.. _examples-miml-classification-rda:

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Regularized discriminant analysis
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RDA is a compromise between LDA and QDA, which allows one to shrink the separate covariances 
of QDA toward a common variance as in LDA. This method is very similar in flavor to ridge 
regression. The regularized covariance matrices of each class is Σk(α) = α Σk + (1 - α) Σ. 
The quadratic discriminant function is defined using the shrunken covariance matrices Σk(α). 
The parameter α in [0, 1] controls the complexity of the model. When α is one, RDA becomes QDA. 
While α is zero, RDA is equivalent to LDA. Therefore, the regularization factor α allows a 
continuum of models between LDA and QDA.

::

    from miml import datasets
    from miml.classification import RegularizedDiscriminantAnalysis

    fn = os.path.join(datasets.get_data_home(), 'classification', 'toy', 
        'toy-test.txt')
    df = DataFrame.read_table(fn, header=None, names=['x1','x2'], 
        format='%2f', index_col=0)

    X = df.values
    y = array(df.index.data)

    alpha = 0.5
    model = RegularizedDiscriminantAnalysis(alpha)
    model.fit(X, y)

    # Plot the decision boundary. For that, we will assign a color to each
    # point in the mesh [x_min, x_max]x[y_min, y_max].
    x_min, x_max = X[:, 0].min() - 1, X[:, 0].max() + 1
    y_min, y_max = X[:, 1].min() - 1, X[:, 1].max() + 1
    n = 50  # size in the mesh
    xx, yy = np.meshgrid(np.linspace(x_min, x_max, n),
                         np.linspace(y_min, y_max, n))
    data = np.vstack((xx.flatten(), yy.flatten())).T
    Z = model.predict(data)

    # Put the result into a color plot
    Z = Z.reshape(xx.shape)

    #Plot
    # Create color maps
    cmap_light = ['#FFAAAA', '#AAAAFF']
    cmap_bold = ['#FF0000', '#0000FF']
    imshow(xx[0,:], yy[:,0], Z, colors=cmap_light)
    # Plot also the training points
    ls = plt.scatter(X[:, 0], X[:, 1], c=y,
                edgecolor=None, s=3, levels=[0,1], colors=cmap_bold)
    plt.contour(xx[0,:], yy[:,0], Z, [0.5], color='k', smooth=False)
    plt.xlim(xx.min(), xx.max())
    plt.ylim(yy.min(), yy.max())
    plt.title("Regularized Discriminant Analysis example (alpha=%.1f)" % alpha)
    
.. image:: ../../../_static/miml/rda_1.png