def set_activation(self, method, sig=None, d_sig=None, sig_0=None, d_sig_0=None):
"""
This method sets the activation functions.
Parameters
----------
method : str
'logistic' , 'htanget', or 'custom'
sig, d_sig, sig_0, and d_sig_0 : function objects
Optional arguments intended for use with the
'custom' method option. They should be functions.
sig_0 and d_sig_0 are the output layer activation functions.
"""
method = method.lower()
if method == 'logistic':
self.sig = lambda z: twod(1 / (1 + np.exp(-z)))
self.d_sig = lambda z: twod(np.multiply(self.sig(z), (1 - self.sig(z))))
self.sig_0 = self.sig
self.d_sig_0 = self.d_sig
elif method == 'htangent':
self.sig = lambda z: twod(np.tanh(z))
self.d_sig = lambda z: twod(1 - np.power(np.tanh(z), 2))
self.sig_0 = self.sig
self.d_sig_0 = self.d_sig
elif method == 'custom':
self.sig = sig
self.d_sig = d_sig
self.sig_0 = sig_0
self.d_sig_0 = d_sig_0
else:
raise ValueError('NNetClassify.set_activation: ' + str(method) + ' is not a valid option for method')
self.activation = method
def train(self, base, n, X, Y, *args, **kargs): """ Learn n new instances of base class. Refer to constructor docstring for descriptions of arguments. """ self.base = base N,D = twod(X).shape n_init = self.n_use step = 1 if n_init == 0: # skip training, use constant predictor; set to local var self.const = np.mean(Y) # (specialized to quadratic loss) y_hat = np.zeros(N) + self.const # figure out current prediction value for i in range(n_init): # if we already have learners... yi = self[i].predict(X).flatten() # ...figure out prediction for the y_hat += (self.alpha[i] * yi) # training data for i in range(n_init, n_init + n): Ri = (Y - y_hat) + 1e-64 # compute residuals (specialized to quadratic loss) self.ensemble.append(base(X, Ri, *args, **kargs)) # fit a model to the gradient residual yi = self[-1].predict(X) # minimize loss over alpha (specialized to quadratic loss) min_loss = step * np.divide((Ri.dot(yi)), (twod(yi).T.dot(yi))) self.alpha.append(min_loss.flatten()[0]) y_hat = (twod(y_hat).T + self.alpha[-1] * yi).flatten() self.n_use += 1
def fsvd(X, K, T=None):
"""
Reduce the dimensionality of X to K features using singular value
decomposition.
Parameters
----------
X : numpy array
M x N array of data.
K : int
Number of desired output features.
T : numpy array (optional)
Transform matrix. Including T will use T instead of computing the
SVD.
Returns
-------
Xsvd : numpy array
N x K matrix of data.
T : numpy array (optional)
Transform matrix
"""
n, m = twod(X).shape
if type(T) is type(None):
U, S, V = np.linalg.svd(X, full_matrices=False) # compute SVD (Ihler uses svds here)
U = U[:, :K]
S = np.diag(S[:K])
V = V.T[:, :K]
Xsvd = U.dot(np.sqrt(S)) # new data coefficients
T = np.sqrt(S[0:K, 0:K]).dot(twod(V).T) # new bases for data
return (Xsvd, T)
Xsvd = np.divide(X, T) # or, use given set of bases
return Xsvd, T
def setActivation(self, method, sig=None, sig0=None):
"""
This method sets the activation functions.
Parameters
----------
method : string, {'logistic' , 'htangent', 'custom'} -- which activation type
Optional arguments for "custom" activation:
sig : function object F(z) returns activation function & its derivative at z (as a tuple)
sig0: activation function object F(z) for final layer of the nnet
"""
raise NotImplementedError # unfinished / tested
method = method.lower()
if method == 'logistic':
self.sig = lambda z: twod(1 / (1 + np.exp(-z)))
self.d_sig = lambda z: twod(np.multiply(self.sig(z), (1 - self.sig(z))))
self.sig_0 = self.sig
self.d_sig_0 = self.d_sig
elif method == 'htangent':
self.sig = lambda z: twod(np.tanh(z))
self.d_sig = lambda z: twod(1 - np.power(np.tanh(z), 2))
self.sig_0 = self.sig
self.d_sig_0 = self.d_sig
elif method == 'custom':
self.sig = sig
self.d_sig = d_sig
self.sig_0 = sig_0
self.d_sig_0 = d_sig_0
else:
raise ValueError('nnetRegress.set_activation: ' + str(method) + ' is not a valid option for method')
self.activation = method
def fkitchensink(X, K, typ, W=None):
"""
Random kitchen sink features from data. Selects K random "kitchen sink"
features of X.
Parameters
----------
X : numpy array
M x N numpy array containing data.
K : int
Number of features to select.
typ : str
One of: 'stump', 'sigmoid', 'sinuoid', or 'linear'.
W : numpy array (optional)
N x K array of parameters. If provided, W uses fixed params.
Returns
-------
Z : numpy array
M x K array of features selected from X.
W : numpy array (optional)
N x K array of random parameters. Only returned if the argument W
isn't provided.
"""
to_return = ()
N, M = twod(X).shape
typ = typ.lower()
if type(W) is type(None): # numpy complains about truth value of arrays
if typ == "stump":
W = np.zeros((2, K))
s = np.sqrt(np.var(X, axis=0))
# random feature index 1..M
W[0, :] = np.floor(np.random.rand(K) * M)
W = W.astype(int)
W[0, :] = W[0, :].astype(int)
W[1, :] = np.random.randn(K) * s[W[0, :]] # random threshold (w/ same variance as that feature)
elif typ in ["sigmoid", "sinusoid", "linear"]:
# random direction for sigmodal ridge, random freq for sinusoids, random linear projections
W = np.random.randn(M, K)
to_return = (W,)
Z = np.zeros((N, K))
if typ == "stump": # decision stump w/ random threshold
for i in range(K):
Z[:, i] = X[:, W[0, i]] >= W[1, i]
elif typ == "sigmoid": # sigmoidal ridge w/ random direction
Z = twod(X).dot(W)
Z = 1 / (1 + np.exp(Z))
elif typ == "sinusoid": # sinusoid w/ random frequency
Z = np.sin(twod(X).dot(W))
elif typ == "linear": # straight linear projection
Z = twod(X).dot(W)
return Z if len(to_return) == 0 else (Z,) + to_return
def log_likelihood(self, X, Y): """ Compute the emperical avg. log likelihood of 'obj' on test data (X,Y). See constructor doc string for argument descriptions. """ r,c = twod(Y).shape if r == 1 and c != 1: Y = twod(Y).T soft = self.predict_soft(X) return np.mean(np.sum(np.log(np.power(soft, Y, )), 1), 0)
def __logistic(self, X): """ This is a helper method that evaluates the logistic function for weights self.wts (1 x d + 1) on data X (n x d). Used in: __gradient_descent predict """ n,d = twod(X).shape X_train = cols((np.ones((n,1)), twod(X))) f = twod(X_train).dot(twod(self.wts).T) return 1 / (1 + np.exp(-f))
def plot_classify_2D(learner, X, Y, pre=lambda x: x):
"""
Plot data and classifier outputs on two-dimensional data.
This function plot data (X,Y) and learner.predict(X, Y)
together. The learner is is predicted on a dense grid
covering data X, to show its decision boundary.
Parameters
----------
learner : learner object
A trained learner object that inherits from one of
the 'Classify' or 'Regressor' base classes.
X : numpy array
N x M array of data; N = number of data, M = dimension
(number of features) of data.
Y : numpy array
1 x N array containing labels corresponding to data points
in X.
pre : function object (optional)
Function that is applied to X before prediction.
"""
if twod(X).shape[1] != 2:
raise ValueError('plot_classify_2d: function can only be called using two-dimensional data (features)')
plt.plot(X[:,0], X[:,1], 'k.')
ax = plt.xlim() + plt.ylim() # get current axis limits
N = 256 # density of evaluation
# evaluate each point of feature space and predict the class
X1 = np.linspace(ax[0], ax[1], N)
X1sp = twod(X1).T * np.ones(N)
X2 = np.linspace(ax[2], ax[3], N)
X2sp = np.ones((N,1)) * X2
Xfeat = cols((twod(X1sp.flatten()).T, twod(X2sp.flatten()).T))
# preprocess/create feature vector if necessary
Xfeat = pre(Xfeat)
# predict using learner
pred = learner.predict(Xfeat)
# plot decision values for space in 'faded' color
clim = np.unique(Y)
clim = [clim[0], clim[0] + 1] if len(clim) == 1 else list(clim)
plt.imshow(np.reshape(pred, (N,N)).T, extent=[X1[0], X1[-1], X2[0], X2[-1]], cmap=plt.cm.Pastel2)
plt.clim(*clim)
plt.show()
def flda(X, Y, K, T=None):
"""
Reduce the dimension of X to K features using (multiclass) discriminant
analysis.
Parameters
----------
X : numpy array
M x N array of data.
Y : numpy array
M x 1 array of labels corresponding to data in X.
K : int
New dimension (number of features) of X.
T : numpy array (optional)
The transform matrix. If this argument is provided, function uses T
instead of computing the LDA.
Returns
-------
Xlda : numpy array
T : numpy array (optional)
TODO: Test; check/test Matlab version?
