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Point Processes Point process: discrete set of points (events) on the real numbers (or some interval on the reals) Usually we are thinking of times of otherwise identical events. (but sometimes space or space-time)
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Lecture 2: Everything you need to know to know about point processes
Outline:• basic ideas• homogeneous (stationary) Poisson processes
• Poisson distribution• inter-event interval distribution• coefficient of variance (CV)• correlation function
• stationary renewal process• relation between IEI distribution and correlation function• Fano factor F• relation between F and CV
• nonstationary (inhomogeneous) Poisson process• time rescaling
Point Processes
Point process: discrete set of points (events) on the real numbers (or some interval on the reals)
€
−∞< t1 < t2L tN < ∞
Point Processes
Point process: discrete set of points (events) on the real numbers (or some interval on the reals)
Usually we are thinking of times of otherwise identical events.(but sometimes space or space-time)
€
−∞< t1 < t2L tN < ∞
Point Processes
Point process: discrete set of points (events) on the real numbers (or some interval on the reals)
Usually we are thinking of times of otherwise identical events.(but sometimes space or space-time)
Examples: radioactive decay, arrival times, earthquakes,neuronal spike trains, …
€
−∞< t1 < t2L tN < ∞
Point Processes
Point process: discrete set of points (events) on the real numbers (or some interval on the reals)
Usually we are thinking of times of otherwise identical events.(but sometimes space or space-time)
Examples: radioactive decay, arrival times, earthquakes,neuronal spike trains, …
Stochastic: characterized by the probability (density) of every set{t1, t2, … tN}
€
−∞< t1 < t2L tN < ∞
Neuronal spike trains
Action potential:
Neuronal spike trains
spike trains evoked by manypresentations of the same stimulus:
Action potential:
Neuronal spike trains
spike trains evoked by manypresentations of the same stimulus:
Action potential:
(apparently) stochastic
Homogeneous Poisson process
Homogeneous Poisson process
Homogeneous Poisson process: r = rate = prob of event per unit time,i.e., rΔt = prob of event in interval [t, t + Δt) (Δt 0)
Homogeneous Poisson process
Homogeneous Poisson process: r = rate = prob of event per unit time,i.e., rΔt = prob of event in interval [t, t + Δt) (Δt 0)
Survivor function: probability of no event in [0,t): S(t)
Homogeneous Poisson process
Homogeneous Poisson process: r = rate = prob of event per unit time,i.e., rΔt = prob of event in interval [t, t + Δt) (Δt 0)
Survivor function: probability of no event in [0,t): S(t)
€
dSdt
= −rS ⇒ S(t) = e−rt
Homogeneous Poisson process
Homogeneous Poisson process: r = rate = prob of event per unit time,i.e., rΔt = prob of event in interval [t, t + Δt) (Δt 0)
Survivor function: probability of no event in [0,t): S(t)
Probability /unit time of first event in [t, t + t)) :
€
dSdt
= −rS ⇒ S(t) = e−rt
Homogeneous Poisson process
Homogeneous Poisson process: r = rate = prob of event per unit time,i.e., rΔt = prob of event in interval [t, t + Δt) (Δt 0)
Survivor function: probability of no event in [0,t): S(t)
Probability /unit time of first event in [t, t + t)) :
€
dSdt
= −rS ⇒ S(t) = e−rt
€
P(t) = − dS(t)dt
= re−rt
Homogeneous Poisson process
Homogeneous Poisson process: r = rate = prob of event per unit time,i.e., rΔt = prob of event in interval [t, t + Δt) (Δt 0)
Survivor function: probability of no event in [0,t): S(t)
Probability /unit time of first event in [t, t + t)) :
(inter-event interval distribution)
€
dSdt
= −rS ⇒ S(t) = e−rt
€
P(t) = − dS(t)dt
= re−rt
Homogeneous Poisson process (2)
Probability of exactly 1 event in [0,T):
€
PT (1) = dt re−rt
0
T∫ ⋅e−r(T −t ) = rTe−rT
Homogeneous Poisson process (2)
Probability of exactly 1 event in [0,T):
€
PT (1) = dt re−rt
0
T∫ ⋅e−r(T −t ) = rTe−rT
Probability of exactly 2 events in [0,T):
€
PT (2) = dt2 dt10
t2∫ re−rt10
T∫ ⋅ re−r( t2 − t1 ) ⋅e−r(T − t2 ) = 12 (rT)2e−rT
Homogeneous Poisson process (2)
Probability of exactly 1 event in [0,T):
€
PT (1) = dt re−rt
0
T∫ ⋅e−r(T −t ) = rTe−rT
Probability of exactly 2 events in [0,T):
€
PT (2) = dt2 dt10
t2∫ re−rt10
T∫ ⋅ re−r( t2 − t1 ) ⋅e−r(T − t2 ) = 12 (rT)2e−rT
… Probability of exactly n events in [0,T):
€
PT (n) = 1n!
