and (has_missing(x)):
            return np.asarray([np.nan])
        else:
            return np.polyfit(x, y, 1)[0]

    slopes = xr.apply_ufunc(
        _linear_slope,
        x,
        y,
        nan_policy,
        vectorize=True,
        dask='parallelized',
        input_core_dims=[[dim], [dim], []],
        output_dtypes=['float64'],
    )
    _warn_if_not_converted_to_original_time_units(x)
    return slopes * slope_factor


@is_xarray([0, 1])
def linregress(x, y, dim='time', nan_policy='none'):
    """Vectorized applciation of ``scipy.stats.linregress``.

    .. note::

        This function will try to infer the time freqency of sampling if ``x`` is in
        datetime units. The final slope and standard error will be returned in the
        original units per that frequency (e.g. SST per year). If the frequency
        cannot be inferred (e.g. because the sampling is irregular), it will return in
        the original units per day (e.g. SST per day).

    Args:
        x (xarray object): Independent variable (predictor) for linear regression.
        y (xarray object): Dependent variable (predictand) for linear regression.
        dim (str, optional): Dimension to apply linear regression over.
            Defaults to "time".
        nan_policy (str, optional): Policy to use when handling nans. Defaults to
            "none".

            * 'none', 'propagate': If a NaN exists anywhere on the given dimension,
                return nans for that whole dimension.
            * 'raise': If a NaN exists at all in the datasets, raise an error.
            * 'omit', 'drop': If a NaN exists in `x` or `y`, drop that index and
                compute the slope without it.

    Returns:
        xarray object: Slope, intercept, correlation, p value, and standard error for
            the linear regression. These 5 parameters are added as a new dimension
            "parameter".

    """
    has_dims(x, dim, 'predictor (x)')
    has_dims(y, dim, 'predictand (y)')
    _check_y_not_independent_variable(y, dim)
    x, slope_factor = _convert_time_and_return_slope_factor(x, dim)

    def _linregress(x, y, slope_factor, nan_policy):
        x, y = _handle_nans(x, y, nan_policy)
        # This catches cases where a given grid cell is full of nans, like in
        # land masking.
        if (nan_policy in ['omit', 'drop']) and (x.size == 0):
            return np.full(5, np.nan)
        else:
            m, b, r, p, e = scipy.stats.linregress(x, y)
            # Multiply slope and standard error by factor. If time indices were
            # converted to numeric units, this gets them back to the original units.
            m *= slope_factor
            e *= slope_factor
            return np.array([m, b, r, p, e])

    results = xr.apply_ufunc(
        _linregress,
        x,
        y,
        slope_factor,
        nan_policy,
        vectorize=True,
        dask='parallelized',
        input_core_dims=[[dim], [dim], [], []],
        output_core_dims=[['parameter']],
        output_dtypes=['float64'],
        output_sizes={'parameter': 5},
    )
    results = results.assign_coords(
        parameter=['slope', 'intercept', 'rvalue', 'pvalue', 'stderr']
    )
    _warn_if_not_converted_to_original_time_units(x)
    return results


@is_xarray(0)
def nanmean(ds, dim='time'):
    """Compute mean of data with NaNs and suppress warning from numpy.

    Args:
        ds (xarray object): Dataset to compute mean over.
        dim (str, optional): Dimension to compute mean over.

    Returns
        xarray object: Reduced by ``dim`` via mean operation.
    """
    if 'time' in ds.dims:
        mask = ds.isnull().isel(time=0)
    else:
        mask = ds.isnull()
    ds = ds.fillna(0).mean(dim)
    ds = ds.where(~mask)
    return ds


@is_xarray(0)
def polyfit(x, y, order, dim='time', nan_policy='none'):
    """Returns the fitted polynomial line of ``y`` regressed onto ``x``.

    .. note::

        This will be released as a standard ``xarray`` func in 0.15.2.

    Args:
        x, y (xr.DataArray or xr.Dataset): Independent and dependent variables used in
            the polynomial fit.
        order (int): Order of polynomial fit to perform.
        dim (str, optional): Dimension to apply polynomial fit over.
            Defaults to "time".
        nan_policy (str, optional): Policy to use when handling nans. Defaults to
            "none".

