def test_lambdify():
# Test lambdify with implemented functions
# first test basic (sympy) lambdify
f = sympy.cos
yield assert_equal(lambdify(x, f(x))(0), 1)
yield assert_equal(lambdify(x, 1 + f(x))(0), 2)
yield assert_equal(lambdify((x, y), y + f(x))(0, 1), 2)
# make an implemented function and test
f = implemented_function("f", lambda x : x+100)
yield assert_equal(lambdify(x, f(x))(0), 100)
yield assert_equal(lambdify(x, 1 + f(x))(0), 101)
yield assert_equal(lambdify((x, y), y + f(x))(0, 1), 101)
# Error for functions with same name and different implementation
f2 = implemented_function("f", lambda x : x+101)
yield assert_raises(ValueError, lambdify, x, f(f2(x)))
# our lambdify, like sympy's lambdify, can also handle tuples,
# lists, dicts as expressions
lam = lambdify(x, (f(x), x))
yield assert_equal(lam(3), (103, 3))
lam = lambdify(x, [f(x), x])
yield assert_equal(lam(3), [103, 3])
lam = lambdify(x, [f(x), (f(x), x)])
yield assert_equal(lam(3), [103, (103, 3)])
lam = lambdify(x, {f(x): x})
yield assert_equal(lam(3), {103: 3})
lam = lambdify(x, {f(x): x})
yield assert_equal(lam(3), {103: 3})
lam = lambdify(x, {x: f(x)})
yield assert_equal(lam(3), {3: 103})
def test_lambdify():
# Test lambdify with implemented functions
# first test basic (sympy) lambdify
f = sympy.cos
assert_equal(lambdify(x, f(x))(0), 1)
assert_equal(lambdify(x, 1 + f(x))(0), 2)
assert_equal(lambdify((x, y), y + f(x))(0, 1), 2)
# make an implemented function and test
f = implemented_function("f", lambda x: x + 100)
assert_equal(lambdify(x, f(x))(0), 100)
assert_equal(lambdify(x, 1 + f(x))(0), 101)
assert_equal(lambdify((x, y), y + f(x))(0, 1), 101)
# Error for functions with same name and different implementation
f2 = implemented_function("f", lambda x: x + 101)
assert_raises(ValueError, lambdify, x, f(f2(x)))
# our lambdify, like sympy's lambdify, can also handle tuples,
# lists, dicts as expressions
lam = lambdify(x, (f(x), x))
assert_equal(lam(3), (103, 3))
lam = lambdify(x, [f(x), x])
assert_equal(lam(3), [103, 3])
lam = lambdify(x, [f(x), (f(x), x)])
assert_equal(lam(3), [103, (103, 3)])
lam = lambdify(x, {f(x): x})
assert_equal(lam(3), {103: 3})
lam = lambdify(x, {f(x): x})
assert_equal(lam(3), {103: 3})
lam = lambdify(x, {x: f(x)})
assert_equal(lam(3), {3: 103})
def test_2d():
B1, B2 = [gen_BrownianMotion() for _ in range(2)]
B1s = implemented_function("B1", B1)
B2s = implemented_function("B2", B2)
s, t = sympy.symbols('s', 't')
e = B1s(s)+B2s(t)
ee = lambdify((s,t), e)
yield assert_almost_equal(ee(B1.x, B2.x), B1.y + B2.y)
def test_alias():
x = F.Term('x')
f = implemented_function('f', lambda x: 2*x)
g = implemented_function('g', lambda x: np.sqrt(x))
ff = F.Formula([f(x), g(x)**2])
n = F.make_recarray([2,4,5], 'x')
assert_almost_equal(ff.design(n)['f(x)'], n['x']*2)
assert_almost_equal(ff.design(n)['g(x)**2'], n['x'])
def test_alias():
x = F.Term('x')
f = implemented_function('f', lambda x: 2*x)
g = implemented_function('g', lambda x: np.sqrt(x))
ff = F.Formula([f(x), g(x)**2])
n = F.make_recarray([2,4,5], 'x')
assert_almost_equal(ff.design(n)['f(x)'], n['x']*2)
assert_almost_equal(ff.design(n)['g(x)**2'], n['x'])
def test_2d():
B1, B2 = [gen_BrownianMotion() for _ in range(2)]
B1s = implemented_function("B1", B1)
B2s = implemented_function("B2", B2)
s, t = sympy.symbols(('s', 't'))
e = B1s(s) + B2s(t)
ee = lambdify((s, t), e)
assert_almost_equal(ee(B1.x, B2.x), B1.y + B2.y)
def natural_spline(t, knots=None, order=3, intercept=False):
""" Return a Formula containing a natural spline
Spline for a Term with specified `knots` and `order`.
