def shut_times_between_burst_pdf_components(mec):
"""
Calculate time constants and amplitudes for a PDF of gaps between bursts.
Parameters
----------
mec : dcpyps.Mechanism
The mechanism to be analysed.
Returns
-------
eigs : ndarray, shape(k, 1)
Time constants.
w : ndarray, shape(k, 1)
Component amplitudes.
"""
uA = np.ones((mec.kA, 1))
eigsB, AmatB = qml.eigs(-mec.QBB)
eigsF, AmatF = qml.eigs(-mec.QFF)
pA = qml.pinf(mec.Q)[:mec.kA]
end = np.dot(-mec.QAA, endBurst(mec))
start = pA / np.dot(pA, end)
rowB = np.dot(start, mec.QAB)
rowF = np.dot(start, mec.QAF)
colB = np.dot(mec.QBA, uA)
colF = np.dot(mec.QFA, uA)
wB = -np.dot(np.dot(rowB, AmatB), colB)
wF = np.dot(np.dot(rowF, AmatF), colF)
w = np.append(wB, wF)
eigs = np.append(eigsB, eigsF)
return eigs, w
def length_no_single_openings_pdf_components(mec):
"""
Calculate time constants and amplitudes for an ideal (no missed events)
exponential burst length probability density function for bursts with
two or more openings.
Parameters
----------
mec : dcpyps.Mechanism
The mechanism to be analysed.
Returns
-------
eigs : ndarray, shape(k, 1)
Time constants.
w : ndarray, shape(k, 1)
Component amplitudes.
"""
eigsA, AmatA = qml.eigs(-mec.QAA)
eigsE, AmatE = qml.eigs(-mec.QEE)
eigs = np.append(eigsE, eigsA)
A = np.append(AmatE[:,:mec.kA, :mec.kA], -AmatA, axis=0)
w = np.zeros(mec.kA + mec.kE)
endB = endBurst(mec)
start = phiBurst(mec)
norm = 1 - np.dot(start, endB)[0]
for i in range(mec.kA + mec.kE):
w[i] = np.dot(np.dot(np.dot(start,
A[i]), (-mec.QAA)), endB) / norm
return eigs, w
def corr_decay_amplitude_A(phiA, QAA, XAA, kA):
"""
Calculate scalar coefficients for correlation coefficien decay (Eq. 2.11,
CH83).
Parameters
----------
Returns
-------
w : ndarray, shape (1, k)
eigs : ndarray, shape (1, k)
"""
varA = corr_variance_A(phiA, QAA, kA)
eigs, A = qml.eigs(XAA)
uA = np.ones((kA))[:,np.newaxis]
invQAA = -nplin.inv(QAA)
row = np.dot(phiA, invQAA)
col = np.dot(invQAA, uA)
ncA = np.rank(XAA) - 1
w = np.zeros((ncA))
n = 0
for i in range(kA):
if fabs(eigs[i]) > 1e-12 and fabs(eigs[i] - 1) > 1e-12:
w[n] = np.dot(np.dot(row, A[i, :, :]), col)[0,0] / varA
n += 1
return w, eigs
def first_opening_length_pdf_components(mec):
"""
Calculate time constants and amplitudes for an ideal (no missed events)
pdf of first opening in a burst with 2 or more openings.
Parameters
----------
mec : dcpyps.Mechanism
The mechanism to be analysed.
Returns
-------
eigs : ndarray, shape(k, 1)
Time constants.
w : ndarray, shape(k, 1)
Component amplitudes.
"""
uA = np.ones((mec.kA, 1))
eigs, A = qml.eigs(-mec.QAA)
GAB, GBA = qml.iGs(mec.Q, mec.kA, mec.kB)
GG = np.dot(GAB, GBA)
norm = np.dot(np.dot(phiBurst(mec), GG), uA)[0]
w = np.zeros(mec.kA)
for i in range(mec.kA):
w[i] = np.dot(np.dot(np.dot(np.dot(phiBurst(mec),
A[i]), (-mec.QAA)), GG), uA) / norm
return eigs, w
def shut_time_total_pdf_components_2more_openings(mec):
"""
Calculate time constants and amplitudes for a PDF of total shut time
per burst (Eq. 3.40, CH82) for bursts with at least 2 openings.
