def Fourier_series(x, f, n = 0):
"""Construit la série de Fourier comme modèle de référence pour le fitting."""
a0, *cos_a = parameters(','.join(['a{}'.format(i) for i in range(0, n + 1)]))
sin_b = parameters(','.join(['b{}'.format(i) for i in range(1, n + 1)]))
series = a0 + sum(ai * cos(i * f * x) + bi * sin(i * f * x) for i, (ai, bi) in enumerate(zip(cos_a, sin_b), start=1))
return series
def fourier_series(x, f, n=0):
# Make the parameter objects for all the terms
a0, *A = parameters(','.join([f'a{i}' for i in range(0, n + 1)]))
B = parameters(','.join([f'b{i}' for i in range(1, n + 1)]))
# Construct the series
series = a0 + sum(ai*cos(i*f*x) + bi*sin(i*f*x)
for i, (ai, bi) in enumerate(zip(A, B), start=1))
return series
def fourier(self) :
n=self.n
w = self.w
lst = range(n+1)
self.a_n = parameters(','.join(['a{}'.format(i) for i in lst]))
self.b_n = parameters(','.join(['b{}'.format(i) for i in lst]))
self.coeff = self.a_n + self.b_n
self.eqn = sum([i * sp.cos(k * w * Symbol('x')) + j * sp.sin(k * w * Symbol('x')) for k,(i,j) in enumerate(zip(self.a_n,self.b_n))])
return self.eqn
def fourier_series(self, x, f, n=2):
a0, *cos_a = parameters(','.join(
['a{}'.format(i) for i in range(0, n + 1)]))
sin_b = parameters(','.join(['b{}'.format(i)
for i in range(1, n + 1)]))
series = a0 + sum(
ai * cos(i * f * x) + bi * sin(i * f * x)
for i, (ai, bi) in enumerate(zip(cos_a, sin_b), start=1))
return series
def test_LogLikelihood_global():
"""
This is a test for global likelihood fitting to multiple data sets.
Based on SO question 56006357.
"""
# creating the data
mu1, mu2 = .05, -.05
sigma1, sigma2 = 3.5, 2.5
n1, n2 = 80, 90
np.random.seed(42)
x1 = np.random.vonmises(mu1, sigma1, n1)
x2 = np.random.vonmises(mu2, sigma2, n2)
n = 2 # number of components
xs = variables('x,' + ','.join('x_{}'.format(i) for i in range(1, n + 1)))
x, xs = xs[0], xs[1:]
ys = variables(','.join('y_{}'.format(i) for i in range(1, n + 1)))
mu, kappa = parameters('mu, kappa')
kappas = parameters(','.join('k_{}'.format(i) for i in range(1, n + 1)),
min=0,
max=10)
mu.min, mu.max = -np.pi, np.pi
template = exp(kappa * cos(x - mu)) / (2 * pi * besseli(0, kappa))
model = Model({
y_i: template.subs({
kappa: k_i,
x: x_i
})
for y_i, x_i, k_i in zip(ys, xs, kappas)
})
all_data = {xs[0]: x1, xs[1]: x2, ys[0]: None, ys[1]: None}
all_params = {'mu': 1}
all_params.update({k_i.name: 1 for k_i in kappas})
# Evaluate the loglikelihood and its jacobian and hessian
logL = LogLikelihood(model, data=all_data)
eval_numerical = logL(**all_params)
jac_numerical = logL.eval_jacobian(**all_params)
hess_numerical = logL.eval_hessian(**all_params)
# Test the types and shapes of the components.
assert isinstance(eval_numerical, float)
assert isinstance(jac_numerical, np.ndarray)
assert isinstance(hess_numerical, np.ndarray)
assert eval_numerical.shape == tuple() # Empty tuple -> scalar
assert jac_numerical.shape == (3, )
assert hess_numerical.shape == (
3,
3,
)
def fourier_series(x, f, n=0):
"""
Returns a symbolic fourier series of order `n`.
:param n: Order of the fourier series.
:param x: Independent variable
:param f: Frequency of the fourier series
"""
# Make the parameter objects for all the terms
a0, *cos_a = parameters(','.join(['a{}'.format(i) for i in range(0, n + 1)]))
sin_b = parameters(','.join(['b{}'.format(i) for i in range(1, n + 1)]))
# Construct the series
series = a0 + sum(ai * cos(i * f * x) + bi * sin(i * f * x)
for i, (ai, bi) in enumerate(zip(cos_a, sin_b), start=1))
return series
def test_simple_kinetics(self):
"""
Simple kinetics data to test fitting
"""
tdata = np.array([10, 26, 44, 70, 120])
adata = 10e-4 * np.array([44, 34, 27, 20, 14])
a, b, t = variables('a, b, t')
k, a0 = parameters('k, a0')
k.value = 0.01
# a0.value, a0.min, a0.max = 54 * 10e-4, 40e-4, 60e-4
a0 = 54 * 10e-4
model_dict = {
D(a, t): -k * a**2,
D(b, t): k * a**2,
}
ode_model = ODEModel(model_dict, initial={t: 0.0, a: a0, b: 0.0})
# Generate some data
tvec = np.linspace(0, 500, 1000)
fit = NumericalLeastSquares(ode_model, t=tdata, a=adata, b=None)
fit_result = fit.execute()
# print(fit_result)
self.assertAlmostEqual(fit_result.value(k), 4.302875e-01, 4)
self.assertAlmostEqual(fit_result.stdev(k), 6.447068e-03, 4)
fit = Fit(ode_model, t=tdata, a=adata, b=None)
fit_result = fit.execute()
# print(fit_result)
self.assertAlmostEqual(fit_result.value(k), 4.302875e-01, 4)
self.assertTrue(np.isnan(fit_result.stdev(k)))
def test_polgar(self):
"""
Analysis of data published here:
This whole ODE support was build to do this analysis in the first place
"""
a, b, c, d, t = variables('a, b, c, d, t')
k, p, l, m = parameters('k, p, l, m')
a0 = 10
b = a0 - d + a
model_dict = {
D(d, t): l * c * b - m * d,
D(c, t): k * a * b - p * c - l * c * b + m * d,
D(a, t): -k * a * b + p * c,
}
ode_model = ODEModel(model_dict,
initial={
t: 0.0,
a: a0,
c: 0.0,
d: 0.0
})
# Generate some data
tdata = np.linspace(0, 3, 1000)
# Eval
AA, AAB, BAAB = ode_model(t=tdata, k=0.1, l=0.2, m=.3, p=0.3)
def test_simple_kinetics(self):
"""
Simple kinetics data to test fitting
"""
tdata = np.array([10, 26, 44, 70, 120])
adata = 10e-4 * np.array([44, 34, 27, 20, 14])
a, b, t = variables('a, b, t')
k, a0 = parameters('k, a0')
k.value = 0.01
# a0.value, a0.min, a0.max = 54 * 10e-4, 40e-4, 60e-4
a0 = 54 * 10e-4
model_dict = {
D(a, t): - k * a**2,
D(b, t): k * a**2,
}
ode_model = ODEModel(model_dict, initial={t: 0.0, a: a0, b: 0.0})
# Analytical solution
model = GradientModel({a: 1 / (k * t + 1 / a0)})
fit = Fit(model, t=tdata, a=adata)
fit_result = fit.execute()
fit = Fit(ode_model, t=tdata, a=adata, b=None, minimizer=MINPACK)
ode_result = fit.execute()
self.assertAlmostEqual(ode_result.value(k) / fit_result.value(k), 1.0, 4)
self.assertAlmostEqual(ode_result.stdev(k) / fit_result.stdev(k), 1.0, 4)
self.assertAlmostEqual(ode_result.r_squared / fit_result.r_squared, 1, 4)
fit = Fit(ode_model, t=tdata, a=adata, b=None)
ode_result = fit.execute()
self.assertAlmostEqual(ode_result.value(k) / fit_result.value(k), 1.0, 4)
self.assertAlmostEqual(ode_result.stdev(k) / fit_result.stdev(k), 1.0, 4)
self.assertAlmostEqual(ode_result.r_squared / fit_result.r_squared, 1, 4)
def test_known_solution(self):
p, c1 = parameters('p, c1')
y, t = variables('y, t')
p.value = 3.0
model_dict = {
D(y, t): - p * y,
}
# Lets say we know the exact solution to this problem
sol = Model({y: exp(- p * t)})
# Generate some data
tdata = np.linspace(0, 3, 10001)
ydata = sol(t=tdata, p=3.22)[0]
ydata += np.random.normal(0, 0.005, ydata.shape)
ode_model = ODEModel(model_dict, initial={t: 0.0, y: ydata[0]})
fit = Fit(ode_model, t=tdata, y=ydata)
ode_result = fit.execute()
c1.value = ydata[0]
fit = Fit(sol, t=tdata, y=ydata)
fit_result = fit.execute()
self.assertAlmostEqual(ode_result.value(p) / fit_result.value(p), 1, 2)
self.assertAlmostEqual(ode_result.r_squared / fit_result.r_squared, 1, 4)
self.assertAlmostEqual(ode_result.stdev(p) / fit_result.stdev(p), 1, 3)
def test_full_eval_range(self):
"""
Test if ODEModels can be evaluated at t < t_initial.