Parameters
----------
t : ``Term``
knots : None or sequence, optional
Sequence of float. Default None (same as empty list)
order : int, optional
Order of the spline. Defaults to a cubic (==3)
intercept : bool, optional
If True, include a constant function in the natural
spline. Default is False
Returns
-------
formula : Formula
A Formula with (len(knots) + order) Terms (if intercept=False,
otherwise includes one more Term), made up of the natural spline
functions.
Examples
--------
>>> x = Term('x')
>>> n = natural_spline(x, knots=[1,3,4], order=3)
>>> xval = np.array([3,5,7.]).view(np.dtype([('x', np.float)]))
>>> n.design(xval, return_float=True)
array([[ 3., 9., 27., 8., 0., -0.],
[ 5., 25., 125., 64., 8., 1.],
[ 7., 49., 343., 216., 64., 27.]])
>>> d = n.design(xval)
>>> print d.dtype.descr
[('ns_1(x)', '<f8'), ('ns_2(x)', '<f8'), ('ns_3(x)', '<f8'), ('ns_4(x)', '<f8'), ('ns_5(x)', '<f8'), ('ns_6(x)', '<f8')]
"""
if knots is None:
knots = {}
fns = []
for i in range(order+1):
n = 'ns_%d' % i
def f(x, i=i):
return x**i
s = implemented_function(n, f)
fns.append(s(t))
for j, k in enumerate(knots):
n = 'ns_%d' % (j+i+1,)
def f(x, k=k, order=order):
return (x-k)**order * np.greater(x, k)
s = implemented_function(n, f)
fns.append(s(t))
if not intercept:
fns.pop(0)
ff = Formula(fns)
return ff
def test_implemented_function():
# Here we check if the default returned functions are anonymous - in
# the sense that we can have more than one function with the same name
f = implemented_function('f', lambda x: 2 * x)
g = implemented_function('f', lambda x: np.sqrt(x))
l1 = lambdify(x, f(x))
l2 = lambdify(x, g(x))
assert_equal(str(f(x)), str(g(x)))
assert_equal(l1(3), 6)
assert_equal(l2(3), np.sqrt(3))
# check that we can pass in a sympy function as input
func = sympy.Function('myfunc')
assert_false(hasattr(func, '_imp_'))
f = implemented_function(func, lambda x: 2 * x)
assert_true(hasattr(func, '_imp_'))
def test_implemented_function():
# Here we check if the default returned functions are anonymous - in
# the sense that we can have more than one function with the same name
f = implemented_function('f', lambda x: 2*x)
g = implemented_function('f', lambda x: np.sqrt(x))
l1 = lambdify(x, f(x))
l2 = lambdify(x, g(x))
yield assert_equal(str(f(x)), str(g(x)))
yield assert_equal(l1(3), 6)
yield assert_equal(l2(3), np.sqrt(3))
# check that we can pass in a sympy function as input
func = sympy.Function('myfunc')
yield assert_false(hasattr(func, 'alias'))
f = implemented_function(func, lambda x: 2*x)
yield assert_true(hasattr(func, 'alias'))
def interp(times, values, fill=0, name=None, **kw):
""" Generic interpolation function of t given `times` and `values`
Imterpolator such that:
f(times[i]) = values[i]
if t < times[0] or t > times[-1]:
f(t) = fill
See ``scipy.interpolate.interp1d`` for details of interpolation
types and other keyword arguments. Default is 'kind' is linear,
making this function, by default, have the same behavior as
``linear_interp``.
