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本讲介绍模型复杂度怎样影响预测精度和计算性能。我们使用的数据集仍然是波士顿房价数据集。对于模型的每一类，我们通过选择有关的模型参数，度量计算性能和预测功效的影响，以此考察模型的复杂度。下面，我们用Python代码解释原理。
代码详解
首先，加载必须的Python函数库。
print(__doc__)
# Author: Eustache Diemert <[email protected]>
# License: BSD 3 clause
import time
import numpy as np
import matplotlib.pyplot as plt
from mpl_toolkits.axes_grid1.parasite_axes import host_subplot
from mpl_toolkits.axisartist.axislines import Axes
from scipy.sparse.csr import csr_matrix
from sklearn import datasets
from sklearn.utils import shuffle
from sklearn.metrics import mean_squared_error
from sklearn.svm.classes import NuSVR
from sklearn.ensemble.gradient_boosting import GradientBoostingRegressor
from sklearn.linear_model.stochastic_gradient import SGDClassifier
from sklearn.metrics import hamming_loss
为了产生模拟数据集，初始化随机数生成器。
np.random.seed(0)
定义函数generate_data, 用来产生用于回归或分类问题的模拟数据。该函数有两个输入参数，其中，case指定回归还是分类问题，sparse是一个逻辑参数，用来指定数据是否是稀疏的，该参数默认值为False. 函数输出模拟的数据集。
def generate_data(case, sparse=False):
"""Generate regression/classification data."""
bunch = None
if case == 'regression':
bunch = datasets.load_boston()
elif case == 'classification':
bunch = datasets.fetch_20newsgroups_vectorized(subset='all')
X, y = shuffle(bunch.data, bunch.target)
offset = int(X.shape[0] * 0.8)
X_train, y_train = X[:offset], y[:offset]
X_test, y_test = X[offset:], y[offset:]
if sparse:
X_train = csr_matrix(X_train)
X_test = csr_matrix(X_test)
else:
X_train = np.array(X_train)
X_test = np.array(X_test)
y_test = np.array(y_test)
y_train = np.array(y_train)
data = {'X_train': X_train, 'X_test': X_test, 'y_train': y_train,
'y_test': y_test}
return data
定义函数benchmark_influence, 它的作用是确立一个影响标准，用以评价模型复杂度。在这里，我们用latency度量计算性能，用MSE度量预测功效。改变这两个参数的值，评价模型的复杂度影响。因此，该函数有一个字典型输入参数conf, 指定要改变的参数类型和参数值。函数输出预测功效prediction_powers, 程序运行时间prediction_times, 模型复杂度complexities.
def benchmark_influence(conf):
"""
Benchmark influence of :changing_param: on both MSE and latency.
"""
prediction_times = []
prediction_powers = []
complexities = []
for param_value in conf['changing_param_values']:
conf['tuned_params'][conf['changing_param']] = param_value
estimator = conf['estimator'](**conf['tuned_params'])
print("Benchmarking %s" % estimator)
estimator.fit(conf['data']['X_train'], conf['data']['y_train'])
conf['postfit_hook'](estimator)
complexity = conf['complexity_computer'](estimator)
complexities.append(complexity)
start_time = time.time()
for _ in range(conf['n_samples']):
y_pred = estimator.predict(conf['data']['X_test'])
elapsed_time = (time.time() - start_time) / float(conf['n_samples'])
prediction_times.append(elapsed_time)
pred_score = conf['prediction_performance_computer'](
conf['data']['y_test'], y_pred)
prediction_powers.append(pred_score)
print("Complexity: %d | %s: %.4f | Pred. Time: %fs\n" % (
complexity, conf['prediction_performance_label'], pred_score,
elapsed_time))
return prediction_powers, prediction_times, complexities
函数plot_influence以accuracy, latency为变量，可视化模型复杂度。
def plot_influence(conf, mse_values, prediction_times, complexities):
"""
Plot influence of model complexity on both accuracy and latency.
"""
plt.figure(figsize=(12, 6))
host = host_subplot(111, axes_class=Axes)
plt.subplots_adjust(right=0.75)
par1 = host.twinx()
host.set_xlabel('Model Complexity (%s)' % conf['complexity_label'])
y1_label = conf['prediction_performance_label']
y2_label = "Time (s)"
host.set_ylabel(y1_label)
par1.set_ylabel(y2_label)
p1, = host.plot(complexities, mse_values, 'b-', label="prediction error")
p2, = par1.plot(complexities, prediction_times, 'r-',
label="latency")
host.legend(loc='upper right')
host.axis["left"].label.set_color(p1.get_color())
par1.axis["right"].label.set_color(p2.get_color())
plt.title('Influence of Model Complexity - %s' % conf['estimator'].__name__)
plt.show()
函数_count_nonzero_coefficients统计估计量的非零系数。
def _count_nonzero_coefficients(estimator):
a = estimator.coef_.toarray()
return np.count_nonzero(a)
下面，我们分别模拟波士顿房价回归和分类的数据集，评价回归和分类预测模型的复杂度影响。
regression_data = generate_data('regression')
classification_data = generate_data('classification', sparse=True)
configurations = [
{'estimator': SGDClassifier,
'tuned_params': {'penalty': 'elasticnet', 'alpha': 0.001, 'loss':
'modified_huber', 'fit_intercept': True, 'tol': 1e-3},
'changing_param': 'l1_ratio',
'changing_param_values': [0.25, 0.5, 0.75, 0.9],
'complexity_label': 'non_zero coefficients',
'complexity_computer': _count_nonzero_coefficients,
'prediction_performance_computer': hamming_loss,
'prediction_performance_label': 'Hamming Loss (Misclassification Ratio)',
'postfit_hook': lambda x: x.sparsify(),
'data': classification_data,
'n_samples': 30},
{'estimator': NuSVR,
'tuned_params': {'C': 1e3, 'gamma': 2 ** -15},
'changing_param': 'nu',
'changing_param_values': [0.1, 0.25, 0.5, 0.75, 0.9],
'complexity_label': 'n_support_vectors',
'complexity_computer': lambda x: len(x.support_vectors_),
'data': regression_data,
'postfit_hook': lambda x: x,
'prediction_performance_computer': mean_squared_error,
'prediction_performance_label': 'MSE',
'n_samples': 30},
{'estimator': GradientBoostingRegressor,
'tuned_params': {'loss': 'ls'},
'changing_param': 'n_estimators',
'changing_param_values': [10, 50, 100, 200, 500],
'complexity_label': 'n_trees',
'complexity_computer': lambda x: x.n_estimators,
'data': regression_data,
'postfit_hook': lambda x: x,
'prediction_performance_computer': mean_squared_error,
'prediction_performance_label': 'MSE',
'n_samples': 30},
]
for conf in configurations:
prediction_performances, prediction_times, complexities = \
benchmark_influence(conf)
plot_influence(conf, prediction_performances, prediction_times,
complexities)
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