"""
if type(T) is not type(None):
return np.divide(X, T)
n, m = twod(X).shape
c = np.unique(Y)
nc = np.zeros(len(c))
mu = np.zeros((len(c), n))
sig = np.zeros((len(c), n, n))
for i in range(len(c)):
idx = np.where(Y == c[i])[0]
nc[i] = len(idx)
mu[i, :] = np.mean(X[:, idx], axis=0)
sig[i, :, :] = np.cov(X[:, idx])
S = (nc / n).dot(np.reshape(sig, (len(c), n * n)))
S = np.reshape(S, (n, n))
U, S, V = np.linalg.svd(X, K) # compute SVD (Ihler uses svds here)
Xlda = U.dot(np.sqrt(S)) # new data coefficients
T = np.sqrt(S[0:K, 0:K]).dot(twod(V).T) # new bases for data
return Xlda, T
def bootstrapData(X, Y=None, n_boot=None):
"""
Bootstrap resample a data set (with replacement):
draw data points (x_i,y_i) from (X,Y) n_boot times.
Parameters
----------
X : MxN numpy array of data points to be resampled.
Y : Mx1 numpy array of labels associated with each datum (optional)
n_boot : int, number of samples to draw (default: N)
Returns
-------
Xboot, Yboot : (tuple of) numpy arrays for the resampled data set
If Y is not present or None, returns only Xboot (non-tuple)
"""
nx,dx = twod(X).shape
Y = Y.flatten()
if n_boot is None: n_boot = nx
idx = np.floor(np.random.rand(n_boot) * nx).astype(int)
X = X[idx,:]
ny = len(Y)
assert ny > 0, 'bootstrapData: Y must contain data'
assert nx == ny, 'bootstrapData: X and Y should have the same length'
Y = Y[idx]
return (X,Y)
def shuffle_data(X, Y): """ Shuffle data in X and Y. Parameters ---------- X : numpy array N x M array of data to shuffle. Y : numpy arra 1 x N array of labels that correspond to data in X. Returns ------- X or (X,Y) : numpy array or tuple of arrays Shuffled data (only returns X and Y if Y contains data). TODO: test more """ nx,dx = twod(X).shape Y = arr(Y).flatten() ny = len(Y) pi = np.random.permutation(nx) X = X[pi,:] if ny > 0: assert ny == nx, 'shuffle_data: X and Y must have the same length' Y = Y[pi] return X,Y return X
def bootstrap_data(X, Y, n_boot): """ Function that resamples (bootstrap) data set: it resamples data points (x_i,y_i) with replacement n_boot times. Parameters ---------- X : numpy array N x M numpy array that contains data points to be sampled. Y : numpy array 1 x N numpy arra that contains labels that map to data points in X. n_boot : int The number of samples to take. Returns ------- (array,array) Tuple containing samples from X and Y. TODO: test more """ Y = Y.flatten() nx,dx = twod(X).shape idx = np.floor(np.random.rand(n_boot) * nx).astype(int) X = X[idx,:] ny = len(Y) assert ny > 0, 'bootstrap_data: Y must contain data' assert nx == ny, 'bootstrap_data: X and Y should have the same length' Y = Y[idx] return (X,Y)
def splitData(X, Y=None, train_fraction=0.80):
"""
Split data into training and test data.
Parameters
----------
X : MxN numpy array of data to split
Y : Mx1 numpy array of associated target values
train_fraction : float, fraction of data used for training (default 80%)
Returns
-------
to_return : (Xtr,Xte,Ytr,Yte) or (Xtr,Xte)
A tuple containing the following arrays (in order): training
data from X, testing data from X, training labels from Y
(if Y contains data), and testing labels from Y (if Y
contains data).
"""
nx,dx = twod(X).shape
ne = round(train_fraction * nx)
Xtr,Xte = X[:ne,:], X[ne:,:]
to_return = (Xtr,Xte)
if Y is not None:
Y = arr(Y).flatten()
ny = len(Y)
if ny > 0:
assert ny == nx, 'splitData: X and Y must have the same length'
Ytr,Yte = Y[:ne], Y[ne:]
to_return += (Ytr,Yte)
return to_return
def fsubset(X, K, feat=None):
"""
Select subset of features from data. Selects a fixed or random subset
of K features from X.
Parameters
----------
X : numpy array
M x N array of data.
K : int
Number of features in output.
feat : array like (optional)
Flat array of indices specifying which features to select.
Returns
-------
X_sub : numpy array
M x K numpy array of data.
feat : numpy array (optional)
1 x N array of indices of selected features. Only returned if feat
argument isn't provided.
"""
n, m = twod(X).shape
to_return = ()
if type(feat) is type(None):
feat = np.random.permutation(m)
feat = feat[0:K]
to_return = (feat,)
X_sub = X[:, feat]
return X_sub if len(to_return) == 0 else (X_sub,) + to_return
def shuffleData(X, Y=None):
"""
Shuffle (randomly reorder) data in X and Y.
Parameters
----------
X : MxN numpy array: N feature values for each of M data points
Y : Mx1 numpy array (optional): target values associated with each data point
Returns
-------
X,Y : (tuple of) numpy arrays of shuffled features and targets
only returns X (not a tuple) if Y is not present or None
Ex:
X2 = shuffleData(X) : shuffles the rows of the data matrix X
X2,Y2 = shuffleData(X,Y) : shuffles rows of X,Y, preserving correspondence
"""
nx,dx = twod(X).shape
Y = arr(Y).flatten()
ny = len(Y)
pi = np.random.permutation(nx)
X = X[pi,:]
if ny > 0:
assert ny == nx, 'shuffleData: X and Y must have the same length'
Y = Y[pi] if Y.ndim <= 1 else Y[pi,:]
return X,Y
return X
def fproject(X, K, proj=None):
"""
Random projection of features from data. Selects a fixed or random linear
projection of K features from X.
Parameters
----------
X : numpy array
M x N array of data.
K : int
Number of features in output.
proj : numpy array (optional)
The projection matrix. If this argument is provided, function uses proj
instead of random matrix.
Returns
-------
X : numpy array
M x K array of projecjtion of data in X.
proj : numpy array (optional)
N x K numpy array that is the project matrix. Only returned if proj
argument isn't provided.
"""
n, m = twod(X).shape
to_return = ()
if type(proj) is type(None):
proj = np.random.randn(m, K)
to_return = (proj,)
X2 = X.dot(proj)
return X2 if len(to_return) == 0 else (X2,) + to_return
def fpoly_mono(X, degree, bias=True):
"""
Create polynomial features of each individual feature (no cross products).
Parameters
----------
X : MxN numpy array of data (each row one data point)
degree : int, the polynomial degree
bias : bool, include constant feature if true (default)
Returns
-------
Xext : MxN' numpy array with each data point's higher order features
"""
m, n = twod(X).shape
if bias:
Xext = np.zeros((m, n * degree + 1))
Xext[:, 0] = 1
k = 1
else:
Xext = np.zeros((m, n * degree))
k = 0
for p in range(degree):
for j in range(n):
Xext[:, k] = np.power(X[:, j], p + 1)
k += 1
return Xext
def kmeans(X, K, init='random', max_iter=100, do_plot=False, to_return=[1, 0, 0]):
"""
Perform K-means clustering on data X.
Parameters
----------
X : numpy array
N x M array containing data to be clustered.
K : int
Number of clusters.
init : str or array (optional)
Either a K x N numpy array containing initial clusters, or
one of the following strings that specifies a cluster init
method: 'random' (K random data points (uniformly) as clusters),
'farthest' (choose cluster 1 uniformly, then the point farthest
from all cluster so far, etc.), or 'k++' (choose cluster 1
uniformly, then points randomly proportional to distance from
current clusters).
max_iter : int (optional)
Maximum number of optimization iterations.
do_plot : bool (optional)
Plot 2D data?
to_return : [bool] (optional)
Array of bools that specifies which values to return. The bool
at to_return[0] indicates whether z should be returned; the bool
at to_return[1] indicates whether c should be returned, etc.
Returns
-------
z : numpy array
N x 1 array containing cluster numbers of data at indices in X.
c : numpy array (optional)
K x M array of cluster centers.
sumd : scalar (optional)
Sum of squared euclidean distances.
TODO: test more
"""
n,d = twod(X).shape # get data size
if type(init) is str:
init = init.lower()
if init == 'random':
pi = np.random.permutation(n)
c = X[pi[0:K],:]
elif init == 'farthest':
c = k_init(X, K, True)
elif init == 'k++':
c = k_init(X, K, False)
else:
raise ValueError('kmeans: value for "init" ( ' + init + ') is invalid')
else:
c = init
z,c,sumd = __optimize(X, n, K, c, max_iter)
return optional_return(to_return, z - 1, c, sumd)
def checkDataShape(X,Y):
"""
Simple helper function to convert vectors to matrices and check the shape of
the data matrices X,Y
"""
X = twod(X).T if X.ndim < 2 else X
#Y = twod(Y).T if Y.ndim < 2 else Y
if X.shape[0] != Y.shape[0]:
raise ValueError("X and Y do not have the same number of data points!")
return X,Y
def data_gauss(N0, N1=None, mu0=arr([0, 0]), mu1=arr([1, 1]), sig0=np.eye(2), sig1=np.eye(2)):
"""
Sample data from a Gaussian model.
Parameters
----------
N0 : int
Number of data to sample for class -1.