Parameters
----------
times : array-like
Increasing sequence of times
values : array-like
Values at the specified times
fill : None or float, optional
Value on the interval (-np.inf, times[0]). Default 0. If None, raises
error outside bounds
name : None or str, optional
Name of symbolic expression to use. If None, a default is used.
\*\*kw : keyword args, optional
passed to ``interp1d``
Returns
-------
f : sympy expression
A Function of t.
Examples
--------
>>> s = interp([0,4,5.],[2.,4,6])
>>> tval = np.array([-0.1,0.1,3.9,4.1,5.1])
>>> res = lambdify_t(s)(tval)
0 outside bounds by default
>>> np.allclose(res, [0, 2.05, 3.95, 4.2, 0])
True
"""
if not fill is None:
if kw.get('bounds_error') is True:
raise ValueError('fill conflicts with bounds error')
fv = kw.get('fill_value')
if not (fv is None or fv is fill or fv == fill): # allow for fill=np.nan
raise ValueError('fill conflicts with fill_value')
kw['bounds_error'] = False
kw['fill_value'] = fill
interpolator = interp1d(times, values, **kw)
# make a new name if none provided
if name is None:
name = 'interp%d' % interp.counter
interp.counter += 1
s = implemented_function(name, interpolator)
return s(T)
def interp(times, values, fill=0, name=None, **kw):
""" Generic interpolation function of t given `times` and `values`
Imterpolator such that:
f(times[i]) = values[i]
if t < times[0]:
f(t) = fill
See ``scipy.interpolate.interp1d`` for details of interpolation
types and other keyword arguments. Default is 'kind' is linear,
making this function, by default, have the same behavior as
``linear_interp``.
Parameters
----------
times : array-like
Increasing sequence of times
values : array-like
Values at the specified times
fill : float, optional
Value on the interval (-np.inf, times[0]). Default 0.
name : None or str, optional
Name of symbolic expression to use. If None, a default is used.
**kw : keyword args, optional
passed to ``interp1d``
Returns
-------
f : sympy expression
A Function of t.
Examples
--------
>>> s = interp([0,4,5.],[2.,4,6], bounds_error=False)
>>> tval = np.array([-0.1,0.1,3.9,4.1,5.1])
>>> res = lambdify_t(s)(tval)
>>> # nans outside bounds
>>> np.isnan(res)
array([ True, False, False, False, True], dtype=bool)
>>> # interpolated values otherwise
>>> np.allclose(res[1:-1], [2.05, 3.95, 4.2])
True
"""
interpolator = interp1d(times, values, **kw)
# make a new name if none provided
if name is None:
name = "interp%d" % interp.counter
interp.counter += 1
s = implemented_function(name, interpolator)
return s(T)
def test_1d():
B = gen_BrownianMotion()
Bs = implemented_function("B", B)
t = sympy.Symbol('t')
expr = 3*sympy.exp(Bs(t)) + 4
expected = 3*np.exp(B.y)+4
ee_vec = lambdify(t, expr)
yield assert_almost_equal(ee_vec(B.x), expected)
# with any arbitrary symbol
b = sympy.Symbol('b')
expr = 3*sympy.exp(Bs(b)) + 4
ee_vec = lambdify(b, expr)
yield assert_almost_equal(ee_vec(B.x), expected)
def test_1d():
B = gen_BrownianMotion()
Bs = implemented_function("B", B)
t = sympy.Symbol('t')
expr = 3 * sympy.exp(Bs(t)) + 4
expected = 3 * np.exp(B.y) + 4
ee_vec = lambdify(t, expr, "numpy")
assert_almost_equal(ee_vec(B.x), expected)
# with any arbitrary symbol
b = sympy.Symbol('b')
expr = 3 * sympy.exp(Bs(b)) + 4
ee_vec = lambdify(b, expr, "numpy")
assert_almost_equal(ee_vec(B.x), expected)
def step_function(times, values, name=None, fill=0):
""" Right-continuous step function of time t
Function of t such that
f(times[i]) = values[i]
if t < times[0]:
f(t) = fill
Parameters
----------
times : (N,) sequence
Increasing sequence of times
values : (N,) sequence
Values at the specified times
fill : float
Value on the interval (-np.inf, times[0])
name : str
Name of symbolic expression to use. If None, a default is used.
