def rotation_matrix_from_rpy(roll, pitch, yaw):
"""
Conversion of roll, pitch, yaw to 4x4 rotation matrix according to:
https://github.com/orocos/orocos_kinematics_dynamics/blob/master/orocos_kdl/src/frames.cpp#L167
:type roll: Union[float, Symbol]
:type pitch: Union[float, Symbol]
:type yaw: Union[float, Symbol]
:return: 4x4 Matrix
:rtype: Matrix
"""
# TODO don't split this into 3 matrices
rx = se.Matrix([[1, 0, 0, 0],
[0, se.cos(roll), -se.sin(roll), 0],
[0, se.sin(roll), se.cos(roll), 0],
[0, 0, 0, 1]])
ry = se.Matrix([[se.cos(pitch), 0, se.sin(pitch), 0],
[0, 1, 0, 0],
[-se.sin(pitch), 0, se.cos(pitch), 0],
[0, 0, 0, 1]])
rz = se.Matrix([[se.cos(yaw), -se.sin(yaw), 0, 0],
[se.sin(yaw), se.cos(yaw), 0, 0],
[0, 0, 1, 0],
[0, 0, 0, 1]])
return (rz * ry * rx)
def __init__(self, settings, args, times_to_use):
par = Params()
self.times_to_use = times_to_use
self.starting_parameters = [2.26E-04, 0.0699, 3.45E-05, 0.05462, 0.0873, 8.92E-03, 5.150E-3, 0.03158, 0.1524]
# Create symbols for symbolic functions
p, y, v = CreateSymbols(settings)
# Choose starting parameters (from J Physiol paper)
para = [2.26E-04, 0.0699, 3.45E-05, 0.05462, 0.0873, 8.92E-03, 5.150E-3, 0.03158, 0.1524]
# Create symbols for symbolic functions
p, y, v = CreateSymbols(par)
# Define system equations and initial conditions
k1 = p[0] * se.exp(p[1] * v)
k2 = p[2] * se.exp(-p[3] * v)
k3 = p[4] * se.exp(p[5] * v)
k4 = p[6] * se.exp(-p[7] * v)
# Write in matrix form taking y = ([C], [O], [I])^T
A = se.Matrix([[-k1 - k3 - k4, k2 - k4, -k4], [k1, -k2 - k3, k4], [-k1, k3 - k1, -k2 - k4 - k1]])
B = se.Matrix([k4, 0, k1])
rhs = np.array(A * y + B)
self.funcs = GetSensitivityEquations(par, p, y, v, A, B, para, times_to_use, sine_wave=args.sine_wave)
def test_sens_limits():
par = Params()
p, y, v = CreateSymbols(par)
reversal_potential = par.Erev
para = np.array([
2.07, 7.17E1, 3.44E-2, 6.18E1, 4.18E2, 2.58E1, 4.75E1, 2.51E1, 3.33E1
])
para = para * 1E-3
# Define system equations and initial conditions
k1 = p[0] * se.exp(p[1] * v)
k2 = p[2] * se.exp(-p[3] * v)
k3 = p[4] * se.exp(p[5] * v)
k4 = p[6] * se.exp(-p[7] * v)
current_limit = (p[-1] * (par.holding_potential - reversal_potential) *
k1 / (k1 + k2) * k4 / (k3 + k4)).subs(
v, par.holding_potential)
print("{} Current limit computed as {}".format(
__file__,
current_limit.subs(p, para).evalf()))
sens_inf = [
float(se.diff(current_limit, p[j]).subs(p, para).evalf())
for j in range(0, par.n_params)
]
print("{} sens_inf calculated as {}".format(__file__, sens_inf))
k = se.symbols('k1, k2, k3, k4')
# Notation is consistent between the two papers
A = se.Matrix([[-k1 - k3 - k4, k2 - k4, -k4], [k1, -k2 - k3, k4],