(rT)n e−rT
Homogeneous Poisson process (2)
Probability of exactly 1 event in [0,T):
€
PT (1) = dt re−rt
0
T∫ ⋅e−r(T −t ) = rTe−rT
Probability of exactly 2 events in [0,T):
€
PT (2) = dt2 dt10
t2∫ re−rt10
T∫ ⋅ re−r( t2 − t1 ) ⋅e−r(T − t2 ) = 12 (rT)2e−rT
… Probability of exactly n events in [0,T):
€
PT (n) = 1n!
(rT)n e−rT Poisson distribution
Poisson distribution
Probability of n events in interval of duration T:
Poisson distribution
Probability of n events in interval of duration T:
mean count: <n> = rT
Poisson distribution
Probability of n events in interval of duration T:
mean count: <n> = rT
variance: <(n -<n>)2> = rT, i.e. <n> ± <n>1/2
Poisson distribution
Probability of n events in interval of duration T:
mean count: <n> = rT
variance: <(n -<n>)2> = rT, i.e. <n> ± <n>1/2
large rT: Gaussian
Poisson distribution
Probability of n events in interval of duration T:
mean count: <n> = rT
variance: <(n -<n>)2> = rT, i.e. <n> ± <n>1/2
large rT: Gaussian
Characteristic functionPoisson distribution with mean a:
€
P(n) = e−a an
n!
Characteristic functionPoisson distribution with mean a:
Characteristic function
€
G(k) = e ikn = e−a e ikn
n∑ an
n!= e−a e ika( )
n
n!n∑
€
P(n) = e−a an
n!
Characteristic functionPoisson distribution with mean a:
Characteristic function
€
G(k) = e ikn = e−a e ikn
n∑ an
n!= e−a e ika( )
n
n!n∑
= e−a exp(e ika) = exp (e ik −1)a[ ]€
P(n) = e−a an
n!
Characteristic functionPoisson distribution with mean a:
Characteristic function
Cumulant generating function
€
G(k) = e ikn = e−a e ikn
n∑ an
n!= e−a e ika( )
n
n!n∑
= e−a exp(e ika) = exp (e ik −1)a[ ]€
P(n) = e−a an
n!
Characteristic functionPoisson distribution with mean a:
Characteristic function
Cumulant generating function
€
G(k) = e ikn = e−a e ikn
n∑ an
n!= e−a e ika( )
n
n!n∑
= e−a exp(e ika) = exp (e ik −1)a[ ]€
P(n) = e−a an
n!
€
logG(k) = e ik −1( )a = ika − 12 k 2a +L
Characteristic functionPoisson distribution with mean a:
Characteristic function
Cumulant generating function
€
G(k) = e ikn = e−a e ikn
n∑ an
n!= e−a e ika( )
n
n!n∑
= e−a exp(e ika) = exp (e ik −1)a[ ]€
P(n) = e−a an
n!
