8 Random Processes: Continuous-time Markov Chains Flashcards
continuous-time Markov chains
In many applications time flows continuously and events (births, deaths, …) can happen at continuous random time t∈ [0, ∞)
(time increments are random, have a distribution)
consider finite/countable state space
X(t)∈ S = {0, 1, 2, . . . }
CTMC
CTMC memoryless
satisfy Markov’s property
Prob{X(tₙ₊₁) = iₙ₊₁ | X(t₀)= i₀,…,X(tₙ)=iₙ}
= Prob{X(tₙ₊₁)=iₙ₊₁|X(tₙ)=iₙ}
the probability of being in state iₙ₊₁ at tₙ₊₁ depends ONLY on being in state iₙ at tₙ
not on entire history
CTMC probability distribution
A CTMC
X(t) ∈ S = {0, 1, 2, . . . }
s characterized by its probability distribution
{p₀(t), p₁(t), ₂(t). . .}
where
pᵢ(t) = Prob{X(t) = i ∈ S}
(process at time t has value i)
and is the vector of probability
p(t) = (p₀(t), p₁(t), ₂(t). . .)^T
pᵢⱼ
Transition probability
pⱼᵢ(t,s)
gives the probability to move from state i at time s at state j at time t>s
pⱼᵢ(t,s)= pⱼᵢ(t-s) =Prob{X(t)=j|X(s)=i}
=Prob{X(t-s)=j|X(0)=i} with t>s
writing
pⱼᵢ(t,s)=pⱼᵢ(t-s)=p(∆t)
with ∆t=t-s >0
we assume time is homogenous
assume time is homogeneous meaning only time difference t-s>0 matters
the probability only dep on the time difference
stochastic transition matrix
As for DTMCs, the transition probability between state i at time s to state j at time t>s is specified by
the element pⱼᵢ(t,s)=pⱼᵢ(t-s)=pⱼᵢ(∆t) of the stochastic transition matrix
P(∆t)=(pⱼᵢ(∆t))
probability conservation
Σ_{k∈S} pₖⱼ(∆t)=1
each column of P(∆t) adds up to 1
Relation between transition probability and probability distribution is
product:
p(∆t)
= P(∆t) x p(0)
prob. vector after ∆t=
transition matrix prob. x vector at t = 0
if initial state X(0)= i ∈ S
then pₖ(0) =δₖ,ᵢ
thus p(0)=
(0)
….
(1)
(0) element i
…
(0)
p_{j∈S}(∆t) = Σ_{k∈S} pⱼₖ(∆t)δₖ,ᵢ
= pⱼᵢ(∆t)
δₖ,ᵢ =
δₖ,ᵢ =
{1 if i = k
{0 if i ̸= k
The Chapman-Kolmogorov equation for CTMCs in matrix form
P(s)P(t) = P(t + s)
All this is similar to what we have seen for discrete-time Markov chains
However, in CTMCs:
- Events can occur at random time that flows continuously (there is no fixed, discrete time increment)
=>Time between two events is randomly distributed
- CTMCs are now analyzed in terms of transition rates
8.2
transition rates
qⱼᵢ
are defined in terms of transition prob. as
qⱼᵢ= lim_{∆t→0} of
[pⱼᵢ(∆t)-pⱼᵢ(0)]/ ∆t
=
{lim_{∆t→0}pⱼᵢ(∆t)/∆t if i ̸= j
{lim_{∆t→0} (pⱼⱼ (∆t)−1)/∆t if i = j,
Since pⱼᵢ(∆t) ≥ 0 and ∆t > 0
⇒ qⱼᵢ ≥ 0 if i ̸= j
where pⱼᵢ(0)= δᵢ,ⱼ
P(0)= I identity matrix
means
pⱼᵢ(∆t) = δⱼ,ᵢ+ qⱼᵢ∆t + o(∆t)
h.o.t negligible/ inifinitesimally small, such that lim_{∆t→0} o(∆t)/∆t = 0
When ∆t → 0 (with o(∆t) negligible):
When ∆t → 0 (with o(∆t) negligible):
Since
(kronecker delta function used, taking sum over j)
Σ_{j∈S} pⱼᵢ
= 1
= 1 + ∆tΣ_{j∈S}qⱼᵢ
= 1 + ∆t[ Σ_{j∈S/i}qⱼᵢ + qⱼⱼ]
(here we demand [] sums to 0 for j not equal to i)
giving
qⱼᵢ = − Σ_{j∈S/i} qⱼᵢ
GENERATOR MATRIX
Q=
(qⱼᵢ)
whose columns add up to zero and
whose off-diagonal element is non-negative and is the rate of the transition
i to j
(qⱼᵢ)
Since P(0) = I, for a CTMC of range S={0,1,2…} , the generator matrix explicitly reads:
(qⱼᵢ)=Q=
lim_{∆t→0} [P(∆t) − I]/((∆t)
=
[ () q₀₁ q₀₂ ….]
[ q₁₀ ( .) q₁₂…]
[ q₂₀ q₂₁ (*..) …]
[… ]
s.t all ≥ 0
from state i=0 from state i=1…
to state
j=0
j=1
…
- = -Σ_{k∈S/0} qₖ₀
*.= -Σ_{k∈S/1} qₖ₁
*..= -Σ_{k∈S/1} qₖ₂
note: each element columns add to 0
except for diagonal all elements correspond to transition rates
gen matrix doesnt dep on delta t
The Chapman-Kolmogorov equation says
P(t + ∆t) = P(∆t)P(t)
pⱼᵢ(t + ∆t) = Σ_{k∈S} pⱼₖ(∆t)pₖᵢ(t)
= Σ_{k∈S} pₖᵢ(t)[δⱼₖ + qⱼₖ∆t + o(∆t)]
now [δⱼₖ ] is 0 or 1 dep on if j=k
REARRANGING
in the limit ∆t → 0
we obtain the master equation
lim_{∆t→0} of
pⱼᵢ(t + ∆t) - pⱼᵢ(t)/∆t
=
dpⱼᵢ(t)/dt
=
Σ_{k∈S} qⱼₖpₖᵢ(t)
matrix form, the master equation
formal sol
d/dt P(t) = Q P(t)
giving
P(t)=
exp(Qt)P(0)
= exp(Qt)
where P(0) =I
and we clearly have
P(t₁ + t₂) = P(t₁)P(t₂)
(exponential of a matrix (generally
is nonsymmetric and large)
which come from (diagonal?) elements
using
p(t + ∆t) = P(∆t)p(t)
as
[p(t + ∆t) − p(t)]/∆t =
[P(∆t) − I]p(t)/∆t
in the limit∆t → 0
obtain
MASTER EQUATION FOR THE PROBABILITY VECTOR
(d/dt) p(t) = Q p(t)
which gives the equation of motion of the probability vector
p(t) = exp(Qt)p(0)
solution
relationship between probability vector and transition matrix
when given IC….
