__docformat__ = "restructuredtext en"
from mdp import numx, numx_rand, utils, graph, Node
class _NGNodeData(object):
"""Data associated to a node in a Growing Neural Gas graph."""
def __init__(self, pos, error=0.0, hits=0, label=None):
# reference vector (spatial position)
self.pos = pos
# cumulative error
self.cum_error = error
self.hits = hits
self.label = label
class _NGEdgeData(object):
"""Data associated to an edge in a Growing Neural Gas graph."""
def __init__(self, age=0):
self.age = age
def inc_age(self):
self.age += 1
class GrowingNeuralGasNode(Node):
"""Learn the topological structure of the input data by building a
corresponding graph approximation.
The algorithm expands on the original Neural Gas algorithm
(see mdp.nodes NeuralGasNode) in that the algorithm adds new nodes are
added to the graph as more data becomes available. Im this way,
if the growth rate is appropriate, one can avoid overfitting or
underfitting the data.
More information about the Growing Neural Gas algorithm can be found in
B. Fritzke, A Growing Neural Gas Network Learns Topologies, in G. Tesauro,
D. S. Touretzky, and T. K. Leen (editors), Advances in Neural Information
Processing Systems 7, pages 625-632. MIT Press, Cambridge MA, 1995.
**Attributes and methods of interest**
- graph -- The corresponding `mdp.graph.Graph` object
"""
def __init__(self, start_poss=None, eps_b=0.2, eps_n=0.006, max_age=50,
lambda_=100, alpha=0.5, d=0.995, max_nodes=2147483647,
input_dim=None, dtype=None):
"""Growing Neural Gas algorithm.
:Parameters:
start_poss
sequence of two arrays containing the position of the
first two nodes in the GNG graph. If unspecified, the
initial nodes are chosen with a random position generated
from a gaussian distribution with zero mean and unit
variance.
eps_b
coefficient of movement of the nearest node to a new data
point. Typical values are 0 < eps_b << 1 .
Default: 0.2
eps_n
coefficient of movement of the neighbours of the nearest
node to a new data point. Typical values are
0 < eps_n << eps_b .
Default: 0.006
max_age
remove an edge after `max_age` updates. Typical values are
10 < max_age < lambda.
Default: 50
`lambda_`
insert a new node after `lambda_` steps. Typical values are O(100).
Default: 100
alpha
when a new node is inserted, multiply the error of the
nodes from which it generated by 0<alpha<1. A typical value
is 0.5.
Default: 0.5
d
each step the error of the nodes are multiplied by 0<d<1.
Typical values are close to 1.
Default: 0.995
max_nodes
maximal number of nodes in the graph.
Default: 2^31 - 1
"""
self.graph = graph.Graph()
self.tlen = 0
#copy parameters
(self.eps_b, self.eps_n, self.max_age, self.lambda_, self.alpha,
self.d, self.max_nodes) = (eps_b, eps_n, max_age, lambda_, alpha,
d, max_nodes)
super(GrowingNeuralGasNode, self).__init__(input_dim, None, dtype)
if start_poss is not None:
if self.dtype is None:
self.dtype = start_poss[0].dtype
node1 = self._add_node(self._refcast(start_poss[0]))
node2 = self._add_node(self._refcast(start_poss[1]))
self._add_edge(node1, node2)
def _set_input_dim(self, n):
self._input_dim = n
self.output_dim = n
def _add_node(self, pos):
node = self.graph.add_node(_NGNodeData(pos))
return node
def _add_edge(self, from_, to_):
self.graph.add_edge(from_, to_, _NGEdgeData())
def _get_nearest_nodes(self, x):
"""Return the two nodes in the graph that are nearest to x and their
squared distances. (Return ([node1, node2], [dist1, dist2])"""
# distance function
def _distance_from_node(node):
#return norm(node.data.pos-x)**2
tmp = node.data.pos - x
return utils.mult(tmp, tmp)
g = self.graph
# distances of all graph nodes from x
distances = numx.array(map(_distance_from_node, g.nodes))
ids = distances.argsort()[:2]
#nearest = [g.nodes[idx] for idx in ids]
#return nearest, distances[ids]
return (g.nodes[ids[0]], g.nodes[ids[1]]), distances.take(ids)
def _move_node(self, node, x, eps):
"""Move a node by eps in the direction x."""