N1 : int
Number of data to sample for class 1.
mu0 : numpy array
mu1 : numpy array
sig0 : numpy array
sig1 : numpy array
Returns
-------
X : numpy array
Array of sampled data.
Y : numpy array
Array of class values that correspond to the data points in X.
TODO: test more
"""
if not N1:
N1 = N0
d1,d2 = twod(mu0).shape[1],twod(mu1).shape[1]
if d1 != d2 or np.any(twod(sig0).shape != arr([d1, d1])) or np.any(twod(sig1).shape != arr([d1, d1])):
raise ValueError('data_gauss: dimensions should agree')
X0 = np.dot(np.random.randn(N0, d1), sqrtm(sig0))
X0 += np.ones((N0,1)) * mu0
Y0 = -np.ones(N0)
X1 = np.dot(np.random.randn(N1, d1), sqrtm(sig1))
X1 += np.ones((N1,1)) * mu1
Y1 = np.ones(N1)
X = np.row_stack((X0,X1))
Y = np.concatenate((Y0,Y1))
return X,Y
def plotGauss2D(mu,cov,*args,**kwargs):
"""
Plot an ellipsoid indicating (one std deviation of) a 2D Gaussian distribution
All additional arguments are passed into plot(.)
"""
from scipy.linalg import sqrtm
theta = np.linspace(0,2*np.pi,50)
circle = np.array([np.sin(theta),np.cos(theta)])
ell = sqrtm(cov).dot(circle)
ell += twod(mu).T
plt.plot( mu[0],mu[1], 'x', ell[0,:],ell[1,:], **kwargs)
def predict(self, X): """ Predict on X. Refer to constructor docstring for description of X. """ N,D = twod(X).shape Y_te = np.zeros((N,1)) + self.const for i,l in enumerate(self): yi = l.predict(X) # figure out current prediction value # if we already have learners, figure out the prediction on the training data Y_te = Y_te + self.alpha[i] * yi return Y_te
def roc(self, X, Y):
"""
This method computes the "receiver operating characteristic" curve on
test data. This method is only defined for binary classifiers. Refer
to the auc doc string for descriptions of X and Y. Method returns
(fpr, tpr, tnr), where
fpr = false positive rate (1xN numpy vector)
tpr = true positive rate (1xN numpy vector)
tnr = true negative rate (1xN numpy vector)
Plot fpr vs. tpr to see the ROC curve.
Plot tpr vs. tnr to see the sensitivity/specificity curve.
"""
if len(self.classes) > 2:
raise ValueError('This method can only supports binary classification ')
try: # compute 'response' (soft binary classification score)
soft = self.predictSoft(X)[:,1] # p(class = 2nd)
except (AttributeError, IndexError):
soft = self.predict(X) # or we can use 'hard' binary prediction if soft is unavailable
n,d = twod(soft).shape
if n == 1:
soft = soft.flatten()
else:
soft = soft.T.flatten()
# number of true negatives and positives
n0 = np.sum(Y == self.classes[0])
n1 = np.sum(Y == self.classes[1])
if n0 == 0 or n1 == 0:
raise ValueError('Data of both class values not found')
# sort data by score value
sorted_soft = np.sort(soft)
indices = np.argsort(soft)
Y = Y[indices]
# compute false positives and true positive rates
tpr = np.divide(np.cumsum(Y[::-1] == self.classes[1]), n1)
fpr = np.divide(np.cumsum(Y[::-1] == self.classes[0]), n0)
tnr = np.divide(np.cumsum(Y == self.classes[0]), n0)[::-1]
# find ties in the sorting score
same = np.append(np.asarray(sorted_soft[0:-1] == sorted_soft[1:]), 0)
tpr = np.append([0], tpr[np.logical_not(same)])
fpr = np.append([0], fpr[np.logical_not(same)])
tnr = np.append([1], tnr[np.logical_not(same)])
return [tpr, fpr, tnr]
def plotClassify2D(learner, X, Y, pre=lambda x: x, axis=None, nGrid=128, **kwargs):
"""
Plot data and classifier outputs on two-dimensional data.
This function plot data (X,Y) and learner.predict(X, Y)
together. The learner is is predicted on a dense grid
covering data X, to show its decision boundary.
Parameters
----------
learner : learner object
A trained learner object that inherits from one of
the 'Classify' or 'Regressor' base classes.
X : numpy array
N x M array of data; N = number of data, M = dimension
(number of features) of data.
Y : numpy array
1 x N arra containing labels corresponding to data points
in X.
pre : function object (optional)
Function that is applied to X before prediction.
axis : a matplotlib axis / plottable object (optional)
nGrid : density of 2D grid points (default 128)
"""
if twod(X).shape[1] != 2:
raise ValueError('plotClassify2D: function can only be called using two-dimensional data (features)')
# TODO: Clean up code
if axis == None: axis = plt
axis.plot( X[:,0],X[:,1], 'k.', visible=False )
# TODO: can probably replace with final dot plot and use transparency for image (?)
ax = axis.axis()
xticks = np.linspace(ax[0],ax[1],nGrid)
yticks = np.linspace(ax[2],ax[3],nGrid)
grid = np.meshgrid( xticks, yticks )
XGrid = np.column_stack( (grid[0].flatten(), grid[1].flatten()) )
if learner is not None:
YGrid = learner.predict( pre(XGrid) )
#axis.contourf( xticks,yticks,YGrid.reshape( (len(xticks),len(yticks)) ), nClasses )
axis.imshow( YGrid.reshape( (len(xticks),len(yticks)) ), extent=axis.axis(), interpolation='nearest',origin='lower',alpha=0.5 )
cmap = plt.cm.get_cmap()
# TODO: if Soft: predictSoft; get colors for each class from cmap; blend pred with colors & show
#
classes = np.unique(Y)
cvals = (classes - min(classes))/(max(classes)-min(classes)+1e-100)
for i,c in enumerate(classes):
axis.plot( X[Y==c,0],X[Y==c,1], 'ko', color=cmap(cvals[i]), **kwargs )
def auc(self, X, Y):
"""
This method computes the area under the roc curve on the given test data.
This method only works on binary classifiers.
Paramters
---------
X : N x M numpy array
N = number of data points; M = number of features.
Y : 1 x N numpy array
Array of classes that refer to the data points in X.
"""
if len(self.classes) > 2:
raise ValueError('This method can only supports binary classification ')
try: # compute 'response' (soft binary classification score)
soft = self.predict_soft(X)[:,1] # p(class = 2nd)
except (AttributeError, IndexError): # or we can use 'hard' binary prediction if soft is unavailable
soft = self.predict(X)
n,d = twod(soft).shape
if n == 1:
soft = soft.flatten()
else:
soft = soft.T.flatten()
sorted_soft = np.sort(soft) # sort data by score value
indices = np.argsort(soft)
Y = Y[indices]
# find ties in the sorting score
same = np.append(np.asarray(sorted_soft[0:-1] == sorted_soft[1:]), 0)
n = len(soft)
rnk = self.__compute_ties(n, same) # compute tied rank values
# number of true negatives and positives
n0 = sum(Y == self.classes[0])
n1 = sum(Y == self.classes[1])
if n0 == 0 or n1 == 0:
raise ValueError('Data of both class values not found')
# compute AUC using Mann-Whitney U statistic
result = (np.sum(rnk[Y == self.classes[1]]) - n1 * (n1 + 1) / 2) / n1 / n0
return result
def from1ofK(Y, values=None):
"""
Function that converts Y from 1-of-K ("1-hot") rep back to single col/row form.
Parameters
----------
Y : arraylike
Matrix to convert from 1-of-k rep.
values : list (optional)
List that specifies which values to use for which index.
Returns
-------
array
Y in single row/col form.
"""
return Y.argmax(1) if not values else twod([values[i] for i in Y.argmax(1)]).T
def train(self, X, Y, initStep=1.0, stopTol=1e-4, stopIter=5000, plot=None):
""" Train the logistic regression using stochastic gradient descent """
## First do some bookkeeping and setup:
self.theta,X,Y = twod(self.theta), arr(X), arr(Y) # convert to numpy arrays
M,N = X.shape
if Y.shape[0] != M:
raise ValueError("Y must have the same number of data (rows) as X")
self.classes = np.unique(Y)
if len(self.classes) != 2:
raise ValueError("Y should have exactly two classes (binary problem expected)")
if self.theta.shape[1] != N+1: # if self.theta is empty, initialize it!
self.theta = np.random.randn(1,N+1)
# Some useful modifications of the data matrices:
X1 = np.hstack((np.ones((M,1)),X)) # make data array with constant feature
Y01 = toIndex(Y, self.classes) # convert Y to canonical "0 vs 1" classes
it = 0
done = False
Jsur = []
J01 = []
while not done:
step = (2.0 * initStep) / (2.0 + it) # common 1/iter step size change
for i in range(M): # for each data point i:
## TODO: compute zi = linear response of X[i,:]
## TODO: compute prediction yi
## TODO: compute soft response si = logistic( zi )
## TODO: compute gradient of logistic loss wrt data point i:
# Take a step down the gradient
self.theta = self.theta - step * gradi
# each pass, compute surrogate loss & error rates:
J01.append( self.err(X,Y) )
## TODO: compute surrogate loss (logistic negative log-likelihood)
## Jsur = sum_i [ (si log si) if yi==1 else ((1-si)log(1-si)) ]
Jsur.append( NotImplemented ) ## TODO ...