A bit of a no news is good news test.
"""
tdata = np.array([0, 10, 26, 44, 70, 120])
adata = 10e-4 * np.array([54, 44, 34, 27, 20, 14])
a, b, t = variables('a, b, t')
k, a0 = parameters('k, a0')
k.value = 0.01
t0 = tdata[2]
a0 = adata[2]
b0 = 0.02729855 # Obtained from evaluating from t=0.
model_dict = {
D(a, t): - k * a**2,
D(b, t): k * a**2,
}
ode_model = ODEModel(model_dict, initial={t: t0, a: a0, b: b0})
fit = Fit(ode_model, t=tdata, a=adata, b=None)
ode_result = fit.execute()
self.assertGreater(ode_result.r_squared, 0.95, 4)
# Now start from a timepoint that is not in the t-array such that it
# triggers another pathway to be taken in integrating it.
# Again, no news is good news.
ode_model = ODEModel(model_dict, initial={t: t0 + 1e-5, a: a0, b: b0})
fit = Fit(ode_model, t=tdata, a=adata, b=None)
ode_result = fit.execute()
self.assertGreater(ode_result.r_squared, 0.95, 4)
def test_simple_kinetics(self):
"""
Simple kinetics data to test fitting
"""
tdata = np.array([10, 26, 44, 70, 120])
adata = 10e-4 * np.array([44, 34, 27, 20, 14])
a, b, t = variables('a, b, t')
k, a0 = parameters('k, a0')
k.value = 0.01
# a0.value, a0.min, a0.max = 54 * 10e-4, 40e-4, 60e-4
a0 = 54 * 10e-4
model_dict = {
D(a, t): -k * a**2,
D(b, t): k * a**2,
}
ode_model = ODEModel(model_dict, initial={t: 0.0, a: a0, b: 0.0})
fit = ConstrainedNumericalLeastSquares(ode_model,
t=tdata,
a=adata,
b=None)
fit_result = fit.execute(tol=1e-9)
self.assertAlmostEqual(fit_result.value(k), 4.302875e-01, 4)
self.assertTrue(fit_result.stdev(k) is None)
def fit_data(data, guess=10.0, use_err=False):
x_data = data[:, 0]
y_data = 1 - data[:, 1]
if use_err:
data_err = data[:, 2]
x, y = sf.variables('x, y')
pKa, n = sf.parameters('pKa, n')
model = sf.Model({y: (10**(n * (pKa - x)) + 1)**-1.0})
pKa.value = guess
n.value = 1
if use_err:
fit = sf.Fit(model,
x=x_data,
y=y_data,
sigma_y=data_err,
minimizer=Powell,
absolute_sigma=True)
else:
fit = sf.Fit(model, x=x_data, y=y_data, minimizer=Powell)
result = fit.execute()
print("pKa.....................................", result.value(pKa), '+/-',
result.stdev(pKa))
print("n.......................................", result.value(n), '+/-',
result.stdev(n))
print("Regression coefficent:................", result.r_squared, '\n')
x_out = np.arange(min(x_data), max(x_data), 10**-3.0)
y_out = fit.model(x=x_out, **result.params)[0]
return x_out, y_out, result.value(pKa), result.stdev(
pKa), result.r_squared, result.value(n), result.stdev(n)
def test_full_eval_range(self):
"""
Test if ODEModels can be evaluated at t < t_initial.
A bit of a no news is good news test.
"""
tdata = np.array([0, 10, 26, 44, 70, 120])
adata = 10e-4 * np.array([54, 44, 34, 27, 20, 14])
a, b, t = variables('a, b, t')
k, a0 = parameters('k, a0')
k.value = 0.01
t0 = tdata[2]
a0 = adata[2]
b0 = 0.02729855 # Obtained from evaluating from t=0.
model_dict = {
D(a, t): - k * a**2,
D(b, t): k * a**2,
}
ode_model = ODEModel(model_dict, initial={t: t0, a: a0, b: b0})
fit = Fit(ode_model, t=tdata, a=adata, b=None)
ode_result = fit.execute()
self.assertGreater(ode_result.r_squared, 0.95, 4)
# Now start from a timepoint that is not in the t-array such that it
# triggers another pathway to be taken in integrating it.
# Again, no news is good news.
ode_model = ODEModel(model_dict, initial={t: t0 + 1e-5, a: a0, b: b0})
fit = Fit(ode_model, t=tdata, a=adata, b=None)
ode_result = fit.execute()
self.assertGreater(ode_result.r_squared, 0.95, 4)
def test_simple_kinetics(self):
"""
Simple kinetics data to test fitting
"""
tdata = np.array([10, 26, 44, 70, 120])
adata = 10e-4 * np.array([44, 34, 27, 20, 14])
a, b, t = variables('a, b, t')
k, a0 = parameters('k, a0')
k.value = 0.01
# a0.value, a0.min, a0.max = 54 * 10e-4, 40e-4, 60e-4
a0 = 54 * 10e-4
model_dict = {
D(a, t): - k * a**2,
D(b, t): k * a**2,
}
ode_model = ODEModel(model_dict, initial={t: 0.0, a: a0, b: 0.0})
# Analytical solution
model = Model({a: 1 / (k * t + 1 / a0)})
fit = Fit(model, t=tdata, a=adata)
fit_result = fit.execute()
fit = Fit(ode_model, t=tdata, a=adata, b=None, minimizer=MINPACK)
ode_result = fit.execute()
self.assertAlmostEqual(ode_result.value(k) / fit_result.value(k), 1.0, 4)
self.assertAlmostEqual(ode_result.stdev(k) / fit_result.stdev(k), 1.0, 4)
self.assertAlmostEqual(ode_result.r_squared / fit_result.r_squared, 1, 4)
fit = Fit(ode_model, t=tdata, a=adata, b=None)
ode_result = fit.execute()
self.assertAlmostEqual(ode_result.value(k) / fit_result.value(k), 1.0, 4)
self.assertAlmostEqual(ode_result.stdev(k) / fit_result.stdev(k), 1.0, 4)
self.assertAlmostEqual(ode_result.r_squared / fit_result.r_squared, 1, 4)
def test_pickle(self):
"""
Make sure models can be pickled are preserved when pickling
"""
a, b = parameters('a, b')
x, y = variables('x, y')
exact_model = Model({y: a * x ** b})
constraint = Model.as_constraint(Eq(a, b), exact_model)
num_model = CallableNumericalModel(
{y: a * x ** b}, independent_vars=[x], params=[a, b]
)
connected_num_model = CallableNumericalModel(
{y: a * x ** b}, connectivity_mapping={y: {x, a, b}}
)
# Test if lsoda args and kwargs are pickled too
ode_model = ODEModel({D(y, x): a * x + b}, {x: 0.0}, 3, 4, some_kwarg=True)
models = [exact_model, constraint, num_model, ode_model,
connected_num_model]
for model in models:
new_model = pickle.loads(pickle.dumps(model))
# Compare signatures
self.assertEqual(model.__signature__, new_model.__signature__)