Returns
-------
f_t : sympy expr
Sympy expression f(t) where f is a sympy implemented anonymous
function of time that implements the step function. To get the
numerical version of the function, use ``lambdify_t(f_t)``
Examples
--------
>>> s = step_function([0,4,5],[2,4,6])
>>> tval = np.array([-0.1,3.9,4.1,5.1])
>>> lam = lambdify_t(s)
>>> lam(tval)
array([ 0., 2., 4., 6.])
"""
if name is None:
name = 'step%d' % step_function.counter
step_function.counter += 1
def _imp(x):
x = np.asarray(x)
f = np.zeros(x.shape) + fill
for time, val in zip(times, values):
f[x >= time] = val
return f
s = implemented_function(name, _imp)
return s(T)
def step_function(times, values, name=None, fill=0):
""" Right-continuous step function of time t
Function of t such that
f(times[i]) = values[i]
if t < times[0]:
f(t) = fill
Parameters
----------
times : (N,) sequence
Increasing sequence of times
values : (N,) sequence
Values at the specified times
fill : float
Value on the interval (-np.inf, times[0])
name : str
Name of symbolic expression to use. If None, a default is used.
Returns
-------
f_t : sympy expr
Sympy expression f(t) where f is a sympy implemented anonymous
function of time that implements the step function. To get the
numerical version of the function, use ``lambdify_t(f_t)``
Examples
--------
>>> s = step_function([0,4,5],[2,4,6])
>>> tval = np.array([-0.1,3.9,4.1,5.1])
>>> lam = lambdify_t(s)
>>> lam(tval)
array([ 0., 2., 4., 6.])
"""
if name is None:
name = 'step%d' % step_function.counter
step_function.counter += 1
def _imp(x):
x = np.asarray(x)
f = np.zeros(x.shape) + fill
for time, val in zip(times, values):
f[x >= time] = val
return f
s = implemented_function(name, _imp)
return s(T)
def linBspline(knots):
""" Create a linear B spline that is zero outside [knots[0],
knots[-1]] (knots is assumed to be sorted).
"""
fns = []
knots = np.array(knots)
for i in range(knots.shape[0]-2):
name = 'bs_%s' % i
k1, k2, k3 = knots[i:i+3]
d1 = k2-k1
def anon(x,k1=k1,k2=k2,k3=k3):
return ((x-k1) / d1 * np.greater(x, k1) * np.less_equal(x, k2) +
(k3-x) / d1 * np.greater(x, k2) * np.less(x, k3))
fns.append(implemented_function(name, anon))
return fns
def linBspline(knots):
""" Create linear B spline that is zero outside [knots[0], knots[-1]]
(knots is assumed to be sorted).
"""
fns = []
knots = np.array(knots)
for i in range(knots.shape[0] - 2):
name = 'bs_%s' % i
k1, k2, k3 = knots[i:i + 3]
d1 = k2 - k1
def anon(x, k1=k1, k2=k2, k3=k3):
return ((x - k1) / d1 * np.greater(x, k1) * np.less_equal(x, k2) +
(k3 - x) / d1 * np.greater(x, k2) * np.less(x, k3))
fns.append(implemented_function(name, anon))
return fns
def define(name, expr):
""" Create function of t expression from arbitrary expression `expr`
Take an arbitrarily complicated expression `expr` of 't' and make it
an expression that is a simple function of t, of form ``'%s(t)' %
name`` such that when it evaluates (via ``lambdify``) it has the
right values.