often the initital distribution p(0) is known and the relationship between transition probabilities and probability distribution is simple.
For instance, if initially,
X(0) = k ∈ S
gives
pᵢ(0) = δᵢ,ₖ
and
pᵢₖ(t) = Prob {X(t) =i | X(0)=k}
=
Prob {X(t)=i)= pᵢ(t)
so given IC
we work out Q
depeneds on the model of the process
then use sol
continuous-time Markov chains with discrete state space
S={0,1,2,…}
states
continuous time t ∈ [0, ∞)
with randomly distributed time between events
when time is continuous the delta t is infinitely small and we have randomly distributed,
probability of each states given by master equation, giving the distribution of the probabilities
master equation
continuous-time MC
d/dt p(t) =Qp(t)
Q= (qⱼᵢ)
where qⱼᵢ≥0
is the rate of i −qⱼᵢ→ j for i ̸= j
each state of system
Q generator matrix, tranisiton rates, each col sums to 0
A Poisson process is
discrete state
continuous time
e.g. number of goals in a football game
a CTMC with and de {X(t) ∈ S : t ∈ [0, ∞)} S = {0, 1, 2, . . .}, λ > 0, ∆t → 0 defined by:
*At t = 0: X(0) = 0
* pᵢ₊₁(∆t) = Prob {X(t + ∆t) = i + 1|X(t) = i} = λ∆t + o(∆t)
(∆t so small we can only move +1,0 in ∆t)
- pᵢᵢ(∆t) = Prob {X(t + ∆t) = i|X(t) = i} = 1 − λ∆t + o(∆t)
*pⱼᵢ(∆t) = Prob {X(t + ∆t) = j ≥ i + 2|X(t) = i} = o(∆t) if j ≥ i + 2
* pⱼᵢ(∆t) = 0 if j < i
where o(∆t) is infinitesimally small s.t lim_{∆t→0} o(∆t)/∆t = 0
, i.e. that decrease to zero faster
than ∆t
In the Poisson process, there are only transitions into higher states (never down): summary
i −λ→ i + 1,
i −o(∆t)→ j > i + 1,
and i −prob. 0→ j < i
For the Poisson process , the transition prob. in is ∆t
pⱼₖ(∆t)
related to
qⱼₖ
pⱼₖ(∆t) = δⱼ,ₖ + qⱼₖ∆t + o(∆t)
when ∆t → 0
qⱼₖ= λ(δⱼ,ₖ₊₁ - δⱼ,ₖ)
Generator matrix Poisson process
Q= (qⱼₖ) =
[−λ 0 0 0 0 . . .]
[λ −λ 0 0 0 . . .]
[0 λ −λ 0 0 . . .]
[0 0 λ −λ 0 . . .]
[. … … …]
0 except in 2 cases
j=k+1
or
j=k
sum of columns is 0
guarantees probability conservation
Given IC X(0)=0
pᵢ(0)=
pᵢ(0)=δᵢ,₀ = pᵢ,₀(0) and
pᵢ(t)= pᵢ,₀(t)
master eq
d/dt p(t) =Qp(t)
with probability vector
p(t)= (p₀(t),p₁(t),…)^T
system of coupled linear ODES:
d/dt p₀(t)
master equation
d/dt p₀(t)= −λp₀(t)
with p₀(0)=1
giving
p₀(t)=exp(−λt)
(p₀)
(p₁)
(…)
=
[−λ 0 0 0 0 . . .]
[λ −λ 0 0 0 . . .]
[0 λ −λ 0 0 . . .]
[0 0 λ −λ 0 . . .]
[. … … …]
*
(p₀)
(p₁)
(…)
with pᵢ(0)=
{1 if i=0
{0 if i not equal to 0
uses the time derivative of each component
d/dt p₁(t)
system of coupled linear ODES:
master equation
d/dt p₁(t)
= λp₀ − λp₁ = −λp₁(t) + λexp(−λt)
with p₁(0) = 0
⇒
p₁(t) = λtexp(−λt)
so probability vanishes exponentially in time
d/dt p₂(t)
system of coupled linear ODES:
…
master equation
d/dt p₂(t)
= λp₁ −λp₂
= −λp₂(t) +λ²texp(−λt),
with p₂(0) = 0 ⇒
p₂(t) = ((λt)²/2)exp(−λt)
…
d/dt pᵢ(t)
= λ(pᵢ₋₁ − pᵢ)
= −λpᵢ(t) + λ(λt)ᶦ⁻¹/(i − 1)! exp(−λt)
with IC
pᵢ(0) = 0 for i > 0
⇒ pᵢ(t) = ((λt)ᶦ/i!) exp(−λt)
We recognize the Poisson distribution of parameter λt
mean E(X(t)]=Var(X(t))=λt
standard dev square root (λt)
as time goes on more pronounced deviation
probability to remain in state 0
This is an example of an important general results: in CTMCs, the time between 2 events is
vanishes exponentially in t
p_0(t)=exp(-λt)
Prob{{i = 0 → i = 1 at time > t} = exp(−λt)
Prob {i = 0 → i = 1 at time ≤ t}
= 1 − exp(−λt)
This is an example of an important general results: in CTMCs, the time between 2 events is exponentially
distributed
Stationary distribution
continuous time MC has unique SD
Under generic conditions the CTMC
{X(t) ∈ S : t ∈ [0, ∞)}
with state space
S={0,1,2,…}
has a unique stationary distribution
lim_{t→∞} p(t)= π
= (π₀,π₁,π₂,…)^T
with
Σ_{i∈S} πᵢ = 1
(probability conservation)
Stationarity
Qπ=0
P(t)π = π
Q= lim_{∆t→0} [P(∆t) - I]/∆t
identity matrix I
Hence the stationary distribution is the right eigenvector of the generator matrix Q with eigenvalue 0
In general:
At stationarity
P(t) π = π
with lim_[t→∞} p(t) = π
⇒** ⃗π** is also the right eigenvector of P with eigenvalue 1
birth and death processes
Often the number of individuals in an evolutionary process changes at most by in each small time
increment; these are called birth (+1) or death (-1) events.