# Trigger the cached vars because we compare `__dict__` s
model.vars
new_model.vars
# Explicitly make sure the connectivity mapping is identical.
self.assertEqual(model.connectivity_mapping,
new_model.connectivity_mapping)
if not isinstance(model, ODEModel):
model.function_dict
model.vars_as_functions
new_model.function_dict
new_model.vars_as_functions
self.assertEqual(model.__dict__, new_model.__dict__)
def test_simple_kinetics(self):
"""
Simple kinetics data to test fitting
"""
tdata = np.array([10, 26, 44, 70, 120])
adata = 10e-4 * np.array([44, 34, 27, 20, 14])
a, b, t = variables('a, b, t')
k, a0 = parameters('k, a0')
k.value = 0.01
# a0.value, a0.min, a0.max = 54 * 10e-4, 40e-4, 60e-4
a0 = 54 * 10e-4
model_dict = {
D(a, t): - k * a**2,
D(b, t): k * a**2,
}
ode_model = ODEModel(model_dict, initial={t: 0.0, a: a0, b: 0.0})
# Generate some data
tvec = np.linspace(0, 500, 1000)
fit = NumericalLeastSquares(ode_model, t=tdata, a=adata, b=None)
fit_result = fit.execute()
# print(fit_result)
self.assertAlmostEqual(fit_result.value(k), 4.302875e-01, 4)
self.assertAlmostEqual(fit_result.stdev(k), 6.447068e-03, 4)
fit = Fit(ode_model, t=tdata, a=adata, b=None)
fit_result = fit.execute()
# print(fit_result)
self.assertAlmostEqual(fit_result.value(k), 4.302875e-01, 4)
self.assertTrue(np.isnan(fit_result.stdev(k)))
def test_multi_indep():
'''
Tests the case with multiple components, multiple parameters and
multiple independent variables
'''
w, x, y, z = sf.variables('w, x, y, z')
a, b, c = sf.parameters('a, b, c')
model = sf.Model({
y: 3 * a * x**2 + b * x * w - c,
z: sf.exp(a * x - b) + c * w
})
x_data = np.arange(10) / 10
w_data = np.arange(10)
exact = model.eval_jacobian(x=x_data, w=w_data, a=3.5, b=2, c=5)
approx = model.finite_difference(x=x_data, w=w_data, a=3.5, b=2, c=5)
_assert_equal(exact, approx)
exact = model.eval_jacobian(x=0.3, w=w_data, a=3.5, b=2, c=5)
approx = model.finite_difference(x=0.3, w=w_data, a=3.5, b=2, c=5)
_assert_equal(exact, approx)
exact = model.eval_jacobian(x=0.3, w=5, a=3.5, b=2, c=5)
approx = model.finite_difference(x=0.3, w=5, a=3.5, b=2, c=5)
_assert_equal(exact, approx)
def fit_plane(xyz):
"""
Fit a plane to the point coordinates in xyz.
Dev note: An alternative implementation is possible that omits the `f`
variable, and thus has one fewer degree of freedom. This means the fit is
easier and maybe more precise. This could be tested. The notebook
req4.1_fit_plane.ipynb in the explore repository
(https://github.com/sundial-pointcloud-geometry/explore) has some notes on
this. The problem with those models where f is just zero and the named
symfit model is created for one of x, y or z instead is that you have to
divide by one of the a, b or c parameters respectively. If one of these
turns out to be zero, symfit will not find a fit. A solution would be
to actually create three models and try another if one of them fails to
converge.
"""
a, b, c, d = sf.parameters('a, b, c, d')
x, y, z, f = sf.variables('x, y, z, f')
plane_model = {f: x * a + y * b + z * c + d}
plane_fit = sf.Fit(plane_model, x=xyz[0], y=xyz[1], z=xyz[2],
f=np.zeros_like(xyz[0]),
constraints=[sf.Equality(a**2 + b**2 + c**2, 1)]) # keep plane normal a unit vector
plane_fit_result = plane_fit.execute()
return plane_fit_result
def test_vector_fitting_guess(self):
"""
Tests fitting to a 3 component vector valued function, with guesses.
"""
a, b, c = parameters('a, b, c')
a.value = 10
b.value = 100
a_i, b_i, c_i = variables('a_i, b_i, c_i')
model = {a_i: a, b_i: b, c_i: c}
xdata = np.array([
[10.1, 9., 10.5, 11.2, 9.5, 9.6, 10.],
[102.1, 101., 100.4, 100.8, 99.2, 100., 100.8],
[71.6, 73.2, 69.5, 70.2, 70.8, 70.6, 70.1],
])
fit = Fit(
model=model,
a_i=xdata[0],
b_i=xdata[1],
c_i=xdata[2],
minimizer = MINPACK
)
fit_result = fit.execute()
self.assertAlmostEqual(fit_result.value(a), np.mean(xdata[0]), 4)
self.assertAlmostEqual(fit_result.value(b), np.mean(xdata[1]), 4)
self.assertAlmostEqual(fit_result.value(c), np.mean(xdata[2]), 4)
def test_likelihood_fitting_gaussian():
"""
Fit using the likelihood method.
"""
mu, sig = parameters('mu, sig')
sig.min = 0.01
sig.value = 3.0
mu.value = 50.
x = Variable('x')
pdf = GradientModel(Gaussian(x, mu, sig))
np.random.seed(10)
# TODO: Do we really need 1k points?
xdata = np.random.normal(51., 3.5, 10000)
# Expected parameter values
mean = np.mean(xdata)
stdev = np.std(xdata)
mean_stdev = stdev / np.sqrt(len(xdata))
fit = Fit(pdf, xdata, objective=LogLikelihood)
fit_result = fit.execute()
assert fit_result.value(mu) == pytest.approx(mean, 1e-6)
assert fit_result.stdev(mu) == pytest.approx(mean_stdev, 1e-3)
assert fit_result.value(sig) == pytest.approx(np.std(xdata), 1e-6)
def test_data_for_constraint(self):
"""
Test the signature handling when constraints are at play. Constraints
should take seperate data, but still kwargs that are not found in either
the model nor the constraints should raise an error.
"""
A, mu, sig = parameters('A, mu, sig')
x, y, Y = variables('x, y, Y')
model = Model({y: A * Gaussian(x, mu=mu, sig=sig)})
constraint = Model.as_constraint(Y, model, constraint_type=Eq)
np.random.seed(2)
xdata = np.random.normal(1.2, 2, 10)
ydata, xedges = np.histogram(xdata, bins=int(np.sqrt(len(xdata))),
density=True)
# Allowed
fit = Fit(model, x=xdata, y=ydata, Y=2, constraints=[constraint])
fit = Fit(model, x=xdata, y=ydata)
fit = Fit(model, x=xdata, objective=LogLikelihood)
# Not allowed
with self.assertRaises(TypeError):
fit = Fit(model, x=xdata, y=ydata, Y=2)
with self.assertRaises(TypeError):
fit = Fit(model, x=xdata, y=ydata, Y=2, Z=3, constraints=[constraint])
def test_pickle():
"""
Make sure models can be pickled are preserved when pickling
"""
a, b = parameters('a, b')
x, y = variables('x, y')
exact_model = Model({y: a * x ** b})
constraint = Model.as_constraint(Eq(a, b), exact_model)
num_model = CallableNumericalModel(
{y: a * x ** b}, independent_vars=[x], params=[a, b]
)
connected_num_model = CallableNumericalModel(
{y: a * x ** b}, connectivity_mapping={y: {x, a, b}}
)
# Test if lsoda args and kwargs are pickled too
ode_model = ODEModel({D(y, x): a * x + b}, {x: 0.0}, 3, 4, some_kwarg=True)
models = [exact_model, constraint, num_model, ode_model, connected_num_model]
for model in models:
new_model = pickle.loads(pickle.dumps(model))
# Compare signatures
assert model.__signature__ == new_model.__signature__
# Trigger the cached vars because we compare `__dict__` s
model.vars
new_model.vars
# Explicitly make sure the connectivity mapping is identical.
assert model.connectivity_mapping == new_model.connectivity_mapping
if not isinstance(model, ODEModel):
model.function_dict
model.vars_as_functions
new_model.function_dict
new_model.vars_as_functions
assert model.__dict__ == new_model.__dict__
def test_likelihood_fitting_gaussian(self):
"""
Fit using the likelihood method.