Parameters
----------
expr : sympy expression
with only 't' as a Symbol
name : str
Returns
-------
nexpr: sympy expression
Examples
--------
>>> t = Term('t')
>>> expr = t**2 + 3*t
>>> print expr #doctest: +SYMPY_EQUAL
3*t + t**2
>>> newexpr = define('f', expr)
>>> print newexpr
f(t)
>>> f = lambdify_t(newexpr)
>>> f(4)
28
>>> 3*4+4**2
28
"""
# make numerical implementation of expression
v = lambdify(T, expr, "numpy")
# convert numerical implementation to sympy function
f = implemented_function(name, v)
# Return expression that is function of time
return f(T)
def define(name, expr):
""" Create function of t expression from arbitrary expression `expr`
Take an arbitrarily complicated expression `expr` of 't' and make it
an expression that is a simple function of t, of form ``'%s(t)' %
name`` such that when it evaluates (via ``lambdify``) it has the
right values.
Parameters
----------
expr : sympy expression
with only 't' as a Symbol
name : str
Returns
-------
nexpr: sympy expression
Examples
--------
>>> t = Term('t')
>>> expr = t**2 + 3*t
>>> print expr #doctest: +SYMPY_EQUAL
3*t + t**2
>>> newexpr = define('f', expr)
>>> print newexpr
f(t)
>>> f = lambdify_t(newexpr)
>>> f(4)
28
>>> 3*4+4**2
28
"""
# make numerical implementation of expression
v = lambdify(T, expr, "numpy")
# convert numerical implementation to sympy function
f = implemented_function(name, v)
# Return expression that is function of time
return f(T)
def taylor_approx(hrf2decompose,
time=None,
delta=None):
""" A Taylor series approximation of an HRF shifted by times `delta`
Returns original HRF and gradient of HRF
Parameters
----------
hrf2decompose : sympy expression
An expression that can be lambdified as a function of 't'. This
is the HRF to be expanded in PCA
time : None or np.ndarray, optional
None gives default value of np.linspace(-15,50,3251) chosen to
match fMRIstat implementation. This corresponds to a time
interval of 0.02. Presumed to be equally spaced.
delta : None or np.ndarray, optional
None results in default value of np.arange(-4.5, 4.6, 0.1)
chosen to match fMRIstat implementation.
Returns
-------
hrf : [sympy expressions]
Sequence length 2 comprising (`hrf2decompose`, ``dhrf``) where
``dhrf`` is the first derivative of `hrf2decompose`.
approx :
TODO
References
----------
Liao, C.H., Worsley, K.J., Poline, J-B., Aston, J.A.D., Duncan, G.H.,
Evans, A.C. (2002). \'Estimating the delay of the response in fMRI
data.\' NeuroImage, 16:593-606.
"""
if time is None:
time = np.linspace(-15,50,3251)
dt = time[1] - time[0]
if delta is None:
delta = np.arange(-4.5, 4.6, 0.1)
# make numerical implementation from hrf function and symbol t.
# hrft returns function values when called with values for time as
# input.
hrft = lambdify_t(hrf2decompose(T))
# interpolator for negative gradient of hrf
dhrft = interp1d(time, -np.gradient(hrft(time), dt), bounds_error=False,
fill_value=0.)
