Computes correlation matrix for sample inputs x.

The value on the position $Corr_{ij}$ in the $Corr$ matrix is calculated using the formula: $$ Corr(X\_i, X\_j) = \frac{Cov(X\_i, X\_j)}{\sqrt{Cov(X\_i, X\_i)Cov(X\_j, X\_j)}} $$ Where:

• $X_i$ is a $i$th row of input

• $Cov(X\_i, X\_j)$ is covariance between features $X_i$ and $X_j$

Time complexity of correlation estimation is $O(N * K^2)$ where $N$ is the number of samples and $K$ is the number of features.

Example

iex> Scholar.Stats.correlation_matrix(Nx.tensor([[3, 6, 5], [26, 75, 3], [23, 4, 1]]))
#Nx.Tensor<
  f32[3][3]
  [
    [1.0, 0.580316960811615, -0.7997867465019226],
    [0.580316960811615, 1.0, 0.024736011400818825],
    [-0.7997867465019226, 0.024736011400818825, 1.0]
  ]
>

iex> Scholar.Stats.correlation_matrix(Nx.tensor([[3, 6], [2, 3], [7, 9], [5, 3]]))
#Nx.Tensor<
  f32[2][2]
  [
    [1.0, 0.6673083305358887],
    [0.6673083305358887, 1.0]
  ]
>

iex> x = Nx.tensor([[3, 6, 5], [26, 75, 3], [23, 4, 1]])
iex> means = Nx.mean(x, axes: [-2])
iex> Scholar.Stats.correlation_matrix(x, means)
#Nx.Tensor<
  f32[3][3]
  [
    [1.0, 0.580316960811615, -0.7997867465019226],
    [0.580316960811615, 1.0, 0.024736011400818825],
    [-0.7997867465019226, 0.024736011400818825, 1.0]
  ]
>

Computes the kurtosis (Fisher or Pearson) of a dataset.

Options

• :axes - Axes to calculate the operation. If set to nil then the operation is performed on the whole tensor. The default value is [0].

• :keep_axes (boolean/0) - If set to true, the axes which are reduced are left. The default value is false.

• :bias (boolean/0) - If false, then the calculations are corrected for statistical bias. The default value is true.

• :variant - If :fisher then Fisher's definition is used, if :pearson then Pearson's definition is used. The default value is :fisher.

Examples

iex> x = Nx.tensor([[3, 5, 3], [2, 6, 1], [9, 3, 2], [1, 6, 8]])
iex> Scholar.Stats.kurtosis(x)
#Nx.Tensor<
  f32[3]
  [-0.7980852127075195, -1.0, -0.8394768238067627]
>

Calculates the nth moment about the mean for a sample.

Options

• :axes - Axes to calculate the operation. If set to nil then the operation is performed on the whole tensor. The default value is [0].

• :keep_axes (boolean/0) - If set to true, the axes which are reduced are left. The default value is false.

Examples

iex> x = Nx.tensor([[3, 5, 3], [2, 6, 1], [9, 3, 2], [1, 6, 8]])
iex> Scholar.Stats.moment(x, 2)
#Nx.Tensor<
  f32[3]
  [9.6875, 1.5, 7.25]
>

Computes the sample skewness of a data set.

Options

• :axes - Axes to calculate the operation. If set to nil then the operation is performed on the whole tensor. The default value is [0].

• :keep_axes (boolean/0) - If set to true, the axes which are reduced are left. The default value is false.

• :bias (boolean/0) - If false, then the calculations are corrected for statistical bias. The default value is true.

Examples

iex> x = Nx.tensor([[3, 5, 3], [2, 6, 1], [9, 3, 2], [1, 6, 8]])
iex> Scholar.Stats.skew(x)
#Nx.Tensor<
  f32[3]
  [0.9794093370437622, -0.8164965510368347, 0.9220733642578125]
>