"""
mu, sig = parameters('mu, sig')
sig.min = 0.01
sig.value = 3.0
mu.value = 50.
x = Variable()
pdf = Gaussian(x, mu, sig)
np.random.seed(10)
xdata = np.random.normal(51., 3.5, 10000)
# Expected parameter values
mean = np.mean(xdata)
stdev = np.std(xdata)
mean_stdev = stdev/np.sqrt(len(xdata))
fit = Fit(pdf, xdata, objective=LogLikelihood)
fit_result = fit.execute()
self.assertAlmostEqual(fit_result.value(mu) / mean, 1, 6)
self.assertAlmostEqual(fit_result.stdev(mu) / mean_stdev, 1, 3)
self.assertAlmostEqual(fit_result.value(sig) / np.std(xdata), 1, 6)
def test_vector_fitting(self):
"""
Tests fitting to a 3 component vector valued function, without bounds
or guesses.
"""
a, b, c = parameters('a, b, c')
a_i, b_i, c_i = variables('a_i, b_i, c_i')
model = {a_i: a, b_i: b, c_i: c}
xdata = np.array([
[10.1, 9., 10.5, 11.2, 9.5, 9.6, 10.],
[102.1, 101., 100.4, 100.8, 99.2, 100., 100.8],
[71.6, 73.2, 69.5, 70.2, 70.8, 70.6, 70.1],
])
fit = Fit(
model=model,
a_i=xdata[0],
b_i=xdata[1],
c_i=xdata[2],
minimizer = MINPACK
)
fit_result = fit.execute()
self.assertAlmostEqual(fit_result.value(a) / 9.985691, 1.0, 5)
self.assertAlmostEqual(fit_result.value(b) / 1.006143e+02, 1.0, 4)
self.assertAlmostEqual(fit_result.value(c) / 7.085713e+01, 1.0, 5)
def test_taylor_model(self):
a, b = parameters('a, b')
x, y, z = variables('x, y, z')
model = Model({y: a * x + b})
appr = TaylorModel(model)
self.assertEqual(set([a, b]), set(appr.params))
appr.p0 = {a: 2.0, b: 5.0}
self.assertEqual(set(appr.p0.keys()),
set(appr.params_0[p] for p in appr.params))
self.assertTrue(LinearLeastSquares.is_linear(appr))
model = Model({z: a * x**2 + b * y**2})
appr = TaylorModel(model)
appr.p0 = {a: 2, b: 5}
model = Model({z: a * x**2 + b * y**2})
appr_2 = TaylorModel(model)
appr_2.p0 = {a: 1, b: 1}
self.assertTrue(appr == appr_2)
model = Model({y: a * sympy.exp(x * b)})
appr = TaylorModel(model)
appr.p0 = {a: 2.0, b: 5.0}
self.assertTrue(LinearLeastSquares.is_linear(appr))
model = Model({y: sympy.sin(a * x)})
appr = TaylorModel(model)
appr.p0 = {a: 0.0}
self.assertTrue(LinearLeastSquares.is_linear(appr))
def test_vector_fitting():
"""
Tests fitting to a 3 component vector valued function, without bounds
or guesses.
"""
a, b, c = parameters('a, b, c')
a_i, b_i, c_i = variables('a_i, b_i, c_i')
model = {a_i: a, b_i: b, c_i: c}
xdata = np.array([
[10.1, 9., 10.5, 11.2, 9.5, 9.6, 10.],
[102.1, 101., 100.4, 100.8, 99.2, 100., 100.8],
[71.6, 73.2, 69.5, 70.2, 70.8, 70.6, 70.1],
])
fit = Fit(model=model,
a_i=xdata[0],
b_i=xdata[1],
c_i=xdata[2],
minimizer=MINPACK)
fit_result = fit.execute()
assert fit_result.value(a) == pytest.approx(np.mean(xdata[0]), 1e-5)
assert fit_result.value(b) == pytest.approx(np.mean(xdata[1]), 1e-4)
assert fit_result.value(c) == pytest.approx(np.mean(xdata[2]), 1e-5)
def test_taylor_model(self):
a, b = parameters('a, b')
x, y, z = variables('x, y, z')
model = Model({y: a * x + b})
appr = TaylorModel(model)
self.assertEqual(set([a, b]), set(appr.params))
appr.p0 = {a: 2.0, b: 5.0}
self.assertEqual(set(appr.p0.keys()), set(appr.params_0[p] for p in appr.params))
self.assertTrue(LinearLeastSquares.is_linear(appr))
model = Model({z: a * x**2 + b * y**2})
appr = TaylorModel(model)
appr.p0 = {a: 2, b: 5}
model = Model({z: a * x**2 + b * y**2})
appr_2 = TaylorModel(model)
appr_2.p0 = {a: 1, b: 1}
self.assertTrue(appr == appr_2)
model = Model({y: a * sympy.exp(x * b)})
appr = TaylorModel(model)
appr.p0 = {a: 2.0, b: 5.0}
self.assertTrue(LinearLeastSquares.is_linear(appr))
model = Model({y: sympy.sin(a * x)})
appr = TaylorModel(model)
appr.p0 = {a: 0.0}
self.assertTrue(LinearLeastSquares.is_linear(appr))
def test_fixed_parameters():
"""
Make sure fixed parameters don't change on fitting
"""
a, b, c, d = parameters('a, b, c, d')
x, y = variables('x, y')
c.value = 4.0
a.min, a.max = 1.0, 5.0 # Bounds are needed for DifferentialEvolution
b.min, b.max = 1.0, 5.0
c.min, c.max = 1.0, 5.0
d.min, d.max = 1.0, 5.0
c.fixed = True
model = Model({y: a * exp(-(x - b)**2 / (2 * c**2)) + d})
# Generate data
xdata = np.linspace(0, 100)
ydata = model(xdata, a=2, b=3, c=2, d=2).y
for minimizer in subclasses(BaseMinimizer):
if minimizer is ChainedMinimizer:
continue
else:
fit = Fit(model, x=xdata, y=ydata, minimizer=minimizer)
fit_result = fit.execute()
# Should still be 4.0, not 2.0!
assert 4.0 == fit_result.params['c']
def test_straight_line_analytical(self):
"""
Test symfit against a straight line, for which the parameters and their
uncertainties are known analytically. Assuming equal weights.
"""
data = [[0, 1], [1, 0], [3, 2], [5, 4]]
xdata, ydata = (np.array(i, dtype='float64') for i in zip(*data))
# x = np.arange(0, 100, 0.1)
# np.random.seed(10)
# y = 3.0*x + 105.0 + np.random.normal(size=x.shape)
dx = xdata - xdata.mean()
dy = ydata - ydata.mean()
mean_squared_x = np.mean(xdata**2) - np.mean(xdata)**2
mean_xy = np.mean(xdata * ydata) - np.mean(xdata)*np.mean(ydata)
a = mean_xy/mean_squared_x
b = ydata.mean() - a * xdata.mean()
self.assertAlmostEqual(a, 0.694915, 6) # values from Mathematica
self.assertAlmostEqual(b, 0.186441, 6)
S = np.sum((ydata - (a*xdata + b))**2)
var_a_exact = S/(len(xdata) * (len(xdata) - 2) * mean_squared_x)
var_b_exact = var_a_exact*np.mean(xdata**2)
a_exact = a
b_exact = b
# We will now compare these exact results with values from symfit, numerically
a, b = parameters('a, b')
x, y = variables('x, y')
model = {y: a*x + b}
fit = NumericalLeastSquares(model, x=xdata, y=ydata)#, absolute_sigma=False)
fit_result = fit.execute()
popt, pcov = curve_fit(lambda z, c, d: c * z + d, xdata, ydata,
jac=lambda z, c, d: np.transpose([xdata, np.ones_like(xdata)]))
# jac=lambda p, x, y, func: np.transpose([x, np.ones_like(x)]))
# Dfun=lambda p, x, y, func: print(p, func, x, y))
# curve_fit
self.assertAlmostEqual(a_exact, popt[0], 4)
self.assertAlmostEqual(b_exact, popt[1], 4)
self.assertAlmostEqual(var_a_exact, pcov[0][0], 6)
self.assertAlmostEqual(var_b_exact, pcov[1][1], 6)
self.assertAlmostEqual(a_exact, fit_result.value(a), 4)
self.assertAlmostEqual(b_exact, fit_result.value(b), 4)
self.assertAlmostEqual(var_a_exact, fit_result.variance(a), 6)
self.assertAlmostEqual(var_b_exact, fit_result.variance(b), 6)
# Do the fit with the LinearLeastSquares object
fit = LinearLeastSquares(model, x=xdata, y=ydata)
fit_result = fit.execute()
self.assertAlmostEqual(a_exact, fit_result.value(a), 4)
self.assertAlmostEqual(b_exact, fit_result.value(b), 4)
self.assertAlmostEqual(var_a_exact, fit_result.variance(a), 6)
self.assertAlmostEqual(var_b_exact, fit_result.variance(b), 6)
# Lets also make sure the entire covariance matrix is the same
for cov1, cov2 in zip(fit_result.params.covariance_matrix.flatten(), pcov.flatten()):
self.assertAlmostEqual(cov1, cov2)
def test_vector_none_fitting(self):
"""
Fit to a 3 component vector valued function with one variables data set
to None, without bounds or guesses.
