def __init__(self, *args, **kwargs):
"""
use the optional key-word arg 'transform' to supply a
2x2 transform-matrix
"""
try:
transform = kwargs.pop('transform')
except KeyError:
transform = Matrix.eye(2)
if isinstance(transform, GridHelperCurveLinear):
assert 'Itransform' not in kwargs, (
'no Itransform when transform is a %s' % type(transform))
grid_helper = transform
elif callable(transform):
grid_helper = GridHelperCurveLinear(
[transform, kwargs.pop('Itransform')])
else:
transform = Matrix(transform)
try:
Itransform = kwargs.pop('Itransform')
except KeyError:
Itransform = transform**(-1)
grid_helper = GridHelperCurveLinear([
self.makeTransform(transform),
self.makeTransform(Itransform)
])
kwargs['grid_helper'] = grid_helper
mpl_toolkits.axisartist.Axes.__init__(self, *args, **kwargs)
def _subSquare(vectors, var, full=False):
"""
given a series of vectors, this function calculates:
(variances,vectors)=numpy.linalg.eigh(vectors.H*vectors)
it's a seperate function because if there are less vectors
than dimensions the process can be accelerated, it just takes some dancing
it is based on this:
>>> vectors=Matrix(helpers.ascomplex(numpy.random.randn(
... numpy.random.randint(1,10),numpy.random.randint(1,10),2
... )))
>>> cov = vectors.H*vectors
>>> Xcov = vectors*vectors.H
>>> (Xval,Xvec) = numpy.linalg.eigh(Xcov)
>>> vec = Xvec.H*vectors
>>> assert vec.H*vec == cov
"""
vectors = Matrix(vectors)
shape = vectors.shape
if not all(shape):
val = numpy.zeros([0])
vec = numpy.zeros([0, shape[1]])
return (val, vec)
eig = numpy.linalg.eigh
if shape[0] >= shape[1] or full or not vectors.any() or (var < 0).any():
scaled = Matrix(var[:, None]*numpy.array(vectors))
cov = vectors.H*scaled
(val, vec) = eig(cov)
vec = vec.H
elif not var.any():
cov = vectors.H*vectors
(_,vec) = eig(cov)
vec = vec.H
val = numpy.zeros(vec.shape[0])
else:
scaled = Matrix(scipy.sqrt(var)[:, None]*numpy.array(vectors))
Xcov = scaled*scaled.H
#Xcov = var[:,None]*numpy.array(vectors)*vectors.H
(_, Xvec) = eig(Xcov)
Xscaled = (Xvec.H*scaled)
val = helpers.mag2(Xscaled)
vec = numpy.array(Xscaled)/scipy.sqrt(val[:, numpy.newaxis])
return (val, vec)
def _subSquare(vectors, var, full=False):
"""
given a series of vectors, this function calculates:
(variances,vectors)=numpy.linalg.eigh(vectors.H*vectors)
it's a seperate function because if there are less vectors
than dimensions the process can be accelerated, it just takes some dancing
it is based on this:
>>> vectors=Matrix(helpers.ascomplex(numpy.random.randn(
... numpy.random.randint(1,10),numpy.random.randint(1,10),2
... )))
>>> cov = vectors.H*vectors
>>> Xcov = vectors*vectors.H
>>> (Xval,Xvec) = numpy.linalg.eigh(Xcov)
>>> vec = Xvec.H*vectors
>>> assert vec.H*vec == cov
"""
vectors = Matrix(vectors)