Parameters
----------
mec : dcpyps.Mechanism
The mechanism to be analysed.
Returns
-------
eigs : ndarray, shape(k, 1)
Time constants.
w : ndarray, shape(k, 1)
Component amplitudes.
"""
GAB, GBA = qml.iGs(mec.Q, mec.kA, mec.kB)
WBB = mec.QBB + np.dot(mec.QBA, GAB)
eigs, A = qml.eigs(-WBB)
norm = 1 - np.dot(phiBurst(mec), endBurst(mec))[0]
w = np.zeros(mec.kB)
for i in range(mec.kB):
w[i] = np.dot(np.dot(np.dot(np.dot(phiBurst(mec), GAB),
A[i]), (mec.QBA)), endBurst(mec)) / norm
return eigs, w
def shut_times_inside_burst_pdf_components(mec):
"""
Calculate time constants and amplitudes for a PDF of all gaps within
bursts (Eq. 3.75, CH82).
Parameters
----------
mec : dcpyps.Mechanism
The mechanism to be analysed.
Returns
-------
eigs : ndarray, shape(k, 1)
Time constants.
w : ndarray, shape(k, 1)
Component amplitudes.
"""
uA = np.ones((mec.kA, 1))
eigs, A = qml.eigs(-mec.QBB)
GAB, GBA = qml.iGs(mec.Q, mec.kA, mec.kB)
interm = nplin.inv(np.eye(mec.kA) - np.dot(GAB, GBA))
norm = openings_mean(mec) - 1
w = np.zeros(mec.kB)
for i in range(mec.kB):
w[i] = np.dot(np.dot(np.dot(np.dot(np.dot(np.dot(phiBurst(mec), interm),
GAB), A[i]), (-mec.QBB)), GBA), uA) / norm
return eigs, w
def open_time_total_pdf_components(mec):
"""
Eq. 3.23, CH82
Parameters
----------
mec : dcpyps.Mechanism
The mechanism to be analysed.
Returns
-------
eigs : ndarray, shape(k, 1)
Time constants.
w : ndarray, shape(k, 1)
Component amplitudes.
"""
GAB, GBA = qml.iGs(mec.Q, mec.kA, mec.kB)
VAA = mec.QAA + np.dot(mec.QAB, GBA)
eigs, A = qml.eigs(-VAA)
uA = np.ones((mec.kA, 1))
w = np.zeros(mec.kA)
for i in range(mec.kA):
w[i] = np.dot(np.dot(np.dot(phiBurst(mec), A[i]), (-VAA)), uA)
return eigs, w
def openings_distr_components(mec):
"""
Calculate coeficients for geometric ditribution P(r)- probability of
seeing r openings (Eq. 3.9 CH82):
P(r) = sum(W * rho^(r-1))
where w
wm = phiB * Am * endB (Eq. 3.10 CH82)
and rho- eigenvalues of GAB * GBA.
Parameters
----------
mec : dcpyps.Mechanism
The mechanism to be analysed.
r : int
Number of openings per burst.
Returns
-------
rho : ndarray, shape (kA,)
w : ndarray, shape (kA,)
"""
GAB, GBA = qml.iGs(mec.Q, mec.kA, mec.kB)
GG = np.dot(GAB, GBA)
rho, A = qml.eigs(GG)
w = np.dot(np.dot(phiBurst(mec), A), endBurst(mec)).transpose()[0]
return rho, w
def HJC_adjacent_mean_open_to_shut_time_pdf(sht, tres, Q, QAA, QAF, QFF, QFA):
"""
Calculate theoretical HJC (with missed events correction) mean open time
given previous/next gap length (continuous function; CHS96 Eq.3.5).
Parameters
----------
sht : array of floats
Shut time interval.
tres : float
Time resolution.
Q : array, shape (k,k)
Q matrix.
QAA, QAF, QFF, QFA : array_like
Submatrices of Q.
Returns
-------
mp : ndarray of floats
Mean open time given previous gap length.
mn : ndarray of floats
Mean open time given next gap length.