€
logG(k) = e ik −1( )a = ika − 12 k 2a +L
⇒ n = a, n − n( )2
= a, L
Characteristic functionPoisson distribution with mean a:
Characteristic function
Cumulant generating function
All cumulants = a
€
G(k) = e ikn = e−a e ikn
n∑ an
n!= e−a e ika( )
n
n!n∑
= e−a exp(e ika) = exp (e ik −1)a[ ]€
P(n) = e−a an
n!
€
logG(k) = e ik −1( )a = ika − 12 k 2a +L
⇒ n = a, n − n( )2
= a, L
Homogeneous Poisson process (3): inter-event interval distribution
rtrtP e)(Exponential distribution: (like radioactive Decay)
Homogeneous Poisson process (3): inter-event interval distribution
rtrtP e)(
€
t = 1r
Exponential distribution: (like radioactive Decay)
mean IEI:
Homogeneous Poisson process (3): inter-event interval distribution
rtrtP e)(
€
t = 1r
€
(t − t )2 = 1r2 = t 2
Exponential distribution: (like radioactive Decay)
mean IEI:
variance:
Homogeneous Poisson process (3): inter-event interval distribution
rtrtP e)(
€
t = 1r
€
(t − t )2 = 1r2 = t 2
€
CV = std devmean
=1
Exponential distribution: (like radioactive Decay)
mean IEI:
variance:
Coefficient of variation:
Homogeneous Poisson process (4): correlation function
)()( f
ftttS notation:
Homogeneous Poisson process (4): correlation function
)()( f
ftttS notation:
Homogeneous Poisson process (4): correlation function
)()( f
ftttS
€
S(t) = r
notation:
mean:
Homogeneous Poisson process (4): correlation function
)())()()(()( rrtSrtSC
)()( f
ftttS
€
S(t) = r
notation:
mean:
correlation function:
Proof:(1):
€
S(t)S( ′ t ) , t ≠ ′ t :
Proof:(1):
S(t) and S(t’) are independent, so
€
S(t)S( ′ t ) , t ≠ ′ t :
€
S(t)S( ′ t ) = S(t) S( ′ t ) = r2
Proof:
Use finite-width, finite-height delta functions:
€
ε (t) = 1ε
, −ε /2 < t < ε /2
(1):
S(t) and S(t’) are independent, so
(2): €
S(t)S( ′ t ) , t ≠ ′ t :
€
S(t)S( ′ t ) = S(t) S( ′ t ) = r2
€
t = ′ t :
Proof:
Use finite-width, finite-height delta functions:
€
ε (t) = 1ε
, −ε /2 < t < ε /2
(1):
S(t) and S(t’) are independent, so
(2): €
S(t)S( ′ t ) , t ≠ ′ t :
€
S(t)S( ′ t ) = S(t) S( ′ t ) = r2
€
t = ′ t :
Proof:
Use finite-width, finite-height delta functions:
€
ε (t) = 1ε
, −ε /2 < t < ε /2
(1):
S(t) and S(t’) are independent, so
(2): €
S(t)S( ′ t ) , t ≠ ′ t :
€
S(t)S( ′ t ) = S(t) S( ′ t ) = r2
€
t = ′ t :
€
S2(t) = rε ⋅ 1ε ⎛ ⎝ ⎜
⎞ ⎠ ⎟2
= rε
= rδε (0)
Proof:
Use finite-width, finite-height delta functions:
€
ε (t) = 1ε
, −ε /2 < t < ε /2
(1):
S(t) and S(t’) are independent, so
(2): €
S(t)S( ′ t ) , t ≠ ′ t :
€
S(t)S( ′ t ) = S(t) S( ′ t ) = r2
€
t = ′ t :
€
S2(t) = rε ⋅ 1ε ⎛ ⎝ ⎜