shape = vectors.shape
if not all(shape):
val = numpy.zeros([0])
vec = numpy.zeros([0, shape[1]])
return (val, vec)
eig = numpy.linalg.eigh
if shape[0] >= shape[1] or full or not vectors.any() or (var < 0).any():
scaled = Matrix(var[:, None] * numpy.array(vectors))
cov = vectors.H * scaled
(val, vec) = eig(cov)
vec = vec.H
elif not var.any():
cov = vectors.H * vectors
(_, vec) = eig(cov)
vec = vec.H
val = numpy.zeros(vec.shape[0])
else:
scaled = Matrix(scipy.sqrt(var)[:, None] * numpy.array(vectors))
Xcov = scaled * scaled.H
#Xcov = var[:,None]*numpy.array(vectors)*vectors.H
(_, Xvec) = eig(Xcov)
Xscaled = (Xvec.H * scaled)
val = helpers.mag2(Xscaled)
vec = numpy.array(Xscaled) / scipy.sqrt(val[:, numpy.newaxis])
return (val, vec)
def makeObjects(flat=None, ndim=None, seed=None):
if seed is None:
seed = randint(1, 1e6)
numpy.random.seed(seed)
randn = numpy.random.randn
if ndim is None:
ndim = randint(0, 20)
shapes = {
None: lambda: max(randint(-ndim, ndim), 0),
True: lambda: randint(1, ndim),
False: lambda: 0,
}
triple = lambda x: [x, x, x]
if flat in shapes:
flat = [item() for item in triple(shapes[flat])]
elif isinstance(flat, int):
flat = triple(flat)
assert all(f <= ndim
for f in flat), "flatness can't be larger than ndim"
rvec = lambda n=1, ndim=ndim: Matrix(randn(n, ndim))
A, B, C = [Mvn.rand([ndim - F, ndim]) for F in flat]
n = randint(1, 2 * ndim)
M = rvec(n).H
M2 = rvec(n).H
E = Matrix.eye(ndim)
K1 = (numpy.random.randn())
K2 = (numpy.random.randn())
N = randint(-5, 5)
return {
'ndim': ndim,
'A': A,
'B': B,
'C': C,
'M': M,
'M2': M2,
'E': E,
'K1': K1,
'K2': K2,
'N': N,
}
def main():
M1=Mvn.rand(2)
M2=Mvn.rand(2)
data1 = M1.sample(100)
data2 = M2.sample(100)
M1 = Mvn.fromData(data1)
M2 = Mvn.fromData(data2)
M3 = Mvn.fromData([M1,M2])
data3 = Matrix.stack([[data1],[data2]])
assert M3 == Mvn.fromData(data3)
A=pylab.gca()
M3.plot(A,facecolor='m',minalpha=0.1,zorder = -1)
M1.plot(A,facecolor='b',minalpha = 0.1,zorder = 0)
M2.plot(A,facecolor='r',minalpha = 0.1,zorder = 1)
pylab.scatter(data1[:,0],data1[:,1],facecolor='b',zorder = 2)
pylab.scatter(data2[:,0],data2[:,1],facecolor='r',zorder = 3)
pylab.show()
def newAx(fig, transform = Matrix.eye(2)):
fig.clear()
axgrid = GridSpec(1, 1)
#get axes
ax = pylab.subplot(
axgrid[:, :],
projection = 'custom',
transform = transform,
)
ax.autoscale(False)
# ax.set_xticks(numpy.arange(-10., 35., 5.))
# ax.set_yticks(numpy.arange(-10., 35., 5.))
ax.set_xlim([-5, 20])
ax.set_ylim([-5, 10])
ax.xaxis.set_major_locator(MultipleLocator(5))
ax.grid('on')
drawLegend(ax)
return ax
def newAx(fig, transform=Matrix.eye(2)):
fig.clear()
axgrid = GridSpec(1, 1)
#get axes
ax = pylab.subplot(
axgrid[:, :],
projection='custom',
transform=transform,
)
ax.autoscale(False)