"""
kA, kF = QAA.shape[0], QFF.shape[0]
uA = np.ones((kA))[:,np.newaxis]
uF = np.ones((kF))[:,np.newaxis]
expQFF = qml.expQt(QFF, tres)
expQAA = qml.expQt(QAA, tres)
GAF, GFA = qml.iGs(Q, kA, kF)
eGAF = qml.eGs(GAF, GFA, kA, kF, expQFF)
eGFA = qml.eGs(GFA, GAF, kF, kA, expQAA)
phiA = qml.phiHJC(eGAF, eGFA, kA)
phiF = qml.phiHJC(eGFA, eGAF, kF)
DARS = qml.dARSdS(tres, QAA, QFF, GAF, GFA, expQFF, kA, kF)
eigs, A = qml.eigs(-Q)
Feigvals, FZ00, FZ10, FZ11 = qml.Zxx(Q, eigs, A, kA, QAA, QFA, QAF, expQAA, False)
Froots = asymptotic_roots(tres, QFF, QAA, QFA, QAF, kF, kA)
FR = qml.AR(Froots, tres, QFF, QAA, QFA, QAF, kF, kA)
Q1 = np.dot(np.dot(DARS, QAF), expQFF)
col1 = np.dot(Q1, uF)
row1 = np.dot(phiA, Q1)
mp = []
mn = []
for t in sht:
eGFAt = qml.eGAF(t, tres, Feigvals, FZ00, FZ10, FZ11, Froots,
FR, QFA, expQAA)
denom = np.dot(np.dot(phiF, eGFAt), uA)[0]
nom1 = np.dot(np.dot(phiF, eGFAt), col1)[0]
nom2 = np.dot(np.dot(row1, eGFAt), uA)[0]
mp.append(nom1 / denom)
mn.append(nom2 / denom)
return np.array(mp), np.array(mn)
def HJC_dependency(top, tsh, tres, Q, QAA, QAF, QFF, QFA):
"""
Calculate normalised joint distribution (CHS96, Eq. 3.22) of an open time
and the following shut time as proposed by Magleby & Song 1992.
Parameters
----------
top, tsh : array_like of floats
Open and shut tims.
tres : float
Time resolution.
Q : array, shape (k,k)
Q matrix.
QAA, QAF, QFF, QFA : array_like
Submatrices of Q.
Returns
-------
dependency : ndarray
"""
kA, kF = QAA.shape[0], QFF.shape[0]
uA = np.ones((kA))[:,np.newaxis]
uF = np.ones((kF))[:,np.newaxis]
expQFF = qml.expQt(QFF, tres)
expQAA = qml.expQt(QAA, tres)
GAF, GFA = qml.iGs(Q, kA, kF)
eGAF = qml.eGs(GAF, GFA, kA, kF, expQFF)
eGFA = qml.eGs(GFA, GAF, kF, kA, expQAA)
phiA = qml.phiHJC(eGAF, eGFA, kA)
phiF = qml.phiHJC(eGFA, eGAF, kF)
eigs, A = qml.eigs(-Q)
Feigvals, FZ00, FZ10, FZ11 = qml.Zxx(Q, eigs, A, kA, QAA, QFA, QAF, expQAA, False)
Froots = asymptotic_roots(tres, QFF, QAA, QFA, QAF, kF, kA)
FR = qml.AR(Froots, tres, QFF, QAA, QFA, QAF, kF, kA)
Aeigvals, AZ00, AZ10, AZ11 = qml.Zxx(Q, eigs, A, kA, QFF, QAF, QFA, expQFF, True)
Aroots = asymptotic_roots(tres, QAA, QFF, QAF, QFA, kA, kF)
AR = qml.AR(Aroots, tres, QAA, QFF, QAF, QFA, kA, kF)
dependency = np.zeros((top.shape[0], tsh.shape[0]))
for i in range(top.shape[0]):
eGAFt = qml.eGAF(top[i], tres, Aeigvals, AZ00, AZ10, AZ11, Aroots,
AR, QAF, expQFF)
fo = np.dot(np.dot(phiA, eGAFt), uF)[0]
for j in range(tsh.shape[0]):
eGFAt = qml.eGAF(tsh[j], tres, Feigvals, FZ00, FZ10, FZ11, Froots,
FR, QFA, expQAA)
fs = np.dot(np.dot(phiF, eGFAt), uA)[0]
fos = np.dot(np.dot(np.dot(phiA, eGAFt), eGFAt), uA)[0]
dependency[i, j] = (fos - (fo * fs)) / (fo * fs)
return dependency
def adjacent_open_to_shut_range_pdf_components(u1, u2, QAA, QAF, QFF, QFA, phiA):
"""
Calculate time constants and areas for an ideal (no missed events)
exponential probability density function of open times adjacent to a
specified shut time range.