⎞ ⎠ ⎟2
= rε
= rδε (0)
so
€
S(t)S( ′ t ) = rδε (t − ′ t ) + r2 1−εδε (t − ′ t )( )
Proof:
Use finite-width, finite-height delta functions:
€
ε (t) = 1ε
, −ε /2 < t < ε /2
(1):
S(t) and S(t’) are independent, so
(2): €
S(t)S( ′ t ) , t ≠ ′ t :
€
S(t)S( ′ t ) = S(t) S( ′ t ) = r2
€
t = ′ t :
€
S2(t) = rε ⋅ 1ε ⎛ ⎝ ⎜
⎞ ⎠ ⎟2
= rε
= rδε (0)
so
€
S(t)S( ′ t ) = rδε (t − ′ t ) + r2 1−εδε (t − ′ t )( )
= r(1−εr)δε (t − ′ t ) + r2
Proof:
Use finite-width, finite-height delta functions:
€
ε (t) = 1ε
, −ε /2 < t < ε /2
(1):
S(t) and S(t’) are independent, so
(2): €
S(t)S( ′ t ) , t ≠ ′ t :
€
S(t)S( ′ t ) = S(t) S( ′ t ) = r2
€
t = ′ t :
€
S2(t) = rε ⋅ 1ε ⎛ ⎝ ⎜
⎞ ⎠ ⎟2
= rε
= rδε (0)
so
€
S(t)S( ′ t ) = rδε (t − ′ t ) + r2 1−εδε (t − ′ t )( )
= r(1−εr)δε (t − ′ t ) + r2
S(t) − S(t)( ) S( ′ t ) − S( ′ t )( ) = δS(t)δS( ′ t ) = r(1−εr)δε (t − ′ t )
Proof:
Use finite-width, finite-height delta functions:
€
ε (t) = 1ε
, −ε /2 < t < ε /2
(1):
S(t) and S(t’) are independent, so
(2): €
S(t)S( ′ t ) , t ≠ ′ t :
€
S(t)S( ′ t ) = S(t) S( ′ t ) = r2
€
t = ′ t :
€
S2(t) = rε ⋅ 1ε ⎛ ⎝ ⎜
⎞ ⎠ ⎟2
= rε
= rδε (0)
so
€
S(t)S( ′ t ) = rδε (t − ′ t ) + r2 1−εδε (t − ′ t )( )
= r(1−εr)δε (t − ′ t ) + r2
S(t) − S(t)( ) S( ′ t ) − S( ′ t )( ) = δS(t)δS( ′ t ) = r(1−εr)δε (t − ′ t ) ε →0 ⏐ → ⏐ ⏐ rδ(t − ′ t )
General stationary point process:
For any point process,
€
C(t) = rδ(t) + A(t)
General stationary point process:
For any point process,
with A(t) continuous,
€
C(t) = rδ(t) + A(t)
€
A(t) t →∞ ⏐ → ⏐ ⏐ 0
General stationary point process:
For any point process,
with A(t) continuous,
(because S(t) is composed of delta-functions) €
C(t) = rδ(t) + A(t)
€
A(t) t →∞ ⏐ → ⏐ ⏐ 0
Stationary renewal processDefined by ISI distribution P(t)
Stationary renewal processDefined by ISI distribution P(t)
Relation between P(t) and C(t):
Stationary renewal process
€
C+(t) ≡ 1r
(C(t) + r2)Θ(t)
Defined by ISI distribution P(t)
Relation between P(t) and C(t): define
Stationary renewal processDefined by ISI distribution P(t)
Relation between P(t) and C(t): define
Stationary renewal process
€
C+(t) ≡ 1r
(C(t) + r2)Θ(t)
€
C+(t) = P(t) + d ′ t P( ′ t )0
t
∫ P(t − ′ t ) +L
= P(t) + d ′ t P( ′ t )0
t
∫ C+(t − ′ t )
Defined by ISI distribution P(t)
Relation between P(t) and C(t): define
Stationary renewal process
€
C+(t) ≡ 1r
(C(t) + r2)Θ(t)
€
C+(t) = P(t) + d ′ t P( ′ t )0
t
∫ P(t − ′ t ) +L
= P(t) + d ′ t P( ′ t )0
t
∫ C+(t − ′ t )
€
C+(λ ) = P(λ ) + P(λ )C+(λ )
Defined by ISI distribution P(t)