# ax.set_xticks(numpy.arange(-10., 35., 5.))
# ax.set_yticks(numpy.arange(-10., 35., 5.))
ax.set_xlim([-5, 20])
ax.set_ylim([-5, 10])
ax.xaxis.set_major_locator(MultipleLocator(5))
ax.grid('on')
drawLegend(ax)
return ax
def __init__(
self,
vectors=Matrix.eye,
mean=numpy.zeros,
):
mean = mean if callable(mean) else numpy.array(mean).flatten()[None, :]
vectors = vectors if callable(vectors) else Matrix(vectors)
stack = helpers.autoshape([
[vectors],
[mean],
], default=1)
#unpack the stack into the object's parameters
self.vectors = Matrix(numpy.real_if_close(stack[0, 0]))
self.mean = Matrix(numpy.real_if_close(stack[1, 0]))
def main():
M1 = Mvn.rand(2)
M2 = Mvn.rand(2)
data1 = M1.sample(100)
data2 = M2.sample(100)
M1 = Mvn.fromData(data1)
M2 = Mvn.fromData(data2)
M3 = Mvn.fromData([M1, M2])
data3 = Matrix.stack([[data1], [data2]])
assert M3 == Mvn.fromData(data3)
A = pylab.gca()
M3.plot(A, facecolor='m', minalpha=0.1, zorder=-1)
M1.plot(A, facecolor='b', minalpha=0.1, zorder=0)
M2.plot(A, facecolor='r', minalpha=0.1, zorder=1)
pylab.scatter(data1[:, 0], data1[:, 1], facecolor='b', zorder=2)
pylab.scatter(data2[:, 0], data2[:, 1], facecolor='r', zorder=3)
pylab.show()
def main():
#generate data
N = 75
red = Mvn.rand(2)
blue = Matrix(red.sample(N))
red = Mvn.fromData(blue)
#create figure
fig = pylab.figure(1, figsize=(7, 7))
fig.suptitle('Mahalabois Distance')
ax = setupAxes(red.transform(-1))
# scatter plot of the origional data
scatter(ax[0, 0], red, blue)
# scatter plot of the normalized data
scatter(ax[1, 0], red / red, blue / red)
# draw the cumulative distribution
cumulative(ax[0, 1], red, blue)
# draw the histogram
hist(red.mah().pdf, red.mah(blue), ax[1, 1])
ax[0, 1].set_xlim([0, None])
ax[1, 1].set_ylim([0, None])
pylab.show()
def reverseTransform(center, M, x, y):
center = center.squeeze()
x = Matrix(x)
y = Matrix(y)
xy = numpy.hstack([x.T, y.T])
xy = xy - center[None, :]
xy = numpy.array(xy * M)
mags = (numpy.array(xy)**2).sum(1)[:, None]**0.5
dirs = xy / mags
xy = dirs * mags**0.5 + center
return xy[:, 0].squeeze(), xy[:, 1].squeeze()
def makeObjects(flat=None, ndim=None, seed=None):
if seed is None:
seed = randint(1, 1e6)
numpy.random.seed(seed)
randn = numpy.random.randn
if ndim is None:
ndim = randint(0, 20)
shapes = {
None:lambda :max(randint(-ndim, ndim), 0),
True:lambda :randint(1, ndim),
False:lambda :0,
}
triple = lambda x:[x, x, x]
if flat in shapes:
flat = [item() for item in triple(shapes[flat])]
elif isinstance(flat, int):
flat = triple(flat)
assert all(f <= ndim for f in flat), "flatness can't be larger than ndim"
rvec = lambda n=1, ndim=ndim:Matrix(randn(n ,ndim))
A,B,C = [
Mvn.rand([ndim-F, ndim])
for F in flat
]
n = randint(1, 2*ndim)
M = rvec(n).H
M2 = rvec(n).H
E = Matrix.eye(ndim)
K1 = (numpy.random.randn())
K2 = (numpy.random.randn())
N = randint(-5, 5)
return {
'ndim' : ndim,
'A' : A,
'B' : B,
'C' : C,
'M' : M,
'M2' : M2,
'E' : E,
'K1' : K1,
'K2' : K2,
'N' : N,
}
def __init__(
self,
vectors= Matrix.eye,
mean= numpy.zeros,
):
mean = mean if callable(mean) else numpy.array(mean).flatten()[None, :]
vectors = vectors if callable(vectors) else Matrix(vectors)
stack=helpers.autoshape([
[vectors],
[mean ],
],default= 1)
#unpack the stack into the object's parameters
self.vectors = Matrix(numpy.real_if_close(stack[0, 0]))
self.mean = Matrix(numpy.real_if_close(stack[1, 0]))
def __init__(self,*args,**kwargs):
"""
use the optional key-word arg 'transform' to supply a
2x2 transform-matrix
"""
try:
transform = kwargs.pop('transform')
except KeyError:
transform = Matrix.eye(2)
if isinstance(transform,GridHelperCurveLinear):
assert 'Itransform' not in kwargs,(
'no Itransform when transform is a %s' %
type(transform)
)
grid_helper = transform
elif callable(transform):
grid_helper = GridHelperCurveLinear([
transform,
kwargs.pop('Itransform')
])
else:
transform = Matrix(transform)