            * 'none', 'propagate': If a NaN exists anywhere on the given dimension,
                return nans for that whole dimension.
            * 'raise': If a NaN exists at all in the datasets, raise an error.
            * 'omit', 'drop': If a NaN exists in `x` or `y`, drop that index and
                compute the slope without it.

    Returns:
        xarray object: The polynomial fit for ``y`` regressed onto ``x``. Has the same
            dimensions as ``y``.
    """
    has_dims(x, dim, 'predictor (x)')
    has_dims(y, dim, 'predictand (y)')
    _check_y_not_independent_variable(y, dim)
    x, _ = _convert_time_and_return_slope_factor(x, dim)

    return xr.apply_ufunc(
        _polyfit,
        x,
        y,
        order,
        nan_policy,
        vectorize=True,
        dask='parallelized',
        input_core_dims=[[dim], [dim], [], []],
        output_core_dims=[[dim]],
        output_dtypes=['float'],
    )


@is_xarray(0)
def rm_poly(x, y, order, dim='time', nan_policy='none'):
    """Removes a polynomial fit from ``y`` regressed onto ``x``.

    Args:
        x, y (xr.DataArray or xr.Dataset): Independent and dependent variables used in
            the polynomial fit.
        order (int): Order of polynomial fit to perform.
        dim (str, optional): Dimension to apply polynomial fit over.
            Defaults to "time".
        nan_policy (str, optional): Policy to use when handling nans. Defaults to
            "none".

            * 'none', 'propagate': If a NaN exists anywhere on the given dimension,
                return nans for that whole dimension.
            * 'raise': If a NaN exists at all in the datasets, raise an error.
            * 'omit', 'drop': If a NaN exists in `x` or `y`, drop that index and
                compute the slope without it.

    Returns:
        xarray object: ``y`` with polynomial fit of order ``order`` removed.
    """
    has_dims(x, dim, 'predictor (x)')
    has_dims(y, dim, 'predictand (y)')
    _check_y_not_independent_variable(y, dim)
    x, _ = _convert_time_and_return_slope_factor(x, dim)

    def _rm_poly(x, y, order, nan_policy):
        fit = _polyfit(x, y, order, nan_policy)
        return y - fit

    return xr.apply_ufunc(
        _rm_poly,
        x,
        y,
        order,
        nan_policy,
        vectorize=True,
        dask='parallelized',
        input_core_dims=[[dim], [dim], [], []],
        output_core_dims=[[dim]],
        output_dtypes=['float64'],
    )


@is_xarray(0)
def rm_trend(x, y, dim='time', nan_policy='none'):
    """Removes a linear fit from ``y`` regressed onto ``x``.

    Args:
        x, y (xr.DataArray or xr.Dataset): Independent and dependent variables used in
            the linear fit.
        dim (str, optional): Dimension to apply linear fit over.
            Defaults to "time".
        nan_policy (str, optional): Policy to use when handling nans. Defaults to
            "none".

            * 'none', 'propagate': If a NaN exists anywhere on the given dimension,
                return nans for that whole dimension.
            * 'raise': If a NaN exists at all in the datasets, raise an error.
            * 'omit', 'drop': If a NaN exists in `x` or `y`, drop that index and
                compute the slope without it.

    Returns:
        xarray object: ``y`` with linear fit removed.
    """
    return rm_poly(x, y, 1, dim=dim, nan_policy=nan_policy)


@is_xarray(0)
def standardize(ds, dim='time'):
    """Standardize Dataset/DataArray

    .. math::
        \\frac{x - \\mu_{x}}{\\sigma_{x}}

    Args:
        ds (xarray object): Dataset or DataArray with variable(s) to standardize.
        dim (optional str): Which dimension to standardize over (default 'time').

    Returns:
        stdized (xarray object): Standardized variable(s).
    """
    stdized = (ds - ds.mean(dim)) / ds.std(dim)
    return stdized