## For debugging: print current parameters & losses
# print self.theta, ' => ', Jsur[-1], ' / ', J01[-1]
# raw_input() # pause for keystroke
# check stopping criteria:
it += 1
done = (it > stopIter) or ( (it>1) and (abs(Jsur[-1]-Jsur[-2])<stopTol) )
def from1ofK(Y, values=None):
"""
Function that converts Y from 1-of-K ("1-hot") rep back to single col/row form.
Parameters
----------
Y : arraylike
Matrix to convert from 1-of-k rep.
values : list (optional)
List that specifies which values to use for which index.
Returns
-------
array
Y in single row/col form.
"""
return Y.argmax(1) if not values else twod(
[values[i] for i in Y.argmax(1)]).T
def cross_validate(X, Y, n_folds, i_fold):
"""
Function that splits data for n-fold cross validation.
Parameters
----------
X : numpy array
N x M numpy array that contains data points.
Y : numpy array
1 x N numpy array that contains labels that correspond to
data points in X.
n_folds : int
Total number of data folds.
i_fold : int
The fold for which the current call of cross_validate
will partition.
Returns
-------
to_return : (Xtr,Xte,Ytr,Yte)
Tuple that contains (in this order) training data from X, testing
data from X, training labels from Y, and testing labels from Y.
"""
Y = arr(Y).flatten()
nx, dx = twod(X).shape
ny = len(Y)
idx = range(nx)
if ny > 0:
assert nx == ny, 'cross_validate: X and Y must have the same length'
n = np.fix(nx / n_folds)
te_start = int((i_fold - 1) * n)
te_end = int((i_fold * n)) if (i_fold * n) <= nx else int(nx)
test_range = list(range(te_start, te_end))
train_range = sorted(set(idx) - set(test_range))
to_return = (X[train_range, :], X[test_range, :])
if ny > 0:
to_return += (Y[train_range], Y[test_range])
return to_return
def k_init(X, K, determ): """ Distance based initialization. Randomly choose a start point, then: if determ == True: choose point farthest from the clusters chosen so far, otherwise: randomly choose new points proportionally to their distance. Parameters ---------- X : numpy array See kmeans docstring. K : int See kmeans docstring. determ : bool See description. Returns ------- c : numpy array K x M array of cluster centers. """ m,n = twod(X).shape clusters = np.zeros((K,n)) clusters[0,:] = X[np.floor(np.random.rand() * m),:] # take random point as first cluster dist = np.sum(np.power((X - np.ones((m,1)) * clusters[0,:]), 2), axis=1).ravel() #print 'dist:',dist for i in range(1,K): #print dist #print np.argmax(dist) if determ: j = np.argmax(dist) # choose farthest point... else: pr = np.cumsum(np.array(dist)); # ...or choose a random point by distance pr = pr / pr[-1] j = np.where(np.random.rand() < pr)[0][0] clusters[i,:] = X[j,:] # update that cluster # update min distances new_dist = np.sum(np.power((X - np.ones((m,1)) * clusters[i,:]), 2), axis=1).ravel() dist = np.minimum(dist, new_dist) #print "dist",dist return clusters
def cross_validate(X, Y, n_folds, i_fold):
"""
Function that splits data for n-fold cross validation.
Parameters
----------
X : numpy array
N x M numpy array that contains data points.
Y : numpy array
1 x N numpy array that contains labels that correspond to
data points in X.
n_folds : int
Total number of data folds.
i_fold : int
The fold for which the current call of cross_validate
will partition.
Returns
-------
to_return : (Xtr,Xte,Ytr,Yte)
Tuple that contains (in this order) training data from X, testing
data from X, training labels from Y, and testing labels from Y.
"""
Y = arr(Y).flatten()
nx, dx = twod(X).shape
ny = len(Y)
idx = range(nx)
if ny > 0:
assert nx == ny, "cross_validate: X and Y must have the same length"
n = np.fix(nx / n_folds)
te_start = int((i_fold - 1) * n)
te_end = int((i_fold * n)) if (i_fold * n) <= nx else int(nx)
test_range = list(range(te_start, te_end))
train_range = sorted(set(idx) - set(test_range))
to_return = (X[train_range, :], X[test_range, :])
if ny > 0:
to_return += (Y[train_range], Y[test_range])
return to_return
def k_init(X, K, determ): """ Distance based initialization. Randomly choose a start point, then: if determ == True: choose point farthest from the clusters chosen so far, otherwise: randomly choose new points proportionally to their distance. Parameters ---------- X : numpy array See kmeans docstring. K : int See kmeans docstring. determ : bool See description. Returns ------- c : numpy array K x M array of cluster centers. """ m,n = twod(X).shape clusters = np.zeros((K,n)) clusters[0,:] = X[int(np.floor(np.random.rand()) * m),:] # take random point as first cluster dist = np.sum(np.power((X - np.ones((m,1)) * clusters[0,:]), 2), axis=1).ravel() #print 'dist:',dist for i in range(1,K): #print dist #print np.argmax(dist) if determ: j = np.argmax(dist) # choose farthest point... else: pr = np.cumsum(np.array(dist)); # ...or choose a random point by distance pr = pr / pr[-1] j = np.where(np.random.rand() < pr)[0][0] clusters[i,:] = X[j,:] # update that cluster # update min distances new_dist = np.sum(np.power((X - np.ones((m,1)) * clusters[i,:]), 2), axis=1).ravel() dist = np.minimum(dist, new_dist) #print "dist",dist return clusters
def auc(self, X, Y):
"""
This method computes the area under the roc curve on the given test data.
This method only works on binary classifiers.
Parameters
---------
X : M x N numpy array
M = number of data points; N = number of features.
Y : M x 1 numpy array
Array of classes (targets) corresponding to the data points in X.
"""
if len(self.classes) != 2:
raise ValueError('This method can only supports binary classification ')
try: # compute 'response' (soft binary classification score)
soft = self.predictSoft(X)[:,1] # p(class = 2nd)
except (AttributeError, IndexError): # or we can use 'hard' binary prediction if soft is unavailable
soft = self.predict(X)
n,d = twod(soft).shape # ensure soft is the correct shape
soft = soft.flatten() if n==1 else soft.T.flatten()
indices = np.argsort(soft) # sort data by score value
Y = Y[indices]
sorted_soft = soft[indices]
# compute rank (averaged for ties) of sorted data
dif = np.hstack( ([True],np.diff(sorted_soft),[True]) )
r1 = np.argwhere(dif).flatten()
r2 = r1[0:-1] + 0.5*(r1[1:]-r1[0:-1]) + 0.5
rnk = r2[np.cumsum(dif[:-1])-1]
# number of true negatives and positives
n0,n1 = sum(Y == self.classes[0]), sum(Y == self.classes[1])
if n0 == 0 or n1 == 0:
raise ValueError('Data of both class values not found')
# compute AUC using Mann-Whitney U statistic
result = (np.sum(rnk[Y == self.classes[1]]) - n1 * (n1 + 1) / 2) / n1 / n0
return result
def auc(self, X, Y):
"""Compute the area under the roc curve on the given test data.
Args:
X (arr): M,N array of M data points with N features each
Y (arr): M, or M,1 array of target class values for each data point
Returns:
float: Area under the ROC curve
This method only works on binary classifiers.
"""
if len(self.classes) != 2:
raise ValueError('This method can only supports binary classification ')
try: # compute 'response' (soft binary classification score)
soft = self.predictSoft(X)[:,1] # p(class = 2nd)
except (AttributeError, IndexError): # or we can use 'hard' binary prediction if soft is unavailable
soft = self.predict(X)
n,d = twod(soft).shape # ensure soft is the correct shape
soft = soft.flatten() if n==1 else soft.T.flatten()
indices = np.argsort(soft) # sort data by score value
Y = Y[indices]
sorted_soft = soft[indices]
# compute rank (averaged for ties) of sorted data
dif = np.hstack( ([True],np.diff(sorted_soft)!=0,[True]) )
r1 = np.argwhere(dif).flatten()
r2 = r1[0:-1] + 0.5*(r1[1:]-r1[0:-1]) + 0.5
rnk = r2[np.cumsum(dif[:-1])-1]
# number of true negatives and positives
n0,n1 = sum(Y == self.classes[0]), sum(Y == self.classes[1])
if n0 == 0 or n1 == 0:
raise ValueError('Data of both class values not found')
# compute AUC using Mann-Whitney U statistic
result = (np.sum(rnk[Y == self.classes[1]]) - n1 * (n1 + 1.0) / 2.0) / n1 / n0
return result
def predictSoft(self, X):
"""
This method makes a "soft" linear classification predition on the data
Uses a (multi)-logistic function to convert linear response to [0,1] confidence
Parameters
----------
X : M x N numpy array
M = number of testing instances; N = number of features.
"""
theta, X = twod(self.theta), arr(X) # convert to numpy if needed
resp = theta[:, 0].T + X.dot(theta[:, 1:].T) # linear response (MxC)
prob = np.exp(resp)
if resp.shape[1] == 1: # binary classification (C=1)
prob /= prob + 1.0 # logistic transform (binary classification; C=1)
prob = np.hstack((1 - prob, prob)) # make a column for each class
else:
prob /= np.sum(prob,
axis=1) # normalize each row (for multi-class)
return prob
def crossValidate(X, Y=None, K=5, i=0):
"""
Create a K-fold cross-validation split of a data set:
crossValidate(X,Y, 5, i) : return the ith of 5 80/20 train/test splits
Parameters
----------
X : MxN numpy array of data points to be resampled.