The class of CTMCs with this property are called “birth-and-death” processes (BDPs)
time is continuous states are discrete
birth and death
rates
{X(t) ∈ S : t ∈ [0, ∞)} with state space S = {0, 1, 2, . . . }
birth: i → i + 1 at rate λᵢ
death: i → i − 1 at rate µᵢ
e.g start at 2 can move in delta t to 1 or 3
birth and death derive probability of transitions:
{X(t) ∈ S : t ∈ [0, ∞)} with state space S = {0, 1, 2, . . . }
birth: i → i + 1 at rate λᵢ
death: i → i − 1 at rate µᵢ
i → i + k in ∆t :
pᵢ₊ₖ ᵢ(∆t) = Prob {∆X(t) = k|X(t) = i} =
{λᵢ∆t + o(∆t) if k = 1 (birth in state i)
{µᵢ∆t + o(∆t) if k = −1(death state i)
{1 − (λi + µi)∆t + o(∆t) if k = 0 (no birth and no death)
{o(∆t) otherwise (negligible)
Probability of transition
i
Since there are no deaths in state 0 (no individuals), we have µ₀
µ₀ = 0
If the state space is
S = {0, 1, . . . , N − 1, N}
λₙ=
λₙ=0
since there cannot
be a birth with N− λₙ=0→ N + 1
here the population is said to not be able to exceed N. In this case
can only reach state N from N-1
at state N
either absorbing stays there
or cant go N+1 but can go right
matrix q_n_ will tell us
When the state space is bounded and
S = {0, 1, . . . , N − 1, N} =⇒
Generator matrix:
qⱼ₊ₖ ⱼ =
lim_{∆t→0}
[pⱼ₊ₖ ⱼ (∆t) − pⱼ₊ₖ ⱼ(0)]/∆t
=
{λⱼ if k = 1
{µⱼ if k = −1
{−(λⱼ + µⱼ) if k = 0
{0 otherwise
Boundaries:
We need to consider the case j+k=0 and j+k=N separately since we have
and µ_0 = 0 λ_N = 0
p₀ⱼ(∆t)=
pₙⱼ(∆t)=
p₀ⱼ(∆t)=
{µ₁∆t + o(∆t) if j = 1
{1 − λ₀∆t + o(∆t) if j = 0
{o(∆t) otherwise
pₙⱼ(∆t)=
{λₙ₋₁∆t + o(∆t) if j = N − 1
{1 − µₙ ∆t + o(∆t) if j = N
{o(∆t) otherwise.
p₀ⱼ(∆t)=
{µ₁∆t + o(∆t) if j = 1
{1 − λ₀∆t + o(∆t) if j = 0
{o(∆t) otherwise
pₙⱼ(∆t)=
{λₙ₋₁∆t + o(∆t) if j = N − 1
{1 − µₙ ∆t + o(∆t) if j = N
{o(∆t) otherwise.
giving q₀ⱼ=
q₀ⱼ=
lim_{∆t→0}
[p₀ⱼ (∆t) − p₀ⱼ (0)]/∆t =
{µ₁ if j = 1
{−λ₀ if j = 0
{0 otherwise.
qₙⱼ=
lim_{∆t→0}
[pₙⱼ (∆t) − pₙⱼ (0)]/∆t =
{λₙ₋₁ if j = N − 1
{−µₙ if j = N
{0 otherwise.
SUMMARY:
class of continuous time MCs
We look at birth death processes
here only change in states are either go up by unit in time increment delta t BIRTH
go down one state DEATH
rates are lambda_i and u_i
index i depends on state we are in
So we wrote down the mastic equation
the probabilities
derive the generator matrix Q
in general we have three non zero rates (birth, death, neither) the sum of elements of a column has to add to one
we have to take care about what happens at boundaries.
e.g for states 0
We could reach state 0 only if there was a death in state 1
i=1 j=0 and there is a death
corresponding rates differ here e.g birth rate =0 at 0
meaning
p₀ⱼ(∆t)=
{µ₁∆t if j = 1
{1 − λ₀∆t if j = 0
{o(∆t) otherwise
if state 0 is absorbing: λ₀=0 and we stay there
we might assume λ₀>0
RECAP:
Let X(t) be a birth and death process
{X(t) ∈ S : t ∈ [0, ∞)}
on state space
{1,2,3,,….,N}
RATES
(BIRTH AND DEATH)
This means the state space is bounded
birth
i → i + 1 at rate λᵢ
death
i → i − 1 at rate µᵢ
from 0 to 1: λ_1
from 1 to 0:µ_1
from 1 to 2:λ_2
etc
In general the birth and death rates vary with i
only death from j = 1 into j = 0
no birth from j = 0
only birth from j = N − 1 into j = N
RECAP:
Let X(t) be a birth and death process
{X(t) ∈ S : t ∈ [0, ∞)}
on state space
{1,2,3,,….,N}
TRANSITION RATES
qⱼ₊ₖ,ⱼ
for j + k ̸= 0, N
qⱼ₊ₖ,ⱼ=
{λⱼ if k = 1
{µⱼ if k = −1
{−(λⱼ+ µⱼ) if k = 0
{0 otherwise
his notation is backwards to what im used to grrrr
RECAP:
Let X(t) be a birth and death process
{X(t) ∈ S : t ∈ [0, ∞)}
on state space
{1,2,3,,….,N}
TRANSITION RATES
q₀ⱼ
q₀ⱼ=
{µ₁ if j = 1
{−λ₀ if j = 0
{0 otherwise.