dhrft.y *= 2
# Create stack of time-shifted HRFs. Time varies over row, delta
# over column.
ts_hrf_vals = np.array([hrft(time - d) for d in delta]).T
# hrf, dhrf
W = np.array([hrft(time), dhrft(time)]).T
# regress hrf, dhrf at times against stack of time-shifted hrfs
WH = np.dot(npl.pinv(W), ts_hrf_vals)
# put these into interpolators to get estimated coefficients for any
# value of delta
coef = [interp1d(delta, w, bounds_error=False,
fill_value=0.) for w in WH]
def approx(time, delta):
value = (coef[0](delta) * hrft(time)
+ coef[1](delta) * dhrft(time))
return value
approx.coef = coef
approx.components = [hrft, dhrft]
(approx.theta,
approx.inverse,
approx.dinverse,
approx.forward,
approx.dforward) = invertR(delta, approx.coef)
dhrf = implemented_function('d%s' % str(hrf2decompose), dhrft)
return [hrf2decompose, dhrf], approx
# Glover canonical HRF models
# they are both Sympy objects
def _getint(f, dt=0.02, t=50):
# numerical integral of function
lf = lambdify_t(f)
tt = np.arange(dt,t+dt,dt)
return lf(tt).sum() * dt
_gexpr = gamma_expr(5.4, 5.2) - 0.35 * gamma_expr(10.8, 7.35)
_gexpr = _gexpr / _getint(_gexpr)
_glover = lambdify_t(_gexpr)
glover = implemented_function('glover', _glover)
glovert = lambdify_t(glover(T))
# Derivative of Glover HRF
_dgexpr = _gexpr.diff(T)
dpos = sympy.Derivative((T >= 0), T)
_dgexpr = _dgexpr.subs(dpos, 0)
_dgexpr = _dgexpr / _getint(sympy_abs(_dgexpr))
_dglover = lambdify_t(_dgexpr)
dglover = implemented_function('dglover', _dglover)
dglovert = lambdify_t(dglover(T))
del(_glover); del(_gexpr); del(dpos); del(_dgexpr); del(_dglover)
# AFNI's HRF
def interp(times, values, fill=0, name=None, **kw):
""" Generic interpolation function of t given `times` and `values`
Imterpolator such that:
f(times[i]) = values[i]
if t < times[0] or t > times[-1]:
f(t) = fill
See ``scipy.interpolate.interp1d`` for details of interpolation
types and other keyword arguments. Default is 'kind' is linear,
making this function, by default, have the same behavior as
``linear_interp``.
Parameters
----------
times : array-like
Increasing sequence of times
values : array-like
Values at the specified times
fill : None or float, optional
Value on the interval (-np.inf, times[0]). Default 0. If None, raises
error outside bounds
name : None or str, optional
Name of symbolic expression to use. If None, a default is used.
\*\*kw : keyword args, optional
passed to ``interp1d``
Returns
-------
f : sympy expression
A Function of t.
Examples
--------
>>> s = interp([0,4,5.],[2.,4,6])
>>> tval = np.array([-0.1,0.1,3.9,4.1,5.1])
>>> res = lambdify_t(s)(tval)
0 outside bounds by default
>>> np.allclose(res, [0, 2.05, 3.95, 4.2, 0])
True
"""
if not fill is None:
if kw.get('bounds_error') is True:
raise ValueError('fill conflicts with bounds error')
fv = kw.get('fill_value')
if not (fv is None or fv is fill
or fv == fill): # allow for fill=np.nan
raise ValueError('fill conflicts with fill_value')
kw['bounds_error'] = False
kw['fill_value'] = fill
interpolator = interp1d(times, values, **kw)
# make a new name if none provided
if name is None:
name = 'interp%d' % interp.counter
interp.counter += 1
s = implemented_function(name, interpolator)
return s(T)
def spectral_decomposition(hrf2decompose,
time=None,
delta=None,
ncomp=2):
""" PCA decomposition of symbolic HRF shifted over time
Perform a PCA expansion of a symbolic HRF, time shifted over the
values in delta, returning the first ncomp components.
This smooths out the HRF as compared to using a Taylor series
approximation.
Parameters
----------
hrf2decompose : sympy expression
An expression that can be lambdified as a function of 't'. This
is the HRF to be expanded in PCA
time : None or np.ndarray, optional
None gives default value of np.linspace(-15,50,3251) chosen to
match fMRIstat implementation. This corresponds to a time
interval of 0.02. Presumed to be equally spaced.
delta : None or np.ndarray, optional
None results in default value of np.arange(-4.5, 4.6, 0.1)
chosen to match fMRIstat implementation.
ncomp : int, optional
Number of principal components to retain.
