def constrained_oasisAR2(y, g, sn, optimize_b=True, b_nonneg=True, optimize_g=0, decimate=5,
shift=100, window=None, tol=1e-9, max_iter=1, penalty=1):
""" Infer the most likely discretized spike train underlying an AR(2) fluorescence trace
Solves the noise constrained sparse non-negative deconvolution problem
min |s|_1 subject to |c-y|^2 = sn^2 T and s_t = c_t-g1 c_{t-1}-g2 c_{t-2} >= 0
Parameters:
----------
y : array of float
One dimensional array containing the fluorescence intensities (with baseline
already subtracted) with one entry per time-bin.
g : (float, float)
Parameters of the AR(2) process that models the fluorescence impulse response.
sn : float
Standard deviation of the noise distribution.
optimize_b : bool, optional, default True
Optimize baseline if True else it is set to 0, see y.
b_nonneg: bool, optional, default True
Enforce strictly non-negative baseline if True.
optimize_g : int, optional, default 0
Number of large, isolated events to consider for optimizing g.
No optimization if optimize_g=0.
decimate : int, optional, default 5
Decimation factor for estimating hyper-parameters faster on decimated data.
shift : int, optional, default 100
Number of frames by which to shift window from on run of NNLS to the next.
window : int, optional, default None (200 or larger dependend on g)
Window size.
tol : float, optional, default 1e-9
Tolerance parameter.
max_iter : int, optional, default 1
Maximal number of iterations.
penalty : int, optional, default 1
Sparsity penalty. 1: min |s|_1 0: min |s|_0
Returns:
-------
c : array of float
The inferred denoised fluorescence signal at each time-bin.
s : array of float
Discretized deconvolved neural activity (spikes).
b : float
Fluorescence baseline value.
(g1, g2) : tuple of float
Parameters of the AR(2) process that models the fluorescence impulse response.
lam : float
Sparsity penalty parameter lambda of dual problem.
References:
----------
* Friedrich J and Paninski L, NIPS 2016
* Friedrich J, Zhou P, and Paninski L, arXiv 2016
"""
T = len(y)
d = (g[0] + sqrt(g[0] * g[0] + 4 * g[1])) / 2
r = (g[0] - sqrt(g[0] * g[0] + 4 * g[1])) / 2
if window is None:
window = int(min(T, max(200, -5 / log(d))))
if not optimize_g:
g11 = (np.exp(log(d) * np.arange(1, T + 1)) * np.arange(1, T + 1)) if d == r else \
(np.exp(log(d) * np.arange(1, T + 1)) -
np.exp(log(r) * np.arange(1, T + 1))) / (d - r)
g12 = np.append(0, g[1] * g11[:-1])
g11g11 = np.cumsum(g11 * g11)
g11g12 = np.cumsum(g11 * g12)
Sg11 = np.cumsum(g11)
f_lam = 1 - g[0] - g[1]
elif decimate == 0: # need to run AR1 anyways for estimating AR coeffs
decimate = 1
thresh = sn * sn * T
# get initial estimate of b and lam on downsampled data using AR1 model
if decimate > 0:
from caiman.source_extraction.cnmf.oasis import oasisAR1, constrained_oasisAR1
_, s, b, aa, lam = constrained_oasisAR1(
y[:len(y) // decimate * decimate].reshape(-1, decimate).mean(1),
d**decimate, sn / sqrt(decimate),
optimize_b=optimize_b, b_nonneg=b_nonneg, optimize_g=optimize_g)
if optimize_g:
from scipy.optimize import minimize
d = aa**(1. / decimate)
if decimate > 1:
s = oasisAR1(y - b, d, lam=lam * (1 - aa) / (1 - d))[1]
r = estimate_time_constant(s, 1, fudge_factor=.98)[0]
g[0] = d + r
g[1] = -d * r
g11 = (np.exp(log(d) * np.arange(1, T + 1)) -
np.exp(log(r) * np.arange(1, T + 1))) / (d - r)
