def exercise_polynomial_fit():
def do_polynomial_fit(x, params):
n_terms = len(params)
y = flex.double(x.size())
for i in range(len(params)):
y += params[i] * flex.pow(x, i)
fit = curve_fitting.univariate_polynomial_fit(x, y, degree=n_terms - 1)
assert approx_equal(params, fit.params, eps=1e-4)
x = flex.double(range(-50, 50))
do_polynomial_fit(x, (2, 3, 5)) # y = 2 + 3x + 5x^2
do_polynomial_fit(x, (-0.0002, -1000)) # y = -0.0002 -1000x
for n_terms in range(1, 6):
params = [100 * random.random() for i in range(n_terms)]
x = flex.double(
frange(-random.randint(1, 10), random.randint(1, 10), 0.1))
functor = curve_fitting.univariate_polynomial(*params)
fd_grads = finite_differences(functor, x)
assert approx_equal(functor.partial_derivatives(x), fd_grads, 1e-4)
do_polynomial_fit(x, params)
def exercise_polynomial_fit():
def do_polynomial_fit(x, params):
n_terms = len(params)
y = flex.double(x.size())
for i in range(len(params)):
y += params[i] * flex.pow(x, i)
fit = curve_fitting.univariate_polynomial_fit(x, y, degree=n_terms-1)
assert approx_equal(params, fit.params, eps=1e-4)
x = flex.double(range(-50,50))
do_polynomial_fit(x, (2,3,5)) # y = 2 + 3x + 5x^2
do_polynomial_fit(x, (-0.0002, -1000)) # y = -0.0002 -1000x
for n_terms in range(1, 6):
params = [100*random.random() for i in range(n_terms)]
x = flex.double(frange(-random.randint(1,10), random.randint(1,10), 0.1))
functor = curve_fitting.univariate_polynomial(*params)
fd_grads = finite_differences(functor, x)
assert approx_equal(functor.partial_derivatives(x), fd_grads, 1e-4)
do_polynomial_fit(x, params)
def exercise_savitzky_golay_smoothing():
plot = False
def rms(flex_double):
return math.sqrt(flex.mean(flex.pow2(flex_double)))
for sigma_frac in (0.005, 0.01, 0.05, 0.1):
mean = random.randint(-5,5)
scale = flex.random_double() * 10
sigma = flex.random_double() * 5 + 1
gaussian = curve_fitting.gaussian(scale, mean, sigma)
x = flex.double(frange(-20,20,0.1))
y = gaussian(x)
rand_norm = scitbx.random.normal_distribution(
mean=0, sigma=sigma_frac*flex.max_absolute(y))
g = scitbx.random.variate(rand_norm)
noise = g(y.size())
y_noisy = y + noise
# according to numerical recipes the best results are obtained where the
# full window width is between 1 and 2 times the number of points at fwhm
# for polynomials of degree 4
half_window = int(round(0.5 * 2.355 * sigma * 10))
y_filtered = savitzky_golay_filter(x, y_noisy, half_window=half_window, degree=4)[1]
extracted_noise = y_noisy - y_filtered
rms_noise = rms(noise)
rms_extracted_noise = rms(extracted_noise)
assert is_below_limit(
value=abs(rand_norm.sigma - rms_noise)/rand_norm.sigma,
limit=0.15)
assert is_below_limit(
value=abs(rand_norm.sigma - rms_extracted_noise)/rand_norm.sigma,
limit=0.15)
diff = y_filtered - y
assert is_below_limit(
value=(rms(diff)/ rand_norm.sigma),
limit=0.4)
if plot:
from matplotlib import pyplot
pyplot.plot(x, y)
pyplot.plot(x, noise)
pyplot.scatter(x, y_noisy, marker="x")
pyplot.plot(x, y_filtered)
pyplot.show()
pyplot.plot(x, extracted_noise)
pyplot.plot(x, noise)
pyplot.show()
return
def exercise_gaussian_fit():
# test fitting of a gaussian
def do_gaussian_fit(scale, mu, sigma):
start = mu - 6 * sigma
stop = mu + 6 * sigma
step = (stop - start) / 1000
