def root_mean_square_error(y_real, y_pred):
#print "...root_mean_square_error"
"""
It computes the root mean squared difference (RMSE)
between predicted and actual ratings for users.
Parameters
----------
y_real : array-like
y_pred : array-like
Returns
-------
Positive floating point value: the best value is 0.0.
return the mean square error
"""
#istart = time.time()
y_real, y_pred = check_arrays(y_real, y_pred)
rmse = np.sqrt((np.sum((y_pred - y_real) ** 2)) / y_real.shape[0])
#isend = time.time()
#print "rmse: ",rmse , "seconds: ", isend-istart
return rmse
def mean_absolute_error(y_real, y_pred):
#print "...mean_absolute_error"
"""
It computes the average absolute difference (MAE)
between predicted and actual ratings for users.
Parameters
----------
y_real : array-like
y_pred : array-like
Returns
-------
Positive floating point value: the best value is 0.0.
return the mean absolute error
"""
istart = time.time()
y_real, y_pred = check_arrays(y_real, y_pred)
mae = np.sum(np.abs(y_pred - y_real)) / y_real.size
isend = time.time()
#print "mae: ",mae , "seconds: ", isend-istart
return mae
def normalized_mean_absolute_error(y_real, y_pred, max_rating, min_rating):
"""
It computes the normalized average absolute difference (NMAE)
between predicted and actual ratings for users.
Parameters
----------
y_real : array-like
The real ratings.
y_pred : array-like
The predicted ratings.
max_rating:
The maximum rating of the model.
min_rating:
The minimum rating of the model.
Returns
-------
Positive floating point value: the best value is 0.0.
return the normalized mean absolute error
"""
y_real, y_pred = check_arrays(y_real, y_pred)
mae = mean_absolute_error(y_real, y_pred)
return mae / (max_rating - min_rating)
def lcss_dist(X, Y, delta, epsilon):
"""Compute the LCSS distance between X and Y using Dynamic Programming.
:param X (array): time series feature array denoted by X
:param Y (array): time series feature array denoted by Y
:param delta (int): time sample matching threshold
:param epsilon (float): amplitude matching threshold
:returns: distance between X and Y with the best alignment
:Reference: M Vlachos et al., "Discovering Similar Multidimensional Trajectories", 2002
"""
X, Y = check_arrays(X, Y)
dist = _lcss_dist(X, Y, delta, epsilon)
return dist
def edr_dist(X, Y, epsilon):
"""Compute the EDR distance between X and Y using Dynamic Programming.
:param X (array): time series feature array denoted by X
:param Y (array): time series feature array denoted by Y
:param epsilon (float): matching threshold
:returns: distance between X and Y with the best alignment
:Reference: L. Chen et al., "Robust and Fast Similarity Search for Moving Object Trajectories", 2005.
"""
X, Y = check_arrays(X, Y)
X = standardization(X)
Y = standardization(Y)
dist = _edr_dist(X, Y, epsilon)
return dist
def dtw_dist(X, Y, w=_np.inf, mode="dependent"):
"""Compute multidimensional DTW distance between X and Y using Dynamic Programming.
:param X (array): time series feature array denoted by X
:param Y (array): time series feature array denoted by Y
:param w (int): window size (default=Inf)
:param mode (string): "dependent" or "independent" (default="dependent")
:returns: distance between X and Y with the best alignment
:Reference: https://www.cs.unm.edu/~mueen/DTW.pdf
"""
X, Y = check_arrays(X, Y)
if mode == "dependent":
dist = _dtw_dist(X, Y, w)
elif mode == "independent":
n_feature = X.shape[0]
dist = 0
for i in range(n_feature):
dist += _dtw_dist(X[[i], :], Y[[i], :], w)
else:
raise ValueError(
"The mode must be either \"dependent\" or \"independent\".")
return dist
def root_mean_square_error(y_real, y_pred):
"""
It computes the root mean squared difference (RMSE)
between predicted and actual ratings for users.