"""
a, b, c = parameters('a, b, c')
a_i, b_i, c_i = variables('a_i, b_i, c_i')
model = {a_i: a, b_i: b, c_i: c}
xdata = np.array([
[10.1, 9., 10.5, 11.2, 9.5, 9.6, 10.],
[102.1, 101., 100.4, 100.8, 99.2, 100., 100.8],
[71.6, 73.2, 69.5, 70.2, 70.8, 70.6, 70.1],
])
fit_none = NumericalLeastSquares(
model=model,
a_i=xdata[0],
b_i=xdata[1],
c_i=None,
)
fit = NumericalLeastSquares(
model=model,
a_i=xdata[0],
b_i=xdata[1],
c_i=xdata[2],
)
fit_none_result = fit_none.execute()
fit_result = fit.execute()
self.assertAlmostEqual(fit_none_result.value(a), fit_result.value(a), 4)
self.assertAlmostEqual(fit_none_result.value(b), fit_result.value(b), 4)
# the parameter without data should be unchanged.
self.assertAlmostEqual(fit_none_result.value(c), 1.0)
def test_vector_fitting_bounds(self):
"""
Tests fitting to a 3 component vector valued function, with bounds.
"""
a, b, c = parameters('a, b, c')
a.min = 0
a.max = 25
b.min = 0
b.max = 500
a_i, b_i, c_i = variables('a_i, b_i, c_i')
model = {a_i: a, b_i: b, c_i: c}
xdata = np.array([
[10.1, 9., 10.5, 11.2, 9.5, 9.6, 10.],
[102.1, 101., 100.4, 100.8, 99.2, 100., 100.8],
[71.6, 73.2, 69.5, 70.2, 70.8, 70.6, 70.1],
])
fit = NumericalLeastSquares(
model=model,
a_i=xdata[0],
b_i=xdata[1],
c_i=xdata[2],
)
fit_result = fit.execute()
self.assertAlmostEqual(fit_result.value(a), np.mean(xdata[0]), 4)
self.assertAlmostEqual(fit_result.value(b), np.mean(xdata[1]), 4)
self.assertAlmostEqual(fit_result.value(c), np.mean(xdata[2]), 4)
def test_vector_none_fitting():
"""
Fit to a 3 component vector valued function with one variables data set
to None, without bounds or guesses.
"""
a, b, c = parameters('a, b, c')
a_i, b_i, c_i = variables('a_i, b_i, c_i')
model = {a_i: a, b_i: b, c_i: c}
xdata = np.array([
[10.1, 9., 10.5, 11.2, 9.5, 9.6, 10.],
[102.1, 101., 100.4, 100.8, 99.2, 100., 100.8],
[71.6, 73.2, 69.5, 70.2, 70.8, 70.6, 70.1],
])
fit_none = Fit(model=model,
a_i=xdata[0],
b_i=xdata[1],
c_i=None,
minimizer=MINPACK)
fit = Fit(model=model,
a_i=xdata[0],
b_i=xdata[1],
c_i=xdata[2],
minimizer=MINPACK)
fit_none_result = fit_none.execute()
fit_result = fit.execute()
assert fit_none_result.value(b) == pytest.approx(fit_result.value(b), 1e-4)
assert fit_none_result.value(a) == pytest.approx(fit_result.value(a), 1e-4)
# the parameter without data should be unchanged.
assert fit_none_result.value(c) == pytest.approx(1.0)
def test_likelihood_fitting_gaussian(self):
"""
Fit using the likelihood method.
"""
mu, sig = parameters('mu, sig')
sig.min = 0.01
sig.value = 3.0
mu.value = 50.
x = Variable()
pdf = Gaussian(x, mu, sig)
np.random.seed(10)
xdata = np.random.normal(51., 3.5, 10000)
# Expected parameter values
mean = np.mean(xdata)
stdev = np.std(xdata)
mean_stdev = stdev/np.sqrt(len(xdata))
fit = Fit(pdf, xdata, objective=LogLikelihood)
fit_result = fit.execute()
self.assertAlmostEqual(fit_result.value(mu) / mean, 1, 6)
self.assertAlmostEqual(fit_result.stdev(mu) / mean_stdev, 1, 3)
self.assertAlmostEqual(fit_result.value(sig) / np.std(xdata), 1, 6)
def __init__(self, data_dir, degree):
# get directory
self.data_dir = data_dir
self.degree = degree
x, y = variables('x, y')
w, = parameters('w')
self.model_dict = {y: self.fourier_series(x, f=w, n=self.degree)}
def test_initial_parameters():
"""
Identical to test_polgar, but with a0 as free Parameter.
"""
a, b, c, d, t = variables('a, b, c, d, t')
k, p, l, m = parameters('k, p, l, m')
a0 = Parameter('a0', min=0, value=10, fixed=True)
c0 = Parameter('c0', min=0, value=0.1)
b = a0 - d + a
model_dict = {
D(d, t): l * c * b - m * d,
D(c, t): k * a * b - p * c - l * c * b + m * d,
D(a, t): - k * a * b + p * c,
}
ode_model = ODEModel(model_dict, initial={t: 0.0, a: a0, c: c0, d: 0.0})
# Generate some data
tdata = np.linspace(0, 3, 1000)
# Eval
AA, AAB, BAAB = ode_model(t=tdata, k=0.1, l=0.2, m=.3, p=0.3, a0=10, c0=0)
fit = Fit(ode_model, t=tdata, a=AA, c=AAB, d=BAAB)
results = fit.execute()
print(results)
assert results.value(a0) == pytest.approx(10, abs=1e-8)
assert results.value(c0) == pytest.approx(0, abs=1e-8)
assert ode_model.params == [a0, c0, k, l, m, p]
assert ode_model.initial_params == [a0, c0]
assert ode_model.model_params == [a0, k, l, m, p]
def test_single_eval(self):
"""
Eval an ODEModel at a single value rather than a vector.
"""
x, y, t = variables('x, y, t')
k, = parameters('k') # C is the integration constant.
# The harmonic oscillator as a system, >1st order is not supported yet.
harmonic_dict = {
D(x, t): - k * y,
D(y, t): k * x,
}
# Make a second model to prevent caching of integration results.
# This also means harmonic_dict should NOT be a Model object.
harmonic_model_array = ODEModel(harmonic_dict, initial={t: 0.0, x: 1.0, y: 0.0})
harmonic_model_points = ODEModel(harmonic_dict, initial={t: 0.0, x: 1.0, y: 0.0})
tdata = np.linspace(0, 100, 101)
X, Y = harmonic_model_array(t=tdata, k=0.1)
# Shuffle the data to prevent using the result at time t to calculate
# t+dt
random_order = np.random.permutation(len(tdata))
for idx in random_order:
t = tdata[idx]
X_val = X[idx]
Y_val = Y[idx]
X_point, Y_point = harmonic_model_points(t=t, k=0.1)
self.assertAlmostEqual(X_point[0], X_val)
self.assertAlmostEqual(Y_point[0], Y_val)
def test_global_fitting(self):
"""
In case of shared parameters between the components of the model, `Fit`
should automatically use `ConstrainedLeastSquares`.