Relation between P(t) and C(t): define
Laplace transform:
Stationary renewal process
€
C+(t) ≡ 1r
(C(t) + r2)Θ(t)
€
C+(t) = P(t) + d ′ t P( ′ t )0
t
∫ P(t − ′ t ) +L
= P(t) + d ′ t P( ′ t )0
t
∫ C+(t − ′ t )
€
C+(λ ) = P(λ ) + P(λ )C+(λ )
€
C+(λ ) = P(λ )1− P(λ )
Defined by ISI distribution P(t)
Relation between P(t) and C(t): define
Laplace transform:
Solve:
Fano factor
€
F =(n − n )2
nspike count variance / mean spike count
Fano factor
€
F =(n − n )2
nF =1
spike count variance / mean spike count
for stationary Poisson process
Fano factor
€
F =(n − n )2
nF =1
€
n = S(t) dt =0
T
∫ rT
δn2 = dt1 dt2 δS(t1)δS(t2)0
T
∫0
T
∫
spike count variance / mean spike count
for stationary Poisson process
Fano factor
€
F =(n − n )2
nF =1
€
n = S(t) dt =0
T
∫ rT
δn2 = dt1 dt2 δS(t1)δS(t2)0
T
∫0
T
∫
spike count variance / mean spike count
for stationary Poisson process
t2 t1
Fano factor
€
F =(n − n )2
nF =1
€
n = S(t) dt =0
T
∫ rT
δn2 = dt1 dt2 δS(t1)δS(t2)0
T
∫0
T
∫ = T C(τ−∞
∞
∫ )dτ
spike count variance / mean spike count
for stationary Poisson process
τ t
Fano factor
€
F =(n − n )2
nF =1
€
n = S(t) dt =0
T
∫ rT
δn2 = dt1 dt2 δS(t1)δS(t2)0
T
∫0
T
∫ = T C(τ−∞
∞
∫ )dτ
€
⇒ F =C(τ )dτ
−∞
∞
∫r
spike count variance / mean spike count
for stationary Poisson process
τ t
F=CV2 for a stationary renewal process
F=CV2 for a stationary renewal process
F depends on integral of C(t), CV depends on moments of P(t).
F=CV2 for a stationary renewal process
F depends on integral of C(t), CV depends on moments of P(t).Use relation between C(λ) and P(λ) to relate F and CV.
F=CV2 for a stationary renewal process
F depends on integral of C(t), CV depends on moments of P(t).Use relation between C(λ) and P(λ) to relate F and CV.
The algebra:
€
F = 1r
dt C(t) =−∞
∞∫ 1r
dt rδ(t) + A(t)[ ]−∞
∞∫
F=CV2 for a stationary renewal process
F depends on integral of C(t), CV depends on moments of P(t).Use relation between C(λ) and P(λ) to relate F and CV.
The algebra:
€
F = 1r
dt C(t) =−∞
∞∫ 1r
dt rδ(t) + A(t)[ ] =1+ 2r
rC+(t) − r2[ ]0
∞∫ dt−∞
∞∫
F=CV2 for a stationary renewal process
F depends on integral of C(t), CV depends on moments of P(t).Use relation between C(λ) and P(λ) to relate F and CV.
The algebra:
€
F = 1r
dt C(t) =−∞
∞∫ 1r
dt rδ(t) + A(t)[ ] =1+ 2r
rC+(t) − r2[ ]0
∞∫ dt−∞
∞∫
=1+ 2limλ →0
dt e−λ t C+(t) − r[ ] =10
∞∫ + 2limλ →0
C+(λ ) − rλ
⎡ ⎣ ⎢
⎤ ⎦ ⎥
F=CV2 for a stationary renewal process
F depends on integral of C(t), CV depends on moments of P(t).Use relation between C(λ) and P(λ) to relate F and CV.