g12 = np.append(0, g[1] * g11[:-1])
g11g11 = np.cumsum(g11 * g11)
g11g12 = np.cumsum(g11 * g12)
Sg11 = np.cumsum(g11)
f_lam = 1 - g[0] - g[1]
elif decimate > 1:
s = oasisAR1(y - b, d, lam=lam * (1 - aa) / (1 - d))[1]
lam *= (1 - d**decimate) / f_lam
# this window size seems necessary and sufficient
possible_spikes = [x + np.arange(-2, 3)
for x in np.where(s > s.max() / 10.)[0]]
ff = np.array(possible_spikes, dtype=np.int).ravel()
ff = np.unique(ff[(ff >= 0) * (ff < T)])
mask = np.zeros(T, dtype=bool)
mask[ff] = True
else:
b = np.percentile(y, 15) if optimize_b else 0
lam = 2 * sn * np.linalg.norm(g11)
mask = None
if b_nonneg:
b = max(b, 0)
# run ONNLS
c, s = onnls(y - b, g, lam=lam, mask=mask,
shift=shift, window=window, tol=tol)
if not optimize_b: # don't optimize b, just the dual variable lambda
for _ in range(max_iter - 1):
res = y - c
RSS = res.dot(res)
if np.abs(RSS - thresh) < 1e-4 * thresh:
break
# calc shift dlam, here attributed to sparsity penalty
tmp = np.empty(T)
ls = np.append(np.where(s > 1e-6)[0], T)
l = ls[0]
tmp[:l] = (1 + d) / (1 + d**l) * \
np.exp(log(d) * np.arange(l)) # first pool
for i, f in enumerate(ls[:-1]): # all other pools
l = ls[i + 1] - f - 1
# if and elif correct last 2 time points for |s|_1 instead |c|_1
if i == len(ls) - 2: # last pool
tmp[f] = (1. / f_lam if l == 0 else
(Sg11[l] + g[1] / f_lam * g11[l - 1]
+ (g[0] + g[1]) / f_lam * g11[l]
- g11g12[l] * tmp[f - 1]) / g11g11[l])
# secondlast pool if last one has length 1
elif i == len(ls) - 3 and ls[-2] == T - 1:
tmp[f] = (Sg11[l] + g[1] / f_lam * g11[l]
- g11g12[l] * tmp[f - 1]) / g11g11[l]
else: # all other pools
tmp[f] = (Sg11[l] - g11g12[l] * tmp[f - 1]) / g11g11[l]
l += 1
tmp[f + 1:f + l] = g11[1:l] * tmp[f] + g12[1:l] * tmp[f - 1]
aa = tmp.dot(tmp)
bb = res.dot(tmp)
cc = RSS - thresh
try:
db = (-bb + sqrt(bb * bb - aa * cc)) / aa
except:
db = -bb / aa
# perform shift
b += db
c, s = onnls(y - b, g, lam=lam, mask=mask,
shift=shift, window=window, tol=tol)
db = np.mean(y - c) - b
b += db
lam -= db / f_lam
else: # optimize b
db = max(np.mean(y - c), 0 if b_nonneg else -np.inf) - b
b += db
lam -= db / (1 - g[0] - g[1])
g_converged = False
for _ in range(max_iter - 1):
res = y - c - b
RSS = res.dot(res)
if np.abs(RSS - thresh) < 1e-4 * thresh:
break
# calc shift db, here attributed to baseline
tmp = np.empty(T)
ls = np.append(np.where(s > 1e-6)[0], T)
l = ls[0]
tmp[:l] = (1 + d) / (1 + d**l) * \
np.exp(log(d) * np.arange(l)) # first pool
for i, f in enumerate(ls[:-1]): # all other pools
l = ls[i + 1] - f
tmp[f] = (Sg11[l - 1] - g11g12[l - 1]
* tmp[f - 1]) / g11g11[l - 1]
tmp[f + 1:f + l] = g11[1:l] * tmp[f] + g12[1:l] * tmp[f - 1]
tmp -= tmp.mean()
aa = tmp.dot(tmp)
bb = res.dot(tmp)
cc = RSS - thresh
try:
db = (-bb + sqrt(bb * bb - aa * cc)) / aa
except:
db = -bb / aa
# perform shift
if b_nonneg:
db = max(db, -b)
b += db
c, s = onnls(y - b, g, lam=lam, mask=mask,
shift=shift, window=window, tol=tol)
# update b and lam
db = max(np.mean(y - c), 0 if b_nonneg else -np.inf) - b
b += db
lam -= db / f_lam
# update g and b
if optimize_g and (not g_converged):
def getRSS(y, opt):
b, ld, lr = opt
if ld < lr:
return 1e3 * thresh
d, r = exp(ld), exp(lr)
g1, g2 = d + r, -d * r
tmp = b + onnls(y - b, [g1, g2], lam,
mask=(s > 1e-2 * s.max()))[0] - y