[-k1, k3 - k1, -k2 - k4 - k1]])
B = se.Matrix([k4, 0, k1])
# Use results from HH equations
current_limit = (p[-1] * (par.holding_potential - reversal_potential) *
k1 / (k1 + k2) * k4 / (k3 + k4)).subs(
v, par.holding_potential)
funcs = GetSensitivityEquations(par, p, y, v, A, B, para, [0])
sens_inf = [
float(se.diff(current_limit, p[j]).subs(p, para).evalf())
for j in range(0, par.n_params)
]
sens = funcs.SimulateForwardModelSensitivities(para)[1]
# Check sens = sens_inf
error = np.abs(sens_inf - sens)
equal = np.all(error < 1e-10)
assert (equal)
return
def rotation3_rpy(roll, pitch, yaw):
""" Conversion of roll, pitch, yaw to 4x4 rotation matrix according to:
https://github.com/orocos/orocos_kinematics_dynamics/blob/master/orocos_kdl/src/frames.cpp#L167
"""
rx = sp.Matrix([[1, 0, 0, 0], [0, sp.cos(roll), -sp.sin(roll), 0],
[0, sp.sin(roll), sp.cos(roll), 0], [0, 0, 0, 1]])
ry = sp.Matrix([[sp.cos(pitch), 0, sp.sin(pitch), 0], [0, 1, 0, 0],
[-sp.sin(pitch), 0, sp.cos(pitch), 0], [0, 0, 0, 1]])
rz = sp.Matrix([[sp.cos(yaw), -sp.sin(yaw), 0, 0],
[sp.sin(yaw), sp.cos(yaw), 0, 0], [0, 0, 1, 0],
[0, 0, 0, 1]])
return (rz * ry * rx)
def __init__(self, settings, args, times_to_use):
par = Params()
self.times_to_use = times_to_use
self.starting_parameters = [
2.26E-04, 0.0699, 3.45E-05, 0.05462, 0.0873, 8.92E-03, 5.150E-3,
0.03158, 0.1524
]
# Create symbols for symbolic functions
p, y, v = CreateSymbols(settings)
# Choose starting parameters (from J Physiol paper)
para = [
2.26E-04, 0.0699, 3.45E-05, 0.05462, 0.0873, 8.92E-03, 5.150E-3,
0.03158, 0.1524
]
# Define system equations and initial conditions
k1 = p[0] * se.exp(p[1] * v)
k2 = p[2] * se.exp(-p[3] * v)
k3 = p[4] * se.exp(p[5] * v)
k4 = p[6] * se.exp(-p[7] * v)
# Write in matrix form taking y = ([C], [O], [I])^T
A = se.Matrix([[-k1 - k3 - k4, k2 - k4, -k4], [k1, -k2 - k3, k4],
[-k1, k3 - k1, -k2 - k4 - k1]])
B = se.Matrix([k4, 0, k1])
rhs = np.array(A * y + B)
protocol = pd.read_csv(
os.path.join(
os.path.dirname(os.path.dirname(os.path.realpath(__file__))),
"protocols", "protocol-staircaseramp.csv"))
times = 10000 * protocol["time"].values
voltages = protocol["voltage"].values
staircase_protocol = scipy.interpolate.interp1d(times,
voltages,
kind="linear")
staircase_protocol_safe = lambda t: staircase_protocol(t) if t < times[
-1] else par.holding_potential
self.funcs = GetSensitivityEquations(par,
p,
y,
v,
A,
B,
para,
times_to_use,
voltage=staircase_protocol_safe)
def CreateSymbols(par):
"""
Create SymEngine symbols to contain the parameters, state variables and the voltage.