try:
Itransform = kwargs.pop('Itransform')
except KeyError:
Itransform = transform**(-1)
grid_helper = GridHelperCurveLinear([
self.makeTransform(transform),
self.makeTransform(Itransform)
])
kwargs['grid_helper']=grid_helper
mpl_toolkits.axisartist.Axes.__init__(self,*args,**kwargs)
def makeTransform(self, M, x, y):
x = Matrix(x)
y = Matrix(y)
xy = numpy.hstack([x.T, y.T])
xy = xy * M
return xy[:, 0].squeeze(), xy[:, 1].squeeze()
class Plane(object):
"""
plane class, meant to (eventually) factor out some code,
and utility from the Mvn class
"""
rtol = 1e-5
"""
relative tolerence
see :py:func:`mvn.helpers.approx`
"""
atol = 1e-8
"""
absolute tolerence
see :py:func:`mvn.helpers.approx`
"""
def __init__(
self,
vectors=Matrix.eye,
mean=numpy.zeros,
):
mean = mean if callable(mean) else numpy.array(mean).flatten()[None, :]
vectors = vectors if callable(vectors) else Matrix(vectors)
stack = helpers.autoshape([
[vectors],
[mean],
], default=1)
#unpack the stack into the object's parameters
self.vectors = Matrix(numpy.real_if_close(stack[0, 0]))
self.mean = Matrix(numpy.real_if_close(stack[1, 0]))
def __repr__(self):
"""
print self
"""
return '\n'.join([
'%s(' % self.__class__.__name__,
' mean=',
(' %r,' % self.mean).replace('\n', '\n' + 8 * ' '),
' vectors=',
(' %r' % self.vectors).replace('\n', '\n' + 8 * ' '),
')',
])
__str__ = __repr__
def __getitem__(self, index):
"""
project the plane into the selected dimensions
"""
assert not isinstance(index, tuple), '1-dimensional index only'
return type(self)(
mean=self.mean[:, index],
vectors=self.vectors[:, index],
)
copy = decorate.automath.Automath.__dict__['copy']
@property
def shape(self):
"""
get the shape of the vectors,the first element is the number of
vectors, the second is their lengths: the number of dimensions of
the space they are embedded in
>>> assert A.vectors.shape == A.shape
>>> assert (A.vectors.shape[0],A.mean.size)==A.shape
>>> assert A.shape[0]==A.rank
>>> assert A.shape[1]==A.ndim
"""
return self.vectors.shape
@property
def rank(self):
"""
get the number of dimensions of the space covered by the mvn
>>> assert A.rank == A.vectors.shape[0]
"""
return self.vectors.shape[0]
@property
def ndim(self):
"""
get the number of dimensions of the space the mvn exists in
>>> assert A.ndim==A.mean.size==A.mean.shape[1]
>>> assert A.ndim==A.vectors.shape[1]
"""
return self.mean.size
@property
def flat(self):
"""
>>> assert bool(A.flat) == bool(A.vectors.shape[1] > A.vectors.shape[0])
"""
return max(self.vectors.shape[1] - self.vectors.shape[0], 0)
def __nonzero__(self):
"""
True if not empty
>>> assert A
>>> assert bool(A) == bool(A.ndim)
>>> assert not A[:0]
"""
return bool(self.ndim)
@decorate.MultiMethod
def __add__(self, other):
"""
add two planes together
"""
raise TypeError("No Apropriate Method Found")
@__add__.register(Plane)
def __add__(self, other):
result = self.copy()
result.mean = result.mean + other
return result
@__add__.register(Plane, Plane)
def __add__(self, other):
return Plane(mean=self.mean + other.mean,
vectors=numpy.vstack([self.vectors, other.vectors]))
def approx(self, *args):
return helpers.approx(*args, atol=self.atol, rtol=self.rtol)
def __and__(self, other):
"""
plane intersection
"""
Nself = self.vectors.null()
Nother = other.vectors.null()
#and stack them
null = numpy.vstack([
Nself,
Nother,
])
mean = numpy.hstack([self.mean, other.mean])
#get length of the component of the means along each null vector
r = numpy.vstack([Nself * self.mean.H, Nother * other.mean.H])
mean = (numpy.linalg.pinv(null, 1e-6) * r).H
return type(self)(vectors=null.null(), mean=mean)
def square(vectors, var=None, full=False):
"""
calculates the eigen-vectors and eigen-values of the covariance matrix that
would be produced by multiplying out A.var*numpy.array(A.vectors.H)*A.vectors
without necessarily calculating the covariance matrix itself.