Parameters
----------
t : float
Time (sec).
QAA : array_like, shape (kA, kA)
Submatrix of Q.
phiA : array_like, shape (1, kA)
Initial vector for openings
Returns
-------
taus : ndarray, shape(k, 1)
Time constants.
areas : ndarray, shape(k, 1)
Component relative areas.
"""
kA = QAA.shape[0]
uA = np.ones((kA))[:,np.newaxis]
invQAA, invQFF = -nplin.inv(QAA), nplin.inv(QFF)
expQFFr = qml.expQt(QFF, u2) - qml.expQt(QFF, u1)
col = np.dot(np.dot(np.dot(np.dot(QAF, invQFF), expQFFr), QFA), uA)
w = np.zeros(kA)
eigs, A = qml.eigs(-QAA)
row = np.dot(phiA, invQAA)
den = np.dot(row, col)[0, 0]
#TODO: remove 'for'
for i in range(kA):
w[i] = np.dot(np.dot(phiA, A[i]), col) / den
return eigs, w
def length_pdf_components(mec):
"""
Calculate time constants and amplitudes for an ideal (no missed events)
exponential burst length probability density function.
Parameters
----------
mec : dcpyps.Mechanism
The mechanism to be analysed.
Returns
-------
eigs : ndarray, shape(k, 1)
Time constants.
w : ndarray, shape(k, 1)
Component amplitudes.
"""
w = np.zeros(mec.kE)
eigs, A = qml.eigs(-mec.QEE)
for i in range(mec.kE):
w[i] = np.dot(np.dot(np.dot(phiBurst(mec),
A[i][:mec.kA, :mec.kA]), (-mec.QAA)), endBurst(mec))
return eigs, w
def HJClik(theta, opts):
"""
Calculate likelihood for a series of open and shut times using HJC missed
events probability density functions (first two dead time intervals- exact
solution, then- asymptotic).
Lik = phi * eGAF(t1) * eGFA(t2) * eGAF(t3) * ... * eGAF(tn) * uF
where t1, t3,..., tn are open times; t2, t4,..., t(n-1) are shut times.
Gaps > tcrit are treated as unusable (e.g. contain double or bad bit of
record, or desens gaps that are not in the model, or gaps so long that
next opening may not be from the same channel). However this calculation
DOES assume that all the shut times predicted by the model are present
within each group. The series of multiplied likelihoods is terminated at
the end of the opening before an unusable gap. A new series is then
started, using appropriate initial vector to give Lik(2), ... At end
these are multiplied to give final likelihood.
Parameters
----------
theta : array_like
Guesses.
bursts : dictionary
A dictionary containing lists of open and shut intervals.
opts : dictionary
opts['mec'] : instance of type Mechanism
opts['tres'] : float
Time resolution (dead time).
opts['tcrit'] : float
Ctritical time interval.
opts['isCHS'] : bool
True if CHS vectors should be used (Eq. 5.7, CHS96).
Returns
-------
loglik : float
Log-likelihood.
newrates : array_like
Updated rates/guesses.
