def mfuncC(f, x):
"""
mfuncC(f, x) : matrix function with possibly complex eigenvalues.
Note: Numeric defines (v,u) = eig(x) => x*u.T = u.T * Diag(v)
This function is needed by sqrtm and allows further functions.
"""
x = array(x)
(v, u) = numerix.mlab.eig(x)
uT = transpose(u)
V = numerix.mlab.diag(f(v + 0j))
y = matrixmultiply(uT, matrixmultiply(V, linear_algebra.inverse(uT)))
return approx_real(y)
def mfuncC(f, x):
"""
mfuncC(f, x) : matrix function with possibly complex eigenvalues.
Note: Numeric defines (v,u) = eig(x) => x*u.T = u.T * Diag(v)
This function is needed by sqrtm and allows further functions.
"""
x = array(x)
(v, u) = numerix.mlab.eig(x)
uT = transpose(u)
V = numerix.mlab.diag(f(v+0j))
y = matrixmultiply(
uT, matrixmultiply(
V, linear_algebra.inverse(uT)))
return approx_real(y)
def view_transformation(E, R, V):
n = (E - R)
## new
# n /= mod(n)
# u = nx.cross(V,n)
# u /= mod(u)
# v = nx.cross(n,u)
# Mr = nx.diag([1.]*4)
# Mt = nx.diag([1.]*4)
# Mr[:3,:3] = u,v,n
# Mt[:3,-1] = -E
## end new
## old
n = n / mod(n)
u = cross(V, n)
u = u / mod(u)
v = cross(n, u)
Mr = [
[u[0], u[1], u[2], 0],
[v[0], v[1], v[2], 0],
[n[0], n[1], n[2], 0],
[0, 0, 0, 1],
]
#
Mt = [[1, 0, 0, -E[0]], [0, 1, 0, -E[1]], [0, 0, 1, -E[2]], [0, 0, 0, 1]]
## end old
return nx.matrixmultiply(Mr, Mt)
def get_proj(self):
"""Create the projection matrix from the current viewing
position.
elev stores the elevation angle in the z plane
azim stores the azimuth angle in the x,y plane
dist is the distance of the eye viewing point from the object
point.
"""
relev,razim = nx.pi * self.elev/180, nx.pi * self.azim/180
xmin,xmax = self.get_w_xlim()
ymin,ymax = self.get_w_ylim()
zmin,zmax = self.get_w_zlim()
# transform to uniform world coordinates 0-1.0,0-1.0,0-1.0
worldM = proj3d.world_transformation(xmin,xmax,
ymin,ymax,
zmin,zmax)
# look into the middle of the new coordinates
R = nx.array([0.5,0.5,0.5])
#
xp = R[0] + nx.cos(razim)*nx.cos(relev)*self.dist
yp = R[1] + nx.sin(razim)*nx.cos(relev)*self.dist
zp = R[2] + nx.sin(relev)*self.dist
E = nx.array((xp, yp, zp))
#
self.eye = E
self.vvec = R - E
self.vvec = self.vvec / proj3d.mod(self.vvec)
if abs(relev) > nx.pi/2:
# upside down
V = nx.array((0,0,-1))
else:
V = nx.array((0,0,1))
zfront,zback = -self.dist,self.dist
viewM = proj3d.view_transformation(E,R,V)
perspM = proj3d.persp_transformation(zfront,zback)
M0 = nx.matrixmultiply(viewM,worldM)
M = nx.matrixmultiply(perspM,M0)
return M
def inv_transform(xs, ys, zs, M):
iM = linear_algebra.inverse(M)
vec = vec_pad_ones(xs, ys, zs)
vecr = nx.matrixmultiply(iM, vec)
try:
vecr = vecr / vecr[3]
except OverflowError:
pass
return vecr[0], vecr[1], vecr[2]
def proj_transform_vec_clip(vec, M):
vecw = nx.matrixmultiply(M, vec)
w = vecw[3]