Y : Mx1 numpy array of labels associated with each datum (optional)
K : number of folds of cross-validation
i : current fold to return (0...K-1)
Returns
-------
Xtr,Xva,Ytr,Yva : (tuple of) numpy arrays for the split data set
If Y is not present or None, returns only Xtr,Xva
"""
nx, dx = twod(X).shape
start = round(float(nx) * i / K)
end = round(float(nx) * (i + 1) / K)
Xte = X[start:end, :]
Xtr = np.vstack((X[0:start, :], X[end:, :]))
to_return = (Xtr, Xte)
Y = arr(Y).flatten()
ny = len(Y)
if ny > 0:
assert ny == nx, 'crossValidate: X and Y must have the same length'
if Y.ndim <= 1:
Yte = Y[start:end]
Ytr = np.hstack((Y[0:start], Y[end:]))
else: # in case targets are multivariate
Yte = Y[start:end, :]
Ytr = np.vstack((Y[0:start, :], Y[end:, :]))
to_return += (Ytr, Yte)
return to_return
def fpoly(X, degree, bias=True):
"""
Create expanded polynomial features of up to a given degree.
Parameters
----------
X : MxN numpy array of data (each row one data point)
degree : int, the polynomial degree
bias : bool, include constant feature if true (default)
Returns
-------
Xext : MxN' numpy array with each data point's higher order features
"""
n, m = twod(X).shape
if (degree + 1)**(m) > 1e7:
err_string = 'fpoly: {}**{} = too many potential output features'.format(
degree + 1, m)
raise ValueError(err_string)
if m == 1: # faster shortcut for scalar data
p = arr(range(0, degree + 1))
Xext = np.power(np.tile(X, (1, len(p))), np.tile(p, (n, 1)))
else:
K = 0
for i in range((degree + 1)**(m)):
powers = np.unravel_index(i, (degree + 1, ) * m)
if sum(powers) > degree: continue
K += 1
Xext = np.zeros((n, K))
k = 0
for i in range((degree + 1)**(m)):
powers = np.unravel_index(i, (degree + 1, ) * m)
if sum(powers) > degree: continue
Xext[:, k] = np.prod(X**list(powers), axis=1)
k += 1
return Xext if bias else Xext[:, 1:]
def bootstrapData(X, Y=None, n_boot=None):
"""
Bootstrap resample a data set (with replacement):
draw data points (x_i,y_i) from (X,Y) n_boot times.
Parameters
----------
X : MxN numpy array of data points to be resampled.
Y : Mx1 numpy array of labels associated with each datum (optional)
n_boot : int, number of samples to draw (default: M)
Returns
-------
Xboot, Yboot : (tuple of) numpy arrays for the resampled data set
If Y is not present or None, returns only Xboot (non-tuple)
"""
nx, dx = twod(X).shape
if n_boot is None: n_boot = nx
idx = np.floor(np.random.rand(n_boot) * nx).astype(int)
if Y is None: return X[idx, :]
Y = Y.flatten()
assert nx == len(Y), 'bootstrapData: X and Y should have the same length'
return (X[idx, :], Y[idx])
def fpoly_mono(X, degree, use_constant=True):
"""
Create polynomial features of each individual feature (no cross products).
Parameters
----------
X : numpy array
N x M array of data.
degree : int
The degree.
use_constant : bool (optional)
If True (default), include a constant feature.
Returns
-------
Xext : numpy array
N x M * degree (+ 1 if use_constant) array of polynomial
features from X.
TODO: test more
"""
m, n = twod(X).shape
if use_constant:
Xext = np.zeros((m, n * degree + 1))
Xext[:, 0] = 1
k = 1
else:
Xext = np.zeros((m, n * degree))
k = 0
for p in range(degree):
for j in range(n):
Xext[:, k] = np.power(X[:, j], p + 1)
k += 1
return Xext
def gmm_draw_params(m, c, n=2, scale=.05):
"""Create a random Gaussian mixture model.
Builds a random GMM with C components and draws M data x^{(i)} from a mixture
of Gaussians in D dimensions
Args:
m (int): Number of data to be drawn from a mixture of Gaussians.
c (int): Number of clusters.
n (int): Number of dimensions.
scale (float): relative scale of the inter- to intra-cluster variance (small = clumpy)
Returns:
tuple of tuples, (pi,mu,sig) = mixture weight, mean, and covariance of each component
"""
pi = np.zeros(c)
for cc in range(c):
pi[cc] = gamrand(10, 0.5)
pi = pi / np.sum(pi)
rho = np.random.rand(n, n)
rho = rho + twod(rho).T
rho = rho + n * np.eye(n)
rho = sqrtm(rho)
mu = np.random.randn(c, n).dot(rho)
ccov = []
for i in range(c):
tmp = np.random.rand(n, n)
tmp = tmp + tmp.T
tmp = scale * (tmp + n * np.eye(n))
ccov.append(tmp)
#print(pi,mu,ccov)
return tuple((pi[cc], mu[cc], ccov[cc]) for cc in range(c))
def split_data(X, Y, train_fraction):
"""
Split data into training and test data.
Parameters
----------
X : numpy array
N x M array of data to split.
Y : numpy arra
1 x N array of labels that correspond to data in X.
train_fraction : float
Fraction of data to use for training.
Returns
-------
to_return : (Xtr,Xte,Ytr,Yte) or (Xtr,Xte)
A tuple containing the following arrays (in order): training
data from X, testing data from X, training labels from Y
(if Y contains data), and testing labels from Y (if Y
contains data).
"""
nx, dx = twod(X).shape
ne = round(train_fraction * nx)
Xtr, Xte = X[:ne, :], X[ne:, :]
to_return = (Xtr, Xte)
Y = arr(Y).flatten()
ny = len(Y)
if ny > 0:
assert ny == nx, 'split_data: X and Y must have the same length'
Ytr, Yte = Y[:ne], Y[ne:]
to_return += (Ytr, Yte)
return to_return
def fhash(X, K, hash=None):
"""
Random hash of features from data. Selects a fixed or random hash of features
from X.
Parameters
----------
X : numpy array
M x N numpy array containing data.
K : int
Number of features to select.
hash : function object (optional)
Hash function to use. If provided, 'hash' uses fixed hash.
Returns
-------
Z : numpy array
M x K array of hashed features of X.
hash : hash function (optional)
Hash function used to hash features. Only returned if 'hash' argument
isn't provided.
"""
to_return = ()
n, m = twod(X).shape
if hash is None: # TODO : not what we want ?!
hash = lambda i: np.floor(np.random.rand(m) * K)[i]
to_return = (hash, )
# do the hashing
Z = np.zeros((n, K))
for i in range(m):
Z[:, hash(i)] = Z[:, hash(i)] + X[:, i]
return Z if len(to_return) == 0 else (Z, ) + to_return
def agglomerative(X, K, method='means', join=None):
"""
Perform hierarchical agglomerative clustering.
Parameters
----------
X : numpy array
N x M array of Data to be clustered.
K : int
The number of clusters into which data should be grouped.
method : str (optional)
str that specifies the method to use for calculating distance between
clusters. Can be one of: 'min', 'max', 'means', or 'average'.
join : numpy array (optional)
N - 1 x 3 that contains a sequence of joining operations. Pass to avoid
reclustering for new X.
to_return : [bool] (optional)
Array of bools that specifies which values to return. The bool
at to_return[0] indicates whether z should be returned; the bool
at to_return[1] indicates whether join should be returned.
Returns (tuple)
-------
z : N x 1 array of cluster assignments.
join : N - 1 x 3 array that contains the sequence of joining operations
peformed by the clustering algorithm.
"""
m, n = twod(X).shape # get data size
D = np.zeros(
(m, m)
) + np.inf # store pairwise distances b/w clusters (D is an upper triangular matrix)
z = arr(range(m)) # assignments of data
num = np.ones(m) # number of data in each cluster
mu = arr(X) # centroid of each cluster
method = method.lower()
if type(join) == type(None): # if join not precomputed
join = np.zeros((m - 1, 3)) # keep track of join sequence
# use standard Euclidean distance
dist = lambda a, b: np.sum(np.power(a - b, 2))
for i in range(m): # compute initial distances
for j in range(i + 1, m):
D[i][j] = dist(X[i, :], X[j, :])
opn = np.ones(m) # store list of clusters still in consideration
val, k = np.min(D), np.argmin(
D) # find first join (closest cluster pair)
for c in range(m - 1):
i, j = np.unravel_index(k, D.shape)
join[c, :] = arr([i, j, val])
# centroid of new cluster
mu_new = (num[i] * mu[i, :] + num[j] * mu[j, :]) / (num[i] +
num[j])
# compute new distances to cluster i
for jj in np.where(opn)[0]:
if jj in [i, j]:
continue
# sort indices because D is an upper triangluar matrix
idxi = tuple(sorted((i, jj)))
idxj = tuple(sorted((j, jj)))
if method == 'min':
D[idxi] = min(D[idxi],
D[idxj]) # single linkage (min dist)
elif method == 'max':
D[idxi] = max(D[idxi],
D[idxj]) # complete linkage (max dist)
elif method == 'means':
D[idxi] = dist(
mu_new, mu[jj, :]) # mean linkage (dist b/w centroids)
elif method == 'average':
# average linkage
D[idxi] = (num[i] * D[idxi] + num[j] * D[idxj]) / (num[i] +
num[j])
opn[j] = 0 # close cluster j (fold into i)
num[i] = num[i] + num[
j] # update total membership in cluster i to include j
mu[i, :] = mu_new # update centroid list
# remove cluster j from consideration as min
for ii in range(m):
if ii != j:
# sort indices because D is an upper triangular matrix
idx = tuple(sorted((ii, j)))
D[idx] = np.inf
val, k = np.min(D), np.argmin(D) # find next smallext pair
# compute cluster assignments given sequence of joins
for c in range(m - K):
z[z == join[c, 1]] = join[c, 0]
uniq = np.unique(z)
for c in range(len(uniq)):
z[z == uniq[c]] = c
return z, join
def kmeans(X, K, init='random', max_iter=100):
"""
Perform K-means clustering on data X.