only death from j = 1 into j = 0
no birth from j = 0
INTO 0
birth rate at 0 might be 0 or maybe not… dep on model −λ₀
RECAP:
Let X(t) be a birth and death process
{X(t) ∈ S : t ∈ [0, ∞)}
on state space
{1,2,3,,….,N}
TRANSITION RATES
qₙⱼ
capital N here
qₙⱼ=
{λₙ₋₁ if j = N − 1
{−µₙ if j = N
{0 otherwise.
only birth from j = N − 1 into j = N
INTO N
RECAP:
Let X(t) be a birth and death process
{X(t) ∈ S : t ∈ [0, ∞)}
on state space
{1,2,3,,….,N}
GENERATOR MATRIX
These will have columns/rows with 3 non zero entries except for the first and last (from the boundary transition rates)
columns sum to 0 for first
negative values in diagonal
pattern each row:
birth neither death
negative on diagonal (usually niether)
each column i: at state i where can we transition
column is death niether() birth
columns sum to 0
death rates above diagonal
birth rates below diagonal
Q=
[−λ₀ µ₁ 0 . . . . . . . . . 0]
[λ₀ −(λ₁ + µ₁) µ₂ 0 . . . . . . .]
[0 λ₁ −(λ₂₋ + µ₂) … … 0 ….].
[0 0 λ₂… … … .]
[0 0 0 ,….]
[………]
[. 0 … … … 0 … λₙ₋₂ −(λₙ+₁ + µₙ₋₁ ) µₙ]
[0 . . . . . . . . . 0 λₙ₋₁ −µₙ]
last and first rows differ due to boundary conditions
RECAP:
Let X(t) be a birth and death process
{X(t) ∈ S : t ∈ [0, ∞)}
on state space
{1,2,3,,….,N}
MASTER EQUATION
for probability vector
probability vector
p(t)(p₀(t),p₁(t),p₂(t),…,pₙ(t))
d/dt p(t) = Qp(t)
giving
p(t)= exp(Qt) p(0)
generator matrix times p
if we can take exponential we are done but can be complicated for this matrix
wjhat we find is that for each p_i we have a coupled differentil-difference equation
RECAP:
Let X(t) be a birth and death process
{X(t) ∈ S : t ∈ [0, ∞)}
on state space
{1,2,3,,….,N}
for each p_i we have a differential-difference equation coupled with other such equations
for j ∈ {1, 2, . . . , N − 1}
from the entries of the matrix Qp
(d/dt)pⱼ
= (λⱼ₋₁pⱼ₋₁(t) + µⱼ₊₁pⱼ₊₁(t) )-(λⱼ +µⱼ)pⱼ
GAIN TERMS - LOSS TERMS
for j ∈ {1, 2, . . . , N − 1}
“balance equation” with gain terms
for birth/death reactions flowing
into j, and loss terms for those
flowing out of
ENTER j from j-1
ENTER j from j+1
EXIT J to j+1
EXIT J to j-1
(solving this is non trivial, p_i dep on others!)
RECAP:
Let X(t) be a birth and death process
{X(t) ∈ S : t ∈ [0, ∞)}
on state space
{1,2,3,,….,N}
for each p_i we have a differential-difference equation coupled with other such equations
The equations at the boundaries
equations at the boundaries j=0 and j=N:
(d/dt)p₀(t) = µ₁p₁(t) − λ₀p₀(t)
no death from j=0
no birth from j=-1
dtpN (t) = λ ₙ₋₁pₙ₋₁(t) − µₙpNₙ(t)
no birth from j=N
no death from j=N+1
enter only by birth in n-1
exit only by death state n
IN TTOAL N COUPLED LINEAR ODES
Let X(t) be a birth and death process
{X(t) ∈ S : t ∈ [0, ∞)}
on state space
state space is unbounded
S = {0, 1, 2, . . . , ∞}
the above special equation for
j=N is replaced by the first equation(the general ones?) that is then valid for
j ∈ {1, 2, . . . , ∞}
and coupled to the special
equation for j=0.
How to solve these equations and find p_j∈S (t)
Generating functions (when possible), otherwise get useful info from the moments of X(t)
Simple birth process (Yule process) intro
This is a process with only birth events occurring at a LINEAR rate
“simple birth process” with rate ,
=> λ_j = jλ
and no death =
µ_j = 0
Simple birth process (Yule process)
linear birth rate
λ_j = jλ
state space
S = {N, N + 1, . . . , ∞}
X(0)=N>0
unbounded since the number of individuals can only grow without limit from a given IC N
Simple birth process (Yule process)
linear birth rate
λ_j = jλ
state space
S = {N, N + 1, . . . , ∞}
X(0)=N>0
TRANSITION RATES
q_{j+k, j}
{jλ if k = 1
(−jλ if k = 0
(0 otherwise
initially
X(0) = N ⇒
p_j (0) = δ_{j,N}
=
{1 if j = N
{ 0 otherwise
Simple birth process (Yule process)
linear birth rate
λ_j = jλ
state space
S = {N, N + 1, . . . , ∞}
X(0)=N>0
GENERATOR MATRIX
FROM q_{j+k, j}
{jλ if k = 1
(−jλ if k = 0
(0 otherwise
initially
X(0) = N ⇒
p_j (0) = δ_{j,N}
=
{1 if j = N
{ 0 otherwise
Q=
[0 0…]
[0 -λ 0 …]
[0 λ -2λ 0…]
[0 0 2λ -3λ 0 …]
….]