"""
# TODO: Errors.
mec = opts['mec']
conc = opts['conc']
tres = opts['tres']
tcrit = opts['tcrit']
is_chsvec = opts['isCHS']
bursts = opts['data']
mec.theta_unsqueeze(np.exp(theta))
mec.set_eff('c', conc)
GAF, GFA = qml.iGs(mec.Q, mec.kA, mec.kF)
expQFF = qml.expQt(mec.QFF, tres)
expQAA = qml.expQt(mec.QAA, tres)
eGAF = qml.eGs(GAF, GFA, mec.kA, mec.kF, expQFF)
eGFA = qml.eGs(GFA, GAF, mec.kF, mec.kA, expQAA)
phiF = qml.phiHJC(eGFA, eGAF, mec.kF)
startB = qml.phiHJC(eGAF, eGFA, mec.kA)
endB = np.ones((mec.kF, 1))
eigen, A = qml.eigs(-mec.Q)
Aeigvals, AZ00, AZ10, AZ11 = qml.Zxx(mec.Q, eigen, A, mec.kA, mec.QFF,
mec.QAF, mec.QFA, expQFF, True)
Aroots = asymptotic_roots(tres,
mec.QAA, mec.QFF, mec.QAF, mec.QFA, mec.kA, mec.kF)
AR = qml.AR(Aroots, tres, mec.QAA, mec.QFF, mec.QAF, mec.QFA, mec.kA, mec.kF)
Feigvals, FZ00, FZ10, FZ11 = qml.Zxx(mec.Q, eigen, A, mec.kA, mec.QAA,
mec.QFA, mec.QAF, expQAA, False)
Froots = asymptotic_roots(tres,
mec.QFF, mec.QAA, mec.QFA, mec.QAF, mec.kF, mec.kA)
FR = qml.AR(Froots, tres, mec.QFF, mec.QAA, mec.QFA, mec.QAF, mec.kF, mec.kA)
if is_chsvec:
startB, endB = qml.CHSvec(Froots, tres, tcrit,
mec.QFA, mec.kA, expQAA, phiF, FR)
loglik = 0
for ind in range(len(bursts)):
burst = bursts[ind]
grouplik = startB
for i in range(len(burst)):
t = burst[i]
if i % 2 == 0: # open time
eGAFt = qml.eGAF(t, tres, Aeigvals, AZ00, AZ10, AZ11, Aroots,
AR, mec.QAF, expQFF)
else: # shut
eGAFt = qml.eGAF(t, tres, Feigvals, FZ00, FZ10, FZ11, Froots,
FR, mec.QFA, expQAA)
grouplik = np.dot(grouplik, eGAFt)
if grouplik.max() > 1e50:
grouplik = grouplik * 1e-100
#print 'grouplik was scaled down'
grouplik = np.dot(grouplik, endB)
try:
loglik += log(grouplik[0])
except:
print ('HJClik: Warning: likelihood has been set to 0')
print ('likelihood=', grouplik[0])
print ('rates=', mec.unit_rates())
loglik = 0
break
newrates = np.log(mec.theta())
return -loglik, newrates
def printout_correlations(mec, output=sys.stdout, eff='c'):
"""
"""
str = ('\n\n*************************************\n' +
'CORRELATIONS\n')
kA, kI = mec.kA, mec.kI
str += ('kA, kF = {0:d}, {1:d}\n'.format(kA, kI))
GAF, GFA = qml.iGs(mec.Q, kA, kI)
rGAF, rGFA = np.rank(GAF), np.rank(GFA)
str += ('Ranks of GAF, GFA = {0:d}, {1:d}\n'.format(rGAF, rGFA))
XFF = np.dot(GFA, GAF)
rXFF = np.rank(XFF)
str += ('Rank of GFA * GAF = {0:d}\n'.format(rXFF))
ncF = rXFF - 1
eigXFF, AXFF = qml.eigs(XFF)
str += ('Eigenvalues of GFA * GAF:\n')
str1 = ''
for i in range(kI):
str1 += '\t{0:.5g}'.format(eigXFF[i])
str += str1 + '\n'
XAA = np.dot(GAF, GFA)
rXAA = np.rank(XAA)
str += ('Rank of GAF * GFA = {0:d}\n'.format(rXAA))
ncA = rXAA - 1
eigXAA, AXAA = qml.eigs(XAA)
str += ('Eigenvalues of GAF * GFA:\n')
str1 = ''
for i in range(kA):
str1 += '\t{0:.5g}'.format(eigXAA[i])
str += str1 + '\n'
phiA, phiF = qml.phiA(mec).reshape((1,kA)), qml.phiF(mec).reshape((1,kI))
