# Non Linear Methods for Classification¶

## Parzen-based classifier¶

class mlpy.Parzen(kernel=None)

Parzen based classifier (binary).

Initialization.

Parameters : kernel : None or mlpy.Kernel object. if kernel is None, K and Kt in .learn() and in .pred() methods must be precomputed kernel matricies, else K and Kt must be training (resp. test) data in input space.
alpha()

Return alpha.

b()

Return b.

labels()

Outputs the name of labels.

learn(K, y)

Compute alpha and b.

Parameters:
K: 2d array_like object
precomputed training kernel matrix (if kernel=None); training data in input space (if kernel is a Kernel object)
y : 1d array_like object (N)
target values
pred(Kt)

Compute the predicted class.

Parameters : Kt : 1d or 2d array_like object precomputed test kernel matrix. (if kernel=None); test data in input space (if kernel is a Kernel object). p : integer or 1d numpy array predicted class

Example:

```>>> import numpy as np
>>> import matplotlib.pyplot as plt
>>> import mlpy
>>> np.random.seed(0)
>>> mean1, cov1, n1 = [1, 4.5], [[1,1],[1,2]], 20  # 20 samples of class 1
>>> x1 = np.random.multivariate_normal(mean1, cov1, n1)
>>> y1 = np.ones(n1, dtype=np.int)
>>> mean2, cov2, n2 = [2.5, 2.5], [[1,1],[1,2]], 30 # 30 samples of class 2
>>> x2 = np.random.multivariate_normal(mean2, cov2, n2)
>>> y2 = 2 * np.ones(n2, dtype=np.int)
>>> x = np.concatenate((x1, x2), axis=0) # concatenate the samples
>>> y = np.concatenate((y1, y2))
>>> K = mlpy.kernel_gaussian(x, x, sigma=2) # kernel matrix
>>> parzen = mlpy.Parzen()
>>> parzen.learn(K, y)
>>> xmin, xmax = x[:,0].min()-1, x[:,0].max()+1
>>> ymin, ymax = x[:,1].min()-1, x[:,1].max()+1
>>> xx, yy = np.meshgrid(np.arange(xmin, xmax, 0.02), np.arange(ymin, ymax, 0.02))
>>> xt = np.c_[xx.ravel(), yy.ravel()] # test points
>>> Kt = mlpy.kernel_gaussian(xt, x, sigma=2) # test kernel matrix
>>> yt = parzen.pred(Kt).reshape(xx.shape)
>>> fig = plt.figure(1)
>>> cmap = plt.set_cmap(plt.cm.Paired)
>>> plot1 = plt.pcolormesh(xx, yy, yt)
>>> plot2 = plt.scatter(x[:,0], x[:,1], c=y)
>>> plt.show()
```

## Kernel Fisher Discriminant Classifier¶

class mlpy.KFDAC(lmb=0.001, kernel=None)

Kernel Fisher Discriminant Analysis Classifier (binary classifier).

The bias term (b) is computed as in [Gavin03].

 [Gavin03] Gavin C. et al. Efficient Cross-Validation of Kernel Fisher Discriminant Classifers. ESANN‘2003 proceedings - European Symposium on Artificial Neural Networks, 2003.

Initialization.

Parameters : lmb : float (>= 0.0) regularization parameter kernel : None or mlpy.Kernel object. if kernel is None, K and Kt in .learn() and in .transform() methods must be precomputed kernel matricies, else K and Kt must be training (resp. test) data in input space.
alpha()

Return alpha.

b()

Return b.

labels()

Outputs the name of labels.

learn(K, y)

Learning method.

Parameters : K: 2d array_like object precomputed training kernel matrix (if kernel=None); training data in input space (if kernel is a Kernel object) y : 1d array_like object integer (N) class labels (only two classes)
pred(Kt)

Compute the predicted response.

Parameters : Kt : 1d or 2d array_like object precomputed test kernel matrix. (if kernel=None); test data in input space (if kernel is a Kernel object). p : integer or 1d numpy array the predicted class(es)

## k-Nearest-Neighbor¶

class mlpy.KNN(k)

k-Nearest Neighbor (euclidean distance)

Parameters : k : int number of nearest neighbors
KNN.learn(x, y)

Learn method.