[0…0 λ(n-1) 0]
all columns add to 0
death rates 0
birth rates λ multiplied by n
Simple birth process (Yule process)
linear birth rate
λ_j = jλ
state space
S = {N, N + 1, . . . , ∞}
X(0)=N>0
MASTER EQUATION
(d/dt)p_j
(d/dt)ⱼ
= λ[(j − 1)pⱼ₋₁(t) − jpⱼ ],
for j = N, N + 1, . . . ,
(d/dt)pⱼ(t)=0
for j=0,1,…,N-1
Since X(t) = j ∈ {N, N + 1, . . . },
j can never be < N
Equivalently: p_j<N = 0
Simple birth process (Yule process)
linear birth rate
λ_j = jλ
state space
S = {N, N + 1, . . . , ∞}
X(0)=N>0
FINDING pⱼ using the generating function
pⱼ
using the generating function
G(z, t) = E(zˣ⁽ᵗ⁾)
=Σ_{j=0,∞} pⱼ (t)zʲ
=Σ_{j=N,∞} pⱼ (t)zʲ
Idea: use the ME to find an equation (PDE) for , solve that equation and G find a way to obtain p_i(t) from G
Simple birth process (Yule process)
linear birth rate
λ_j = jλ
state space
S = {N, N + 1, . . . , ∞}
X(0)=N>0
Idea: use the ME to find an equation (PDE) for , solve that equation and G find a way to obtain p_i(t) from G
∂G(z, t)/∂t
(d/dt) (Σ_{j=0,∞} pⱼ (t)zʲ)
from ME
=
λ Σⱼ₌ₙ₊₁∞[(j-1)pⱼ₋₁zʲ] − λ Σⱼ₌ₙ∞ jpⱼzʲ
starting with j=N gives 0
=
λz² Σⱼ₌ₙ∞jpⱼzʲ⁻¹] − λz Σⱼ₌ₙ∞ [jpⱼzʲ⁻¹]
shifting j to j-1
second sum is ∂G/∂z
=
λz(z-1) (∂G(z,t)/∂z)
with G(z,0) = Σⱼ₌ₙ∞ jpⱼ(0)zʲ = z^N
pⱼ(0) = 1 if j=N , 0 otherwise
with G(z,0) = Σⱼ₌ₙ∞ jpⱼ(0)zʲ
with G(z,0) = Σⱼ₌ₙ∞ jpⱼ(0)zʲ = z^N
pⱼ(0) = 1 if j=N , 0 otherwise
Simple birth process (Yule process)
linear birth rate
λ_j = jλ
state space
S = {N, N + 1, . . . , ∞}
X(0)=N>0
The linear PDE for i
∂G(z, t)/∂t
=
λz(z-1) (∂G(z,t)/∂z)
with G(z,0) = Σⱼ₌ₙ∞ jpⱼ(0)zʲ = z^N
for the master eq
G(z,t)
s simple enough to be solvable using the method of the characteristics (outside
the scope of this module) =>
(z^N exp(-λNt))/
[1-z(1-exp(-λt))]^N
We recognize the generating function of a negative binomial distribution (see App. D, Sec.7.4/7.5.)
From the properties of the negative binomial distribution, we obtain the solution of the ME:
p_{j+N}(t)
=
(N +J-1)
( j )
Exp(-λNt) (1- exp(-λt))^j
for j=0,1,2,…
G(z,t)
Simple birth process (Yule process)
linear birth rate
λ_j = jλ
state space
S = {N, N + 1, . . . , ∞}
X(0)=N>0
MEAN
(z^N exp(-λNt))/
[1-z(1-exp(-λt))]^N
E(X(t)) = ∂_z G|_z=1 = NexP(λt)
says grows exponentially
G(z,t)
Simple birth process (Yule process)
linear birth rate
λ_j = jλ
state space
S = {N, N + 1, . . . , ∞}
X(0)=N>0
VARIANCE
(z^N exp(-λNt))/
[1-z(1-exp(-λt))]^N
var(X(t)) =
Nexp(2λt)(1 − exp(−λt)
It is often difficult to obtain an expression for and then to get p_j(t) from it so
COMPUTE THE FIRST MOMENTS OF X(t)
mean
1st (raw) moment is the mean
E(X(t)) = ∂_z G|_z=1
m(t) ≡ E(X(t)) = Σ_j∈S jpⱼ(t)
It is often difficult to obtain an expression for and then to get p_j(t) from it so
COMPUTE THE FIRST MOMENTS OF X(t)
from the master equation:
Variance
var(X(t)) = m_2(t) − m^2(t)
SECOND MOMENT - [FIRST MOMENT]^2
second moment is
m_2(t) ≡ E(X²(t)) = Σ_j∈S j²pⱼ(t)
Hence, when t is large the standard deviation from the mean grows exponentially as exp(λt)
from the master equation:
find the first moment
(d/dt)pⱼ(t)
= λ[(j − 1)pⱼ₋₁(t) − jpⱼ ],
for j = N, N + 1, . . . ,
(d/dt)pⱼ(t)=0
for j=0,1,…,N-1
Since X(t) = j ∈ {N, N + 1, . . . },
j can never be < N
Equivalently: p_j<N = 0
E(X(t)) = ∂_z G|_z=1
dm/dt
=Σⱼ₌₀∞ [j dpⱼ/dt]
from ME subbing dpⱼ/dt
= λΣⱼ₌₀∞[j(j − 1)pⱼ₋₁]− λΣⱼ₌₀∞[j²pⱼ ],
shifting j-1 into j
=λΣⱼ₌ₙ∞[j(j +1)pⱼ]− λE[X²]
=[λ-λ] m_2(t) + λm(t)
thus
m^DOT = λm
so mean
m(t)= Nexp(λt)
on average, the population grows exponentially in time
from the master equation:
find the second moment
m_2(t) ≡ E(X²(t)) = Σ_j∈S j²pⱼ(t)
m^DOT_2
=Σ_j∈S j²(d/dt)pⱼ(t)
from ME
=λΣⱼ₌ₙ∞[j²(j − 1)pⱼ₋₁]− λΣⱼ₌ₙ∞[j³pⱼ ],
shifting j-1 to j
=λΣⱼ₌ₙ∞[(j+1)²jpⱼ₋₁]− λE[X³],
=(λ-λ)m_3 +2λm_2 + λm
thus
m_2^DOT =2λm_2(t) + λm(t)
=λ[2m_2(t) +Nexp(λt)]
with
m_2(0) =N^2
giving
m_2(t) = (Nexp(t))^2 + λNexp(2λt)*integral_[0,t] exp(-λs).ds
=m(t)^2 +N(exp(2λt)-exp(λt))
Hence, when t is large the standard deviation from the mean grows exponentially as exp(λt)
overview
why are boundary conditions important?
conditions at boundaries are important and can lead to population
extinction or to its unbounded growth.