Parameters
----------
X : numpy array
N x M array containing data to be clustered.
K : int
Number of clusters.
init : str or array (optional)
Either a K x N numpy array containing initial clusters, or
one of the following strings that specifies a cluster init
method: 'random' (K random data points (uniformly) as clusters),
'farthest' (choose cluster 1 uniformly, then the point farthest
from all cluster so far, etc.), or 'k++' (choose cluster 1
uniformly, then points randomly proportional to distance from
current clusters).
max_iter : int (optional)
Maximum number of optimization iterations.
Returns (as tuple)
-------
z : N x 1 array containing cluster numbers of data at indices in X.
c : K x M array of cluster centers.
sumd : (scalar) sum of squared euclidean distances.
"""
n, d = twod(X).shape
# First, initialize the clusters to something:
if type(init) is str:
init = init.lower()
if init == 'random':
pi = np.random.permutation(n)
c = X[pi[0:K], :]
elif init == 'farthest':
c = k_init(X, K, True)
elif init == 'k++':
c = k_init(X, K, False)
else:
raise ValueError('kmeans: value for "init" ( ' + init +
') is invalid')
else:
c = init
# Now, optimize the objective using coordinate descent:
iter = 1
done = (iter > max_iter)
sumd = np.inf
sum_old = np.inf
z = np.zeros((n, ))
#print c
while not done:
sumd = 0
for i in range(n):
# compute distances for each cluster center
dists = np.sum((c - twod(X[i, :]))**2, axis=1)
#dists = np.sum(np.power((c - np.tile(X[i,:], (K,1))), 2), axis=1)
val = np.min(dists, axis=0) # assign datum i to nearest cluster
z[i] = np.argmin(dists, axis=0)
sumd = sumd + val
#print z
for j in range(K): # now update each cluster center j...
if np.any(z == j):
c[j, :] = np.mean(
X[(z == j).flatten(), :],
0) # ...to be the mean of the assigned data...
else:
c[j, :] = X[int(np.floor(np.random.rand(
))), :] # ...or random restart if no assigned data
done = (iter > max_iter) or (sumd == sum_old)
sum_old = sumd
iter += 1
return z, c, sumd
def fpoly_pair(X, degree, use_constant=True):
"""
Create polynomial features of each individual feature (too many cross
products).
Parameters
----------
X : numpy array
N x M array of data.
degree : int
The degree.
use_constant : bool (optional)
If True (default), include a constant feature.
Returns
-------
Xext : numpy array
TODO: test more
"""
m, n = twod(X).shape
npoly = np.ceil(
(n**(degree + 1) - 1) / (n - 1)) # ceil to fix possible roundoff error
if use_constant:
Xext = np.zeros((m, npoly))
Xext[:, 0] = 1
Xcur = 1
k = 1
else:
Xext = np.zeros((m, npoly - 1))
Xcur = 1
k = 0
# hard coded to be a shorter length
if degree == 2:
Xext[:, k:k + n] = X
k += n
Z = np.reshape(X, (m, 1, n))
X2 = np.zeros((m, 1))
for i in range(twod(Z).shape[2]):
X2 = cols((X2, X * Z[:, :, i]))
X2 = X2[:, 1:]
idx = np.where((twod(arr(range(1, n + 1))).T >= arr(range(
1, n + 1))).T.ravel())[0]
K = len(idx)
Xext[:, k:k + K] = X2[:, idx]
return Xext[:, 0:k + K]
for p in range(degree):
# workaround to make up for numpy's lack of bsxfun
if type(Xcur) is int:
Xcur = X * Xcur
else:
new_Xcur = np.zeros((m, 1))
for i in range(Xcur.shape[2]):
new_Xcur = cols((new_Xcur, X * Xcur[:, :, i]))
Xcur = new_Xcur[:, 1:]
Xcur = Xcur.reshape((m, np.size(Xcur) / m))
K = Xcur.shape[1]
Xext[:, k:k + K] = Xcur
k = k + K
Xcur = Xcur.reshape((m, 1, K))
return Xext
def train(self,
X,
Y,
init='zeros',
stepsize=.01,
stopTol=1e-4,
stopIter=5000):
"""Train the neural network.
Args:
X : MxN numpy array containing M data points with N features each
Y : Mx1 numpy array of targets (class labels) for each data point in X
sizes : [Nin, Nh1, ... , Nout]
Nin is the number of features, Nout is the number of outputs,
which is the number of classes. Member weights are {W1, ... , WL-1},
where W1 is Nh1 x Nin, etc.
init : str
'none', 'zeros', or 'random'. inits the neural net weights.
stepsize : scalar
The stepsize for gradient descent (decreases as 1 / iter).
stopTol : scalar
Tolerance for stopping criterion.
stopIter : int
The maximum number of steps before stopping.
activation : str
'logistic', 'htangent', or 'custom'. Sets the activation functions.
"""
if self.wts[0].shape[1] - 1 != len(X[0]):
raise ValueError(
'layer[0] must equal the number of columns of X (number of features)'
)
self.classes = self.classes if len(self.classes) else np.unique(Y)
if len(self.classes) != self.wts[-1].shape[
0]: # and (self.wts[-1].shape[0]!=1 or len(self.classes)!=2):
raise ValueError(
'layers[-1] must equal the number of classes in Y, or 1 for binary Y'
)
M, N = mat(X).shape # d = dim of data, n = number of data points
C = len(self.classes) # number of classes
L = len(self.wts) # get number of layers
Y_tr_k = to1ofK(Y, self.classes) # convert Y to 1-of-K format
# outer loop of stochastic gradient descent
it = 1 # iteration number
nextPrint = 1 # next time to print info
done = 0 # end of loop flag
J01, Jsur = [], [] # misclassification rate & surrogate loss values
while not done:
step_i = float(
stepsize) / it # step size evolution; classic 1/t decrease
# stochastic gradient update (one pass)
for j in range(M):
A, Z = self.__responses(twod(
X[j, :])) # compute all layers' responses, then backdrop
delta = (Z[L] - Y_tr_k[j, :]) * arr(self.dSig0(
Z[L])) # take derivative of output layer
for l in range(L - 1, -1, -1):
grad = delta.T.dot(
Z[l]) # compute gradient on current layer wts
delta = delta.dot(self.wts[l]) * arr(self.dSig(
Z[l])) # propagate gradient down
delta = delta[:, 1:] # discard constant feature
self.wts[
l] -= step_i * grad # take gradient step on current layer wts
J01.append(self.err_k(X, Y_tr_k)) # error rate (classification)
Jsur.append(self.mse_k(X, Y_tr_k)) # surrogate (mse on output)
if it >= nextPrint:
print('it {} : Jsur = {}, J01 = {}'.format(
it, Jsur[-1], J01[-1]))
nextPrint *= 2
# check if finished
done = (it > 1) and (np.abs(Jsur[-1] - Jsur[-2]) <
stopTol) or it >= stopIter
it += 1
if Y is not None:
Y = arr(Y).flatten()
ny = len(Y)
if ny > 0:
assert ny == nx, 'splitData: X and Y must have the same length'
Ytr, Yte = Y[:ne], Y[ne:]
to_return += (Ytr, Yte)
return to_return
if __name__ == "__main__":
amazon = np.genfromtxt("amazon_cells_labelled.txt",
delimiter="\t", dtype=None, encoding='utf-8')
# amazon = [(string,0),(string,1)....]
tupleX, tupleY = zip(*amazon)
#tupleX = (string1, string2, ...)
#tupleY = (1,0,1,1,0,...)
X = np.array(tupleX)
Y = np.array(tupleY)
X = twod(X).T
Xtr, Xte, Ytr, Yte = splitData(X, Y, .75)
"""
model = Sequential()
model.add(Dense(256,activation='relu',input_shape=(999,),W_regularizer=12(0.001)))
model.add(Dense(256,activation='relu'))
model.add(Dense(10,activation='softmax'))
model.summary()
model.compile(loss='categorical_crossentropy',optimizer='adam',metrics=['accuracy'])
"""
def plotClassify2D(learner,
X,
Y,
pre=lambda x: x,
axis=None,
nGrid=128,
**kwargs):
"""
Plot data and classifier outputs on two-dimensional data.
This function plot data (X,Y) and learner.predict(X, Y)
together. The learner is is predicted on a dense grid
covering data X, to show its decision boundary.
