@experimental(as_of="0.4.0")
def ca(X, scaling=1):
    r"""Compute correspondence analysis, a multivariate statistical
    technique for ordination.

    In general, rows in the data table will correspond to samples and
    columns to features, but the method is symmetric. In order to
    measure the correspondence between rows and columns, the
    :math:`\chi^2` distance is used, and those distances are preserved
    in the transformed space. The :math:`\chi^2` distance doesn't take
    double zeros into account, and so it is expected to produce better
    ordination that PCA when the data has lots of zero values.

    It is related to Principal Component Analysis (PCA) but it should
    be preferred in the case of steep or long gradients, that is, when
    there are many zeros in the input data matrix.

    Parameters
    ----------
    X : pd.DataFrame
        Samples by features table (n, m). It can be applied to different kinds
        of data tables but data must be non-negative and dimensionally
        homogeneous (quantitative or binary). The rows correspond to the
        samples and the columns correspond to the features.
    scaling : {1, 2}
        For a more detailed explanation of the interpretation, check Legendre &
        Legendre 1998, section 9.4.3. The notes that follow are quick
        recommendations.

        Scaling type 1 maintains :math:`\chi^2` distances between rows
        (samples): in the transformed space, the euclidean distances between
        rows are equal to the :math:`\chi^2` distances between rows in the
        original space. It should be used when studying the ordination of
        samples. Rows (samples) that are near a column (features) have high
        contributions from it.

        Scaling type 2 preserves :math:`\chi^2` distances between columns
        (features), so euclidean distance between columns after transformation
        is equal to :math:`\chi^2` distance between columns in the original
        space. It is best used when we are interested in the ordination of
        features. A column (features) that is next to a row (sample) means that
        it is more abundant there.

        Other types of scalings are currently not implemented, as they're less
        used by ecologists (Legendre & Legendre 1998, p. 456).

        In general, features appearing far from the center of the biplot and
        far from its edges will probably exhibit better relationships than
        features either in the center (may be multimodal features, not related
        to the shown ordination axes...) or the edges (sparse features...).

    Returns
    -------
    OrdinationResults
        Object that stores the computed eigenvalues, the transformed sample
        coordinates and the transformed features coordinates.

    Raises
    ------
    NotImplementedError
        If the scaling value is not either `1` or `2`.
    ValueError
        If any of the input matrix elements are negative.

    Notes
    -----
    The algorithm is based on [1]_, \S 9.4.1., and is expected to give the same
    results as ``cca(X)`` in R's package vegan.

    See Also
    --------
    cca

    References
    ----------
    .. [1] Legendre P. and Legendre L. 1998. Numerical Ecology. Elsevier,
       Amsterdam.

    """

    if scaling not in {1, 2}:
        raise NotImplementedError(
            "Scaling {0} not implemented.".format(scaling))

    short_method_name = 'CA'
    long_method_name = 'Correspondance Analysis'

    # we deconstruct the dataframe to avoid duplicating the data and be able
    # to perform operations on the matrix
    row_ids = X.index
    column_ids = X.columns
    X = np.asarray(X.values, dtype=np.float64)

    # Correspondance Analysis
    r, c = X.shape

    if X.min() < 0:
        raise ValueError("Input matrix elements must be non-negative.")

    # Step 1 (similar to Pearson chi-square statistic)
    grand_total = X.sum()
    Q = X / grand_total

    column_marginals = Q.sum(axis=0)
    row_marginals = Q.sum(axis=1)

    # Formula 9.32 in Lagrange & Lagrange (1998). Notice that it's
    # an scaled version of the contribution of each cell towards
    # Pearson chi-square statistic.
    expected = np.outer(row_marginals, column_marginals)
    Q_bar = (Q - expected) / np.sqrt(expected)  # Eq. 9.32

    # Step 2 (Singular Value Decomposition)
    U_hat, W, Ut = svd(Q_bar, full_matrices=False)
    # Due to the centering, there are at most min(r, c) - 1 non-zero
    # eigenvalues (which are all positive)
    rank = svd_rank(Q_bar.shape, W)
    assert rank <= min(r, c) - 1
    U_hat = U_hat[:, :rank]
    W = W[:rank]
    U = Ut[:rank].T

    # Both scalings are a bit intertwined, so we'll compute both and
    # then choose
    V = column_marginals[:, None]**-0.5 * U
    V_hat = row_marginals[:, None]**-0.5 * U_hat
    F = V_hat * W
    # According to Formula 9.43, this should hold
    # assert np.allclose(F, (row_marginals**-1)[:, None] * Q.dot(V))
    # but it doesn't (notice that W**2==Lambda):
    # (9.43a) F = V_hat W = D(p_i+)^{-1/2} U_hat W
    #           = D(p_i+)^{-1/2} Q_bar U W^{-1} W  (substituting 9.38)
    #           = D(p_i+)^{-1/2} Q_bar U
    # (9.43b) F = D(p_i+)^{-1} Q V
    #           = D(p_i+)^{-1} Q D(p_+j)^{-1/2} U  (substituting 9.41)
    #           = D(p_i+)^{-1/2} D(p_i+)^{-1/2} Q D(p_+j)^{-1/2} U
    #           = D(p_i+)^{-1/2} Q_tilde U         (using 9.40)
    # It holds if we replace Q in 9.43b with Q after centering, ie
    # assert np.allclose(
    #    F,
    #    (row_marginals**-1)[:, None] * (Q - expected).dot(V))
    # Comparing results with vegan and the examples in the book, 9.43a
    # is the right one. The same issue happens in 9.44, where also
    # 9.44a is the one that matches vegan's output.
    # (9.44a) F_hat = V W = D(p_+j)^{-1/2} U W
    #               = D(p_+j)^{-1/2} Q_bar' U_hat W^{-1} W (using 9.39)
    #               = D(p_+j)^{-1/2} Q_bar' U_hat
    # (9.44b) F_hat = D(p_+j)^{-1} Q' V_hat
    #               = D(p_+j)^{-1/2} Q_tilde' U_hat (using 9.40 and 9.42)
    F_hat = V * W

    # Eigenvalues
    eigvals = W**2

    # features scores
    features_scores = [V, F_hat][scaling - 1]
    # sample scores (weighted averages of features scores)
    sample_scores = [F, V_hat][scaling - 1]

    # build the OrdinationResults object
    sample_columns = ['%s%d' % (short_method_name, i+1)
                      for i in range(sample_scores.shape[1])]
    feature_columns = ['%s%d' % (short_method_name, i+1)
                       for i in range(features_scores.shape[1])]

    eigvals = pd.Series(eigvals, ['%s%d' % (short_method_name, i+1) for i in
                                  range(eigvals.shape[0])])
    samples = pd.DataFrame(sample_scores, row_ids, sample_columns)
    features = pd.DataFrame(features_scores, column_ids, feature_columns)

    return OrdinationResults(short_method_name, long_method_name, eigvals,
                             samples=samples, features=features)
dataset/ETHPy150Open biocore/scikit-bio/skbio/stats/ordination/_correspondence_analysis.py/ca/original