The algebra:
€
F = 1r
dt C(t) =−∞
∞∫ 1r
dt rδ(t) + A(t)[ ] =1+ 2r
rC+(t) − r2[ ]0
∞∫ dt−∞
∞∫
=1+ 2limλ →0
dt e−λ t C+(t) − r[ ] =10
∞∫ + 2limλ →0
C+(λ ) − rλ
⎡ ⎣ ⎢
⎤ ⎦ ⎥
=1+ 2limλ →0
P(λ )1− P(λ )
− rλ
⎡ ⎣ ⎢
⎤ ⎦ ⎥
F=CV2 for a stationary renewal process
F depends on integral of C(t), CV depends on moments of P(t).Use relation between C(λ) and P(λ) to relate F and CV.
The algebra:
Now use
€
F = 1r
dt C(t) =−∞
∞∫ 1r
dt rδ(t) + A(t)[ ] =1+ 2r
rC+(t) − r2[ ]0
∞∫ dt−∞
∞∫
=1+ 2limλ →0
dt e−λ t C+(t) − r[ ] =10
∞∫ + 2limλ →0
C+(λ ) − rλ
⎡ ⎣ ⎢
⎤ ⎦ ⎥
=1+ 2limλ →0
P(λ )1− P(λ )
− rλ
⎡ ⎣ ⎢
⎤ ⎦ ⎥
€
P(λ ) =1− λ t + 12 λ2 t 2 +L
F=CV2 for a stationary renewal process
F depends on integral of C(t), CV depends on moments of P(t).Use relation between C(λ) and P(λ) to relate F and CV.
The algebra:
Now use and
€
F = 1r
dt C(t) =−∞
∞∫ 1r
dt rδ(t) + A(t)[ ] =1+ 2r
rC+(t) − r2[ ]0
∞∫ dt−∞
∞∫
=1+ 2limλ →0
dt e−λ t C+(t) − r[ ] =10
∞∫ + 2limλ →0
C+(λ ) − rλ
⎡ ⎣ ⎢
⎤ ⎦ ⎥
=1+ 2limλ →0
P(λ )1− P(λ )
− rλ
⎡ ⎣ ⎢
⎤ ⎦ ⎥
€
P(λ ) =1− λ t + 12 λ2 t 2 +L
€
r = 1t
F=CV2 (continued)
€
F =1+ 2limλ →0
1− λ t + 12 λ2 t 2
λ t − 12 λ2 t 2
− 1λ t
⎡
⎣ ⎢ ⎢
⎤
⎦ ⎥ ⎥
F=CV2 (continued)
€
F =1+ 2limλ →0
1− λ t + 12 λ2 t 2
λ t − 12 λ2 t 2 − 1
λ t
⎡
⎣ ⎢ ⎢
⎤
⎦ ⎥ ⎥
=1+ 2limλ →0
1λ t
1− λ t + 12 λ2 t 2
1− 12 λ t 2 / t
−1 ⎡
⎣ ⎢ ⎢
⎤
⎦ ⎥ ⎥
F=CV2 (continued)
€
F =1+ 2limλ →0
1− λ t + 12 λ2 t 2
λ t − 12 λ2 t 2
− 1λ t
⎡
⎣ ⎢ ⎢
⎤
⎦ ⎥ ⎥
=1+ 2limλ →0
1λ t
1− λ t + 12 λ2 t 2
1− 12 λ t 2 / t
−1 ⎡
⎣ ⎢ ⎢
⎤
⎦ ⎥ ⎥
=1+ 2limλ →0
1λ t
1− λ t + 12 λ2 t 2
1− 12 λ t 2 / t
−1− 1
2 λ t 2 / t
1− 12 λ t 2 / t
⎡
⎣ ⎢ ⎢
⎤
⎦ ⎥ ⎥
F=CV2 (continued)
€
F =1+ 2limλ →0
1− λ t + 12 λ2 t 2
λ t − 12 λ2 t 2
− 1λ t
⎡
⎣ ⎢ ⎢
⎤
⎦ ⎥ ⎥
=1+ 2limλ →0
1λ t
1− λ t + 12 λ2 t 2
1− 12 λ t 2 / t
−1 ⎡
⎣ ⎢ ⎢
⎤
⎦ ⎥ ⎥
=1+ 2limλ →0
1λ t
1− λ t + 12 λ2 t 2
1− 12 λ t 2 / t
−1− 1
2 λ t 2 / t
1− 12 λ t 2 / t
⎡
⎣ ⎢ ⎢
⎤
⎦ ⎥ ⎥
=1+ 2limλ →0
1λ t
−λ t + 12 λ t 2 / t + 1
2 λ2 t 2
1− 12 λ t 2 / t
⎡
⎣ ⎢ ⎢
⎤
⎦ ⎥ ⎥
F=CV2 (continued)
€
F =1+ 2limλ →0
1− λ t + 12 λ2 t 2
λ t − 12 λ2 t 2