Parameters
----------
learner : learner object
A trained learner object that inherits from one of
the 'Classify' or 'Regressor' base classes.
X : numpy array
N x M array of data; N = number of data, M = dimension
(number of features) of data.
Y : numpy array
1 x N arra containing labels corresponding to data points
in X.
pre : function object (optional)
Function that is applied to X before prediction.
axis : a matplotlib axis / plottable object (optional)
nGrid : density of 2D grid points (default 128)
"""
if twod(X).shape[1] != 2:
raise ValueError(
'plotClassify2D: function can only be called using two-dimensional data (features)'
)
# TODO: Clean up code
if axis == None: axis = plt
# hld = axis.ishold();
# axis.hold(True);
axis.plot(X[:, 0], X[:, 1], 'k.', visible=False)
# TODO: can probably replace with final dot plot and use transparency for image (?)
ax = axis.axis()
xticks = np.linspace(ax[0], ax[1], nGrid)
yticks = np.linspace(ax[2], ax[3], nGrid)
grid = np.meshgrid(xticks, yticks)
XGrid = np.column_stack((grid[0].flatten(), grid[1].flatten()))
if learner is not None:
YGrid = learner.predict(pre(XGrid))
#axis.contourf( xticks,yticks,YGrid.reshape( (len(xticks),len(yticks)) ), nClasses )
axis.imshow(YGrid.reshape((len(xticks), len(yticks))),
extent=ax,
interpolation='nearest',
origin='lower',
alpha=0.5,
aspect='auto')
cmap = plt.cm.get_cmap()
# TODO: if Soft: predictSoft; get colors for each class from cmap; blend pred with colors & show
#
try:
classes = np.array(learner.classes)
except Exception:
classes = np.unique(Y)
cvals = (classes - min(classes)) / (max(classes) - min(classes) + 1e-100)
for i, c in enumerate(classes):
axis.plot(X[Y == c, 0],
X[Y == c, 1],
'ko',
color=cmap(cvals[i]),
**kwargs)
axis.axis(ax)
def gmmEM(X, K, init='random', max_iter=100, tol=1e-6):
"""
Perform Gaussian mixture EM (expectation-maximization) clustering on data X.
Parameters
----------
X : numpy array
N x M array containing data to be clustered.
K : int
Number of clusters.
init : str or array (optional)
Either a K x N numpy array containing initial clusters, or
one of the following strings that specifies a cluster init
method: 'random' (K random data points (uniformly) as clusters)
'farthest' (choose cluster 1 uniformly, then the point farthest
from all cluster so far, etc.)
'k++' (choose cluster 1
uniformly, then points randomly proportional to distance from
current clusters).
max_iter : int (optional)
Maximum number of iterations.
tol : scalar (optional)
Stopping tolerance.
Returns
-------
z : 1 x N numpy array of cluster assignments (int indices).
T : {'pi': np.array, 'mu': np.array, 'sig': np.array} : Gaussian component parameters
soft : numpy array; soft assignment probabilities (rounded for assign)
ll : float; Log-likelihood under the returned model.
"""
# init
N, D = twod(X).shape # get data size
if type(init) is str:
init = init.lower()
if init == 'random':
pi = np.random.permutation(N)
mu = X[pi[0:K], :]
elif init == 'farthest':
mu = k_init(X, K, True)
elif init == 'k++':
mu = k_init(X, K, False)
else:
raise ValueError('gmmEM: value for "init" ( ' + init +
') is invalid')
else:
mu = init
sig = np.zeros((D, D, K))
for c in range(K):
sig[:, :, c] = np.eye(D)
alpha = np.ones(K) / K
R = np.zeros((N, K))
iter, ll, ll_old = 1, np.inf, np.inf
done = iter > max_iter
C = np.log(2 * np.pi) * D / 2
while not done:
ll = 0
for c in range(K):
# compute log prob of all data under model c
V = X - np.tile(mu[c, :], (N, 1))
R[:, c] = -0.5 * np.sum(
(V.dot(np.linalg.inv(sig[:, :, c]))) * V,
axis=1) - 0.5 * np.log(np.linalg.det(sig[:, :, c])) + np.log(
alpha[c]) - C
# avoid numberical issued by removing constant 1st
mx = R.max(1)
R -= np.tile(twod(mx).T, (1, K))
# exponentiate and compute sum over components
R = np.exp(R)
nm = R.sum(1)
# update log-likelihood of data
ll = np.sum(np.log(nm) + mx)
R /= np.tile(twod(nm).T,
(1, K)) # normalize to give membership probabilities
alpha = R.sum(0) # total weight for each component
for c in range(K):
# weighted mean estimate
mu[c, :] = (R[:, c] / alpha[c]).T.dot(X)
tmp = X - np.tile(mu[c, :], (N, 1))
# weighted covar estimate
sig[:, :, c] = tmp.T.dot(
tmp * np.tile(twod(R[:, c]).T / alpha[c],
(1, D))) + 1e-32 * np.eye(D)
alpha /= N
# stopping criteria
done = (iter >= max_iter) or np.abs(ll - ll_old) < tol
ll_old = ll
iter += 1
z = from1ofK(R)
soft = R
T = {'pi': alpha, 'mu': mu, 'sig': sig}
return z, T, soft, ll
def data_GMM(N, C, D=2, get_Z=False):
"""
Sample data from a Gaussian mixture model. Draws N data x_i from a mixture
of Gaussians, with C clusters in D dimensions.
Parameters
----------
N : int
Number of data to be drawn from a mixture of Gaussians.
C : int
Number of clusters.
D : int
Number of dimensions.
get_Z : bool
If True, returns a an array indicating the cluster from which each
data point was drawn.
Returns
-------
X : numpy array
N x D array of data.
Z : numpy array (optional)
1 x N array of cluster ids.
TODO: test more
"""
C += 1
pi = np.zeros(C)
for c in range(C):
pi[c] = gamrand(10, 0.5)
pi = pi / np.sum(pi)
cpi = np.cumsum(pi)
rho = np.random.rand(D, D)
rho = rho + twod(rho).T
rho = rho + D * np.eye(D)
rho = sqrtm(rho)
mu = mat(np.random.randn(c, D)) * mat(rho)
ccov = []
for i in range(C):
tmp = np.random.rand(D, D)
tmp = tmp + tmp.T
tmp = 0.5 * (tmp + D * np.eye(D))
ccov.append(sqrtm(tmp))
p = np.random.rand(N)
Z = np.ones(N)
for c in range(C - 1):
Z[p > cpi[c]] = c
Z = Z.astype(int)
X = mu[Z, :]
for c in range(C):
X[Z == c, :] = X[Z == c, :] + mat(np.random.randn(np.sum(Z == c),
D)) * mat(ccov[c])
if get_Z:
return (arr(X), Z)
else:
return arr(X)
def train(self,
X,
Y,
initStep=1.0,
stopTol=1e-4,
stopIter=5000,
plot=None):
""" Train the logistic regression using stochastic gradient descent """
## First do some bookkeeping and setup:
self.theta, X, Y = twod(self.theta), arr(X), arr(
Y) # convert to numpy arrays
M, N = X.shape
if Y.shape[0] != M:
raise ValueError("Y must have the same number of data (rows) as X")
self.classes = np.unique(Y)
if len(self.classes) != 2:
raise ValueError(
"Y should have exactly two classes (binary problem expected)")
if self.theta.shape[
1] != N + 1: # if self.theta is empty, initialize it!
self.theta = np.random.randn(1, N + 1)
# Some useful modifications of the data matrices:
X1 = np.hstack((np.ones(
(M, 1)), X)) # make data array with constant feature
Y01 = toIndex(Y,
self.classes) # convert Y to canonical "0 vs 1" classes
it = 0
done = False
Jsur = []
J01 = []
while not done:
step = (2.0 * initStep) / (2.0 + it
) # common 1/iter step size change
si = []
for i in range(M): # for each data point i:
## Computing the linear response
zi = X1[i, :].dot(self.theta.T)
## Computing the prediction yi
yi = Y01[i]
## Computing soft response
si.append(self.logistic(zi))
## Computing gradient of logistic loss
gradi = (si[i] - yi) * X1[i, :]
# Take a step down the gradient
self.theta = self.theta - step * gradi
# each pass, compute surrogate loss & error rates:
J01.append(self.err(X, Y))
## Computing surrogate loss
sum_i = 0
for i in range(M):
sum_i += Y01[i] * si[i] * np.log(
si[i]) + (1 - Y01[i]) * (1 - si[i]) * np.log(1 - si[i])
Jsur.append(sum_i / M) ## TODO ...
## For debugging: print current parameters & losses
# print self.theta, ' => ', Jsur[-1], ' / ', J01[-1]
# raw_input() # pause for keystroke
# check stopping criteria:
it += 1
done = (it > stopIter) or ((it > 1) and
(abs(Jsur[-1] - Jsur[-2]) < stopTol))
self.numberOfIterations = it
if self.plotFlag == True:
plt.semilogx(range(it), np.abs(Jsur), label='Surrogate Loss')
plt.semilogx(range(it), np.abs(J01), label='Error Rate')
plt.legend(loc='upper right')
plt.xlabel('# of iterations')
plt.ylabel('Losses')
plt.show()
def plotClassify2D(learner,
X,
Y,
pre=lambda x: x,
ax=None,
nGrid=128,
cm=None,
bgalpha=0.3,
soft=False,
**kwargs):
"""
Plot data and classifier outputs on two-dimensional data.
