# Copyright 2001 by Jeffrey Chang. All rights reserved. # This code is part of the Biopython distribution and governed by its # license. Please see the LICENSE file that should have been included # as part of this package. """ Maximum Entropy code. Uses Improved Iterative Scaling: XXX ref # XXX need to define terminology """ import numpy # XXX typecodes for Numeric # XXX multiprocessor MAX_IIS_ITERATIONS = 10000 # Maximum iterations for IIS. IIS_CONVERGE = 1E-5 # Convergence criteria for IIS. MAX_NEWTON_ITERATIONS = 100 # Maximum iterations on Newton's method. NEWTON_CONVERGE = 1E-10 # Convergence criteria for Newton's method. 00027 class MaxEntropy: """Holds information for a Maximum Entropy classifier. Members: classes List of the possible classes of data. alphas List of the weights for each feature. feature_fns List of the feature functions. """ def __init__(self): self.classes = [] self.alphas = [] self.feature_fns = [] def calculate(me, observation): """calculate(me, observation) -> list of log probs Calculate the log of the probability for each class. me is a MaxEntropy object that has been trained. observation is a vector representing the observed data. The return value is a list of unnormalized log probabilities for each class. """ scores = [] for klass in range(len(me.classes)): lprob = 0.0 for fn, alpha in map(None, me.feature_fns, me.alphas): lprob += fn(observation, klass) * alpha scores.append(lprob) return scores def classify(me, observation): """classify(me, observation) -> class Classify an observation into a class. """ scores = calculate(me, observation) max_score, klass = scores[0], me.classes[0] for i in range(1, len(scores)): if scores[i] > max_score: max_score, klass = scores[i], me.classes[i] return klass def _eval_feature_fn(fn, xs, classes): """_eval_feature_fn(fn, xs, classes) -> dict of values Evaluate a feature function on every instance of the training set and class. fn is a callback function that takes two parameters: a training instance and a class. Return a dictionary of (training set index, class index) -> non-zero value. Values of 0 are not stored in the dictionary. """ values = {} for i in range(len(xs)): for j in range(len(classes)): f = fn(xs[i], classes[j]) if f != 0: values[(i, j)] = f return values def _calc_empirical_expects(xs, ys, classes, features): """_calc_empirical_expects(xs, ys, classes, features) -> list of expectations Calculate the expectation of each function from the data. This is the constraint for the maximum entropy distribution. Return a list of expectations, parallel to the list of features. """ # E[f_i] = SUM_x,y P(x, y) f(x, y) # = 1/N f(x, y) class2index = {} for index, key in enumerate(classes): class2index[key] = index ys_i = [class2index[y] for y in ys] expect = [] N = len(xs) for feature in features: s = 0 for i in range(N): s += feature.get((i, ys_i[i]), 0) expect.append(float(s) / N) return expect def _calc_model_expects(xs, classes, features, alphas): """_calc_model_expects(xs, classes, features, alphas) -> list of expectations. Calculate the expectation of each feature from the model. This is not used in maximum entropy training, but provides a good function for debugging. """ # SUM_X P(x) SUM_Y P(Y|X) F(X, Y) # = 1/N SUM_X SUM_Y P(Y|X) F(X, Y) p_yx = _calc_p_class_given_x(xs, classes, features, alphas) expects = [] for feature in features: sum = 0.0 for (i, j), f in feature.items(): sum += p_yx[i][j] * f expects.append(sum/len(xs)) return expects def _calc_p_class_given_x(xs, classes, features, alphas): """_calc_p_class_given_x(xs, classes, features, alphas) -> matrix Calculate P(y|x), where y is the class and x is an instance from the training set. Return a XSxCLASSES matrix of probabilities. """ prob_yx = numpy.zeros((len(xs), len(classes))) # Calculate log P(y, x). for feature, alpha in map(None, features, alphas): for (x, y), f in feature.items(): prob_yx[x][y] += alpha * f # Take an exponent to get P(y, x) prob_yx = numpy.exp(prob_yx) # Divide out the probability over each class, so we get P(y|x). for i in range(len(xs)): z = sum(prob_yx[i]) prob_yx[i] = prob_yx[i] / z #prob_yx = [] #for i in range(len(xs)): # z = 0.0 # Normalization factor for this x, over all classes. # probs = [0.0] * len(classes) # for j in range(len(classes)): # log_p = 0.0 # log of the probability of f(x, y) # for k in range(len(features)): # log_p += alphas[k] * features[k].get((i, j), 0.0) # probs[j] = numpy.exp(log_p) # z += probs[j] # # Normalize the probabilities for this x. # probs = map(lambda x, z=z: x/z, probs) # prob_yx.append(probs) return prob_yx def _calc_f_sharp(N, nclasses, features): """_calc_f_sharp(N, nclasses, features) -> matrix of f sharp values.""" # f#(x, y) = SUM_i feature(x, y) f_sharp = numpy.zeros((N, nclasses)) for feature in features: for (i, j), f in feature.items(): f_sharp[i][j] += f return f_sharp def _iis_solve_delta(N, feature, f_sharp, empirical, prob_yx): # Solve delta using Newton's method for: # SUM_x P(x) * SUM_c P(c|x) f_i(x, c) e^[delta_i * f#(x, c)] = 0 delta = 0.0 iters = 0 while iters < MAX_NEWTON_ITERATIONS: # iterate for Newton's method f_newton = df_newton = 0.0 # evaluate the function and derivative for (i, j), f in feature.items(): prod = prob_yx[i][j] * f * numpy.exp(delta * f_sharp[i][j]) f_newton += prod df_newton += prod * f_sharp[i][j] f_newton, df_newton = empirical - f_newton / N, -df_newton / N ratio = f_newton / df_newton delta -= ratio if numpy.fabs(ratio) < NEWTON_CONVERGE: # converged break iters = iters + 1 else: raise RuntimeError("Newton's method did not converge") return delta def _train_iis(xs, classes, features, f_sharp, alphas, e_empirical): # Do one iteration of hill climbing to find better alphas. # This is a good function to parallelize. # Pre-calculate P(y|x) p_yx = _calc_p_class_given_x(xs, classes, features, alphas) N = len(xs) newalphas = alphas[:] for i in range(len(alphas)): delta = _iis_solve_delta(N, features[i], f_sharp, e_empirical[i], p_yx) newalphas[i] += delta return newalphas def train(training_set, results, feature_fns, update_fn=None): """train(training_set, results, feature_fns[, update_fn]) -> MaxEntropy object Train a maximum entropy classifier on a training set. training_set is a list of observations. results is a list of the class assignments for each observation. feature_fns is a list of the features. These are callback functions that take an observation and class and return a 1 or 0. update_fn is a callback function that's called at each training iteration. It is passed a MaxEntropy object that encapsulates the current state of the training. """ if not len(training_set): raise ValueError("No data in the training set.") if len(training_set) != len(results): raise ValueError("training_set and results should be parallel lists.") # Rename variables for convenience. xs, ys = training_set, results # Get a list of all the classes that need to be trained. classes = list(set(results)) classes.sort() # Cache values for all features. features = [_eval_feature_fn(fn, training_set, classes) for fn in feature_fns] # Cache values for f#. f_sharp = _calc_f_sharp(len(training_set), len(classes), features) # Pre-calculate the empirical expectations of the features. e_empirical = _calc_empirical_expects(xs, ys, classes, features) # Now train the alpha parameters to weigh each feature. alphas = [0.0] * len(features) iters = 0 while iters < MAX_IIS_ITERATIONS: nalphas = _train_iis(xs, classes, features, f_sharp, alphas, e_empirical) diff = map(lambda x, y: numpy.fabs(x-y), alphas, nalphas) diff = reduce(lambda x, y: x+y, diff, 0) alphas = nalphas me = MaxEntropy() me.alphas, me.classes, me.feature_fns = alphas, classes, feature_fns if update_fn is not None: update_fn(me) if diff < IIS_CONVERGE: # converged break else: raise RuntimeError("IIS did not converge") return me if __name__ == "__main__" : #Car data from example Naive Bayes Classifier example by Eric Meisner November 22, 2003 #http://www.inf.u-szeged.hu/~ormandi/teaching/mi2/02-naiveBayes-example.pdf xcar=[ ['Red', 'Sports', 'Domestic'], ['Red', 'Sports', 'Domestic'], ['Red', 'Sports', 'Domestic'], ['Yellow', 'Sports', 'Domestic'], ['Yellow', 'Sports', 'Imported'], ['Yellow', 'SUV', 'Imported'], ['Yellow', 'SUV', 'Imported'], ['Yellow', 'SUV', 'Domestic'], ['Red', 'SUV', 'Imported'], ['Red', 'Sports', 'Imported'] ] ycar=[ 'Yes', 'No', 'Yes', 'No', 'Yes', 'No', 'Yes', 'No', 'No', 'Yes' ] #Requires some rules or features def udf1(ts, cl): if ts[0] =='Red': return 0 else: return 1 def udf2(ts, cl): if ts[1] =='Sports': return 0 else: return 1 def udf3(ts, cl): if ts[2] =='Domestic': return 0 else: return 1 user_functions=[udf1, udf2, udf3] # must be an iterable type xe=train(xcar, ycar, user_functions) for xv,yv in zip(xcar, ycar): xc=classify(xe, xv) print 'Pred:', xv, 'gives', xc, 'y is', yv

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