− 1λ t
⎡
⎣ ⎢ ⎢
⎤
⎦ ⎥ ⎥
=1+ 2limλ →0
1λ t
1− λ t + 12 λ2 t 2
1− 12 λ t 2 / t
−1 ⎡
⎣ ⎢ ⎢
⎤
⎦ ⎥ ⎥
=1+ 2limλ →0
1λ t
1− λ t + 12 λ2 t 2
1− 12 λ t 2 / t
−1− 1
2 λ t 2 / t
1− 12 λ t 2 / t
⎡
⎣ ⎢ ⎢
⎤
⎦ ⎥ ⎥
=1+ 2limλ →0
1λ t
−λ t + 12 λ t 2 / t + 1
2 λ2 t 2
1− 12 λ t 2 / t
⎡
⎣ ⎢ ⎢
⎤
⎦ ⎥ ⎥
=1+ 2 −1+ 12 t 2 / t 2
[ ]
F=CV2 (continued)
€
F =1+ 2limλ →0
1− λ t + 12 λ2 t 2
λ t − 12 λ2 t 2
− 1λ t
⎡
⎣ ⎢ ⎢
⎤
⎦ ⎥ ⎥
=1+ 2limλ →0
1λ t
1− λ t + 12 λ2 t 2
1− 12 λ t 2 / t
−1 ⎡
⎣ ⎢ ⎢
⎤
⎦ ⎥ ⎥
=1+ 2limλ →0
1λ t
1− λ t + 12 λ2 t 2
1− 12 λ t 2 / t
−1− 1
2 λ t 2 / t
1− 12 λ t 2 / t
⎡
⎣ ⎢ ⎢
⎤
⎦ ⎥ ⎥
=1+ 2limλ →0
1λ t
−λ t + 12 λ t 2 / t + 1
2 λ2 t 2
1− 12 λ t 2 / t
⎡
⎣ ⎢ ⎢
⎤
⎦ ⎥ ⎥
=1+ 2 −1+ 12 t 2 / t 2
[ ]
= −1+t 2 + t − t( )
2
t 2
F=CV2 (continued)
€
F =1+ 2limλ →0
1− λ t + 12 λ2 t 2
λ t − 12 λ2 t 2
− 1λ t
⎡
⎣ ⎢ ⎢
⎤
⎦ ⎥ ⎥
=1+ 2limλ →0
1λ t
1− λ t + 12 λ2 t 2
1− 12 λ t 2 / t
−1 ⎡
⎣ ⎢ ⎢
⎤
⎦ ⎥ ⎥
=1+ 2limλ →0
1λ t
1− λ t + 12 λ2 t 2
1− 12 λ t 2 / t
−1− 1
2 λ t 2 / t
1− 12 λ t 2 / t
⎡
⎣ ⎢ ⎢
⎤
⎦ ⎥ ⎥
=1+ 2limλ →0
1λ t
−λ t + 12 λ t 2 / t + 1
2 λ2 t 2
1− 12 λ t 2 / t
⎡
⎣ ⎢ ⎢
⎤
⎦ ⎥ ⎥
=1+ 2 −1+ 12 t 2 / t 2
[ ]
= −1+t 2 + t − t( )
2
t 2 =t − t( )
2
t 2
F=CV2 (continued)
€
F =1+ 2limλ →0
1− λ t + 12 λ2 t 2
λ t − 12 λ2 t 2
− 1λ t
⎡
⎣ ⎢ ⎢
⎤
⎦ ⎥ ⎥
=1+ 2limλ →0
1λ t
1− λ t + 12 λ2 t 2
1− 12 λ t 2 / t
−1 ⎡
⎣ ⎢ ⎢
⎤
⎦ ⎥ ⎥
=1+ 2limλ →0
1λ t
1− λ t + 12 λ2 t 2
1− 12 λ t 2 / t
−1− 1
2 λ t 2 / t
1− 12 λ t 2 / t
⎡
⎣ ⎢ ⎢
⎤
⎦ ⎥ ⎥
=1+ 2limλ →0
1λ t
−λ t + 12 λ t 2 / t + 1
2 λ2 t 2
1− 12 λ t 2 / t
⎡
⎣ ⎢ ⎢
⎤
⎦ ⎥ ⎥
=1+ 2 −1+ 12 t 2 / t 2
[ ]
= −1+t 2 + t − t( )
2
t 2 =t − t( )
2
t 2 = CV 2
Nonstationary Poisson processes
Nonstationary Poisson process: time-dependent rate r(t)
Nonstationary Poisson processes
Nonstationary Poisson process: time-dependent rate r(t)
Time rescaling: instead of t, use
€
s = r( ′ t )d ′ t t∫
Nonstationary Poisson processes
Nonstationary Poisson process: time-dependent rate r(t)
Time rescaling: instead of t, use
Then the event rate per unit s is 1, i.e. the process is stationarywhen viewed as a function of s.