These are used to generate functions for the right hand side and Jacobian
"""
# Create parameter symbols
p = se.Matrix([se.symbols('p%d' % j) for j in range(par.n_params)])
# Create state variable symbols
y = se.Matrix([se.symbols('y%d' % i) for i in range(par.n_state_vars)])
# Create voltage symbol
v = se.symbols('v')
return p, y, v
def subs(self, d):
"""Substitute expressions
Parameters
----------
d : dict
Dictionary of from: to pairs for substitution
Examples
--------
>>> import sympy
>>> from pharmpy import RandomVariable, Parameter
>>> omega = Parameter("OMEGA_CL", 0.1)
>>> rv = RandomVariable.normal("IIV_CL", "IIV", 0, omega.symbol)
>>> rv.subs({omega.symbol: sympy.Symbol("OMEGA_NEW")})
>>> rv
IIV_CL ~ 𝒩 (0, OMEGA_NEW)
"""
if self._mean is not None:
self._mean = self._mean.subs(d)
self._variance = self._variance.subs(d)
self._symengine_variance = symengine.Matrix(
self._variance.rows, self._variance.cols, self._variance)
if self._sympy_rv is not None:
self._sympy_rv = self._sympy_rv.subs(d)
def quaternion_from_rpy(roll, pitch, yaw):
"""
:type roll: Union[float, Symbol]
:type pitch: Union[float, Symbol]
:type yaw: Union[float, Symbol]
:return: 4x1 Matrix
:type: Matrix
"""
roll_half = roll / 2.0
pitch_half = pitch / 2.0
yaw_half = yaw / 2.0
c_roll = se.cos(roll_half)
s_roll = se.sin(roll_half)
c_pitch = se.cos(pitch_half)
s_pitch = se.sin(pitch_half)
c_yaw = se.cos(yaw_half)
s_yaw = se.sin(yaw_half)
cc = c_roll * c_yaw
cs = c_roll * s_yaw
sc = s_roll * c_yaw
ss = s_roll * s_yaw
x = c_pitch * sc - s_pitch * cs
y = c_pitch * ss + s_pitch * cc
z = c_pitch * cs - s_pitch * sc
w = c_pitch * cc + s_pitch * ss
return se.Matrix([x, y, z, w])
def rotation_matrix_from_axis_angle(axis, angle):
"""
Conversion of unit axis and angle to 4x4 rotation matrix according to:
http://www.euclideanspace.com/maths/geometry/rotations/conversions/angleToMatrix/index.htm
:param axis: 3x1 Matrix
:type axis: Matrix
:type angle: Union[float, Symbol]
:return: 4x4 Matrix
:rtype: Matrix
"""
ct = se.cos(angle)
st = se.sin(angle)
vt = 1 - ct
m_vt_0 = vt * axis[0]
m_vt_1 = vt * axis[1]
m_vt_2 = vt * axis[2]
m_st_0 = axis[0] * st
m_st_1 = axis[1] * st
m_st_2 = axis[2] * st
m_vt_0_1 = m_vt_0 * axis[1]
m_vt_0_2 = m_vt_0 * axis[2]
m_vt_1_2 = m_vt_1 * axis[2]
return se.Matrix(
[[ct + m_vt_0 * axis[0], -m_st_2 + m_vt_0_1, m_st_1 + m_vt_0_2, 0],
[m_st_2 + m_vt_0_1, ct + m_vt_1 * axis[1], -m_st_0 + m_vt_1_2, 0],
[-m_st_1 + m_vt_0_2, m_st_0 + m_vt_1_2, ct + m_vt_2 * axis[2], 0],
[0, 0, 0, 1]])
def quatrotateabout(q1, q2):
return sp.Matrix([
(q1[0] * q2[0]) - (q1[1] * q2[1]) - (q1[2] * q2[2]) - (q1[3] * q2[3]),
(q1[0] * q2[1]) + (q1[1] * q2[0]) + (q1[2] * q2[3]) - (q1[3] * q2[2]),
(q1[0] * q2[2]) - (q1[1] * q2[3]) + (q1[2] * q2[0]) + (q1[3] * q2[1]),
(q1[0] * q2[3]) + (q1[1] * q2[2]) - (q1[2] * q2[1]) + (q1[3] * q2[0])
])
def json_decoder(dct):
if '__type__' in dct:
t = dct['__type__']
if t == json_sym_matrix:
return sp.Matrix(dct['data']) # nested_list_to_sym(dct['data'])
elif t == json_sym_expr:
try:
return sp.sympify(dct['data'].encode('utf-8')) if isinstance(
dct['data'], unicode) else sp.sympify(dct['data'])
except NameError as e:
raise Exception(
'NameError Occured while trying to sympify string {}. Error: {}\n{}'
.format(repr(dct['data']), str(e), traceback.format_exc()))
elif t[:8] == '<class \'' and t[-2:] == '\'>':