It is also setup to handle vectors with infinite variances.
origionally the idea came from these two line on wikipedia:
http://en.wikipedia.org/wiki/Square_root_of_a_matrix:
'''if T = A*A.H = B*B.H, then there exists a unitary U s.t. A = B*U'''
http://en.wikipedia.org/wiki/Unitary_matrix
'''In mathematics, a unitary matrix is an nxn complex matrix U satisfying
the condition U.H*U = I, U*U.H = I'''
*********************************
A better description for all this is the compact singular value decomposition.
http://en.wikipedia.org/wiki/Singular_value_decomposition#Compact_SVD
but here I only need one of the two sets of vectors, so I actually calculate the smaller of
the two possible covariance marixes and, and then it's eigen-stuff.
"""
if var is None:
var = numpy.ones(vectors.shape[0])
finite = numpy.isfinite(var) & numpy.isfinite(vectors.asarray()).all(1)
infinite = ~finite
Ivar = numpy.array([])
Ivectors = Matrix(numpy.zeros((0, vectors.shape[1])))
if infinite.any():
#square up the infinite vectors
#Ivar is unused
(Ivar, Ivectors) = _subSquare(vectors=vectors[infinite, :],
var=numpy.ones_like(var[infinite]),
full=True)
#take the finite variances and vectors
var = var[~infinite]
vectors = vectors[~infinite, :]
small = helpers.approx(Ivar)
Ivar = Ivar[~small]
SIvectors = Ivectors[~small, :]
if vectors.any():
#revove the component parallel to each infinite vector
vectors = vectors - vectors * SIvectors.H * SIvectors
elif var.size:
num = helpers.approx(var).sum()
#gab the extra vectors here, because if the vectors are all zeros eig will fail
vectors = Ivectors[small, :]
vectors = vectors[:num, :]
Ivectors = SIvectors
if var.size:
(var, vectors) = _subSquare(vectors, var)
if Ivar.size and var.size:
#sort the finite variances
order = numpy.argsort(abs(var))
var = var[order]
vectors = vectors[order, :]
#if there are more vectors than dimensions
kill = var.size + Ivar.size - vectors.shape[1]
if kill > 0:
#squeeze the vectors with the smallest variances
var = var[kill:]
vectors = vectors[kill:, :]
return (numpy.concatenate((var, numpy.inf * numpy.ones_like(Ivar))),
numpy.vstack([vectors, Ivectors]))
#create figure
fig = pylab.figure(figsize=(6, 6))
## kalman filter parameters
#the actual, hidden state
actual = numpy.array([[0, 5]])
#the sensor
sensor = Mvn(vectors=[[1, 0], [0, 1]], var=[1, numpy.inf])
#the system noise
noise = Mvn(vectors=[[1, 0], [0, 1]], var=numpy.array([0.5, 1])**2)
#the shear transform to move the system forward
transform = Matrix([[1, 0], [0.5, 1]])
filtered = sensor.measure(actual)
## initial plot
ax = newAx(fig)
#plot the initial actual position
ax.plot(actual[:, 0], actual[:, 1], **actualParams)
ax.set_title('Kalman Filtering: Start')
pylab.xlabel('Position')
pylab.ylabel('Velocity')
P.publish(fig)
#measure the actual position, and plot the measurment
class Plane(object):
"""
plane class, meant to (eventually) factor out some code,
and utility from the Mvn class
"""
rtol = 1e-5
"""
relative tolerence
see :py:func:`mvn.helpers.approx`