Goal of this lecture: simple birth and death process of Question 3 of Example Sheet 4
missing recording :(
Q3 EXample sheet 4
We consider consider a population of size given by a birth-and-death process
X(t) ∈ S = {0, 1, 2, . . . , ∞}
where t = [0, ∞)is the continuous time. The population evolves through simple birth and death events
rates
λ_j =jλ
BIRTH RATE λ>0
µ_j=jµ
death rate µ>0
IC X(0)=N>0
p_j(0)= δ_{j,N}
The state space is unbounded (semi-infinite) as there is no upper bound in the number of individuals
in the population. However, since λ_0 = µ_0 = 0
there can be no birth or death from state j=0, the
state j=0 is here an absorbing state;
j=0 is the only absorbing state of this birth-and-death process:
What is the master equation (ME) for this simple birth-and-death process?
INCLUDING CASE AT 0
(d/dt)pⱼ
= (λⱼ₋₁pⱼ₋₁(t) + µⱼ₊₁pⱼ₊₁(t) )-(λⱼ +µⱼ)pⱼ
GAIN TERMS - LOSS TERMS
for j ∈ {1, 2, . . . , N − 1}
if it reaches the state j=0, the population is extinct and does not recover. However, since the state space is
semi-infinite, absorption and extinction are not guaranteed!
As seen in lecture 8.II and section 8.5 of the lecture notes, the ME here is the set of balance quations
(d/dt)pⱼ
= (j-1)λpⱼ₋₁(t) + (j+1)µpⱼ₊₁(t) -j(λ +µ)pⱼ
GAIN TERMS - LOSS TERMS
birth from j-1 death from j+1 - loss term
for j ∈ {1, 2, . . . , N − 1}
subbing in the linear rates
coupled to a special equation at j=0 (absorbing boundary):
(d/dt)p_0 = µp_1(t)
hereλ_0=0 x λ=0
and p_j(t) = 0 of j <0
STATE j=0 ABSORBING so only one possible transition into it 1 into 0 µ_1
Q3 EXample sheet 4
Can we solve the ME and find the p_j(t) , alternatively what can we do?
Finding explicitly is complicated, even though it is possible to find the
generating function. which turns out to be a rather complicated expression
G(z, t) = E(zˣ⁽ᵗ⁾)) = SUM_[j=0,∞]zʲ p_j (t)
Q3 EXample sheet 4
for the mean
compute the first and second moments of the process from the ME.
(d/dt)pⱼ
= (j-1)λpⱼ₋₁(t) + (j+1)µpⱼ₊₁(t) -j(λ +µ)pⱼ
GAIN TERMS - LOSS TERMS
birth from j-1 death from j+1 - loss term
for j ∈ {1, 2, . . . , N − 1}
m(t) ≡ E(X(t)) = Σ_j∈S jpⱼ(t)
By multiplying both sides of the ME by j and summing overj ∈ S
m^dot(t)
= Σⱼ₌₀∞ j(d/dt)pⱼ
by ME
=λΣⱼ₌₀∞ j(j-1)λpⱼ₋₁(t) + µΣⱼ₌₀∞j(j+1)pⱼ₊₁(t) -(λ +µ)Σⱼ₌₀∞j^2pⱼ(t)
shifting second sum j+1 to k
=λΣⱼ₌₀∞(j+1)jpⱼ + µΣⱼ₌₀∞(j-1)jpⱼ - (λ+µ)m_2(t)
m_2 terms cancel out
=λ(m_2(t) +m(t)) + µ(m_2(t)-m(t))- (λ+µ)m_2(t)
so m^dot(t) = ( (λ-µ)m(t)
with m(0)=N
giving m(t)=Nexp( (λ-µ))t
as t tends to infinity
this tends to
infinity λ>µ
0 λ<µ
N λ=µ
dep on λ and µ
Q3 EXample sheet 4
2nd moment
m_2(t)=E[X^2(t)]) = Σ_j∈S j^2pⱼ(t)
working this out in a similar way
NOTES…
m_2^dot (t) = 2(λ-µ)m_2 + (λ+µ)m(t)
using the sol for m(t) helps to see
= 2(λ-µ)m_2(t) + (λ+µ)Nexp((λ-µ)t)
m_2(0)=N^2
Solving this (inhomogeneous) linear ODE with standard method
E.g, using the “variation of the constant method” or a suitable integrating factors
m_2^dot (t) = 2(λ-µ)m_2(t) + (λ+µ)Nexp((λ-µ)t)
m_2(0)=N^2
Solving this (inhomogeneous) linear ODE with standard method
E.g, using the “variation of the constant method” or a suitable integrating factors
Notes show
m_2(t)=
N^2 exp(2(λ−µ)t) + exp(2(λ−µ)t)*integral_[0,t] exp(2(λ−µ)s) N((λ+µ)exp((λ−µ)s).ds
INTEGRATING FACTOR IS FIRST EXP IN INTEGRAL
=
{ m(t)^2 + N[( λ+µ)/(λ -µ) ] exp((λ -µ)t)( exp((λ -µ)t) - 1) if λ /= µ.
{ N^2 +2Nλt if λ = µ.