This function plots data (X,Y) and learner.predict(X, Y)
together. The learner is is predicted on a dense grid
covering the data X, to show its decision boundary.
Parameters
----------
learner : a classifier with "predict" function and optionally "classes" list
X : (m,n) numpy array of data (m points in n=2 dimension)
Y : (m,) or (m,1) int array of class values OR (m,c) array of class probabilities (see predictSoft)
pre : function object (optional) applied to X before learner.predict()
ax : a matplotlib axis / plottable object (optional)
nGrid : density of 2D grid points (default 128)
soft : use predictSoft & blend colors (default: False => use predict() and show decision regions)
bgalpha: alpha transparency (1=opaque, 0=transparent) for decision function image
cm : pyplot colormap (default: None = use default colormap)
[other arguments will be passed through to the pyplot scatter function on the data points]
"""
if twod(X).shape[1] != 2:
raise ValueError(
'plotClassify2D: function can only be called using two-dimensional data (features)'
)
# make robust to differing arguments in scatter vs plot, e.g. "s"/"ms" (marker size)
if "s" not in kwargs and "ms" in kwargs: kwargs["s"] = kwargs.pop("ms")
try:
classes = np.array(learner.classes)
# learner has explicit list of classes; use those
except Exception:
if len(Y.shape) == 1 or Y.shape[1] == 1:
classes = np.unique(
Y) # or, use data points' class values to guess
else:
classes = np.arange(Y.shape[1], dtype=int)
# or, get number of classes from soft predictions
vmin, vmax = classes.min() - .1, classes.max() + .1
# get (slightly expanded) value range for class values
if ax is None:
ax = plt.gca()
# default: use current axes
if cm is None:
cm = plt.cm.get_cmap()
# get the colormap
classvals = (classes - vmin) / (vmax - vmin + 1e-100)
# map class values to [0,1] for colormap
classcolor = cm(classvals)
# and get the RGB values for each class
ax.plot(X[:, 0], X[:, 1], 'k.', visible=False, ms=0)
# invisible plot to set axis range if required
axrng = ax.axis()
if learner is not None: # if we were given a learner to predict with:
xticks, yticks = np.linspace(axrng[0], axrng[1], nGrid), np.linspace(
axrng[2], axrng[3], nGrid)
grid = np.meshgrid(xticks, yticks)
# apply it to a dense grid of points
XGrid = np.column_stack((grid[0].flatten(), grid[1].flatten()))
if soft:
YGrid = learner.predictSoft(pre(XGrid)).dot(classcolor)
# soft prediction: blend class colors
YGrid[YGrid < 0] = 0
YGrid[YGrid > 1] = 1
YGrid = YGrid.reshape((nGrid, nGrid, classcolor.shape[1]))
#axis.contourf( xticks,yticks,YGrid[:,0].reshape( (len(xticks),len(yticks)) ), nContours )
ax.imshow(YGrid,
extent=axrng,
interpolation='nearest',
origin='lower',
alpha=bgalpha,
aspect='auto',
vmin=vmin,
vmax=vmax,
cmap=cm)
else:
YGrid = learner.predict(pre(XGrid)).reshape((nGrid, nGrid))
# hard prediction: use class colors
vmin, vmax = min(YGrid.min() - .1,
vmin), max(YGrid.max() + .1,
vmax) # check outputs for new classes?
classvals = (classes - vmin) / (vmax - vmin + 1e-100)
classcolor = cm(classvals)
# if so, recalc colors?
#axis.contourf( xticks,yticks,YGrid.reshape( (len(xticks),len(yticks)) ), nClasses )
ax.imshow(YGrid,
extent=axrng,
interpolation='nearest',
origin='lower',
alpha=bgalpha,
aspect='auto',
vmin=vmin,
vmax=vmax,
cmap=cm)
if len(Y.shape) == 1 or Y.shape[1] == 1:
data_colors = classcolor[np.searchsorted(classes, Y)]
# use colors if Y is discrete class
else:
data_colors = Y.dot(classcolor)
data_colors[data_colors > 1] = 1
# blend colors if Y is a soft confidence
ax.scatter(X[:, 0], X[:, 1], c=data_colors, **kwargs)
def train(self,
X,
Y,
init='zeros',
stepsize=.01,
stopTol=1e-4,
stopIter=5000):
"""Train the neural network.
Args:
X : MxN numpy array containing M data points with N features each
Y : Mx1 numpy array of targets for each data point in X
sizes (list of int): [Nin, Nh1, ... , Nout]
Nin is the number of features, Nout is the number of outputs,
which is the number of target dimensions (usually 1). Weights are {W1, ... , WL-1},
where W1 is Nh1 x Nin, etc.
init (str): 'none', 'zeros', or 'random'. inits the neural net weights.
stepsize (float): The stepsize for gradient descent (decreases as 1 / iter).
stopTol (float): Tolerance for stopping criterion.
stopIter (int): The maximum number of steps before stopping.
activation (str): 'logistic', 'htangent', or 'custom'. Sets the activation functions.
"""
if self.wts[0].shape[1] - 1 != len(X[0]):
raise ValueError(
'layer[0] must equal the number of columns of X (number of features)'
)
if self.wts[-1].shape[0] > 1 and self.wts[-1].shape[0] != Y.shape[1]:
raise ValueError(
'layers[-1] must equal the number of classes in Y, or 1 for binary Y'
)
M, N = arr(X).shape # d = dim of data, n = number of data points
L = len(self.wts) # get number of layers
Y = arr(Y)
Y2d = Y if len(Y.shape) > 1 else Y[:, np.newaxis]
# outer loop of stochastic gradient descent
it = 1 # iteration number
nextPrint = 1 # next time to print info
done = 0 # end of loop flag
Jsur = [] # misclassification rate & surrogate loss values
while not done:
step_i = (2.0 * stepsize) / (
2.0 + it) # step size evolution; classic 1/t decrease
# stochastic gradient update (one pass)
for j in range(M):
A, Z = self.__responses(twod(
X[j, :])) # compute all layers' responses, then backdrop
delta = (Z[L] - Y2d[j, :]) * arr(self.dSig0(
Z[L])) # take derivative of output layer
for l in range(L - 1, -1, -1):
grad = delta.T.dot(
Z[l]) # compute gradient on current layer wts
delta = delta.dot(self.wts[l]) * arr(self.dSig(
Z[l])) # propagate gradient down
delta = delta[:, 1:] # discard constant feature
self.wts[
l] -= step_i * grad # take gradient step on current layer wts
Jsur.append(self.mse(X, Y2d)) # surrogate (mse on output)
if it >= nextPrint:
print('it {} : J = {}'.format(it, Jsur[-1]))
nextPrint *= 2
# check if finished
done = (it > 1) and (np.abs(Jsur[-1] - Jsur[-2]) <
stopTol) or it >= stopIter
it += 1
def train(self, X, Y, initStep=1.0, stopTol=1e-4, stopIter=5000, plot=None):
""" Train the logistic regression using stochastic gradient descent """
## First do some bookkeeping and setup:
self.theta,X,Y = twod(self.theta), arr(X), arr(Y) # convert to numpy arrays
M,N = X.shape
if Y.shape[0] != M:
raise ValueError("Y must have the same number of data (rows) as X")
self.classes = np.unique(Y)
if len(self.classes) != 2:
raise ValueError("Y should have exactly two classes (binary problem expected)")
if self.theta.shape[1] != N+1: # if self.theta is empty, initialize it!
self.theta = np.random.randn(1,N+1)
# Some useful modifications of the data matrices:
X1 = np.hstack((np.ones((M,1)),X)) # make data array with constant feature
Y01 = toIndex(Y, self.classes) # convert Y to canonical "0 vs 1" classes
it = 0
done = False
Jsur = []
J01 = []
Jloss = 0
while not done:
step = (2.0 * initStep) / (2.0 + it) # common 1/iter step size change
for i in range(M): # for each data point i:
## compute zi = linear response of X[i,:]
zi = np.dot(np.array(X1[i,:]),np.transpose(self.theta))
## compute prediction yi
## compute soft response si = logistic(zi)
si = (math.exp(zi))/(1+math.exp(zi))
if si >= 0.5:
yi = 1
else:
yi = 0
## compute gradient of logistic loss wrt data point i:
gradi = (si - Y01[i])*X1[i,:]
# Take a step down the gradient
self.theta = self.theta - step * gradi
Ji = -yi*math.log(si) - (1-yi)*math.log(1-si)
Jloss = Jloss+Ji
if it == 0:
self.theta = list(self.theta[0])
elif it >= 1:
self.theta = list(self.theta)
# each pass, compute surrogate loss & error rates:
J01.append( self.err(X,Y) )
## compute surrogate loss (logistic negative log-likelihood)
Jloss = Jloss/M
## Jsur = sum_i [ (si log si) if yi==1 else ((1-si)log(1-si)) ]
Jsur.append(Jloss) ## TODO ...
## For debugging: print current parameters & losses
# print(self.theta, ' => ', Jsur[-1], ' / ', J01[-1])
# raw_input() # pause for keystroke
# check stopping criteria:
it += 1
done = (it > stopIter) or ( (it>1) and (abs(Jsur[-1]-Jsur[-2])<stopTol) )
return [it, J01, Jsur]