t = t[8:-2]
if t not in class_registry:
class_registry[t] = import_class(t)
t = class_registry[t]
if not issubclass(t, JSONSerializable):
raise Exception(
'Can not decode JSON object tagged with type "{}". The type is not a subtype of JSONSerializable'
.format(t))
try:
if hasattr(t, 'json_factory'):
return t.json_factory(
**{k: v
for k, v in dct.items() if k != '__type__'})
return t(**{k: v for k, v in dct.items() if k != '__type__'})
except TypeError as e:
raise TypeError(
'Error occurred while instantiating an object of type "{}":\n {}'
.format(t.__name__, e))
return dct
def axisanglerotationmatrix(axis_angle):
R = axisanglemagnitude(axis_angle)
x = Piecewise((axis_angle[0] / R, R > 0), (1, True))
y = Piecewise((axis_angle[1] / R, R > 0), (0, True))
z = Piecewise((axis_angle[2] / R, R > 0), (0, True))
csr = sp.cos(R)
one_minus_csr = (1 - csr)
snr = sp.sin(R)
return sp.Matrix([[
csr + x * x * (1 - csr), x * y * one_minus_csr - z * snr,
x * z * one_minus_csr + y * snr
],
[
y * x * one_minus_csr + z * snr,
csr + y * y * one_minus_csr,
y * z * one_minus_csr - x * snr
],
[
z * x * one_minus_csr - y * snr,
z * y * one_minus_csr + x * snr,
csr + z * z * one_minus_csr
]])
def axisanglerotationmatrix(axis_angle):
R = axisanglemagnitude(axis_angle)
x = axis_angle[0] / R
y = axis_angle[1] / R
z = axis_angle[2] / R
csr = sp.cos(R)
one_minus_csr = (1 - csr)
snr = sp.sin(R)
return sp.Matrix([[
csr + x * x * (1 - csr), x * y * one_minus_csr - z * snr,
x * z * one_minus_csr + y * snr
],
[
y * x * one_minus_csr + z * snr,
csr + y * y * one_minus_csr,
y * z * one_minus_csr - x * snr
],
[
z * x * one_minus_csr - y * snr,
z * y * one_minus_csr + x * snr,
csr + z * z * one_minus_csr
]])
def diffable_slerp(q1, q2, t):
"""
!takes a long time to compile!
:param q1: 4x1 Matrix
:type q1: Matrix
:param q2: 4x1 Matrix
:type q2: Matrix
:param t: float, 0-1
:type t: Union[float, Symbol]
:return: 4x1 Matrix; Return spherical linear interpolation between two quaternions.
:rtype: Matrix
"""
cos_half_theta = dot(q1, q2)
if0 = -cos_half_theta
q2 = diffable_if_greater_zero(if0, -q2, q2)
cos_half_theta = diffable_if_greater_zero(if0, -cos_half_theta, cos_half_theta)
if1 = diffable_abs(cos_half_theta) - 1.0
# enforce acos(x) with -1 < x < 1
cos_half_theta = diffable_min_fast(1, cos_half_theta)
cos_half_theta = diffable_max_fast(-1, cos_half_theta)
half_theta = acos(cos_half_theta)
sin_half_theta = sqrt(1.0 - cos_half_theta * cos_half_theta)
# prevent /0
sin_half_theta = diffable_if_eq_zero(sin_half_theta, 1, sin_half_theta)
if2 = 0.001 - diffable_abs(sin_half_theta)
ratio_a = sin((1.0 - t) * half_theta) / sin_half_theta
ratio_b = sin(t * half_theta) / sin_half_theta
return diffable_if_greater_eq_zero(if1, se.Matrix(q1),
diffable_if_greater_zero(if2, 0.5 * q1 + 0.5 * q2, ratio_a * q1 + ratio_b * q2))
def axis_angle_from_matrix(rotation_matrix):
"""
:param rotation_matrix: 4x4 or 3x3 Matrix
:type rotation_matrix: Matrix
:return: 3x1 Matrix, angle
:rtype: (Matrix, Union[float, Symbol])
"""
rm = rotation_matrix
angle = (trace(rm[:3, :3]) - 1) / 2
angle = se.Min(angle, 1)
angle = se.Max(angle, -1)
angle = se.acos(angle)
x = (rm[2, 1] - rm[1, 2])
y = (rm[0, 2] - rm[2, 0])
z = (rm[1, 0] - rm[0, 1])
n = se.sqrt(x * x + y * y + z * z)
m = if_eq_zero(n, 1, n)
axis = se.Matrix([
if_eq_zero(n, 0, x / m),
