class_docstr: Computes a covariance matrix based on the Spectral Mixture Kernel
  between inputs :math:`\mathbf{x_1}` and :math:`\mathbf{x_2}`.
  
  It was proposed in `Gaussian Process Kernels for Pattern Discovery and Extrapolation`_.
  
  .. note::
      Unlike other kernels,
  
          * ard_num_dims **must equal** the number of dimensions of the data.
          * This kernel should not be combined with a :class:`gpytorch.kernels.ScaleKernel`.
  
  :param int num_mixtures: The number of components in the mixture.
  :param int ard_num_dims: Set this to match the dimensionality of the input.
      It should be `d` if x1 is a `... x n x d` matrix. (Default: `1`.)
  :param batch_shape: Set this if the data is batch of input data. It should
      be `b_1 x ... x b_j` if x1 is a `b_1 x ... x b_j x n x d` tensor. (Default: `torch.Size([])`.)
  :type batch_shape: torch.Size, optional
  :param active_dims: Set this if you want to compute the covariance of only
      a few input dimensions. The ints corresponds to the indices of the dimensions. (Default: `None`.)
  :type active_dims: float, optional
  :param eps: The minimum value that the lengthscale can take (prevents divide by zero errors). (Default: `1e-6`.)
  :type eps: float, optional
  
  :param mixture_scales_prior: A prior to set on the mixture_scales parameter
  :type mixture_scales_prior: ~gpytorch.priors.Prior, optional
  :param mixture_scales_constraint: A constraint to set on the mixture_scales parameter
  :type mixture_scales_constraint: ~gpytorch.constraints.Interval, optional
  :param mixture_means_prior: A prior to set on the mixture_means parameter
  :type mixture_means_prior: ~gpytorch.priors.Prior, optional
  :param mixture_means_constraint: A constraint to set on the mixture_means parameter
  :type mixture_means_constraint: ~gpytorch.constraints.Interval, optional
  :param mixture_weights_prior: A prior to set on the mixture_weights parameter
  :type mixture_weights_prior: ~gpytorch.priors.Prior, optional
  :param mixture_weights_constraint: A constraint to set on the mixture_weights parameter
  :type mixture_weights_constraint: ~gpytorch.constraints.Interval, optional
  
  :ivar torch.Tensor mixture_scales: The lengthscale parameter. Given
      `k` mixture components, and `... x n x d` data, this will be of size `... x k x 1 x d`.
  :ivar torch.Tensor mixture_means: The mixture mean parameters (`... x k x 1 x d`).
  :ivar torch.Tensor mixture_weights: The mixture weight parameters (`... x k`).
  
  Example:
  
      >>> # Non-batch
      >>> x = torch.randn(10, 5)
      >>> covar_module = gpytorch.kernels.SpectralMixtureKernel(num_mixtures=4, ard_num_dims=5)
      >>> covar = covar_module(x)  # Output: LazyVariable of size (10 x 10)
      >>>
      >>> # Batch
      >>> batch_x = torch.randn(2, 10, 5)
      >>> covar_module = gpytorch.kernels.SpectralMixtureKernel(num_mixtures=4, batch_size=2, ard_num_dims=5)
      >>> covar = covar_module(x)  # Output: LazyVariable of size (10 x 10)
  
  .. _Gaussian Process Kernels for Pattern Discovery and Extrapolation:
      https://arxiv.org/pdf/1302.4245.pdf
docstr: Initialize mixture components based on the empirical spectrum of the data.
  This will often be better than the standard initialize_from_data method, but it assumes
  that your inputs are evenly spaced.
  
  :param torch.Tensor train_x: Training inputs
  :param torch.Tensor train_y: Training outputs
docstr_tok: ["Initialize","mixture","components","based","on","the","empirical","spectrum","of","the","data",".","This","will","often","be","better","than","the","standard","initialize_from_data","method",",","but","it","assumes","that","your","inputs","are","evenly","spaced",".",":","param","torch.Tensor","train_x",":","Training","inputs",":","param","torch.Tensor","train_y",":","Training","outputs"]
code: def initialize_from_data_empspect(
      self, train_x: torch.Tensor, train_y: torch.Tensor
  ):
      """
      Initialize mixture components based on the empirical spectrum of the data.
      This will often be better than the standard initialize_from_data method, but it assumes
      that your inputs are evenly spaced.
  
      :param torch.Tensor train_x: Training inputs
      :param torch.Tensor train_y: Training outputs
      """
  
      import numpy as np
      from scipy.fftpack import fft
      from scipy.integrate import cumtrapz
  
      with torch.no_grad():
          if not torch.is_tensor(train_x) or not torch.is_tensor(
              train_y
          ):
              raise RuntimeError(
                  "train_x and train_y should be tensors"
              )
          if train_x.ndimension() == 1:
              train_x = train_x.unsqueeze(-1)
          if self.active_dims is not None:
              train_x = train_x[..., self.active_dims]
  
          # Flatten batch dimensions
          train_x = train_x.view(-1, train_x.size(-1))
          train_y = train_y.view(-1)
  
          N = train_x.size(-2)
          emp_spect = (
              np.abs(fft(train_y.cpu().detach().numpy())) ** 2 / N
          )
          M = math.floor(N / 2)
  
          freq1 = np.arange(M + 1)
          freq2 = np.arange(-M + 1, 0)
          freq = np.hstack((freq1, freq2)) / N
          freq = freq[: M + 1]
          emp_spect = emp_spect[: M + 1]
  
          total_area = np.trapz(emp_spect, freq)
          spec_cdf = np.hstack((np.zeros(1), cumtrapz(emp_spect, freq)))
          spec_cdf = spec_cdf / total_area
  
          a = np.random.rand(1000, self.ard_num_dims)
          p, q = np.histogram(a, spec_cdf)
          bins = np.digitize(a, q)
          slopes = (spec_cdf[bins] - spec_cdf[bins - 1]) / (
              freq[bins] - freq[bins - 1]
          )
          intercepts = spec_cdf[bins - 1] - slopes * freq[bins - 1]
          inv_spec = (a - intercepts) / slopes
  
          from sklearn.mixture import GaussianMixture
  
          GMM = GaussianMixture(
              n_components=self.num_mixtures, covariance_type="diag"
          ).fit(inv_spec)
          means = GMM.means_
          varz = GMM.covariances_
          weights = GMM.weights_
  
          dtype = self.raw_mixture_means.dtype
          device = self.raw_mixture_means.device
          self.mixture_means = torch.tensor(
              means, dtype=dtype, device=device
          ).unsqueeze(-2)
          self.mixture_scales = torch.tensor(
              varz, dtype=dtype, device=device
          ).unsqueeze(-2)
          self.mixture_weights = torch.tensor(
              weights, dtype=dtype, device=device
          )
  
imports: import logging
  import math
  from typing import Optional, Tuple, Union
  import torch
  from ..constraints import Interval, Positive
  from ..priors import Prior
  from .kernel import Kernel
label_desc: Use image node_id 11 for calling the SpectralMixtureKernel obj's underlying member method code with example usage: obj.initialize_from_data_empspect(train_x, train_y) without return types