.. _examples-miml-classification-lda:

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Linear discriminant analysis
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Linear discriminant analysis (LDA) is based on the Bayes decision theory and assumes that the 
conditional probability density functions are normally distributed. LDA also makes the simplifying 
homoscedastic assumption (i.e. that the class covariances are identical) and that the covariances 
have full rank. With these assumptions, the discriminant function of an input being in a class is 
purely a function of this linear combination of independent variables.

::

    from miml import datasets
    from miml.classification import LinearDiscriminantAnalysis

    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)

    model = LinearDiscriminantAnalysis()
    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("Linear Discriminant Analysis example")
    
.. image:: ../../../_static/miml/lda_1.png