if_eq_zero(n, 0, y / m),
if_eq_zero(n, 1, z / m)
])
sign = diffable_sign(angle)
axis *= sign
angle = sign * angle
return axis, angle
def rotation_of(frame):
"""
:param frame: 4x4 Matrix
:type frame: Matrix
:return: 4x4 Matrix; sets the translation part of a frame to 0
:rtype: Matrix
"""
return frame[:4, :3].row_join(se.Matrix([0, 0, 0, 1]))
def translation_of(frame):
"""
:param frame: 4x4 Matrix
:type frame: Matrix
:return: 4x4 Matrix; sets the rotation part of a frame to identity
:rtype: Matrix
"""
return se.eye(3).col_join(se.Matrix([[0] * 3])).row_join(frame[:4, 3:])
def vector3(x, y, z):
"""
:param x: Union[float, Symbol]
:param y: Union[float, Symbol]
:param z: Union[float, Symbol]
:rtype: Matrix
"""
return se.Matrix([x, y, z, 0])
def quaternion_conjugate(quaternion):
"""
:param quaternion: 4x1 Matrix
:type quaternion: Matrix
:return: 4x1 Matrix
:rtype: Matrix
"""
return se.Matrix([-quaternion[0], -quaternion[1], -quaternion[2], quaternion[3]])
def quatrotationmatrix(q):
qw, qi, qj, qk = q
s = quatmagnitude(q)
return sp.Matrix(
[[1 - 2 * s * (qj * qj + qk * qk), 2 * s * (qi * qj - qk * qw), 2 * s * (qi * qk + qj * qw)],
[2 * s * (qi * qj + qk * qw), 1 - 2 * s * (qi * qi + qk * qk), 2 * s * (qj * qk - qi * qw)],
[2 * s * (qi * qk - qj * qw), 2 * s * (qj * qk + qi * qw), 1 - 2 * s * (qi * qi + qj * qj)]
])
def point3(x, y, z):
"""
:param x: Union[float, Symbol]
:param y: Union[float, Symbol]
:param z: Union[float, Symbol]
:rtype: Matrix
"""
return se.Matrix([x, y, z, 1])
def axisanglecompose(axis_angle, axis_angle2):
a = axisanglemagnitude(axis_angle)
ah = axisanglenormalize(axis_angle)
b = axisanglemagnitude(axis_angle2)
bh = axisanglenormalize(axis_angle2)
sina = sin(a / 2)
asina = [ah[0] * sina, ah[1] * sina, ah[2] * sina]
sinb = sin(b / 2)
bsinb = [bh[0] * sinb, bh[1] * sinb, bh[2] * sinb]
c = 2 * acos(cos(a / 2)*cos(b / 2) - dot3d(asina, bsinb))
d = axisanglenormalize(cos(a / 2) * sp.Matrix(bsinb) + cos(b / 2) * sp.Matrix(asina) + cross3d(asina, bsinb))
return sp.Matrix([
c * d[0],
c * d[1],
c * d[2]
])
def cross(u, v):
"""
:param u: 1d matrix
:type u: Matrix
:param v: 1d matrix
:type v: Matrix
:return: 1d Matrix. If u and v have length 4, it ignores the last entry and adds a zero to the result.
:rtype: Matrix
"""
if len(u) != len(v):
raise ValueError('lengths {} and {} don\'t align'.format(len(u), len(v)))
if len(u) == 3:
return se.Matrix([u[1] * v[2] - u[2] * v[1],
u[2] * v[0] - u[0] * v[2],
u[0] * v[1] - u[1] * v[0]])
if len(u) == 4:
return se.Matrix([u[1] * v[2] - u[2] * v[1],
u[2] * v[0] - u[0] * v[2],
u[0] * v[1] - u[1] * v[0],
0])
def rotation3_quaternion(x, y, z, w):
""" Unit quaternion to 4x4 rotation matrix according to:
https://github.com/orocos/orocos_kinematics_dynamics/blob/master/orocos_kdl/src/frames.cpp#L167
"""
x2 = x * x
y2 = y * y
z2 = z * z
w2 = w * w
return sp.Matrix(
[[w2 + x2 - y2 - z2, 2 * x * y - 2 * w * z, 2 * x * z + 2 * w * y, 0],
[2 * x * y + 2 * w * z, w2 - x2 + y2 - z2, 2 * y * z - 2 * w * x, 0],
[2 * x * z - 2 * w * y, 2 * y * z + 2 * w * x, w2 - x2 - y2 + z2, 0],
[0, 0, 0, 1]])
def imu_correct_up(mu, imu_rot, up_in_obj):
rot = imu_rot[0:4]
rotv = imu_rot[4:7]
up = quatrotatevector(rot, up_in_obj)
G = [ 0, 0, 1]
N = cross(up, G)
angle = atan2(sqrt(up[0] * up[0] + up[1] * up[1]), up[2])
for i in range(3):
N[i] = N[i] * angle * mu / 2.