Q3 EXample sheet 4 variance
var(X(t))
m_2 - m^2
=
{N([λ+µ]/[λ−µ]) exp(λ−µ)(exp((λ−µ)t) -1) if λ ̸= µ
{2Nλt if λ = µ
This mans that as t tends to infinity Var tends to
infinity when λ> µ
0 when λ <µ
q3 ps4 summary behaviour
λ<µ:
λ>µ:
λ=µ:
cases:
both the average population size and fluctuations grow in time when λ> µ:
mean and standard deviations = √variance grow exponentially as ∼ exp((λ−µ)t)
⇒ possible extinction, but in general population grows when λ > µ and N is finite
but not certain, and extinction is unlikely (can occur with a very small probability) when N is finite but large
graph shows variance about exponential shape, growing as time increases, possible extinction for some paths
This confirms that the population eventually goes extinct λ <µ:
When λ = µ, mean is constant but standard deviations grow ∼ √λt =⇒ extinction is certain
occurs with probability 1
graph of X(t) for this case shows noisy about m(t), which is constant but then decrease to 0 as time goes on
LEVEL 5 INTEREVENT TIMES
INTRO
In continuous-time Markov chains, the time between two successive events (intervent of waiting time is a random variable: how are the interevent times distributed? In the simple birth-and-death process, the state j=0 is absorbing=> Does this imply that extinction is guaranteed?
Goal of this lecture: general results on the interevent times in CTMCs and a theorem on the
extinction probability in birth-and-death processes
LEVEL 5 INTEREVENT TIMES
T_i
In continuous-time Markov chains (CTMCs), the time between 2 sucessive events (birth, death, …)
i and i+1 is a random variable
Tᵢ
{Tᵢ}_i≥0
is the set of interevent, or waiting, times between events i and i+1
T_0 time to first event
LEVEL 5 INTEREVENT TIMES
W_i
Wᵢ time at which event or transition i occurs
ie a birth or death happens at precisely these times
Tᵢ=W_{i+1} - Wᵢ
figure for sample path in BD process
Times between events
steps some longer some shorter some ascent some descend birth v death
T_1=W_2-W_1
L5
What is the intervent times distribution ?
General result: the interevent (waiting) times in CTMCs are exponentially distributed
random variables
For the Poisson process (Sec. 8.3) we found that the interevent times were exponentially distributed.
p_0 = exp(-lambda t) when starting at state 0
meaning the probability that you stay at state 0 vanishes exponentially with time
L5
General result: the interevent (waiting) times in CTMCs are exponentially distributed
random variables
PROBABILITY OF TRANISTION OF X moving
n → j ̸= n in ∆t is
Prob{Tᵢ ≤ ∆t}
considering a CTMC X(t) of range A
X(Wᵢ) =n ∈ S
PROBABILITY OF TRANISTION OF X moving
n → j ̸= n in ∆t is
Prob{Tᵢ ≤ ∆t} = Σ_{j̸=n,j∈S} pⱼₙ(∆t)
(sum of probabilities of all the moves away)
we have that
pⱼₙ(∆t) = qⱼₙ∆t + o(∆t) (something vanishingly small we neglect)
qₙₙ(∆t)=- Σ_{j̸=n,j∈S} qⱼₙ
(transition probability of staying at n, negative of the sum of the other as have to add to 0?
PROPERTY OF GENERATOR MATRIX SUM OF ALL ELEMENT OF A OLUMN ADD TO 0)
so this becomes
Prob{Tᵢ ≤ ∆t} = Σ_{j̸=n,j∈S} qⱼₙ∆t + o(∆t) = |qₙₙ|∆t+o(∆t)
L5
General result: the interevent (waiting) times in CTMCs are exponentially distributed
random variables
PROBABILITY OF TRANISTION OF X not moving in ∆t
Prob{Tᵢ > ∆t}
Prob{Tᵢ > ∆t} = 1- Prob{Tᵢ ≤ ∆t}
=1- |qₙₙ|∆t+o(∆t)
+term that is negligible as ∆t tends to 0
L5
General result: the interevent (waiting) times in CTMCs are exponentially distributed
random variables
pₙₙ(2∆t)
probability of___
pₙₙ(2∆t)
probability of being in state n still after 2∆t
with ∆t → 0 is
pₙₙ(2∆t) = pₙₙ(∆t)pₙₙ(∆t)= p²ₙₙ(∆t)
( prob. to be in state n after ∆t multiplied by the probability
of staying in that state for another ∆t)
AS TIME IS HOMOGENEOUS
L5
General result: the interevent (waiting) times in CTMCs are exponentially distributed
random variables
slice time t
into t/∆t
pₙₙ(t)=exp(-|qₙₙ|t)
Waiting time in CTMCs is an exponentially distributed random variable
probability that you remain in state n is vanishingly small with exponential term
L5
General result: the interevent (waiting) times in CTMCs are exponentially distributed
random variables
slice time t
into t/∆t
intervals ∆t
pₙₙ(t)=
pₙₙ((t/∆t)∆t)=
pₙₙ(∆t). pₙₙ(∆t)……..pₙₙ(∆t)
product of (t/∆t) times
=(pₙₙ(∆t))^{(t/∆t)}
= (1- |qₙₙ|∆t+o(∆t))^{t/∆t}
as ∆t→0
limit converges to
exp(-|qₙₙ|t)
so
pₙₙ(t)=exp(-|qₙₙ|t)
L5
General result: the interevent (waiting) times in CTMCs are exponentially distributed
random variables
if pₙₙ(t)=exp(-|qₙₙ|t)
then
cumulative waiting times distribution
Fᵢ(t) ≡ Prob{Tᵢ ≤ t}
Fᵢ(t) ≡ Prob{Tᵢ ≤ t}
= 1 − Prob{Tᵢ = Wᵢ₊₁ − Wᵢ > t}
= 1 - exp(-|qₙₙ|t)
because
Prob{Tᵢ = Wᵢ₊₁ − Wᵢ > t}=
pₙₙ(t)=e−|qnn|t
Fᵢ is the cumulative waiting times distribution, with
Fᵢ(0)=0
Fᵢ(∞) = 1
L5
General result: the interevent (waiting) times in CTMCs are exponentially distributed
random variables
cumulative waiting times distribution
Fᵢ(t) ≡ Prob{Tᵢ ≤ t}
= 1 - exp(-|qₙₙ|t)
OBTAIN THE PROBABILITY DENSITY
By defn differentiate wrt t
giving
PROBABILITY DENSITY
dFᵢ/dt
= (d/dt) [Prob{Tᵢ ≤ t}]
= (d/dt) [1 - exp(-|qₙₙ|t)]
= |qₙₙ| exp(-|qₙₙ|t)]
=> The probability to remain in state n, if X(0)=n, after time t decays exponentially with t
=> The probability to move into another state j not equal to n, if X(0)=n, after time t is exponentially close to 1
eventually will move away
L5
General result: the interevent (waiting) times in CTMCs are exponentially distributed
random variables
probability density function
= |qₙₙ| exp(-|qₙₙ|t)]
cumulative waiting times distribution
Fᵢ(t) ≡ Prob{Tᵢ ≤ t}
= 1 - exp(-|qₙₙ|t)
what is
Prob{t₁ ≤ Tᵢ ≤ t₂}
Prob{t₁ ≤ Tᵢ ≤ t₂}=
Fᵢ(t₂)- Fᵢ(t₁)
=
exp(-|qₙₙ|t₁) - exp(-|qₙₙ|t₂)
difference between two exoonentials from cumulative
L5 our findings are summarised in this thm 8.1
SHOWN in lectures and in notes
Exponential distribution of waiting times).
