code: # Copyright The PyTorch Lightning team.
  #
  # Licensed under the Apache License, Version 2.0 (the "License");
  # you may not use this file except in compliance with the License.
  # You may obtain a copy of the License at
  #
  #     http://www.apache.org/licenses/LICENSE-2.0
  #
  # Unless required by applicable law or agreed to in writing, software
  # distributed under the License is distributed on an "AS IS" BASIS,
  # WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
  # See the License for the specific language governing permissions and
  # limitations under the License.
  from typing import Any, List, Optional, Tuple
  
  import torch
  from torch import Tensor
  
  from torchmetrics.functional.regression.pearson import _pearson_corrcoef_compute, _pearson_corrcoef_update
  from torchmetrics.metric import Metric
  
  
  def _final_aggregation(
      means_x: Tensor,
      means_y: Tensor,
      vars_x: Tensor,
      vars_y: Tensor,
      corrs_xy: Tensor,
      nbs: Tensor,
  ) -> Tuple[Tensor, Tensor, Tensor, Tensor]:
      """Aggregate the statistics from multiple devices.
  
      Formula taken from here: `Aggregate the statistics from multiple devices`_
      """
      # assert len(means_x) > 1 and len(means_y) > 1 and len(vars_x) > 1 and len(vars_y) > 1 and len(corrs_xy) > 1
      mx1, my1, vx1, vy1, cxy1, n1 = means_x[0], means_y[0], vars_x[0], vars_y[0], corrs_xy[0], nbs[0]
      for i in range(1, len(means_x)):
          mx2, my2, vx2, vy2, cxy2, n2 = means_x[i], means_y[i], vars_x[i], vars_y[i], corrs_xy[i], nbs[i]
  
          nb = n1 + n2
          mean_x = (n1 * mx1 + n2 * mx2) / nb
          mean_y = (n1 * my1 + n2 * my2) / nb
          var_x = 1 / (n1 + n2 - 1) * ((n1 - 1) * vx1 + (n2 - 1) * vx2 + ((n1 * n2) / (n1 + n2)) * (mx1 - mx2) ** 2)
          var_y = 1 / (n1 + n2 - 1) * ((n1 - 1) * vy1 + (n2 - 1) * vy2 + ((n1 * n2) / (n1 + n2)) * (my1 - my2) ** 2)
  
          corr1 = n1 * cxy1 + n1 * (mx1 - mean_x) * (my1 - mean_y)
          corr2 = n2 * cxy2 + n2 * (mx2 - mean_x) * (my2 - mean_y)
          corr_xy = (corr1 + corr2) / (n1 + n2)
  
          mx1, my1, vx1, vy1, cxy1, n1 = mean_x, mean_y, var_x, var_y, corr_xy, nb
  
      return var_x, var_y, corr_xy, nb
  
  
  class PearsonCorrcoef(Metric):
      r"""
      Computes `Pearson Correlation Coefficient`_:
  
      .. math::
          P_{corr}(x,y) = \frac{cov(x,y)}{\sigma_x \sigma_y}
  
      Where :math:`y` is a tensor of target values, and :math:`x` is a
      tensor of predictions.
  
      Forward accepts
  
      - ``preds`` (float tensor): ``(N,)``
      - ``target``(float tensor): ``(N,)``
  
      Args:
          compute_on_step:
              Forward only calls ``update()`` and return None if this is set to False. default: True
          dist_sync_on_step:
              Synchronize metric state across processes at each ``forward()``
              before returning the value at the step. default: False
          process_group:
              Specify the process group on which synchronization is called. default: None (which selects the entire world)
  
      Example:
          >>> from torchmetrics import PearsonCorrcoef
          >>> target = torch.tensor([3, -0.5, 2, 7])
          >>> preds = torch.tensor([2.5, 0.0, 2, 8])
          >>> pearson = PearsonCorrcoef()
          >>> pearson(preds, target)
          tensor(0.9849)
  
      """
      is_differentiable = True
      higher_is_better = None  # both -1 and 1 are optimal
      preds: List[Tensor]
      target: List[Tensor]
      mean_x: Tensor
      mean_y: Tensor
      var_x: Tensor
      var_y: Tensor
      corr_xy: Tensor
      n_total: Tensor
  
      def __init__(
          self,
          compute_on_step: bool = True,
          dist_sync_on_step: bool = False,
          process_group: Optional[Any] = None,
      ) -> None:
          super().__init__(
              compute_on_step=compute_on_step,
              dist_sync_on_step=dist_sync_on_step,
              process_group=process_group,
          )
  
          self.add_state("mean_x", default=torch.tensor(0.0), dist_reduce_fx=None)
          self.add_state("mean_y", default=torch.tensor(0.0), dist_reduce_fx=None)
          self.add_state("var_x", default=torch.tensor(0.0), dist_reduce_fx=None)
          self.add_state("var_y", default=torch.tensor(0.0), dist_reduce_fx=None)
          self.add_state("corr_xy", default=torch.tensor(0.0), dist_reduce_fx=None)
          self.add_state("n_total", default=torch.tensor(0.0), dist_reduce_fx=None)
  
      def update(self, preds: Tensor, target: Tensor) -> None:  # type: ignore
          """Update state with predictions and targets.
  
          Args:
              preds: Predictions from model
              target: Ground truth values
          """
          self.mean_x, self.mean_y, self.var_x, self.var_y, self.corr_xy, self.n_total = _pearson_corrcoef_update(
              preds, target, self.mean_x, self.mean_y, self.var_x, self.var_y, self.corr_xy, self.n_total
          )
  
      def compute(self) -> Tensor:
          """Computes pearson correlation coefficient over state."""
          if self.mean_x.numel() > 1:  # multiple devices, need further reduction
              var_x, var_y, corr_xy, n_total = _final_aggregation(
                  self.mean_x, self.mean_y, self.var_x, self.var_y, self.corr_xy, self.n_total
              )
          else:
              var_x = self.var_x
              var_y = self.var_y
              corr_xy = self.corr_xy
              n_total = self.n_total
  
          return _pearson_corrcoef_compute(var_x, var_y, corr_xy, n_total)
  
apis: torch.tensor