# `Scholar.Covariance.LedoitWolf`
[🔗](https://github.com/elixir-nx/scholar/blob/main/lib/scholar/covariance/ledoit_wolf.ex#L1)

Ledoit-Wolf is a particular form of shrinkage covariance estimator,
where the shrinkage coefficient is computed using O. Ledoit and M. Wolf’s formula.

Ledoit and M. Wolf's formula as
described in "A Well-Conditioned Estimator for Large-Dimensional
Covariance Matrices", Ledoit and Wolf, Journal of Multivariate
Analysis, Volume 88, Issue 2, February 2004, pages 365-411.

# `fit`
[🔗](https://github.com/elixir-nx/scholar/blob/main/lib/scholar/covariance/ledoit_wolf.ex#L109)

Estimate the shrunk Ledoit-Wolf covariance matrix.

## Options

* `:assume_centered?` (`t:boolean/0`) - If `true`, data will not be centered before computation.
    Useful when working with data whose mean is almost, but not exactly
    zero.
    If `false`, data will be centered before computation. The default value is `false`.

## Return Values

  The function returns a struct with the following parameters:

  * `:covariance` - Tensor of shape `{num_features, num_features}`. Estimated covariance matrix.

  * `:shrinkage` - Coefficient in the convex combination used for the computation of the shrunken estimate. Range is `[0, 1]`.

  * `:location` - Tensor of shape `{num_features,}`.
    Estimated location, i.e. the estimated mean.

## Examples

    iex> key = Nx.Random.key(0)
    iex> {x, _new_key} = Nx.Random.multivariate_normal(key, Nx.tensor([0.0, 0.0]), Nx.tensor([[0.4, 0.2], [0.2, 0.8]]), shape: {50}, type: :f32)
    iex> model = Scholar.Covariance.LedoitWolf.fit(x)
    iex> model.covariance
    #Nx.Tensor<
      f32[2][2]
      [
        [0.3557686507701874, 0.17340737581253052],
        [0.17340737581253052, 1.0300586223602295]
      ]
    >
    iex> model.shrinkage
    #Nx.Tensor<
      f32
      0.15034137666225433
    >
    iex> model.location
    #Nx.Tensor<
      f32[2]
      [0.17184630036354065, 0.3276958167552948]
    >

    iex> key = Nx.Random.key(0)
    iex> {x, _new_key} = Nx.Random.multivariate_normal(key, Nx.tensor([0.0, 0.0, 0.0]), Nx.tensor([[3.0, 2.0, 1.0], [1.0, 2.0, 3.0], [1.3, 1.0, 2.2]]), shape: {10}, type: :f32)
    iex> model = Scholar.Covariance.LedoitWolf.fit(x)
    iex> model.covariance
    #Nx.Tensor<
      f32[3][3]
      [
        [2.5945029258728027, 1.5078359842300415, 1.1623677015304565],
        [1.5078359842300415, 2.106797456741333, 1.1812156438827515],
        [1.1623677015304565, 1.1812156438827515, 1.4606266021728516]
      ]
    >
    iex> model.shrinkage
    #Nx.Tensor<
      f32
      0.1908363401889801
    >
    iex> model.location
    #Nx.Tensor<
      f32[3]
      [1.1228725910186768, 0.5419300198554993, 0.8678852319717407]
    >

    iex> key = Nx.Random.key(0)
    iex> {x, _new_key} = Nx.Random.multivariate_normal(key, Nx.tensor([0.0, 0.0, 0.0]), Nx.tensor([[3.0, 2.0, 1.0], [1.0, 2.0, 3.0], [1.3, 1.0, 2.2]]), shape: {10}, type: :f32)
    iex> cov = Scholar.Covariance.LedoitWolf.fit(x, assume_centered?: true)
    iex> cov.covariance
    #Nx.Tensor<
      f32[3][3]
      [
        [3.8574986457824707, 2.2048025131225586, 2.1504499912261963],
        [2.2048025131225586, 2.4572863578796387, 1.7215262651443481],
        [2.1504499912261963, 1.7215262651443481, 2.154898166656494]
      ]
    >

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*Consult [api-reference.md](api-reference.md) for complete listing*