return tmp.dot(tmp)
result = minimize(lambda x: getRSS(y, x), (b, log(d), log(r)),
bounds=((0 if b_nonneg else None, None),
(None, -1e-4), (None, -1e-3)), method='L-BFGS-B',
options={'gtol': 1e-04, 'maxiter': 10, 'ftol': 1e-05})
if abs(result['x'][1] - log(d)) < 1e-3:
g_converged = True
b, ld, lr = result['x']
d, r = exp(ld), exp(lr)
g = (d + r, -d * r)
c, s = onnls(y - b, g, lam=lam, mask=mask,
shift=shift, window=window, tol=tol)
# update b and lam
db = max(np.mean(y - c), 0 if b_nonneg else -np.inf) - b
b += db
lam -= db
if penalty == 0: # get (locally optimal) L0 solution
def c4smin(y, s, s_min):
ls = np.append(np.where(s > s_min)[0], T)
tmp = np.zeros_like(s)
l = ls[0] # first pool
tmp[:l] = max(0, np.exp(log(d) * np.arange(l)).dot(y[:l]) * (1 - d * d)
/ (1 - d**(2 * l))) * np.exp(log(d) * np.arange(l))
for i, f in enumerate(ls[:-1]): # all other pools
l = ls[i + 1] - f
tmp[f] = (g11[:l].dot(y[f:f + l]) - g11g12[l - 1]
* tmp[f - 1]) / g11g11[l - 1]
tmp[f + 1:f + l] = g11[1:l] * tmp[f] + g12[1:l] * tmp[f - 1]
return tmp
spikesizes = np.sort(s[s > 1e-6])
i = len(spikesizes) // 2
l = 0
u = len(spikesizes) - 1
while u - l > 1:
s_min = spikesizes[i]
tmp = c4smin(y - b, s, s_min)
res = y - b - tmp
RSS = res.dot(res)
if RSS < thresh or i == 0:
l = i
i = (l + u) // 2
res0 = tmp
else:
u = i
i = (l + u) // 2
if i > 0:
c = res0
s = np.append([0, 0], c[2:] - g[0] * c[1:-1] - g[1] * c[:-2])
return c, s, b, g, lam
def constrained_oasisAR2(y,
g,
sn,
optimize_b=True,
b_nonneg=True,
optimize_g=0,
decimate=5,
shift=100,
window=None,
tol=1e-9,
max_iter=1,
penalty=1):
""" Infer the most likely discretized spike train underlying an AR(2) fluorescence trace
Solves the noise constrained sparse non-negative deconvolution problem
min (s)_1 subject to (c-y)^2 = sn^2 T and s_t = c_t-g1 c_{t-1}-g2 c_{t-2} >= 0
Args:
y : array of float
One dimensional array containing the fluorescence intensities (with baseline
already subtracted) with one entry per time-bin.
g : (float, float)
Parameters of the AR(2) process that models the fluorescence impulse response.
sn : float
Standard deviation of the noise distribution.
optimize_b : bool, optional, default True
Optimize baseline if True else it is set to 0, see y.
b_nonneg: bool, optional, default True
Enforce strictly non-negative baseline if True.
optimize_g : int, optional, default 0
Number of large, isolated events to consider for optimizing g.
No optimization if optimize_g=0.
decimate : int, optional, default 5
Decimation factor for estimating hyper-parameters faster on decimated data.
shift : int, optional, default 100
Number of frames by which to shift window from on run of NNLS to the next.
window : int, optional, default None (200 or larger dependend on g)
Window size.
tol : float, optional, default 1e-9
Tolerance parameter.
max_iter : int, optional, default 1
Maximal number of iterations.
penalty : int, optional, default 1
Sparsity penalty. 1: min (s)_1 0: min (s)_0
Returns:
c : array of float
The inferred denoised fluorescence signal at each time-bin.
s : array of float
Discretized deconvolved neural activity (spikes).
b : float
Fluorescence baseline value.
(g1, g2) : tuple of float
Parameters of the AR(2) process that models the fluorescence impulse response.
lam : float
Sparsity penalty parameter lambda of dual problem.