# ! make sure that eps already has the right dtype
node.data.pos += eps*(x - node.data.pos)
def _remove_old_edges(self, edges):
"""Remove all edges older than the maximal age."""
g, max_age = self.graph, self.max_age
for edge in edges:
if edge.data.age > max_age:
g.remove_edge(edge)
if edge.head.degree() == 0:
g.remove_node(edge.head)
if edge.tail.degree() == 0:
g.remove_node(edge.tail)
def _insert_new_node(self):
"""Insert a new node in the graph where it is more necessary (i.e.
where the error is the largest)."""
g = self.graph
# determine the node with the highest error
errors = map(lambda x: x.data.cum_error, g.nodes)
qnode = g.nodes[numx.argmax(errors)]
# determine the neighbour with the highest error
neighbors = qnode.neighbors()
errors = map(lambda x: x.data.cum_error, neighbors)
fnode = neighbors[numx.argmax(errors)]
# new node, halfway between the worst node and the worst of
# its neighbors
new_pos = 0.5*(qnode.data.pos + fnode.data.pos)
new_node = self._add_node(new_pos)
# update edges
edges = qnode.get_edges(neighbor=fnode)
g.remove_edge(edges[0])
self._add_edge(qnode, new_node)
self._add_edge(fnode, new_node)
# update errors
qnode.data.cum_error *= self.alpha
fnode.data.cum_error *= self.alpha
new_node.data.cum_error = 0.5*(qnode.data.cum_error+
fnode.data.cum_error)
def get_nodes_position(self):
return numx.array(map(lambda n: n.data.pos, self.graph.nodes),
dtype = self.dtype)
def _train(self, input):
g = self.graph
d = self.d
if len(g.nodes)==0:
# if missing, generate two initial nodes at random
# assuming that the input data has zero mean and unit variance,
# choose the random position according to a gaussian distribution
# with zero mean and unit variance
normal = numx_rand.normal
self._add_node(self._refcast(normal(0.0, 1.0, self.input_dim)))
self._add_node(self._refcast(normal(0.0, 1.0, self.input_dim)))
# loop on single data points
for x in input:
self.tlen += 1
# step 2 - find the nearest nodes
# dists are the squared distances of x from n0, n1
(n0, n1), dists = self._get_nearest_nodes(x)
# step 3 - increase age of the emanating edges
for e in n0.get_edges():
e.data.inc_age()
# step 4 - update error
n0.data.cum_error += numx.sqrt(dists[0])
# step 5 - move nearest node and neighbours
self._move_node(n0, x, self.eps_b)
# neighbors undirected
neighbors = n0.neighbors()
for n in neighbors:
self._move_node(n, x, self.eps_n)
# step 6 - update n0<->n1 edge
if n1 in neighbors:
# should be one edge only
edges = n0.get_edges(neighbor=n1)
edges[0].data.age = 0
else:
self._add_edge(n0, n1)
# step 7 - remove old edges
self._remove_old_edges(n0.get_edges())
# step 8 - add a new node each lambda steps
if not self.tlen % self.lambda_ and len(g.nodes) < self.max_nodes:
self._insert_new_node()
# step 9 - decrease errors
for node in g.nodes:
node.data.cum_error *= d
def nearest_neighbor(self, input):
"""Assign each point in the input data to the nearest node in
the graph. Return the list of the nearest node instances, and
the list of distances.
Executing this function will close the training phase if
necessary."""
super(GrowingNeuralGasNode, self).execute(input)
nodes = []
dists = []
for x in input:
(n0, _), dist = self._get_nearest_nodes(x)
nodes.append(n0)
dists.append(numx.sqrt(dist[0]))
return nodes, dists
class NeuralGasNode(GrowingNeuralGasNode):
"""Learn the topological structure of the input data by building a
corresponding graph approximation (original Neural Gas algorithm).
The Neural Gas algorithm was originally published in Martinetz, T. and
Schulten, K.: A "Neural-Gas" Network Learns Topologies. In Kohonen, T.,
Maekisara, K., Simula, O., and Kangas, J. (eds.), Artificial Neural
Networks. Elsevier, North-Holland., 1991.
**Attributes and methods of interest**
- graph -- The corresponding `mdp.graph.Graph` object
- max_epochs - maximum number of epochs until which to train.
"""
def __init__(self, num_nodes = 10,
start_poss=None,
epsilon_i=0.3, # initial epsilon
epsilon_f=0.05, # final epsilon
lambda_i=30., # initial lambda
lambda_f=0.01, # final lambda
max_age_i=20, # initial edge lifetime
max_age_f=200, # final edge lifetime
max_epochs=100,
n_epochs_to_train=None,
input_dim=None,
dtype=None):
"""Neural Gas algorithm.
Default parameters taken from the original publication.