qc = axisangle2quat(N)
return sp.Matrix([*quatrotateabout(qc, rot), *rotv])
def quaternion_from_axis_angle(axis, angle):
"""
:param axis: 3x1 Matrix
:type axis: Matrix
:type angle: Union[float, Symbol]
:return: 4x1 Matrix
:rtype: Matrix
"""
half_angle = angle / 2
return se.Matrix([axis[0] * se.sin(half_angle),
axis[1] * se.sin(half_angle),
axis[2] * se.sin(half_angle),
se.cos(half_angle)])
def quaternion_multiply(q1, q2):
"""
:param q1: 4x1 Matrix
:type q1: Matrix
:param q2: 4x1 Matrix
:type q2: Matrix
:return: 4x1 Matrix
:rtype: Matrix
"""
x0, y0, z0, w0 = q2
x1, y1, z1, w1 = q1
return se.Matrix([x1 * w0 + y1 * z0 - z1 * y0 + w1 * x0,
-x1 * z0 + y1 * w0 + z1 * x0 + w1 * y0,
x1 * y0 - y1 * x0 + z1 * w0 + w1 * z0,
-x1 * x0 - y1 * y0 - z1 * z0 + w1 * w0])
def axis_angle_from_quaternion(x, y, z, w):
"""
:type x: Union[float, Symbol]
:type y: Union[float, Symbol]
:type z: Union[float, Symbol]
:type w: Union[float, Symbol]
:return: 4x1 Matrix
:rtype: Matrix
"""
# TODO buggy, angle goes from 0 - 2*pi instead of -pi - +pi
w2 = sp.sqrt(1 - w**2)
angle = (2 * sp.acos(w))
x = x / w2
y = y / w2
z = z / w2
return sp.Matrix([x, y, z]), angle
def axis_angle_from_quaternion(x, y, z, w):
"""
:type x: Union[float, Symbol]
:type y: Union[float, Symbol]
:type z: Union[float, Symbol]
:type w: Union[float, Symbol]
:return: 4x1 Matrix
:rtype: Matrix
"""
l = norm([x, y, z, w])
x, y, z, w = x / l, y / l, z / l, w / l
w2 = se.sqrt(1 - w**2)
angle = (2 * se.acos(se.Min(se.Max(-1, w), 1)))
m = if_eq_zero(w2, 1, w2) # avoid /0
x = if_eq_zero(w2, 0, x / m)
y = if_eq_zero(w2, 0, y / m)
z = if_eq_zero(w2, 1, z / m)
return se.Matrix([x, y, z]), angle
def joint_normal(cls, names, level, mu, sigma):
"""Create joint normally distributed random variables
Parameters
----------
names : list
Names of the random variables
level : str
Variability level
mu : matrix or list
Vector of the means of the random variables
sigma : matrix or list of lists
Covariance matrix of the random variables
Example
-------
>>> from pharmpy import RandomVariable, Parameter
>>> omega_cl = Parameter("OMEGA_CL", 0.1)
>>> omega_v = Parameter("OMEGA_V", 0.1)
>>> corr_cl_v = Parameter("OMEGA_CL_V", 0.01)
>>> rv1, rv2 = RandomVariable.joint_normal(["IIV_CL", "IIV_V"], 'IIV', [0, 0],
... [[omega_cl.symbol, corr_cl_v.symbol], [corr_cl_v.symbol, omega_v.symbol]])
>>> rv1
⎡IIV_CL⎤ ⎧⎡0⎤ ⎡ OMEGA_CL OMEGA_CL_V⎤⎫
⎢ ⎥ ~ 𝒩 ⎪⎢ ⎥, ⎢ ⎥⎪
⎣IIV_V ⎦ ⎩⎣0⎦ ⎣OMEGA_CL_V OMEGA_V ⎦⎭
"""
mean = sympy.Matrix(mu)
variance = sympy.Matrix(sigma)
if variance.is_positive_semidefinite is False:
raise ValueError('Sigma matrix is not positive semidefinite')
rvs = []
for name in names:
rv = cls(name, level)
rv._mean = mean.copy()
rv._variance = variance.copy()
rv._symengine_variance = symengine.Matrix(variance.rows,
variance.cols, sigma)
rv._joint_names = names.copy()
rvs.append(rv)
return rvs