References:
Friedrich J and Paninski L, NIPS 2016
Friedrich J, Zhou P, and Paninski L, arXiv 2016
"""
T = len(y)
d = (g[0] + sqrt(g[0] * g[0] + 4 * g[1])) / 2
r = (g[0] - sqrt(g[0] * g[0] + 4 * g[1])) / 2
if window is None:
window = int(min(T, max(200, -5 / log(d))))
if not optimize_g:
g11 = (np.exp(log(d) * np.arange(1, T + 1)) * np.arange(1, T + 1)) if d == r else \
(np.exp(log(d) * np.arange(1, T + 1)) -
np.exp(log(r) * np.arange(1, T + 1))) / (d - r)
g12 = np.append(0, g[1] * g11[:-1])
g11g11 = np.cumsum(g11 * g11)
g11g12 = np.cumsum(g11 * g12)
Sg11 = np.cumsum(g11)
f_lam = 1 - g[0] - g[1]
elif decimate == 0: # need to run AR1 anyways for estimating AR coeffs
decimate = 1
thresh = sn * sn * T
# get initial estimate of b and lam on downsampled data using AR1 model
if decimate > 0:
from caiman.source_extraction.cnmf.oasis import oasisAR1, constrained_oasisAR1
_, s, b, aa, lam = constrained_oasisAR1(
y[:len(y) // decimate * decimate].reshape(-1, decimate).mean(1),
d**decimate,
sn / sqrt(decimate),
optimize_b=optimize_b,
b_nonneg=b_nonneg,
optimize_g=optimize_g)
if optimize_g:
from scipy.optimize import minimize
d = aa**(1. / decimate)
if decimate > 1:
s = oasisAR1(y - b, d, lam=lam * (1 - aa) / (1 - d))[1]
r = estimate_time_constant(s, 1, fudge_factor=.98)[0]
g[0] = d + r
g[1] = -d * r
g11 = (np.exp(log(d) * np.arange(1, T + 1)) -
np.exp(log(r) * np.arange(1, T + 1))) / (d - r)
g12 = np.append(0, g[1] * g11[:-1])
g11g11 = np.cumsum(g11 * g11)
g11g12 = np.cumsum(g11 * g12)
Sg11 = np.cumsum(g11)
f_lam = 1 - g[0] - g[1]
elif decimate > 1:
s = oasisAR1(y - b, d, lam=lam * (1 - aa) / (1 - d))[1]
lam *= (1 - d**decimate) / f_lam
# this window size seems necessary and sufficient
possible_spikes = [
x + np.arange(-2, 3) for x in np.where(s > s.max() / 10.)[0]
]
ff = np.array(possible_spikes, dtype=np.int).ravel()
ff = np.unique(ff[(ff >= 0) * (ff < T)])
mask = np.zeros(T, dtype=bool)
mask[ff] = True
else:
b = np.percentile(y, 15) if optimize_b else 0
lam = 2 * sn * np.linalg.norm(g11)
mask = None
if b_nonneg:
b = max(b, 0)
# run ONNLS
c, s = onnls(y - b,
g,
lam=lam,
mask=mask,
shift=shift,
window=window,
tol=tol)
if not optimize_b: # don't optimize b, just the dual variable lambda
for _ in range(max_iter - 1):
res = y - c
RSS = res.dot(res)
if np.abs(RSS - thresh) < 1e-4 * thresh:
break
# calc shift dlam, here attributed to sparsity penalty
tmp = np.empty(T)
ls = np.append(np.where(s > 1e-6)[0], T)
l = ls[0]
tmp[:l] = (1 + d) / (1 + d**l) * \
np.exp(log(d) * np.arange(l)) # first pool
for i, f in enumerate(ls[:-1]): # all other pools
l = ls[i + 1] - f - 1
# if and elif correct last 2 time points for |s|_1 instead |c|_1
if i == len(ls) - 2: # last pool
tmp[f] = (1. / f_lam if l == 0 else
(Sg11[l] + g[1] / f_lam * g11[l - 1] +
(g[0] + g[1]) / f_lam * g11[l] -
g11g12[l] * tmp[f - 1]) / g11g11[l])
# secondlast pool if last one has length 1
elif i == len(ls) - 3 and ls[-2] == T - 1:
tmp[f] = (Sg11[l] + g[1] / f_lam * g11[l] -
g11g12[l] * tmp[f - 1]) / g11g11[l]
else: # all other pools
tmp[f] = (Sg11[l] - g11g12[l] * tmp[f - 1]) / g11g11[l]