Parameters
----------
y_real : array-like
y_pred : array-like
Returns
-------
Positive floating point value: the best value is 0.0.
return the mean square error
"""
y_real, y_pred = check_arrays(y_real, y_pred)
return np.sqrt((np.sum((y_pred - y_real) ** 2)) / y_real.shape[0])
def mean_absolute_error(y_real, y_pred):
"""
It computes the average absolute difference (MAE)
between predicted and actual ratings for users.
Parameters
----------
y_real : array-like
y_pred : array-like
Returns
-------
Positive floating point value: the best value is 0.0.
return the mean absolute error
"""
y_real, y_pred = check_arrays(y_real, y_pred)
return np.sum(np.abs(y_pred - y_real)) / y_real.size if y_real.size != 0 else None
def normalized_mean_absolute_error(y_real, y_pred, max_rating, min_rating):
#print "...normalized_mean_absolute_error"
"""
It computes the normalized average absolute difference (NMAE)
between predicted and actual ratings for users.
Parameters
----------
y_real : array-like
The real ratings.
y_pred : array-like
The predicted ratings.
max_rating:
The maximum rating of the model.
min_rating:
The minimum rating of the model.
Returns
-------
Positive floating point value: the best value is 0.0.
return the normalized mean absolute error
"""
istart = time.time()
y_real, y_pred = check_arrays(y_real, y_pred)
mae = mean_absolute_error(y_real, y_pred)
isend = time.time()
#print "nmae: ",mae / (max_rating - min_rating), "seconds: ", isend-istart
return mae / (max_rating - min_rating)
def precision_recall_fscore(y_real, y_pred, beta=1.0):
"""Compute precisions, recalls, f-measures
for recommender systems
The precision is the ratio :math:`tp / (tp + fp)` where tp is the number of
true positives and fp the number of false positives. In recommender systems
...
The recall is the ratio :math:`tp / (tp + fn)` where tp is the number of
true positives and fn the number of false negatives. In recommender
systems...
The F_beta score can be interpreted as a weighted harmonic mean of
the precision and recall, where an F_beta score reaches its best
value at 1 and worst score at 0.
The F_beta score weights recall beta as much as precision. beta = 1.0 means
recall and precision are as important.
Parameters
----------
y_real : array, shape = [n_samples]
true recommended items
y_pred : array, shape = [n_samples]
predicted recommended items
beta : float, 1.0 by default
the strength of recall versus precision in the f-score
Returns
-------
precision: array, shape = [n_unique_labels], dtype = np.double
recall: array, shape = [n_unique_labels], dtype = np.double
f1_score: array, shape = [n_unique_labels], dtype = np.double
References
----------
http://en.wikipedia.org/wiki/Precision_and_recall
"""
y_real, y_pred = check_arrays(y_real, y_pred)
assert(beta > 0)
n_users = y_real.shape[0]
precision = np.zeros(n_users, dtype=np.double)
recall = np.zeros(n_users, dtype=np.double)
fscore = np.zeros(n_users, dtype=np.double)
try:
# oddly, we may get an "invalid" rather than a "divide" error here
old_err_settings = np.seterr(divide='ignore', invalid='ignore')
for i, y_items_pred in enumerate(y_pred):
intersection_size = np.intersect1d(y_items_pred, y_real[i]).size
precision[i] = (intersection_size / float(len(y_real[i]))) \
if len(y_real[i]) else 0.0
recall[i] = (intersection_size / float(len(y_items_pred))) \
if len(y_items_pred) else 0.0
# handle division by 0.0 in precision and recall
precision[np.isnan(precision)] = 0.0
recall[np.isnan(precision)] = 0.0
#fbeta Score
beta2 = beta ** 2
fscore = (1 + beta2) * (precision * recall) \
/ (beta2 * precision + recall)
#handle division by 0.0 in fscore
fscore[(precision + recall) == 0.0] = 0.0
finally:
np.seterr(**old_err_settings)
return precision, recall, fscore
def smacof(similarities, metric=True, n_components=2, init=None, n_init=8,
n_jobs=1, max_iter=300, verbose=0, eps=1e-3, random_state=None):
"""
Computes multidimensional scaling using SMACOF (Scaling by Majorizing a
Complicated Function) algorithm
The SMACOF algorithm is a multidimensional scaling algorithm: it minimizes
a objective function, the *stress*, using a majorization technique. The
Stress Majorization, also known as the Guttman Transform, guarantees a
monotone convergence of Stress, and is more powerful than traditional
techniques such as gradient descent.