Let {X(t) ∈ S : t ∈ [0,∞)}
be a CTMC of transition matrix P(t) = (pⱼₖ(t)) and generator matrix Q = (qⱼₖ), with
j, k ∈ S and,
Σ_{ j̸=n} pⱼₙ(∆t)
= |qₙₙ|∆t + o(∆t),
∀n ∈ S.
Given X(Wi) = n, the **waiting time {T_i = W_{i+1} − W_i}i≥0 is an exponential random
variable with parameter |qₙₙ| and
cumulative distribution function
Fᵢ(t) = Prob {Tᵢ ≤ t}
= 1 − exp(−|qₙₙ|t)
The **mean **and variance of Tᵢ are
E(Tᵢ) = |qₙₙ|⁻¹
and
var(Tᵢ) = |qₙₙ|⁻²
difference between transition matrix and generator matrix
transition matrix P(t) = (pⱼₖ(t)) and generator matrix Q = (qⱼₖ), with
j, k ∈ S and,
Generator matrix
ransition rates between different states of the Markov process.
q_jk rate of transitioning from state k to state j?? his notes
with q_jj defined so rows sum to 0
q_jj= - sum_{k /=j} q_jk
rate of change for markov process dX(T)=QX(t)dt
transition marix*
P describes probabilities of moving between states over time
probability of going from state given process starts
pjk from k to j?
satsifies (d/dt)P(t) = QP(t)
P(t)=exp(Qt)
matrix exponential which can be hard to find
L5 our findings are summarised in this thm 8.1
applied to poisson process seen earlier
For the Poisson process, Sec. 8.3, we have
|q_nn| = λ =⇒
E(T_i) = 1/λ,
var(T_i) = 1/λ^2,
F_i(t) = 1 − exp(λt)
L5:
A theorem on the probability of extinction in birth-and-death processes
Is the absorbing state X=0 always reached and the population always goes extinct?
what happens when λ ≥ µ > 0
we have seen that for the simple birth-and-death process
X(t)∈ {0, 1, . . . , ∞} : t ∈ [0, ∞)
birth rate λ_j = jλ
death rate µ_j = jµ
as t→∞
E[X(t)]→ ∞
Var(X(t))→ ∞
when λ ≥ µ > 0
birth capita birth rate exceeds per capita death rate
absorbing state X=0 always reached?
Possible but not certain: due to fluctuations from random birth and death events => standard deviations
grow in time and can exceed the mean of X(t) causing extinction, but unbounded growth is also possible
MEAN GROWS BUT DEVIATIONS ABOUT IT ALSO GROW
L5:
in simple birth and death process
p₀(∞) CASES
also as t → ∞ what happens to X(∞) ?
probability of extinction
depends on whether the state space is bounded and X(0)
when t → ∞:
X(∞) = 0 with prob. p₀(∞)
EXTINCT
and
X(∞) → ∞ with prob. 1 − p₀(∞)
GROWS WITHOUT BOUND
goes extinct or grows without bound
probabilites might not be 0 and 1 might be between
L5:
General result stated as
p₀(∞)
EXTINCTION certain?impossible? possible? cases
if S= {0, 1, 2, . . . , ∞} is UNBOUNDED
µ₀ = λ₀ = 0 the state 0 is absorbing
for X(0)=n≥ 1
we have
EXTINCTION CERTAIN
p₀(∞)=1
if Σᵢ₌₁∞ [µ₁µ₂…µᵢ]/[λ₁λ₂…λᵢ]= ∞ DIVERGES
EXTINCTION NOT CERTAIN but possible
p₀(∞)=
[Σᵢ₌₁∞ [µ₁µ₂…µᵢ]/[λ₁λ₂…λᵢ]]/
[1+Σᵢ₌₁∞ [µ₁µ₂…µᵢ]/[λ₁λ₂…λᵢ]]
if
Σᵢ₌₁∞ [µ₁µ₂…µᵢ]/[λ₁λ₂…λᵢ] < ∞ FINITE
EXTINCTION ALMOST IMPOSSIBLE
p₀(∞)≈0
if Σᵢ₌₁∞ [µ₁µ₂…µᵢ]/[λ₁λ₂…λᵢ] < ∞ AND INITIAL POP LARGE n»1
FINITE AND LARGE POP
L5:Theorem 8.2:
birth and death process
FOR THE SIMPLE BIRTH AND DEATH PROCESSES ON
S= {0, 1, 2, . . . , ∞}
p₀(∞)=
{1 if µ ≥ λ
{(µ/λ)ⁿ if µ < λ
q3 ps4!!
you should know how to sum geometric progressions
L5:Theorem 8.2:
birth and death process
what happens on finite state space? with absorbing state 0?
S = {0, 1, 2, . . . , N}
µ₀ = λ₀ = 0
we have
state N not absorbing then
λₙ = 0, µₙ > 0 (⇒ state N is not absorbing) and X(0) = n ≥ 1
p₀(∞)=1
meaning EXTINCTION ALWAYS CERTAIN
will end up reaching the absorbing state 0
might be after a long time