"""
atol = 1e-8
"""
absolute tolerence
see :py:func:`mvn.helpers.approx`
"""
def __init__(
self,
vectors= Matrix.eye,
mean= numpy.zeros,
):
mean = mean if callable(mean) else numpy.array(mean).flatten()[None, :]
vectors = vectors if callable(vectors) else Matrix(vectors)
stack=helpers.autoshape([
[vectors],
[mean ],
],default= 1)
#unpack the stack into the object's parameters
self.vectors = Matrix(numpy.real_if_close(stack[0, 0]))
self.mean = Matrix(numpy.real_if_close(stack[1, 0]))
def __repr__(self):
"""
print self
"""
return '\n'.join([
'%s(' % self.__class__.__name__,
' mean=',
(' %r,' % self.mean).replace('\n', '\n'+8*' '),
' vectors=',
(' %r' % self.vectors).replace('\n', '\n'+8*' '),
')',
])
__str__ = __repr__
def __getitem__(self, index):
"""
project the plane into the selected dimensions
"""
assert not isinstance(index, tuple),'1-dimensional index only'
return type(self)(
mean= self.mean[:, index],
vectors= self.vectors[:, index],
)
copy = decorate.automath.Automath.__dict__['copy']
@property
def shape(self):
"""
get the shape of the vectors,the first element is the number of
vectors, the second is their lengths: the number of dimensions of
the space they are embedded in
>>> assert A.vectors.shape == A.shape
>>> assert (A.vectors.shape[0],A.mean.size)==A.shape
>>> assert A.shape[0]==A.rank
>>> assert A.shape[1]==A.ndim
"""
return self.vectors.shape
@property
def rank(self):
"""
get the number of dimensions of the space covered by the mvn
>>> assert A.rank == A.vectors.shape[0]
"""
return self.vectors.shape[0]
@property
def ndim(self):
"""
get the number of dimensions of the space the mvn exists in
>>> assert A.ndim==A.mean.size==A.mean.shape[1]
>>> assert A.ndim==A.vectors.shape[1]
"""
return self.mean.size
@property
def flat(self):
"""
>>> assert bool(A.flat) == bool(A.vectors.shape[1] > A.vectors.shape[0])
"""
return max(self.vectors.shape[1] - self.vectors.shape[0], 0)
def __nonzero__(self):
"""
True if not empty
>>> assert A
>>> assert bool(A) == bool(A.ndim)
>>> assert not A[:0]
"""
return bool(self.ndim)
@decorate.MultiMethod
def __add__(self, other):
"""
add two planes together
"""
raise TypeError("No Apropriate Method Found")
@__add__.register(Plane)
def __add__(self, other):
result = self.copy()
result.mean = result.mean+other
return result
@__add__.register(Plane, Plane)
def __add__(self, other):
return Plane(
mean = self.mean+other.mean,
vectors = numpy.vstack([self.vectors, other.vectors])
)
def approx(self, *args):
return helpers.approx(*args, atol = self.atol, rtol = self.rtol)
def __and__(self, other):
"""
plane intersection
"""
Nself = self.vectors.null()
Nother = other.vectors.null()
#and stack them
null = numpy.vstack([
Nself,
Nother,
])
mean = numpy.hstack([self.mean, other.mean])
#get length of the component of the means along each null vector
r = numpy.vstack([Nself*self.mean.H, Nother*other.mean.H])
mean = (numpy.linalg.pinv(null, 1e-6)*r).H
return type(self)(vectors= null.null(), mean=mean)