:return:
"""
x_1, x_2, y_1, y_2 = variables('x_1, x_2, y_1, y_2')
y0, a_1, a_2, b_1, b_2 = parameters('y0, a_1, a_2, b_1, b_2')
# The following vector valued function links all the equations together
# as stated in the intro.
model = Model({
y_1: a_1 * x_1**2 + b_1 * x_1 + y0,
y_2: a_2 * x_2**2 + b_2 * x_2 + y0,
})
self.assertTrue(model.shared_parameters)
# Generate data from this model
xdata1 = np.linspace(0, 10)
xdata2 = xdata1[::2] # Only every other point.
ydata1, ydata2 = model(x_1=xdata1, x_2=xdata2, a_1=101.3, b_1=0.5, a_2=56.3, b_2=1.1111, y0=10.8)
# Add some noise to make it appear like real data
np.random.seed(1)
ydata1 += np.random.normal(0, 2, size=ydata1.shape)
ydata2 += np.random.normal(0, 2, size=ydata2.shape)
xdata = [xdata1, xdata2]
ydata = [ydata1, ydata2]
# Guesses
a_1.value = 100
a_2.value = 50
b_1.value = 1
b_2.value = 1
y0.value = 10
fit = Fit(
model, x_1=xdata[0], x_2=xdata[1], y_1=ydata[0], y_2=ydata[1]
)
self.assertIsInstance(fit.fit, ConstrainedNumericalLeastSquares)
# The next model does not share parameters, but is still a vector
model = Model({
y_1: a_1 * x_1**2 + b_1 * x_1,
y_2: a_2 * x_2**2 + b_2 * x_2,
})
fit = Fit(
model, x_1=xdata[0], x_2=xdata[1], y_1=ydata[0], y_2=ydata[1]
)
self.assertFalse(model.shared_parameters)
self.assertIsInstance(fit.fit, NumericalLeastSquares)
# Scalar model, so it should use NumericalLeastSquares.
model = Model({
y_1: a_1 * x_1**2 + b_1 * x_1,
})
fit = Fit(model, x_1=xdata[0], y_1=ydata[0])
self.assertFalse(model.shared_parameters)
self.assertIsInstance(fit.fit, NumericalLeastSquares)
def test_single_eval(self):
"""
Eval an ODEModel at a single value rather than a vector.
"""
x, y, t = variables('x, y, t')
k, = parameters('k') # C is the integration constant.
# The harmonic oscillator as a system, >1st order is not supported yet.
harmonic_dict = {
D(x, t): - k * y,
D(y, t): k * x,
}
# Make a second model to prevent caching of integration results.
# This also means harmonic_dict should NOT be a Model object.
harmonic_model_array = ODEModel(harmonic_dict, initial={t: 0.0, x: 1.0, y: 0.0})
harmonic_model_points = ODEModel(harmonic_dict, initial={t: 0.0, x: 1.0, y: 0.0})
tdata = np.linspace(-100, 100, 101)
X, Y = harmonic_model_array(t=tdata, k=0.1)
# Shuffle the data to prevent using the result at time t to calculate
# t+dt
random_order = np.random.permutation(len(tdata))
for idx in random_order:
t = tdata[idx]
X_val = X[idx]
Y_val = Y[idx]
X_point, Y_point = harmonic_model_points(t=t, k=0.1)
self.assertAlmostEqual(X_point[0], X_val)
self.assertAlmostEqual(Y_point[0], Y_val)
def test_vector_fitting_guess(self):
"""
Tests fitting to a 3 component vector valued function, with guesses.
"""
a, b, c = parameters('a, b, c')
a.value = 10
b.value = 100
a_i, b_i, c_i = variables('a_i, b_i, c_i')
model = {a_i: a, b_i: b, c_i: c}
xdata = np.array([
[10.1, 9., 10.5, 11.2, 9.5, 9.6, 10.],
[102.1, 101., 100.4, 100.8, 99.2, 100., 100.8],
[71.6, 73.2, 69.5, 70.2, 70.8, 70.6, 70.1],
])
fit = NumericalLeastSquares(
model=model,
a_i=xdata[0],
b_i=xdata[1],
c_i=xdata[2],
)
fit_result = fit.execute()
self.assertAlmostEqual(fit_result.value(a), np.mean(xdata[0]), 4)
self.assertAlmostEqual(fit_result.value(b), np.mean(xdata[1]), 4)
self.assertAlmostEqual(fit_result.value(c), np.mean(xdata[2]), 4)
def test_known_solution(self):
p, c1 = parameters('p, c1')
y, t = variables('y, t')
p.value = 3.0
model_dict = {
D(y, t): - p * y,
}
# Lets say we know the exact solution to this problem
sol = Model({y: exp(- p * t)})
# Generate some data
tdata = np.linspace(0, 3, 10001)
ydata = sol(t=tdata, p=3.22)[0]
ydata += np.random.normal(0, 0.005, ydata.shape)
ode_model = ODEModel(model_dict, initial={t: 0.0, y: ydata[0]})
fit = Fit(ode_model, t=tdata, y=ydata)
ode_result = fit.execute()
c1.value = ydata[0]
fit = Fit(sol, t=tdata, y=ydata)
fit_result = fit.execute()
self.assertAlmostEqual(ode_result.value(p) / fit_result.value(p), 1, 2)
self.assertAlmostEqual(ode_result.r_squared / fit_result.r_squared, 1, 4)
self.assertAlmostEqual(ode_result.stdev(p) / fit_result.stdev(p), 1, 3)
def test_vector_fitting(self):
"""
Tests fitting to a 3 component vector valued function, without bounds
or guesses.
"""
a, b, c = parameters('a, b, c')
a_i, b_i, c_i = variables('a_i, b_i, c_i')
model = {a_i: a, b_i: b, c_i: c}
xdata = np.array([
[10.1, 9., 10.5, 11.2, 9.5, 9.6, 10.],
[102.1, 101., 100.4, 100.8, 99.2, 100., 100.8],
[71.6, 73.2, 69.5, 70.2, 70.8, 70.6, 70.1],
])
fit = NumericalLeastSquares(
model=model,
a_i=xdata[0],
b_i=xdata[1],
c_i=xdata[2],
)
fit_result = fit.execute()
self.assertAlmostEqual(fit_result.value(a), 9.985691, 6)
self.assertAlmostEqual(fit_result.value(b), 1.006143e+02, 4)
self.assertAlmostEqual(fit_result.value(c), 7.085713e+01, 5)
def test_global_fitting(self):
"""
Test a global fitting scenario with datasets of unequal length. In this
scenario, a quartic equation is fitted where the constant term is shared
between the datasets. (e.g. identical background noise)
"""
x_1, x_2, y_1, y_2 = variables('x_1, x_2, y_1, y_2')
y0, a_1, a_2, b_1, b_2 = parameters('y0, a_1, a_2, b_1, b_2')
# The following vector valued function links all the equations together
# as stated in the intro.
model = Model({
y_1: a_1 * x_1**2 + b_1 * x_1 + y0,
y_2: a_2 * x_2**2 + b_2 * x_2 + y0,
})
# Generate data from this model
# xdata = np.linspace(0, 10)
xdata1 = np.linspace(0, 10)
xdata2 = xdata1[::2] # Make the sets of unequal size
ydata1, ydata2 = model(x_1=xdata1,
x_2=xdata2,
a_1=101.3,
b_1=0.5,
a_2=56.3,
b_2=1.1111,
y0=10.8)
# Add some noise to make it appear like real data
np.random.seed(1)
ydata1 += np.random.normal(0, 2, size=ydata1.shape)
ydata2 += np.random.normal(0, 2, size=ydata2.shape)
xdata = [xdata1, xdata2]
ydata = [ydata1, ydata2]
# Guesses
a_1.value = 100
a_2.value = 50
b_1.value = 1
b_2.value = 1
y0.value = 10
sigma_y = np.concatenate((np.ones(20), [2., 4., 5, 7, 3]))
fit = ConstrainedNumericalLeastSquares(model,
x_1=xdata[0],
x_2=xdata[1],
y_1=ydata[0],
y_2=ydata[1],
sigma_y_2=sigma_y)
fit_result = fit.execute()
# fit_curves = model(x_1=xdata[0], x_2=xdata[1], **fit_result.params)
self.assertAlmostEqual(fit_result.value(y0), 1.061892e+01, 3)
self.assertAlmostEqual(fit_result.value(a_1), 1.013269e+02, 3)
self.assertAlmostEqual(fit_result.value(a_2), 5.625694e+01, 3)
self.assertAlmostEqual(fit_result.value(b_1), 3.362240e-01, 3)
self.assertAlmostEqual(fit_result.value(b_2), 1.565253e+00, 3)
def app(self, n):
n = n + 1
x, y = variables('x, y')
w, = parameters('w')
model_dict = {y: self.fourier_series(x, f=w, n=n)}
fit = Fit(model_dict, x=self.xdata, y=self.ydata)
fit_result = fit.execute()
return fit.model(x=self.xdata, **fit_result.params).y
def test_interdependency_constrained(self):
"""
Test a model with interdependent components, and with constraints which
depend on the Model's output.