:Parameters:
start_poss
sequence of two arrays containing the position of the
first two nodes in the GNG graph. In unspecified, the
initial nodes are chosen with a random position generated
from a gaussian distribution with zero mean and unit
variance.
num_nodes
number of nodes to use. Ignored if start_poss is given.
epsilon_i, epsilon_f
initial and final values of epsilon. Fraction of the distance
between the closest node and the presented data point by which the
node moves towards the data point in an adaptation step. Epsilon
decays during training by e(t) = e_i(e_f/e_i)^(t/t_max) with t
being the epoch.
lambda_i, lambda_f
initial and final values of lambda. Lambda influences how the
weight change of nodes in the ranking decreases with lower rank. It
is sometimes called the "neighborhood factor". Lambda decays during
training in the same manner as epsilon does.
max_age_i, max_age_f
Initial and final lifetime, after which an edge will be removed.
Lifetime is measured in terms of adaptation steps, i.e.,
presentations of data points. It decays during training like
epsilon does.
max_epochs
number of epochs to train. One epoch has passed when all data points
from the input have been presented once. The default in the original
publication was 40000, but since this has proven to be impractically
high too high for many real-world data sets, we adopted a default
value of 100.
n_epochs_to_train
number of epochs to train on each call. Useful for batch learning
and for visualization of the training process. Default is to
train once until max_epochs is reached.
"""
self.graph = graph.Graph()
if n_epochs_to_train is None:
n_epochs_to_train = max_epochs
#copy parameters
self.num_nodes = num_nodes
self.start_poss = start_poss
self.epsilon_i = epsilon_i
self.epsilon_f = epsilon_f
self.lambda_i = lambda_i
self.lambda_f = lambda_f
self.max_age_i = max_age_i
self.max_age_f = max_age_f
self.max_epochs = max_epochs
self.n_epochs_to_train = n_epochs_to_train
super(GrowingNeuralGasNode, self).__init__(input_dim, None, dtype)
if start_poss is not None:
if self.num_nodes != len(start_poss):
self.num_nodes = len(start_poss)
if self.dtype is None:
self.dtype = start_poss[0].dtype
for node_ind in range(self.num_nodes):
self._add_node(self._refcast(start_poss[node_ind]))
self.epoch = 0
def _train(self, input):
g = self.graph
if len(g.nodes) == 0:
# if missing, generate num_nodes initial nodes at random
# assuming that the input data has zero mean and unit variance,
# choose the random position according to a gaussian distribution
# with zero mean and unit variance
normal = numx_rand.normal
for _ in range(self.num_nodes):
self._add_node(self._refcast(normal(0.0, 1.0, self.input_dim)))
epoch = self.epoch
e_i = self.epsilon_i
e_f = self.epsilon_f
l_i = self.lambda_i
l_f = self.lambda_f
T_i = float(self.max_age_i)
T_f = float(self.max_age_f)
max_epochs = float(self.max_epochs)
remaining_epochs = self.n_epochs_to_train
while remaining_epochs > 0:
# reset permutation of data points
di = numx.random.permutation(input)
if epoch < max_epochs:
denom = epoch/max_epochs
else:
denom = 1.
epsilon = e_i * ((e_f/e_i)**denom)
lmbda = l_i * ((l_f/l_i)**denom)
T = T_i * ((T_f/T_i)**denom)
epoch += 1
for x in di:
# Step 1 rank nodes according to their distance to random point
ranked_nodes = self._rank_nodes_by_distance(x)
# Step 2 move nodes
for rank,node in enumerate(ranked_nodes):
#TODO: cut off at some rank when using many nodes
#TODO: check speedup by vectorizing
delta_w = epsilon * numx.exp(-rank / lmbda) * \
(x - node.data.pos)
node.data.pos += delta_w
# Step 3 update edge weight
for e in g.edges:
e.data.inc_age()
# Step 4 set age of edge between first two nodes to zero
# or create it if it doesn't exist.
n0 = ranked_nodes[0]
n1 = ranked_nodes[1]
nn = n0.neighbors()
if n1 in nn:
edges = n0.get_edges(neighbor=n1)
edges[0].data.age = 0 # should only be one edge
else:
self._add_edge(n0, n1)
# step 5 delete edges with age > max_age
self._remove_old_edges(max_age=T)
remaining_epochs -= 1
self.epoch = epoch
def _rank_nodes_by_distance(self, x):
"""Return the nodes in the graph in a list ranked by their squared
distance to x. """
#TODO: Refactor together with GNGNode._get_nearest_nodes
# distance function
def _distance_from_node(node):
tmp = node.data.pos - x
return utils.mult(tmp, tmp) # maps to mdp.numx.dot
g = self.graph
# distances of all graph nodes from x
distances = numx.array(map(_distance_from_node, g.nodes))
ids = distances.argsort()
ranked_nodes = [g.nodes[id] for id in ids]
return ranked_nodes
def _remove_old_edges(self, max_age):
"""Remove edges with age > max_age."""
g = self.graph
for edge in self.graph.edges:
if edge.data.age > max_age:
g.remove_edge(edge)