# The splines are functions of t (time)
bsp_fns = linBspline(np.arange(0, 10, 2))
# We're going to evaluate at these specific values of time
tt = np.linspace(0, 50, 101)
tvals = tt.view(np.dtype([('t', np.float)]))
# Some inter-stimulus intervals
isis = np.random.uniform(low=0, high=3, size=(4, )) + 10.
# Made into event onset times
e = np.cumsum(isis)
# Make event onsets into functions of time convolved with the spline functions.
event_funcs = [utils.events(e, f=fn) for fn in bsp_fns]
# Put into a formula.
f = Formula(event_funcs)
# The design matrix
X = f.design(tvals, return_float=True)
# Show the design matrix as line plots
plt.plot(X[:, 0])
plt.plot(X[:, 1])
plt.plot(X[:, 2])
plt.xlabel('time (s)')
plt.title('B spline used as bases for an FIR response model')
plt.show()
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
# The splines are functions of t (time)
bsp_fns = linBspline(np.arange(0,10,2))
# We're going to evaluate at these specific values of time
tt = np.linspace(0,50,101)
tvals= tt.view(np.dtype([('t', np.float)]))
# Some interstimulus intervals
isis = np.random.uniform(low=0, high=3, size=(4,)) + 10.
# Made into event onset times
e = np.cumsum(isis)
f = Formula([utils.events(e, f=fn) for fn in bsp_fns])
# The design matrix
X = f.design(tvals, return_float=True)
plt.plot(X[:,0])
plt.plot(X[:,1])
plt.plot(X[:,2])
plt.show()
from nipy.algorithms.statistics.api import Formula, make_recarray
from nipy.modalities.fmri import utils, hrf
# Inter-stimulus intervals (time between events)
dt = np.random.uniform(low=0, high=2.5, size=(50,))
# Onset times from the ISIs
t = np.cumsum(dt)
# We're going to model the amplitudes ('a') by dt (the time between events)
a = sympy.Symbol('a')
linear = utils.define('linear', utils.events(t, dt, f=hrf.glover))
quadratic = utils.define('quad', utils.events(t, dt, f=hrf.glover, g=a**2))
cubic = utils.define('cubic', utils.events(t, dt, f=hrf.glover, g=a**3))
f1 = Formula([linear, quadratic, cubic])
# Evaluate this time-based formula at specific times to make the design matrix
tval = make_recarray(np.linspace(0,100, 1001), 't')
X1 = f1.design(tval, return_float=True)
# Now we make a model where the relationship of time between events and signal
# is an exponential with a time constant tau
l = sympy.Symbol('l')
exponential = utils.events(t, dt, f=hrf.glover, g=sympy.exp(-l*a))
f3 = Formula([exponential])
# Make a design matrix by passing in time and required parameters
params = make_recarray([(4.5, 3.5)], ('l', '_b0'))
X3 = f3.design(tval, params, return_float=True)
from nipy.algorithms.statistics.api import Term, Formula
data = np.rec.fromarrays(([1,3,4,5,8,10,9], range(1,8)),
... names=('Y', 'X'))
f = Formula([Term("X"), 1])
dmtx = f.design(data, return_float=True)
model = ARModel(dmtx, 2)
We go through the ``model.iterative_fit`` procedure long-hand:
for i in range(6):
... results = model.fit(data['Y'])
... print("AR coefficients:", model.rho)
... rho, sigma = yule_walker(data["Y"] - results.predicted,
... order=2,
... df=model.df_resid)
... model = ARModel(model.design, rho) #doctest: +FP_6DP
...
AR coefficients: [ 0. 0.]