This is done in the MatrixSymbol formalism, using a Tikhonov
regularization as an example. In this, a matrix inverse has to be
calculated and is used multiple times. Therefore we split that term of
into a seperate component, so the inverse only has to be computed once
per model call.
See https://arxiv.org/abs/1901.05348 for a more detailed background.
"""
N = Symbol('N', integer=True)
M = MatrixSymbol('M', N, N)
W = MatrixSymbol('W', N, N)
I = MatrixSymbol('I', N, N)
y = MatrixSymbol('y', N, 1)
c = MatrixSymbol('c', N, 1)
a, = parameters('a')
z, = variables('z')
i = Idx('i')
model_dict = {
W: Inverse(I + M / a ** 2),
c: - W * y,
z: sqrt(c.T * c)
}
# Sympy currently does not support derivatives of matrix expressions,
# so we use CallableModel instead of Model.
model = CallableModel(model_dict)
# Generate data
iden = np.eye(2)
M_mat = np.array([[2, 1], [3, 4]])
y_vec = np.array([[3], [5]])
eval_model = model(I=iden, M=M_mat, y=y_vec, a=0.1)
# Calculate the answers 'manually' so I know it was done properly
W_manual = np.linalg.inv(iden + M_mat / 0.1 ** 2)
c_manual = - np.atleast_2d(W_manual.dot(y_vec))
z_manual = np.atleast_1d(np.sqrt(c_manual.T.dot(c_manual)))
self.assertEqual(y_vec.shape, (2, 1))
self.assertEqual(M_mat.shape, (2, 2))
self.assertEqual(iden.shape, (2, 2))
self.assertEqual(W_manual.shape, (2, 2))
self.assertEqual(c_manual.shape, (2, 1))
self.assertEqual(z_manual.shape, (1, 1))
np.testing.assert_almost_equal(W_manual, eval_model.W)
np.testing.assert_almost_equal(c_manual, eval_model.c)
np.testing.assert_almost_equal(z_manual, eval_model.z)
fit = Fit(model, z=z_manual, I=iden, M=M_mat, y=y_vec)
fit_result = fit.execute()
# See if a == 0.1 was reconstructed properly. Since only a**2 features
# in the equations, we check for the absolute value. Setting a.min = 0.0
# is not appreciated by the Minimizer, it seems.
self.assertAlmostEqual(np.abs(fit_result.value(a)), 0.1)
def test_jacobian_matrix(self):
"""
The jacobian matrix of a model should be a 2D list (matrix) containing
all the partial derivatives.
"""
a, b, c = parameters('a, b, c')
a_i, b_i, c_i = variables('a_i, b_i, c_i')
model = Model({a_i: 2 * a + 3 * b, b_i: 5 * b, c_i: 7 * c})
self.assertEqual([[2, 3, 0], [0, 5, 0], [0, 0, 7]], model.jacobian)
def test_global_fitting(self):
"""
Test a global fitting scenario with datasets of unequal length. In this
scenario, a quartic equation is fitted where the constant term is shared
between the datasets. (e.g. identical background noise)
"""
x_1, x_2, y_1, y_2 = variables('x_1, x_2, y_1, y_2')
y0, a_1, a_2, b_1, b_2 = parameters('y0, a_1, a_2, b_1, b_2')
# The following vector valued function links all the equations together
# as stated in the intro.
model = Model({
y_1: a_1 * x_1**2 + b_1 * x_1 + y0,
y_2: a_2 * x_2**2 + b_2 * x_2 + y0,
})
# Generate data from this model
# xdata = np.linspace(0, 10)
xdata1 = np.linspace(0, 10)
xdata2 = xdata1[::2] # Make the sets of unequal size
ydata1, ydata2 = model(x_1=xdata1, x_2=xdata2, a_1=101.3, b_1=0.5, a_2=56.3, b_2=1.1111, y0=10.8)
# Add some noise to make it appear like real data
np.random.seed(1)
ydata1 += np.random.normal(0, 2, size=ydata1.shape)
ydata2 += np.random.normal(0, 2, size=ydata2.shape)
xdata = [xdata1, xdata2]
ydata = [ydata1, ydata2]
# Guesses
a_1.value = 100
a_2.value = 50
b_1.value = 1
b_2.value = 1
y0.value = 10
eval_jac = model.eval_jacobian(x_1=xdata1, x_2=xdata2, a_1=101.3,
b_1=0.5, a_2=56.3, b_2=1.1111, y0=10.8)
self.assertEqual(len(eval_jac), 2)
for comp in eval_jac:
self.assertEqual(len(comp), len(model.params))
sigma_y = np.concatenate((np.ones(20), [2., 4., 5, 7, 3]))
fit = Fit(model, x_1=xdata[0], x_2=xdata[1],
y_1=ydata[0], y_2=ydata[1], sigma_y_2=sigma_y)
fit_result = fit.execute()
# fit_curves = model(x_1=xdata[0], x_2=xdata[1], **fit_result.params)
self.assertAlmostEqual(fit_result.value(y0), 1.061892e+01, 3)
self.assertAlmostEqual(fit_result.value(a_1), 1.013269e+02, 3)
self.assertAlmostEqual(fit_result.value(a_2), 5.625694e+01, 3)
self.assertAlmostEqual(fit_result.value(b_1), 3.362240e-01, 3)
self.assertAlmostEqual(fit_result.value(b_2), 1.565253e+00, 3)
def test_order(self):
"""
The model has to behave like an OrderedDict. This is of the utmost importance!
"""
x, y_1, y_2 = variables('x, y_1, y_2')
a, b = parameters('a, b')
model_dict = {y_2: a * x**2, y_1: 2 * x * b}
model = Model(model_dict)
self.assertEqual(model.dependent_vars, list(model.keys()))
def test_hessian_matrix(self):
"""
The Hessian matrix of a model should be a 3D list (matrix) containing
all the 2nd partial derivatives.
"""
a, b, c = parameters('a, b, c')
a_i, b_i, c_i = variables('a_i, b_i, c_i')
model = Model({a_i: 2 * a**2 + 3 * b, b_i: 5 * b**2, c_i: 7 * c*b})
self.assertEqual([[[4, 0, 0], [0, 0, 0], [0, 0, 0]],
[[0, 0, 0], [0, 10, 0], [0, 0, 0]],
[[0, 0, 0], [0, 0, 7], [0, 7, 0]]], model.hessian)
def test_basinhopping(self):
func = lambda x: np.cos(14.5 * x - 0.3) + (x + 0.2) * x
x0 = [1.]
np.random.seed(555)
res = basinhopping(func, x0, minimizer_kwargs={"method": "BFGS"}, niter=200)
np.random.seed(555)
x, = parameters('x')
fit = BasinHopping(func, [x], local_minimizer=BFGS)
fit_result = fit.execute(niter=200)
# fit_result = fit.execute(minimizer_kwargs={"method": "BFGS"}, niter=200)
self.assertEqual(res.x, fit_result.value(x))
self.assertEqual(res.fun, fit_result.objective_value)
def test_linear_analytical_fit(self):
a, b = parameters('a, b')
x, y = variables('x, y')
model = {y: a * x + b}
data = [[0, 1], [1, 0], [3, 2], [5, 4]]
xdata, ydata = (np.array(i, dtype='float64') for i in zip(*data))
fit = LinearLeastSquares(model, x=xdata, y=ydata)
fit_result = fit.execute()
self.assertAlmostEqual(fit_result.value(a), 0.694915, 6) # values from Mathematica
self.assertAlmostEqual(fit_result.value(b), 0.186441, 6)
def test_unequal_data(self):
"""
Test to make sure finite differences work with data of unequal length.