varA = corr_variance_A(phiA, mec.QAA, kA)
varF = corr_variance_A(phiF, mec.QII, kI)
# open - open time correlations
str += ('\n OPEN - OPEN TIME CORRELATIONS')
str += ('Variance of open time = {0:.5g}\n'.format(varA))
SDA = sqrt(varA)
str += ('SD of all open times = {0:.5g} ms\n'.format(SDA * 1000))
n = 50
SDA_mean_n = SDA / sqrt(float(n))
str += ('SD of means of {0:d} open times if'.format(n) +
'uncorrelated = {0:.5g} ms\n'.format(SDA_mean_n * 1000))
covAtot = 0
for i in range(1, n):
covA = corr_covariance_A(i+1, phiA, mec.QAA, XAA, kA)
ro = correlation_coefficient(covA, varA, varA)
covAtot += (n - i) * ro * varA
vtot = n * varA + 2. * covAtot
actSDA = sqrt(vtot / (n * n))
str += ('Actual SD of mean = {0:.5g} ms\n'.format(actSDA * 1000))
pA = 100 * (actSDA - SDA_mean_n) / SDA_mean_n
str += ('Percent difference as result of correlation = {0:.5g}\n'.
format(pA))
v2A = corr_limit_A(phiA, mec.QAA, AXAA, eigXAA, kA)
pmaxA = 100 * (sqrt(1 + 2 * v2A / varA) - 1)
str += ('Limiting value of percent difference for large n = {0:.5g}\n'.
format(pmaxA))
str += ('Correlation coefficients, r(k), for up to lag k = 5:\n')
for i in range(5):
covA = corr_covariance_A(i+1, phiA, mec.QAA, XAA, kA)
ro = correlation_coefficient(covA, varA, varA)
str += ('r({0:d}) = {1:.5g}\n'.format(i+1, ro))
# shut - shut time correlations
str += ('\n SHUT - SHUT TIME CORRELATIONS\n')
str += ('Variance of shut time = {0:.5g}\n'.format(varF))
SDF = sqrt(varF)
str += ('SD of all shut times = {0:.5g} ms\n'.format(SDF * 1000))
n = 50
SDF_mean_n = SDF / sqrt(float(n))
str += ('SD of means of {0:d} shut times if'.format(n) +
'uncorrelated = {0:.5g} ms\n'.format(SDF_mean_n * 1000))
covFtot = 0
for i in range(1, n):
covF = corr_covariance_A(i+1, phiF, mec.QII, XFF, kI)
ro = correlation_coefficient(covF, varF, varF)
covFtot += (n - i) * ro * varF
vtotF = 50 * varF + 2. * covFtot
actSDF = sqrt(vtotF / (50. * 50.))
str += ('Actual SD of mean = {0:.5g} ms\n'.format(actSDF * 1000))
pF = 100 * (actSDF - SDF_mean_n) / SDF_mean_n
str += ('Percent difference as result of correlation = {0:.5g}\n'.
format(pF))
v2F = corr_limit_A(phiF, mec.QII, AXFF, eigXFF, kI)
pmaxF = 100 * (sqrt(1 + 2 * v2F / varF) - 1)
str += ('Limiting value of percent difference for large n = {0:.5g}\n'.
format(pmaxF))
str += ('Correlation coefficients, r(k), for up to k = 5 lags:\n')
for i in range(5):
covF = corr_covariance_A(i+1, phiF, mec.QII, XFF, kI)
ro = correlation_coefficient(covF, varF, varF)
str += ('r({0:d}) = {1:.5g}\n'.format(i+1, ro))
# open - shut time correlations
str += ('\n OPEN - SHUT TIME CORRELATIONS\n')
str += ('Correlation coefficients, r(k), for up to k= 5 lags:\n')
for i in range(5):
covAF = corr_covariance_AF(i+1, phiA, mec.QAA, mec.QII, XAA, GAF, kA, kI)
ro = correlation_coefficient(covAF, varA, varF)
str += ('r({0:d}) = {1:.5g}\n'.format(i+1, ro))
return str