AR coefficients: [-0.61530877 -1.01542645]
AR coefficients: [-0.72660832 -1.06201457]
AR coefficients: [-0.7220361 -1.05365352]
AR coefficients: [-0.72229201 -1.05408193]
AR coefficients: [-0.722278 -1.05405838]
results.theta #doctest: +FP_6DP
array([ 1.59564228, -0.58562172])
results.t() #doctest: +FP_6DP
array([ 38.0890515 , -3.45429252])
print(results.Tcontrast([0,1])) #doctest: +FP_6DP
<T contrast: effect=-0.58562172384377043, sd=0.16953449108110835,
t=-3.4542925165805847, df_den=5>
print(results.Fcontrast(np.identity(2))) #doctest: +FP_6DP
<F contrast: F=4216.810299725842, df_den=5, df_num=2>
Reinitialize the model, and do the automated iterative fit
from nipy.algorithms.statistics.api import Formula, make_recarray
from nipy.modalities.fmri import utils, hrf
from nipy.modalities.fmri.fmristat import hrf as delay
# We take event onsets, and a specified HRF model, and make symbolic functions
# of time
c1 = utils.events([3,7,10], f=hrf.glover) # Symbolic function of time
c2 = utils.events([1,3,9], f=hrf.glover) # Symbolic function of time
c3 = utils.events([3,4,6], f=delay.spectral[0]) # Symbolic function of time
# We can also use a Fourier basis for some other onsets - again making symbolic
# functions of time
d = utils.fourier_basis([3,5,7]) # Formula
# Make a formula for all four sets of onsets
f = Formula([c1,c2,c3]) + d
# A contrast is a formula expressed on the elements of the design formula
contrast = Formula([c1-c2, c1-c3])
# Instantiate actual values of time at which to create the design matrix rows
t = make_recarray(np.linspace(0,20,50), 't')
# Make the design matrix, and get contrast matrices for the design
X, c = f.design(t, return_float=True, contrasts={'C':contrast})
# c is a dictionary, containing a 2 by 9 matrix - the F contrast matrix for our
# contrast of interest
assert X.shape == (50, 9)
assert c['C'].shape == (2, 9)
# hrf
h1 = sympy.Function('hrf1')
h2 = sympy.Function('hrf2')
t = Term('t')
# Session 1
t1 = Term('t1')
c11 = utils.events([3,7,10], f=h1); c11 = c11.subs(t, t1)
c21 = utils.events([1,3,9], f=h1); c21 = c21.subs(t, t1)
c31 = utils.events([2,4,8], f=h1); c31 = c31.subs(t, t1)
d1 = utils.fourier_basis([0.3,0.5,0.7]); d1 = d1.subs(t, t1)
tval1 = np.linspace(0,20,101)
f1 = Formula([c11,c21,c31]) + d1
f1 = f1.subs('hrf1', hrf.glover)
# Session 2
t2 = Term('t2')
c12 = utils.events([3.3,7,10], f=h2); c12 = c12.subs(t, t2)
c22 = utils.events([1,3.2,9], f=h2); c22 = c22.subs(t, t2)
c32 = utils.events([2,4.2,8], f=h2); c32 = c32.subs(t, t2)
d2 = utils.fourier_basis([0.3,0.5,0.7]); d2 = d2.subs(t, t2)
tval2 = np.linspace(0,10,51)
f2 = Formula([c12,c22,c32]) + d2
f2 = f2.subs('hrf2', hrf.dglover)
sess_factor = Factor('sess', [1,2])
# run 1
t1 = Term('t1') # Time within run 1
c11 = utils.events([3, 7, 10], f=h1) # Condition 1, run 1
# The events utility returns a formula in terms of 't' - general time
c11 = c11.subs(t, t1) # Now make it in terms of time in run 1
# Same for conditions 2 and 3
c21 = utils.events([1, 3, 9], f=h1)
c21 = c21.subs(t, t1)
c31 = utils.events([2, 4, 8], f=h1)
c31 = c31.subs(t, t1)
# Add also a Fourier basis set for drift with frequencies 0.3, 0.5, 0.7
d1 = utils.fourier_basis([0.3, 0.5, 0.7])
d1 = d1.subs(t, t1)
# Here's our formula for run 1 signal terms of time in run 1 (t1)
f1 = Formula([c11, c21, c31]) + d1
# run 2
t2 = Term('t2') # Time within run 2
# Conditions 1 through 3 in run 2
c12 = utils.events([3.3, 7, 10], f=h2)
c12 = c12.subs(t, t2)
c22 = utils.events([1, 3.2, 9], f=h2)
c22 = c22.subs(t, t2)
c32 = utils.events([2, 4.2, 8], f=h2)
c32 = c32.subs(t, t2)
d2 = utils.fourier_basis([0.3, 0.5, 0.7])
d2 = d2.subs(t, t2)