x = flex.double(frange(start, stop, step))
y = scale * flex.exp(-flex.pow2(x - mu) / (2 * sigma**2))
fit = curve_fitting.single_gaussian_fit(x, y)
assert approx_equal(fit.a, scale, 1e-4)
assert approx_equal(fit.b, mu, eps=1e-4)
assert approx_equal(fit.c, sigma, eps=1e-4)
for i in range(10):
scale = random.random() * 1000
sigma = (random.random() + 0.0001) * 10
mu = (-1)**random.randint(0, 1) * random.random() * 1000
functor = curve_fitting.gaussian(scale, mu, sigma)
start = mu - 6 * sigma
stop = mu + 6 * sigma
step = (stop - start) / 1000
x = flex.double(frange(start, stop, step))
fd_grads = finite_differences(functor, x)
assert approx_equal(functor.partial_derivatives(x), fd_grads, 1e-4)
do_gaussian_fit(scale, mu, sigma)
# if we take the log of a gaussian we can fit a parabola
scale = 123
mu = 3.2
sigma = 0.1
x = flex.double(frange(2, 4, 0.01))
y = scale * flex.exp(-flex.pow2(x - mu) / (2 * sigma**2))
# need to be careful to only use values of y > 0
eps = 1e-15
x = flex.double([x[i] for i in range(x.size()) if y[i] > eps])
y = flex.double([y[i] for i in range(y.size()) if y[i] > eps])
fit = curve_fitting.univariate_polynomial_fit(x, flex.log(y), degree=2)
c, b, a = fit.params
assert approx_equal(mu, -b / (2 * a))
assert approx_equal(sigma * sigma, -1 / (2 * a))
# test multiple gaussian fits
gaussians = [
curve_fitting.gaussian(0.3989538, 3.7499764, 0.7500268),
curve_fitting.gaussian(0.7978957, 6.0000004, 0.5000078)
]
x = flex.double(frange(0, 10, 0.1))
y = flex.double(x.size())
for i in range(len(gaussians)):
g = gaussians[i]
scale, mu, sigma = g.a, g.b, g.c
y += g(x)
starting_gaussians = [
curve_fitting.gaussian(1, 4, 1),
curve_fitting.gaussian(1, 5, 1)
]
fit = curve_fitting.gaussian_fit(x, y, starting_gaussians)
for g1, g2 in zip(gaussians, fit.gaussians):
assert approx_equal(g1.a, g2.a, eps=1e-4)
assert approx_equal(g1.b, g2.b, eps=1e-4)
assert approx_equal(g1.c, g2.c, eps=1e-4)
# use example of 5-gaussian fit from here:
# http://research.stowers-institute.org/efg/R/Statistics/MixturesOfDistributions/index.htm
gaussians = [
curve_fitting.gaussian(0.10516252, 23.32727, 2.436638),
curve_fitting.gaussian(0.46462715, 33.09053, 2.997594),
curve_fitting.gaussian(0.29827916, 41.27244, 4.274585),
curve_fitting.gaussian(0.08986616, 51.24468, 5.077521),
curve_fitting.gaussian(0.04206501, 61.31818, 7.070303)
]
x = flex.double(frange(0, 80, 0.1))
y = flex.double(x.size())
for i in range(len(gaussians)):
g = gaussians[i]
scale, mu, sigma = g.a, g.b, g.c
y += g(x)
termination_params = scitbx.lbfgs.termination_parameters(
min_iterations=500)
starting_gaussians = [
curve_fitting.gaussian(1, 21, 2.1),
curve_fitting.gaussian(1, 30, 2.8),
curve_fitting.gaussian(1, 40, 2.2),
curve_fitting.gaussian(1, 51, 1.2),
curve_fitting.gaussian(1, 60, 2.3)
]
fit = curve_fitting.gaussian_fit(x,
y,
starting_gaussians,
termination_params=termination_params)
y_calc = fit.compute_y_calc()
assert approx_equal(y, y_calc, eps=1e-2)
have_cma_es = libtbx.env.has_module("cma_es")
if have_cma_es:
fit = curve_fitting.cma_es_minimiser(starting_gaussians, x, y)
y_calc = fit.compute_y_calc()
assert approx_equal(y, y_calc, eps=5e-2)
def exercise_gaussian_fit():
# test fitting of a gaussian
def do_gaussian_fit(scale, mu, sigma):