# clip here..
txs, tys, tzs = vecw[0] / w, vecw[1] / w, vecw[2] / w
tis = (vecw[0] >= 0) * (vecw[0] <= 1) * (vecw[1] >= 0) * (vecw[1] <= 1)
if nx.sometrue(tis):
tis = vecw[1] < 1
return txs, tys, tzs, tis
def test_proj_make_M(E=None):
# eye point
E = E or nx.array([1, -1, 2]) * 1000
#E = nx.array([20,10,20])
R = nx.array([1, 1, 1]) * 100
V = nx.array([0, 0, 1])
viewM = view_transformation(E, R, V)
perspM = persp_transformation(100, -100)
M = nx.matrixmultiply(perspM, viewM)
return M
def __init__(self, x, y, dx, dy, width=1.0, **kwargs):
"""Draws an arrow, starting at (x,y), direction and length
given by (dx,dy) the width of the arrow is scaled by width
"""
arrow = array([[0.0, 0.1], [0.0, -0.1], [0.8, -0.1], [0.8, -0.3], [1.0, 0.0], [0.8, 0.3], [0.8, 0.1]])
L = sqrt(dx ** 2 + dy ** 2) or 1 # account for div by zero
arrow[:, 0] *= L
arrow[:, 1] *= width
cx = float(dx) / L
sx = float(dy) / L
M = array([[cx, sx], [-sx, cx]])
verts = matrixmultiply(arrow, M) + [x, y]
Polygon.__init__(self, [tuple(t) for t in verts], **kwargs)
def __init__(self, x, y, dx, dy, width=1.0, **kwargs):
"""Draws an arrow, starting at (x,y), direction and length
given by (dx,dy) the width of the arrow is scaled by width
"""
arrow = array([[0.0, 0.1], [0.0, -0.1], [0.8, -0.1], [0.8, -0.3],
[1.0, 0.0], [0.8, 0.3], [0.8, 0.1]])
L = sqrt(dx**2 + dy**2) or 1 # account for div by zero
arrow[:, 0] *= L
arrow[:, 1] *= width
cx = float(dx) / L
sx = float(dy) / L
M = array([[cx, sx], [-sx, cx]])
verts = matrixmultiply(arrow, M) + [x, y]
Polygon.__init__(self, [tuple(t) for t in verts], **kwargs)
def norm(x, y=2):
"""
Norm of a matrix or a vector according to Matlab.
The description is taken from Matlab:
For matrices...
NORM(X) is the largest singular value of X, max(svd(X)).
NORM(X,2) is the same as NORM(X).
NORM(X,1) is the 1-norm of X, the largest column sum,
= max(sum(abs((X)))).
NORM(X,inf) is the infinity norm of X, the largest row sum,
= max(sum(abs((X')))).
NORM(X,'fro') is the Frobenius norm, sqrt(sum(diag(X'*X))).
NORM(X,P) is available for matrix X only if P is 1, 2, inf or 'fro'.
For vectors...
NORM(V,P) = sum(abs(V).^P)^(1/P).
NORM(V) = norm(V,2).
NORM(V,inf) = max(abs(V)).
NORM(V,-inf) = min(abs(V)).
"""
x = numerix.asarray(x)
if MLab.rank(x) == 2:
if y == 2:
return MLab.max(MLab.svd(x)[1])
elif y == 1:
return MLab.max(MLab.sum(numerix.absolute((x))))
elif y == 'inf':
return MLab.max(MLab.sum(numerix.absolute((MLab.transpose(x)))))
elif y == 'fro':
return MLab.sqrt(
MLab.sum(
MLab.diag(numerix.matrixmultiply(MLab.transpose(x), x))))
else:
verbose.report_error('Second argument not permitted for matrices')
return None
else:
if y == 'inf':
return MLab.max(numerix.absolute(x))
elif y == '-inf':
return MLab.min(numerix.absolute(x))
else:
return numerix.power(
MLab.sum(numerix.power(numerix.absolute(x), y)), 1 / float(y))
def norm(x,y=2):
"""
Norm of a matrix or a vector according to Matlab.
The description is taken from Matlab:
For matrices...
NORM(X) is the largest singular value of X, max(svd(X)).
NORM(X,2) is the same as NORM(X).
NORM(X,1) is the 1-norm of X, the largest column sum,
= max(sum(abs((X)))).
NORM(X,inf) is the infinity norm of X, the largest row sum,
= max(sum(abs((X')))).
NORM(X,'fro') is the Frobenius norm, sqrt(sum(diag(X'*X))).
NORM(X,P) is available for matrix X only if P is 1, 2, inf or 'fro'.
For vectors...
NORM(V,P) = sum(abs(V).^P)^(1/P).
NORM(V) = norm(V,2).
NORM(V,inf) = max(abs(V)).
NORM(V,-inf) = min(abs(V)).
"""
x = numerix.asarray(x)
if MLab.rank(x)==2:
if y==2:
return MLab.max(MLab.svd(x)[1])
elif y==1:
return MLab.max(MLab.sum(numerix.absolute((x))))
elif y=='inf':
return MLab.max(MLab.sum(numerix.absolute((MLab.transpose(x)))))
elif y=='fro':
return MLab.sqrt(MLab.sum(MLab.diag(numerix.matrixmultiply(MLab.transpose(x),x))))
else:
verbose.report_error('Second argument not permitted for matrices')
return None
else:
if y == 'inf':
return MLab.max(numerix.absolute(x))
elif y == '-inf':
return MLab.min(numerix.absolute(x))
else:
return numerix.power(MLab.sum(numerix.power(numerix.absolute(x),y)),1/float(y))
def polyval(p, x):
"""
y = polyval(p,x)
p is a vector of polynomial coeffients and y is the polynomial
evaluated at x.