Parameters : x : 2d array_like object (N,P) training data y : 1d array_like integer class labels
KNN.pred(t)

Predict KNN model on a test point(s).

Parameters : t : 1d or 2d array_like object ([M,] P) test point(s) p : int or 1d numpy array the predicted value(s). Retuns the smallest label minus one (KNN.labels()[0]-1) when the classification is not unique.
KNN.nclasses()

Returns the number of classes.

KNN.labels()

Outputs the name of labels.

Example:

```>>> import numpy as np
>>> import matplotlib.pyplot as plt
>>> import mlpy
>>> np.random.seed(0)
>>> mean1, cov1, n1 = [1, 5], [[1,1],[1,2]], 200  # 200 samples of class 1
>>> x1 = np.random.multivariate_normal(mean1, cov1, n1)
>>> y1 = np.ones(n1, dtype=np.int)
>>> mean2, cov2, n2 = [2.5, 2.5], [[1,0],[0,1]], 300 # 300 samples of class 2
>>> x2 = np.random.multivariate_normal(mean2, cov2, n2)
>>> y2 = 2 * np.ones(n2, dtype=np.int)
>>> mean3, cov3, n3 = [5, 8], [[0.5,0],[0,0.5]], 200 # 200 samples of class 3
>>> x3 = np.random.multivariate_normal(mean3, cov3, n3)
>>> y3 = 3 * np.ones(n3, dtype=np.int)
>>> x = np.concatenate((x1, x2, x3), axis=0) # concatenate the samples
>>> y = np.concatenate((y1, y2, y3))
>>> knn = mlpy.KNN(k=3)
>>> knn.learn(x, y)
>>> xmin, xmax = x[:,0].min()-1, x[:,0].max()+1
>>> ymin, ymax = x[:,1].min()-1, x[:,1].max()+1
>>> xx, yy = np.meshgrid(np.arange(xmin, xmax, 0.1), np.arange(ymin, ymax, 0.1))
>>> xnew = np.c_[xx.ravel(), yy.ravel()]
>>> ynew = knn.pred(xnew).reshape(xx.shape)
>>> ynew[ynew == 0] = 1 # set the samples with no unique classification to 1
>>> fig = plt.figure(1)
>>> cmap = plt.set_cmap(plt.cm.Paired)
>>> plot1 = plt.pcolormesh(xx, yy, ynew)
>>> plot2 = plt.scatter(x[:,0], x[:,1], c=y)
>>> plt.show()
```

## Classification Tree¶

class mlpy.ClassTree(stumps=0, minsize=1)

Classification Tree (gini index)

Parameters : stumps : bool True: compute single split or False: standard tree minsize : int (>=0) minimum number of cases required to split a leaf
ClassTree.learn(x, y)

Learn method.

Parameters : x : 2d array_like object (N x P) training data y : 1d array_like integer class labels
ClassTree.pred(t)

Predict Tree model on a test point(s).

Parameters : t : 1d or 2d array_like object ([M,] P) test point(s) p : int or 1d numpy array the predicted value(s). Retuns the smallest label minus one (ClassTree.labels()[0]-1) when the classification is not unique.
ClassTree.nclasses()

Returns the number of classes.

ClassTree.labels()

Outputs the name of labels.