Returns
-------
hrf : [sympy expressions]
A sequence length `ncomp` of symbolic HRFs that are the
principal components.
approx :
TODO
"""
if time is None:
time = np.linspace(-15,50,3251)
dt = time[1] - time[0]
if delta is None:
delta = np.arange(-4.5, 4.6, 0.1)
# make numerical implementation from hrf function and symbol t.
# hrft returns function values when called with values for time as
# input.
hrft = lambdify_t(hrf2decompose(T))
# Create stack of time-shifted HRFs. Time varies over row, delta
# over column.
ts_hrf_vals = np.array([hrft(time - d) for d in delta]).T
ts_hrf_vals = np.nan_to_num(ts_hrf_vals)
# PCA
U, S, V = npl.svd(ts_hrf_vals, full_matrices=0)
# make interpolators from the generated bases
basis = []
for i in range(ncomp):
b = interp1d(time, U[:, i], bounds_error=False, fill_value=0.)
# normalize components witn integral of abs of first component
if i == 0:
d = np.fabs((b(time) * dt).sum())
b.y /= d
basis.append(b)
# reconstruct time courses for all bases
W = np.array([b(time) for b in basis]).T
# regress basis time courses against original time shifted time
# courses, ncomps by len(delta) parameter matrix
WH = np.dot(npl.pinv(W), ts_hrf_vals)
# put these into interpolators to get estimated coefficients for any
# value of delta
coef = [interp1d(delta, w, bounds_error=False, fill_value=0.) for w in WH]
# swap sign of first component to match that of input HRF. Swap
# other components if we swap the first, to standardize signs of
# components across SVD implementations.
if coef[0](0) < 0: # coefficient at time shift of 0
for i in range(ncomp):
coef[i].y *= -1.
basis[i].y *= -1.
def approx(time, delta):
value = 0
for i in range(ncomp):
value += coef[i](delta) * basis[i](time)
return value
approx.coef = coef
approx.components = basis
(approx.theta,
approx.inverse,
approx.dinverse,
approx.forward,
approx.dforward) = invertR(delta, approx.coef)
# construct aliased functions from bases
symbasis = []
for i, b in enumerate(basis):
symbasis.append(
implemented_function('%s%d' % (str(hrf2decompose), i), b))
return symbasis, approx
def natural_spline(t, knots=None, order=3, intercept=False):
""" Return a Formula containing a natural spline
Spline for a Term with specified `knots` and `order`.
Parameters
----------
t : ``Term``
knots : None or sequence, optional
Sequence of float. Default None (same as empty list)
order : int, optional
Order of the spline. Defaults to a cubic (==3)
intercept : bool, optional
If True, include a constant function in the natural
spline. Default is False
Returns
-------
formula : Formula
A Formula with (len(knots) + order) Terms (if intercept=False,
otherwise includes one more Term), made up of the natural spline
functions.
Examples
--------
>>> x = Term('x')
>>> n = natural_spline(x, knots=[1,3,4], order=3)
>>> xval = np.array([3,5,7.]).view(np.dtype([('x', np.float)]))
>>> n.design(xval, return_float=True)
array([[ 3., 9., 27., 8., 0., -0.],
[ 5., 25., 125., 64., 8., 1.],
[ 7., 49., 343., 216., 64., 27.]])
>>> d = n.design(xval)
>>> print d.dtype.descr
[('ns_1(x)', '<f8'), ('ns_2(x)', '<f8'), ('ns_3(x)', '<f8'), ('ns_4(x)', '<f8'), ('ns_5(x)', '<f8'), ('ns_6(x)', '<f8')]
"""
if knots is None:
knots = {}
fns = []
for i in range(order + 1):
n = 'ns_%d' % i
def f(x, i=i):
return x**i
s = implemented_function(n, f)
fns.append(s(t))
for j, k in enumerate(knots):
n = 'ns_%d' % (j + i + 1, )
def f(x, k=k, order=order):
return (x - k)**order * np.greater(x, k)
s = implemented_function(n, f)
fns.append(s(t))
if not intercept:
fns.pop(0)
ff = Formula(fns)
return ff