# Formula for run 2 signal in terms of time in run 2 (t2)
f2 = Formula([c12, c22, c32]) + d2
# Formula for run 2 signal in terms of time in run 2 (t2)
f2 = Formula([c12, c22, c32]) + d2
# Factor giving constant for run. The [1, 2] means that there are two levels to
# this factor, and that when we get to pass in values for this factor,
# instantiating an actual design matrix (see below), a value of 1 means level
# 1 and a value of 2 means level 2.
run_factor = Factor('run', [1, 2])
run_1_coder = run_factor.get_term(1) # Term coding for level 1
run_2_coder = run_factor.get_term(2) # Term coding for level 2
# The multi run formula will combine the indicator (dummy value) terms from the
# run factor with the formulae for the runs (which are functions of (run1, run2)
# time. The run_factor terms are step functions that are zero when not in the
# run, 1 when in the run.
f = Formula([run_1_coder]) * f1 + Formula([run_2_coder]) * f2 + run_factor
# Now, we evaluate the formula. So far we've been entirely symbolic. Now we
# start to think about the values at which we want to evaluate our symbolic
# formula.
# We'll use these values for time within run 1. The times are in seconds from
# the beginning of run 1. In our case run 1 was 20 seconds long. 101 below
# gives 101 values from 0 to 20 including the endpoints, giving a dt of 0.2.
tval1 = np.linspace(0, 20, 101)
# run 2 lasts 10 seconds. These are the times in terms of the start of run 2.
tval2 = np.linspace(0, 10, 51)
# We pad out the tval1 / tval2 time vectors with zeros corresponding to the
# TRs in run 2 / run 1.
ttval1 = np.hstack([tval1, np.zeros(tval2.shape)])
from nipy.algorithms.statistics.api import Formula, make_recarray
from nipy.modalities.fmri import utils, hrf
from nipy.modalities.fmri.fmristat import hrf as delay
# We take event onsets, and a specified HRF model, and make symbolic functions
# of time
c1 = utils.events([3, 7, 10], f=hrf.glover) # Symbolic function of time
c2 = utils.events([1, 3, 9], f=hrf.glover) # Symbolic function of time
c3 = utils.events([3, 4, 6], f=delay.spectral[0]) # Symbolic function of time
# We can also use a Fourier basis for some other onsets - again making symbolic
# functions of time
d = utils.fourier_basis([3, 5, 7]) # Formula
# Make a formula for all four sets of onsets
f = Formula([c1, c2, c3]) + d
# A contrast is a formula expressed on the elements of the design formula
contrast = Formula([c1 - c2, c1 - c3])
# Instantiate actual values of time at which to create the design matrix rows
t = make_recarray(np.linspace(0, 20, 50), 't')
# Make the design matrix, and get contrast matrices for the design
X, c = f.design(t, return_float=True, contrasts={'C': contrast})
# c is a dictionary, containing a 2 by 9 matrix - the F contrast matrix for our
# contrast of interest
assert X.shape == (50, 9)
assert c['C'].shape == (2, 9)
# Session 2
t2 = Term('t2')
c12 = utils.events([3.3,7,10], f=h2); c12 = c12.subs(t, t2)
c22 = utils.events([1,3.2,9], f=h2); c22 = c22.subs(t, t2)
c32 = utils.events([2,4.2,8], f=h2); c32 = c32.subs(t, t2)
d2 = utils.fourier_basis([0.3,0.5,0.7]); d2 = d2.subs(t, t2)
tval2 = np.linspace(0,10,51)
f2 = Formula([c12,c22,c32]) + d2
sess_factor = Factor('sess', [1,2])
# The multi session formula
f = Formula([sess_factor.terms[0]]) * f1 + Formula([sess_factor.terms[1]]) * f2
# Now, we evaluate the formula
# the arrays now have 152=101+51 rows...
ttval1 = np.hstack([tval1, np.zeros(tval2.shape)])
ttval2 = np.hstack([np.zeros(tval1.shape), tval2])
session = np.array([1]*tval1.shape[0] + [2]*tval2.shape[0])
f = f.subs(h1, hrf.glover)
f = f.subs(h2, hrf.glover)
# Create the recarray that will be used to create the design matrix
rec = np.array([(t1,t2, s) for t1, t2, s in zip(ttval1, ttval2, session)],
np.dtype([('t1', np.float),
