Revert "let fit also return residuals, remove bounds and delete predi…

…ct function"

This reverts commit eff8daf.

mesmer/stats/_auto_regression.py

@@ -635,14 +635,13 @@ def fit_auto_regression_monthly(monthly_data, time_dim="time"):
 
     Returns
     -------
-    obj : ``xr.Dataset``
+    ar_params : ``xr.Dataset``
         Dataset containing the estimated parameters of the AR(1) process, the ``intercept`` and the
-        ``slope`` for each month and gridpoint. Additionally, the ``residuals`` are returned.
+        ``slope`` for each month and gridpoint.
     """
     _check_dataarray_form(monthly_data, "monthly_data", ndim=2, required_dims=time_dim)
     monthly_data = monthly_data.groupby(f"{time_dim}.month")
     ar_params = []
-    residuals = []
 
     for month in range(1, 13):
         if month == 1:
@@ -657,25 +656,21 @@ def fit_auto_regression_monthly(monthly_data, time_dim="time"):
             prev_month = monthly_data[month - 1]
             cur_month = monthly_data[month]
 
-        slope, intercept, resids = xr.apply_ufunc(
+        slope, intercept = xr.apply_ufunc(
             _fit_auto_regression_monthly_np,
             cur_month,
             prev_month,
             input_core_dims=[[time_dim], [time_dim]],
-            output_core_dims=[[], [], [time_dim]],
+            output_core_dims=[[], []],
             exclude_dims={time_dim},
             vectorize=True,
         )
 
         ar_params.append(xr.Dataset({"slope": slope, "intercept": intercept}))
-        residuals.append(resids.assign_coords({time_dim: cur_month[time_dim]}))
-
 
     month = xr.Variable("month", np.arange(1, 13))
-    ar_params = xr.concat(ar_params, dim=month)
-    residuals = xr.concat(residuals, dim=time_dim).rename("residuals")
 
-    return xr.merge([ar_params, residuals])
+    return xr.concat(ar_params, dim=month)
 
 
 def _fit_auto_regression_monthly_np(data_month, data_prev_month):
@@ -705,11 +700,72 @@ def lin_func(indep_var, slope, intercept):
         lin_func,
         data_prev_month,  # independent variable
         data_month,  # dependent variable
+        bounds=([-1, -np.inf], [1, np.inf]),
     )[0]
 
-    residuals = data_month - lin_func(data_prev_month, slope, intercept)
+    return slope, intercept
 
-    return slope, intercept, residuals
+
+def predict_auto_regression_monthly(ar_params, time, buffer):
+    """deterministically predict time series of an auto regression process
+    with lag one (AR(1)) using individual parameters for each month.
+    This function is deterministic, i.e. does not produce random noise!
+
+    Parameters
+    ----------
+    ar_params : xr.Dataset
+        Dataset containing the estimated parameters of the AR1 process. Must contain the
+        following DataArray objects:
+
+        - intercept
+        - slope
+
+        both of shape (12, n_gridpoints).
+    time : xr.DataArray
+        The time coordinates that determines the length of the predicted timeseries and
+        that will be the assigned time dimension of the predictions.
+    buffer : int
+        Buffer to initialize the auto-regressive process (ensures that start at 0 does
+        not influence overall result).
+
+    Returns
+    -------
+    result : xr.DataArray
+        Predicted deterministic time series of the specified AR(1). The array has
+        shape n_timesteps x n_gridpoints.
+
+    """
+    _check_dataset_form(ar_params, "ar_params", required_vars=("intercept", "slope"))
+
+    (month_dim, gridcell_dim) = ar_params.intercept.dims
+    (n_months, n_gridpoints) = ar_params.intercept.shape
+
+    _check_dataarray_form(
+        ar_params.intercept,
+        "intercept",
+        ndim=2,
+        required_dims=(month_dim, gridcell_dim),
+    )
+    _check_dataarray_form(
+        ar_params.slope,
+        "slope",
+        ndim=2,
+        required_dims=(month_dim, gridcell_dim),
+        shape=(n_months, n_gridpoints),
+    )
+    result = _draw_ar_corr_monthly_xr_internal(
+        intercept=ar_params.intercept,
+        slope=ar_params.slope,
+        covariance=None,
+        time=time,
+        realisation=1,
+        seed=0,
+        buffer=buffer,
+        time_dim="time",
+        realisation_dim="realisation",
+    ).squeeze("realisation", drop=True)
+
+    return result
 
 
 def draw_auto_regression_monthly(