Example:

```>>> import numpy as np
>>> import matplotlib.pyplot as plt
>>> import mlpy
>>> np.random.seed(0)
>>> mean1, cov1, n1 = [1, 5], [[1,1],[1,2]], 200  # 200 samples of class 1
>>> x1 = np.random.multivariate_normal(mean1, cov1, n1)
>>> y1 = np.ones(n1, dtype=np.int)
>>> mean2, cov2, n2 = [2.5, 2.5], [[1,0],[0,1]], 300 # 300 samples of class 2
>>> x2 = np.random.multivariate_normal(mean2, cov2, n2)
>>> y2 = 2 * np.ones(n2, dtype=np.int)
>>> mean3, cov3, n3 = [6, 8], [[0.5,0],[0,0.5]], 200 # 200 samples of class 3
>>> x3 = np.random.multivariate_normal(mean3, cov3, n3)
>>> y3 = 3 * np.ones(n3, dtype=np.int)
>>> x = np.concatenate((x1, x2, x3), axis=0) # concatenate the samples
>>> y = np.concatenate((y1, y2, y3))
>>> tree = mlpy.ClassTree(minsize=10)
>>> tree.learn(x, y)
>>> xmin, xmax = x[:,0].min()-1, x[:,0].max()+1
>>> ymin, ymax = x[:,1].min()-1, x[:,1].max()+1
>>> xx, yy = np.meshgrid(np.arange(xmin, xmax, 0.1), np.arange(ymin, ymax, 0.1))
>>> xnew = np.c_[xx.ravel(), yy.ravel()]
>>> ynew = tree.pred(xnew).reshape(xx.shape)
>>> ynew[ynew == 0] = 1 # set the samples with no unique classification to 1
>>> fig = plt.figure(1)
>>> cmap = plt.set_cmap(plt.cm.Paired)
>>> plot1 = plt.pcolormesh(xx, yy, ynew)
>>> plot2 = plt.scatter(x[:,0], x[:,1], c=y)
>>> plt.show()
```

## Maximum Likelihood Classifier¶

class mlpy.MaximumLikelihoodC

Maximum Likelihood Classifier

MaximumLikelihoodC.learn(x, y)

Learn method.

Parameters : x : 2d array_like object (N,P) training data y : 1d array_like integer class labels
MaximumLikelihoodC.pred(t)

Predict Maximum Likelihood model on a test point(s).

Parameters : t : 1d or 2d array_like object ([M,] P) test point(s) p : int or 1d numpy array the predicted value(s). Retuns the smallest label minus one (MaximumLikelihoodC.labels()[0]-1) when the classification is not unique.
MaximumLikelihoodC.nclasses()

Returns the number of classes.

MaximumLikelihoodC.labels()

Outputs the name of labels.

Example:

```>>> import numpy as np
>>> import matplotlib.pyplot as plt
>>> import mlpy
>>> np.random.seed(0)
>>> mean1, cov1, n1 = [1, 5], [[1,1],[1,2]], 200  # 200 samples of class 1
>>> x1 = np.random.multivariate_normal(mean1, cov1, n1)
>>> y1 = np.ones(n1, dtype=np.int)
>>> mean2, cov2, n2 = [2.5, 2.5], [[1,0],[0,1]], 300 # 300 samples of class 2
>>> x2 = np.random.multivariate_normal(mean2, cov2, n2)
>>> y2 = 2 * np.ones(n2, dtype=np.int)
>>> mean3, cov3, n3 = [6, 8], [[0.5,0],[0,0.5]], 200 # 200 samples of class 3
>>> x3 = np.random.multivariate_normal(mean3, cov3, n3)
>>> y3 = 3 * np.ones(n3, dtype=np.int)
>>> x = np.concatenate((x1, x2, x3), axis=0) # concatenate the samples
>>> y = np.concatenate((y1, y2, y3))
>>> ml = mlpy.MaximumLikelihoodC()
>>> ml.learn(x, y)
>>> xmin, xmax = x[:,0].min()-1, x[:,0].max()+1
>>> ymin, ymax = x[:,1].min()-1, x[:,1].max()+1
>>> xx, yy = np.meshgrid(np.arange(xmin, xmax, 0.1), np.arange(ymin, ymax, 0.1))
>>> xnew = np.c_[xx.ravel(), yy.ravel()]
>>> ynew = ml.pred(xnew).reshape(xx.shape)
>>> ynew[ynew == 0] = 1 # set the samples with no unique classification to 1
>>> fig = plt.figure(1)
>>> cmap = plt.set_cmap(plt.cm.Paired)
>>> plot1 = plt.pcolormesh(xx, yy, ynew)
>>> plot2 = plt.scatter(x[:,0], x[:,1], c=y)
>>> plt.show()
```