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# scipy.special.xlogy

scipy.special.xlogy(x, y, out=None) = <ufunc 'xlogy'>

  Compute `x*log(y)` so that the result is 0 if `x = 0`.

  Parameters:
    xarray_like

      Multiplier

    yarray_like

      Argument

    outndarray, optional

      Optional output array for the function results

  Returns:
    zscalar or ndarray

      Computed x*log(y)

  Notes

  The log function used in the computation is the natural log.

  Added in version 0.13.0.

  `xlogy` has experimental support for Python Array API Standard compatible backends in addition to NumPy. Please consider testing these features by setting an environment variable `SCIPY_ARRAY_API=1` and providing CuPy, PyTorch, JAX, or Dask arrays as array arguments. The following combinations of backend and device (or other capability) are supported.

  Library | CPU | GPU
  --- | --- | ---
  NumPy | ✅ | n/a
  CuPy | n/a | ✅
  PyTorch | ✅ | ✅
  JAX | ✅ | ✅
  Dask | ✅ | n/a

  See Support for the array API standard for more information.

  Examples

  We can use this function to calculate the binary logistic loss also known as the binary cross entropy. This loss function is used for binary classification problems and is defined as:

  \[\begin{split}L = 1/n * \\sum_{i=0}^n -(y_i*log(y\\_pred_i) + (1-y_i)*log(1-y\\_pred_i))\end{split}\]

  We can define the parameters x and y as y and y_pred respectively. y is the array of the actual labels which over here can be either 0 or 1. y_pred is the array of the predicted probabilities with respect to the positive class (1).

      >>> import numpy as np
      >>> from scipy.special import xlogy
      >>> y = np.array([0, 1, 0, 1, 1, 0])
      >>> y_pred = np.array([0.3, 0.8, 0.4, 0.7, 0.9, 0.2])
      >>> n = len(y)
      >>> loss = -(xlogy(y, y_pred) + xlogy(1 - y, 1 - y_pred)).sum()
      >>> loss /= n
      >>> loss
      0.29597052165495025

  A lower loss is usually better as it indicates that the predictions are similar to the actual labels. In this example since our predicted probabilities are close to the actual labels, we get an overall loss that is reasonably low and appropriate.