start = mu - 6 * sigma
stop = mu + 6 * sigma
step = (stop - start)/1000
x = flex.double(frange(start, stop, step))
y = scale * flex.exp(-flex.pow2(x - mu) / (2 * sigma**2))
fit = curve_fitting.single_gaussian_fit(x, y)
assert approx_equal(fit.a, scale, 1e-4)
assert approx_equal(fit.b, mu, eps=1e-4)
assert approx_equal(fit.c, sigma, eps=1e-4)
for i in range(10):
scale = random.random() * 1000
sigma = (random.random() + 0.0001) * 10
mu = (-1)**random.randint(0,1) * random.random() * 1000
functor = curve_fitting.gaussian(scale, mu, sigma)
start = mu - 6 * sigma
stop = mu + 6 * sigma
step = (stop - start)/1000
x = flex.double(frange(start, stop, step))
fd_grads = finite_differences(functor, x)
assert approx_equal(functor.partial_derivatives(x), fd_grads, 1e-4)
do_gaussian_fit(scale, mu, sigma)
# if we take the log of a gaussian we can fit a parabola
scale = 123
mu = 3.2
sigma = 0.1
x = flex.double(frange(2, 4, 0.01))
y = scale * flex.exp(-flex.pow2(x - mu) / (2 * sigma**2))
# need to be careful to only use values of y > 0
eps = 1e-15
x = flex.double([x[i] for i in range(x.size()) if y[i] > eps])
y = flex.double([y[i] for i in range(y.size()) if y[i] > eps])
fit = curve_fitting.univariate_polynomial_fit(x, flex.log(y), degree=2)
c, b, a = fit.params
assert approx_equal(mu, -b/(2*a))
assert approx_equal(sigma*sigma, -1/(2*a))
# test multiple gaussian fits
gaussians = [curve_fitting.gaussian(0.3989538, 3.7499764, 0.7500268),
curve_fitting.gaussian(0.7978957, 6.0000004, 0.5000078)]
x = flex.double(frange(0, 10, 0.1))
y = flex.double(x.size())
for i in range(len(gaussians)):
g = gaussians[i]
scale, mu, sigma = g.a, g.b, g.c
y += g(x)
starting_gaussians = [
curve_fitting.gaussian(1, 4, 1),
curve_fitting.gaussian(1, 5, 1)]
fit = curve_fitting.gaussian_fit(x, y, starting_gaussians)
for g1, g2 in zip(gaussians, fit.gaussians):
assert approx_equal(g1.a, g2.a, eps=1e-4)
assert approx_equal(g1.b, g2.b, eps=1e-4)
assert approx_equal(g1.c, g2.c, eps=1e-4)
# use example of 5-gaussian fit from here:
# http://research.stowers-institute.org/efg/R/Statistics/MixturesOfDistributions/index.htm
gaussians = [curve_fitting.gaussian(0.10516252, 23.32727, 2.436638),
curve_fitting.gaussian(0.46462715, 33.09053, 2.997594),
curve_fitting.gaussian(0.29827916, 41.27244, 4.274585),
curve_fitting.gaussian(0.08986616, 51.24468, 5.077521),
curve_fitting.gaussian(0.04206501, 61.31818, 7.070303)]
x = flex.double(frange(0, 80, 0.1))
y = flex.double(x.size())
for i in range(len(gaussians)):
g = gaussians[i]
scale, mu, sigma = g.a, g.b, g.c
y += g(x)
termination_params = scitbx.lbfgs.termination_parameters(
min_iterations=500)
starting_gaussians = [curve_fitting.gaussian(1, 21, 2.1),
curve_fitting.gaussian(1, 30, 2.8),
curve_fitting.gaussian(1, 40, 2.2),
curve_fitting.gaussian(1, 51, 1.2),
curve_fitting.gaussian(1, 60, 2.3)]
fit = curve_fitting.gaussian_fit(
x, y, starting_gaussians, termination_params=termination_params)
y_calc = fit.compute_y_calc()
assert approx_equal(y, y_calc, eps=1e-2)
have_cma_es = libtbx.env.has_module("cma_es")
if have_cma_es:
fit = curve_fitting.cma_es_minimiser(starting_gaussians, x, y)
y_calc = fit.compute_y_calc()
assert approx_equal(y, y_calc, eps=5e-2)