Example code to remove a polynomial (quadratic) trend from y:
p = polyfit(x, y, 2)
trend = polyval(p, x)
resid = y - trend
See also polyfit
"""
x = asarray(x) + 0.
p = reshape(p, (len(p), 1))
X = vander(x, len(p))
y = matrixmultiply(X, p)
return reshape(y, x.shape)
def polyval(p,x):
"""
y = polyval(p,x)
p is a vector of polynomial coeffients and y is the polynomial
evaluated at x.
Example code to remove a polynomial (quadratic) trend from y:
p = polyfit(x, y, 2)
trend = polyval(p, x)
resid = y - trend
See also polyfit
"""
x = asarray(x)+0.
p = reshape(p, (len(p),1))
X = vander(x,len(p))
y = matrixmultiply(X,p)
return reshape(y, x.shape)
def corrcoef(*args):
"""
corrcoef(X) where X is a matrix returns a matrix of correlation
coefficients for each numrows observations and numcols variables.
corrcoef(x,y) where x and y are vectors returns the matrix or
correlation coefficients for x and y.
Numeric arrays can be real or complex
The correlation matrix is defined from the covariance matrix C as
r(i,j) = C[i,j] / sqrt(C[i,i]*C[j,j])
"""
if len(args)==2:
X = transpose(array([args[0]]+[args[1]]))
elif len(args)==1:
X = args[0]
else:
raise RuntimeError, 'Only expecting 1 or 2 arguments'
C = cov(X)
if len(args)==2:
d = resize(diagonal(C), (2,1))
denom = numerix.mlab.sqrt(matrixmultiply(d,transpose(d)))
else:
dc = diagonal(C)
N = len(dc)
shape = N,N
vi = resize(dc, shape)
denom = numerix.mlab.sqrt(vi*transpose(vi)) # element wise multiplication
r = divide(C,denom)
try: return r.real
except AttributeError: return r
def corrcoef(*args):
"""
corrcoef(X) where X is a matrix returns a matrix of correlation
coefficients for each numrows observations and numcols variables.
corrcoef(x,y) where x and y are vectors returns the matrix or
correlation coefficients for x and y.
Numeric arrays can be real or complex
The correlation matrix is defined from the covariance matrix C as
r(i,j) = C[i,j] / sqrt(C[i,i]*C[j,j])
"""
if len(args) == 2:
X = transpose(array([args[0]] + [args[1]]))
elif len(args) == 1:
X = args[0]
else:
raise RuntimeError, 'Only expecting 1 or 2 arguments'
C = cov(X)
if len(args) == 2:
d = resize(diagonal(C), (2, 1))
denom = numerix.mlab.sqrt(matrixmultiply(d, transpose(d)))
else:
dc = diagonal(C)
N = len(dc)
shape = N, N
vi = resize(dc, shape)
denom = numerix.mlab.sqrt(vi *
transpose(vi)) # element wise multiplication
r = divide(C, denom)
try:
return r.real
except AttributeError:
return r
def prepca(P, frac=0):
"""
Compute the principal components of P. P is a numVars x
numObservations numeric array. frac is the minimum fraction of
variance that a component must contain to be included
Return value are
Pcomponents : a num components x num observations numeric array
Trans : the weights matrix, ie, Pcomponents = Trans*P
fracVar : the fraction of the variance accounted for by each
component returned
"""
U,s,v = svd(P)
varEach = s**2/P.shape[1]
totVar = asum(varEach)
fracVar = divide(varEach,totVar)
ind = int(asum(fracVar>=frac))
# select the components that are greater
Trans = transpose(U[:,:ind])
# The transformed data
Pcomponents = matrixmultiply(Trans,P)
return Pcomponents, Trans, fracVar[:ind]
def prepca(P, frac=0):
"""
Compute the principal components of P. P is a numVars x
numObservations numeric array. frac is the minimum fraction of
variance that a component must contain to be included
Return value are
Pcomponents : a num components x num observations numeric array
Trans : the weights matrix, ie, Pcomponents = Trans*P
fracVar : the fraction of the variance accounted for by each
component returned
"""
U, s, v = svd(P)
varEach = s**2 / P.shape[1]
totVar = asum(varEach)
fracVar = divide(varEach, totVar)
ind = int(asum(fracVar >= frac))
# select the components that are greater
Trans = transpose(U[:, :ind])
# The transformed data
Pcomponents = matrixmultiply(Trans, P)
return Pcomponents, Trans, fracVar[:ind]
def rot_x(V, alpha):
cosa, sina = nx.cos(alpha), nx.sin(alpha)
M1 = nx.array([[1, 0, 0, 0], [0, cosa, -sina, 0], [0, sina, cosa, 0],
[0, 0, 0, 0]])
#
return nx.matrixmultiply(M1, V)
def __init__(
self,
x,
y,
dx,
dy,
width=0.001,
length_includes_head=False,
head_width=None,
head_length=None,
shape="full",
overhang=0,
head_starts_at_zero=False,
**kwargs
):
"""Returns a new Arrow.