l += 1
tmp[f + 1:f + l] = g11[1:l] * tmp[f] + g12[1:l] * tmp[f - 1]
aa = tmp.dot(tmp)
bb = res.dot(tmp)
cc = RSS - thresh
try:
db = (-bb + sqrt(bb * bb - aa * cc)) / aa
except:
db = -bb / aa
# perform shift
b += db
c, s = onnls(y - b,
g,
lam=lam,
mask=mask,
shift=shift,
window=window,
tol=tol)
db = np.mean(y - c) - b
b += db
lam -= db / f_lam
else: # optimize b
db = max(np.mean(y - c), 0 if b_nonneg else -np.inf) - b
b += db
lam -= db / (1 - g[0] - g[1])
g_converged = False
for _ in range(max_iter - 1):
res = y - c - b
RSS = res.dot(res)
if np.abs(RSS - thresh) < 1e-4 * thresh:
break
# calc shift db, here attributed to baseline
tmp = np.empty(T)
ls = np.append(np.where(s > 1e-6)[0], T)
l = ls[0]
tmp[:l] = (1 + d) / (1 + d**l) * \
np.exp(log(d) * np.arange(l)) # first pool
for i, f in enumerate(ls[:-1]): # all other pools
l = ls[i + 1] - f
tmp[f] = (Sg11[l - 1] -
g11g12[l - 1] * tmp[f - 1]) / g11g11[l - 1]
tmp[f + 1:f + l] = g11[1:l] * tmp[f] + g12[1:l] * tmp[f - 1]
tmp -= tmp.mean()
aa = tmp.dot(tmp)
bb = res.dot(tmp)
cc = RSS - thresh
try:
db = (-bb + sqrt(bb * bb - aa * cc)) / aa
except:
db = -bb / aa
# perform shift
if b_nonneg:
db = max(db, -b)
b += db
c, s = onnls(y - b,
g,
lam=lam,
mask=mask,
shift=shift,
window=window,
tol=tol)
# update b and lam
db = max(np.mean(y - c), 0 if b_nonneg else -np.inf) - b
b += db
lam -= db / f_lam
# update g and b
if optimize_g and (not g_converged):
def getRSS(y, opt):
b, ld, lr = opt
if ld < lr:
return 1e3 * thresh
d, r = exp(ld), exp(lr)
g1, g2 = d + r, -d * r
tmp = b + onnls(
y - b, [g1, g2], lam, mask=(s > 1e-2 * s.max()))[0] - y
return tmp.dot(tmp)
result = minimize(lambda x: getRSS(y, x), (b, log(d), log(r)),
bounds=((0 if b_nonneg else None, None),
(None, -1e-4), (None, -1e-3)),
method='L-BFGS-B',
options={
'gtol': 1e-04,
'maxiter': 10,
'ftol': 1e-05
})
if abs(result['x'][1] - log(d)) < 1e-3:
g_converged = True
b, ld, lr = result['x']
d, r = exp(ld), exp(lr)
g = (d + r, -d * r)
c, s = onnls(y - b,
g,
lam=lam,
mask=mask,
shift=shift,
window=window,
tol=tol)
# update b and lam
db = max(np.mean(y - c), 0 if b_nonneg else -np.inf) - b
b += db
lam -= db
if penalty == 0: # get (locally optimal) L0 solution
def c4smin(y, s, s_min):
ls = np.append(np.where(s > s_min)[0], T)
tmp = np.zeros_like(s)
l = ls[0] # first pool
tmp[:l] = max(
0,
np.exp(log(d) * np.arange(l)).dot(y[:l]) * (1 - d * d) /
(1 - d**(2 * l))) * np.exp(log(d) * np.arange(l))
for i, f in enumerate(ls[:-1]): # all other pools
l = ls[i + 1] - f
tmp[f] = (g11[:l].dot(y[f:f + l]) -
g11g12[l - 1] * tmp[f - 1]) / g11g11[l - 1]
tmp[f + 1:f + l] = g11[1:l] * tmp[f] + g12[1:l] * tmp[f - 1]
return tmp
spikesizes = np.sort(s[s > 1e-6])
i = len(spikesizes) // 2
l = 0
u = len(spikesizes) - 1
while u - l > 1:
s_min = spikesizes[i]
tmp = c4smin(y - b, s, s_min)
res = y - b - tmp
RSS = res.dot(res)
if RSS < thresh or i == 0:
l = i
i = (l + u) // 2
res0 = tmp
else:
u = i
i = (l + u) // 2
if i > 0:
c = res0
s = np.append([0, 0], c[2:] - g[0] * c[1:-1] - g[1] * c[:-2])
return c, s, b, g, lam