€
s = r( ′ t )d ′ t t∫
Nonstationary Poisson processes
Nonstationary Poisson process: time-dependent rate r(t)
Time rescaling: instead of t, use
Then the event rate per unit s is 1, i.e. the process is stationarywhen viewed as a function of s. In particular, still have• Poisson count distribution in any interval€
s = r( ′ t )d ′ t t∫
Nonstationary Poisson processes
Nonstationary Poisson process: time-dependent rate r(t)
Time rescaling: instead of t, use
Then the event rate per unit s is 1, i.e. the process is stationarywhen viewed as a function of s. In particular, still have• Poisson count distribution in any interval• F =1
€
s = r( ′ t )d ′ t t∫
Nonstationary Poisson processes
Nonstationary Poisson process: time-dependent rate r(t)
Time rescaling: instead of t, use
Then the event rate per unit s is 1, i.e. the process is stationarywhen viewed as a function of s. In particular, still have• Poisson count distribution in any interval• F =1
Nonstationary renewal process: time-dependent inter-event distribution
€
s = r( ′ t )d ′ t t∫
Nonstationary Poisson processes
Nonstationary Poisson process: time-dependent rate r(t)
Time rescaling: instead of t, use
Then the event rate per unit s is 1, i.e. the process is stationarywhen viewed as a function of s. In particular, still have• Poisson count distribution in any interval• F =1
Nonstationary renewal process: time-dependent inter-event distribution
)()(0
tPtP t = inter-event probability starting at t0
€
s = r( ′ t )d ′ t t∫
homework
• Prove that the ISI distribution is exponential for a stationary Poisson process.
• Prove that the CV is 1 for a stationary Poisson process.• Show that the Poisson distribution approaches a Gaussian one for
large mean spike count.• Prove that F = CV2 for a stationary renewal process.• Show why the spike count distribution for an inhomogeneous
Poisson process is the same as that for a homogeneous Poisson process with the same mean spike count.