length_includes_head: True if head is counted in calculating the length.
shape: ['full', 'left', 'right']
overhang: distance that the arrow is swept back (0 overhang means
triangular shape).
head_starts_at_zero: if True, the head starts being drawn at coordinate
0 instead of ending at coordinate 0.
"""
if head_width is None:
head_width = 3 * width
if head_length is None:
head_length = 1.5 * head_width
distance = sqrt(dx ** 2 + dy ** 2)
if length_includes_head:
length = distance
else:
length = distance + head_length
if not length:
verts = [] # display nothing if empty
else:
# start by drawing horizontal arrow, point at (0,0)
hw, hl, hs, lw = head_width, head_length, overhang, width
left_half_arrow = array(
[
[0.0, 0.0], # tip
[-hl, -hw / 2.0], # leftmost
[-hl * (1 - hs), -lw / 2.0], # meets stem
[-length, -lw / 2.0], # bottom left
[-length, 0],
]
)
# if we're not including the head, shift up by head length
if not length_includes_head:
left_half_arrow += [head_length, 0]
# if the head starts at 0, shift up by another head length
if head_starts_at_zero:
left_half_arrow += [head_length / 2.0, 0]
# figure out the shape, and complete accordingly
if shape == "left":
coords = left_half_arrow
else:
right_half_arrow = left_half_arrow * [1, -1]
if shape == "right":
coords = right_half_arrow
elif shape == "full":
coords = concatenate([left_half_arrow, right_half_arrow[::-1]])
else:
raise ValueError, "Got unknown shape: %s" % shape
cx = float(dx) / distance
sx = float(dy) / distance
M = array([[cx, sx], [-sx, cx]])
verts = matrixmultiply(coords, M) + (x + dx, y + dy)
Polygon.__init__(self, map(tuple, verts), **kwargs)
def __init__(self, x, y, dx, dy, width=0.001, length_includes_head=False, \
head_width=None, head_length=None, shape='full', overhang=0, \
head_starts_at_zero=False,**kwargs):
"""Returns a new Arrow.
length_includes_head: True if head is counted in calculating the length.
shape: ['full', 'left', 'right']
overhang: distance that the arrow is swept back (0 overhang means
triangular shape).
head_starts_at_zero: if True, the head starts being drawn at coordinate
0 instead of ending at coordinate 0.
"""
if head_width is None:
head_width = 3 * width
if head_length is None:
head_length = 1.5 * head_width
distance = sqrt(dx**2 + dy**2)
if length_includes_head:
length = distance
else:
length = distance + head_length
if not length:
verts = [] #display nothing if empty
else:
#start by drawing horizontal arrow, point at (0,0)
hw, hl, hs, lw = head_width, head_length, overhang, width
left_half_arrow = array([
[0.0, 0.0], #tip
[-hl, -hw / 2.0], #leftmost
[-hl * (1 - hs), -lw / 2.0], #meets stem
[-length, -lw / 2.0], #bottom left
[-length, 0],
])
#if we're not including the head, shift up by head length
if not length_includes_head:
left_half_arrow += [head_length, 0]
#if the head starts at 0, shift up by another head length
if head_starts_at_zero:
left_half_arrow += [head_length / 2.0, 0]
#figure out the shape, and complete accordingly
if shape == 'left':
coords = left_half_arrow
else:
right_half_arrow = left_half_arrow * [1, -1]
if shape == 'right':
coords = right_half_arrow
elif shape == 'full':
coords = concatenate(
[left_half_arrow, right_half_arrow[::-1]])
else:
raise ValueError, "Got unknown shape: %s" % shape
cx = float(dx) / distance
sx = float(dy) / distance
M = array([[cx, sx], [-sx, cx]])
verts = matrixmultiply(coords, M) + (x + dx, y + dy)
Polygon.__init__(self, map(tuple, verts), **kwargs)
def proj_transform_vec(vec, M):
vecw = nx.matrixmultiply(M, vec)
w = vecw[3]
# clip here..
txs, tys, tzs = vecw[0] / w, vecw[1] / w, vecw[2] / w
return txs, tys, tzs