The SMACOF algorithm for metric MDS can summarized by the following steps:
1. Set an initial start configuration, randomly or not.
2. Compute the stress
3. Compute the Guttman Transform
4. Iterate 2 and 3 until convergence.
The nonmetric algorithm adds a monotonic regression steps before computing
the stress.
Parameters
----------
similarities : symmetric ndarray, shape (n_samples, n_samples)
similarities between the points
metric : boolean, optional, default: True
compute metric or nonmetric SMACOF algorithm
n_components : int, optional, default: 2
number of dimension in which to immerse the similarities
overridden if initial array is provided.
init : {None or ndarray of shape (n_samples, n_components)}, optional
if None, randomly chooses the initial configuration
if ndarray, initialize the SMACOF algorithm with this array
n_init : int, optional, default: 8
Number of time the smacof algorithm will be run with different
initialisation. The final results will be the best output of the
n_init consecutive runs in terms of stress.
n_jobs : int, optional, default: 1
The number of jobs to use for the computation. This works by breaking
down the pairwise matrix into n_jobs even slices and computing them in
parallel.
If -1 all CPUs are used. If 1 is given, no parallel computing code is
used at all, which is useful for debugging. For n_jobs below -1,
(n_cpus + 1 + n_jobs) are used. Thus for n_jobs = -2, all CPUs but one
are used.
max_iter : int, optional, default: 300
Maximum number of iterations of the SMACOF algorithm for a single run
verbose : int, optional, default: 0
level of verbosity
eps : float, optional, default: 1e-6
relative tolerance w.r.t stress to declare converge
random_state : integer or numpy.RandomState, optional
The generator used to initialize the centers. If an integer is
given, it fixes the seed. Defaults to the global numpy random
number generator.
Returns
-------
X : ndarray (n_samples,n_components)
Coordinates of the n_samples points in a n_components-space
stress : float
The final value of the stress (sum of squared distance of the
disparities and the distances for all constrained points)
Notes
-----
"Modern Multidimensional Scaling - Theory and Applications" Borg, I.;
Groenen P. Springer Series in Statistics (1997)
"Nonmetric multidimensional scaling: a numerical method" Kruskal, J.
Psychometrika, 29 (1964)
"Multidimensional scaling by optimizing goodness of fit to a nonmetric
hypothesis" Kruskal, J. Psychometrika, 29, (1964)
"""
similarities, = check_arrays(similarities, sparse_format='dense')
random_state = check_random_state(random_state)
if hasattr(init, '__array__'):
init = np.asarray(init).copy()
if not n_init == 1:
warnings.warn(
'Explicit initial positions passed: '
'performing only one init of the MDS instead of %d'
% n_init)
n_init = 1
best_pos, best_stress = None, None
if n_jobs == 1:
for it in range(n_init):
pos, stress = _smacof_single(similarities, metric=metric,
n_components=n_components, init=init,
max_iter=max_iter, verbose=verbose,
eps=eps, random_state=random_state)
if best_stress is None or stress < best_stress:
best_stress = stress
best_pos = pos.copy()
else:
seeds = random_state.randint(np.iinfo(np.int32).max, size=n_init)
results = Parallel(n_jobs=n_jobs, verbose=max(verbose - 1, 0))(
delayed(_smacof_single)(
similarities, metric=metric, n_components=n_components,
init=init, max_iter=max_iter, verbose=verbose, eps=eps,
random_state=seed)
for seed in seeds)
positions, stress = zip(*results)
best = np.argmin(stress)
best_stress = stress[best]
best_pos = positions[best]
return best_pos, best_stress