"""
x_1, x_2, y_1, y_2 = sf.variables('x_1, x_2, y_1, y_2')
y0, a_1, a_2, b_1, b_2 = sf.parameters('y0, a_1, a_2, b_1, b_2')
model = sf.Model({
y_1: a_1 * x_1**2 + b_1 * x_1 + y0,
y_2: a_2 * x_2**2 + b_2 * x_2 + y0,
})
# Generate data from this model
xdata1 = np.linspace(0, 10)
xdata2 = xdata1[::2] # Only every other point.
exact = model.eval_jacobian(x_1=xdata1, x_2=xdata2,
a_1=101.3, b_1=0.5, a_2=56.3, b_2=1.1111, y0=10.8)
approx = model.finite_difference(x_1=xdata1, x_2=xdata2,
a_1=101.3, b_1=0.5, a_2=56.3, b_2=1.1111, y0=10.8)
# First axis is the number of components
self.assertEqual(len(exact), 2)
self.assertEqual(len(approx), 2)
# Second axis is the number of parameters, same for all components
for exact_comp, approx_comp, xdata in zip(exact, approx, [xdata1, xdata2]):
self.assertEqual(len(exact_comp), len(model.params))
self.assertEqual(len(approx_comp), len(model.params))
for exact_elem, approx_elem in zip(exact_comp, approx_comp):
self.assertEqual(exact_elem.shape, xdata.shape)
self.assertEqual(approx_elem.shape, xdata.shape)
self._assert_equal(exact, approx, rtol=1e-4)
model = sf.Model({
y_1: a_1 * x_1**2 + b_1 * x_1,
y_2: a_2 * x_2**2 + b_2 * x_2,
})
exact = model.eval_jacobian(x_1=xdata1, x_2=xdata2,
a_1=101.3, b_1=0.5, a_2=56.3, b_2=1.1111)
approx = model.finite_difference(x_1=xdata1, x_2=xdata2,
a_1=101.3, b_1=0.5, a_2=56.3, b_2=1.1111)
self._assert_equal(exact, approx, rtol=1e-4)
model = sf.Model({
y_1: a_1 * x_1**2 + b_1 * x_1,
})
exact = model.eval_jacobian(x_1=xdata1, a_1=101.3, b_1=0.5)
approx = model.finite_difference(x_1=xdata1, a_1=101.3, b_1=0.5)
self._assert_equal(exact, approx, rtol=1e-4)
def test_MatrixSymbolModel(self):
"""
Test a model which is defined by ModelSymbols, see #194
"""
N = Symbol('N', integer=True)
M = MatrixSymbol('M', N, N)
W = MatrixSymbol('W', N, N)
I = MatrixSymbol('I', N, N)
y = MatrixSymbol('y', N, 1)
c = MatrixSymbol('c', N, 1)
a, b = parameters('a, b')
z, x = variables('z, x')
model_dict = {
W: Inverse(I + M / a ** 2),
c: - W * y,
z: sqrt(c.T * c)
}
# TODO: This should be a Model in the future, but sympy is not yet
# capable of computing Matrix derivatives at the time of writing.
model = CallableModel(model_dict)
self.assertEqual(model.params, [a])
self.assertEqual(model.independent_vars, [I, M, y])
self.assertEqual(model.dependent_vars, [z])
self.assertEqual(model.interdependent_vars, [W, c])
self.assertEqual(model.connectivity_mapping,
{W: {I, M, a}, c: {W, y}, z: {c}})
# Generate data
iden = np.eye(2)
M_mat = np.array([[2, 1], [3, 4]])
y_vec = np.array([3, 5])
eval_model = model(I=iden, M=M_mat, y=y_vec, a=0.1)
W_manual = np.linalg.inv(iden + M_mat / 0.1 ** 2)
c_manual = - W_manual.dot(y_vec)
z_manual = np.atleast_1d(np.sqrt(c_manual.T.dot(c_manual)))
np.testing.assert_allclose(eval_model.W, W_manual)
np.testing.assert_allclose(eval_model.c, c_manual)
np.testing.assert_allclose(eval_model.z, z_manual)
# Now try to retrieve the value of `a` from a fit
a.value = 0.2
fit = Fit(model, z=z_manual, I=iden, M=M_mat, y=y_vec)
fit_result = fit.execute()
eval_model = model(I=iden, M=M_mat, y=y_vec, **fit_result.params)
self.assertAlmostEqual(0.1, np.abs(fit_result.value(a)))
np.testing.assert_allclose(eval_model.W, W_manual, rtol=1e-5)
np.testing.assert_allclose(eval_model.c, c_manual, rtol=1e-5)
np.testing.assert_allclose(eval_model.z, z_manual, rtol=1e-5)
def test_1_multi_model(self):
'''Tests the case with 1 component and multiple parameters'''
x, y = sf.variables('x, y')
a, b = sf.parameters('a, b')
model = sf.Model({y: 3 * a * x**2 - sf.exp(b) * x})
x_data = np.arange(10)
exact = model.eval_jacobian(x=x_data, a=3.5, b=2)
approx = model.finite_difference(x=x_data, a=3.5, b=2)
np.testing.assert_allclose(exact, approx)
exact = model.eval_jacobian(x=3, a=3.5, b=2)
approx = model.finite_difference(x=3, a=3.5, b=2)
np.testing.assert_allclose(exact, approx)
def test_model_from_dict(self):
"""
Tries to create a model from a dictionary.
"""
x, y_1, y_2 = variables('x, y_1, y_2')
a, b = parameters('a, b')
# This way the test fails rather than errors.
try:
Model({
y_1: 2 * a * x,
y_2: b * x**2
})
except Exception as error:
self.fail('test_model_from_dict raised {}'.format(error))
def test_powell(self):
"""
Powell with a single parameter gave an error because a 0-d array was
returned by scipy. So no error here is winning.
"""
x, y = variables('x, y')
a, b = parameters('a, b')
b.fixed = True
model = Model({y: a * x + b})
xdata = np.linspace(0, 10)
ydata = model(x=xdata, a=5.5, b=15.0).y + np.random.normal(0, 1)
fit = Fit({y: a * x + b}, x=xdata, y=ydata, minimizer=Powell)
fit_result = fit.execute()
self.assertAlmostEqual(fit_result.value(b), 1.0)
def test_multi_multi_model(self):
'''Tests the case with multiple components and multiple parameters'''
x, y, z = sf.variables('x, y, z')
a, b, c = sf.parameters('a, b, c')
model = sf.Model({y: 3 * a * x**2 + b * x - c,
z: sf.exp(a*x - b) * c})
x_data = np.arange(10)
exact = model.eval_jacobian(x=x_data, a=3.5, b=2, c=5)
approx = model.finite_difference(x=x_data, a=3.5, b=2, c=5)
np.testing.assert_allclose(exact, approx, rtol=1e-5)
exact = model.eval_jacobian(x=3, a=3.5, b=2, c=5)
approx = model.finite_difference(x=3, a=3.5, b=2, c=5)
np.testing.assert_allclose(exact, approx, rtol=1e-5)
def test_multi_1_model(self):
'''Tests the case with multiple components and one parameter'''
x, y, z = sf.variables('x, y, z')
a, = sf.parameters('a')
model = sf.Model({y: 3 * a * x**2,
z: sf.exp(a*x)})
x_data = np.arange(10)
exact = model.eval_jacobian(x=x_data, a=3.5)
approx = model.finite_difference(x=x_data, a=3.5)
np.testing.assert_allclose(exact, approx)
exact = model.eval_jacobian(x=3, a=3.5)
approx = model.finite_difference(x=3, a=3.5)
np.testing.assert_allclose(exact, approx)
def test_model_as_dict(self):
x, y_1, y_2 = variables('x, y_1, y_2')
a, b = parameters('a, b')
model_dict = OrderedDict([(y_1, a * x**2), (y_2, 2 * x * b)])
model = Model(model_dict)
self.assertEqual(id(model[y_1]), id(model_dict[y_1]))
self.assertEqual(id(model[y_2]), id(model_dict[y_2]))
self.assertEqual(len(model), len(model_dict))
self.assertEqual(model.items(), model_dict.items())
self.assertEqual(model.keys(), model_dict.keys())
self.assertEqual(list(model.values()), list(model_dict.values()))
self.assertTrue(y_1 in model)
self.assertFalse(model[y_1] in model)
