lecture 9 point processes - stanford university · point processes in time and space point...
TRANSCRIPT
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Lecture 9Point Processes
Dennis SunStats 253
July 21, 2014
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Outline of Lecture
1 Last Words about the Frequency Domain
2 Point Processes in Time and Space
3 Inhomogeneous Poisson Processes
4 Second-Order Properties
5 Wrapping Up
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Last Words about the Frequency Domain
Where are we?
1 Last Words about the Frequency Domain
2 Point Processes in Time and Space
3 Inhomogeneous Poisson Processes
4 Second-Order Properties
5 Wrapping Up
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Last Words about the Frequency Domain
Why is the frequency domain useful?
Time Frequency
=⇒
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Last Words about the Frequency Domain
Why is the frequency domain useful?
Time Frequency
=⇒
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Last Words about the Frequency Domain
Why is the frequency domain useful?
The frequency domain more often captures our intuitionabout when two signals are “similar”.
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Point Processes in Time and Space
Where are we?
1 Last Words about the Frequency Domain
2 Point Processes in Time and Space
3 Inhomogeneous Poisson Processes
4 Second-Order Properties
5 Wrapping Up
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Point Processes in Time and Space
Trinity of Spatial Statistics
Today we complete the “trinity” of spatial statistics...
South African Witwatersrand Gold Reef (grams per ton)
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Lattice Data Geostatistics Point Processes
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Point Processes in Time and Space
Point Processes
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Point Processes in Time and Space
Marked Point Processes
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Point Processes in Time and Space
Point Processes
• The distinguishing feature of point processes is that the locations siare now random.
• There may or may not be labels y(si) associated with the points.
• Basic model: Poisson processes
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Point Processes in Time and Space
Poisson Processes in Time
0 10 20 30 40
• N(t): number of events that have occurred up to time t
• Properties:
1 N(t+ h)−N(t) ∼ Pois(λh)2 If s < t < u < v, then N(t)−N(s) is independent of N(v)−N(u).
• Question: Is this well-defined? (i.e., Can I simulate a process thathas these properties?)
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Point Processes in Time and Space
Simulating a Temporal Poisson Process: Method 1
• Generate waiting times Wiiid∼ Exp(λ).
• Then, the first event occurs at time W1, the second at timeW1 +W2, etc.
• Formally, N(t) = max{k :∑k
i=1Wi ≤ t}.• Claim: N(t) has the desired properties.
1 N(t+ h)−N(t) ∼ Pois(λh)2 If s < t < u < v, then N(t)−N(s) is independent of N(v)−N(u).
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Point Processes in Time and Space
Simulating a Temporal Poisson Process: Method 2
• This method simulates a Poisson process on [0, T ].
• Generate N(T ) ∼ Pois(λT ).
• Generate si ∼ Unif(0, T ) for i = 1, ..., N(T ).
• Question: Why is N(t+ h)−N(t) ∼ Pois(λh)?
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Point Processes in Time and Space
How do we generalize this to space?
D
N(A) ∼ Pois(λ|A|)
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Point Processes in Time and Space
Poisson Processes in Space
D
N(A) and N(B) are independent if A ∩B = ∅.
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Point Processes in Time and Space
Is this well-defined?
• We can generalize Method 2 for temporal processes:
• Generate N(D) ∼ Pois(λ|D|).• Generate si ∼ Unif(D), i = 1, ..., N(D).
• Method 1 doesn’t generalize. /• Once again, the moral of this class:
Temporal processes are easy, spatial processes are hard.
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Point Processes in Time and Space
Parameter Estimation
• How do you estimate λ?
• Easy: λ = N(D)|D| .
• Many ways to justify this, e.g., MLE.
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Inhomogeneous Poisson Processes
Where are we?
1 Last Words about the Frequency Domain
2 Point Processes in Time and Space
3 Inhomogeneous Poisson Processes
4 Second-Order Properties
5 Wrapping Up
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Inhomogeneous Poisson Processes
The Inhomogeneous Poisson Process
• This model is restrictive: assumes points are equally likely to beanywhere in D.
• Generalization: suppose there is a density λ(·) on D. Then
N(A) ∼ Pois
(∫Aλ(s) ds
).
• This is called an inhomogeneous Poisson process.
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Inhomogeneous Poisson Processes
Testing for Homogeneity
• How do we test if our process is homogeneous? (Another term iscomplete spatial randomness.)
• Let’s try to generate a homogeneous process in R.
• General strategy for devising tests:• Come up with some test statistic, e.g.,
G(r) =1
N(D)#{si whose nearest neighbor is closer than r}
K(r) =1
λ
#{(i, j) : d(si, sj) ≤ r}N(D)
• Simulate test statistic under null hypothesis.• Compare with observed value of test statistic.
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Inhomogeneous Poisson Processes
Estimating λ(·)Kernel density estimation: Choose a kernel K that integrates to 1:∫
K(s, s′) ds = 1.
s′
Place a kernel centered at each si. Then the estimator is
λ(s) =
N(D)∑i=1
K(s, si).
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Inhomogeneous Poisson Processes
Estimating λ(·)
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Inhomogeneous Poisson Processes
Estimating λ(·)
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Inhomogeneous Poisson Processes
Estimating λ(·)
May need to correct for edge effects:
λ(s) =
N(D)∑i=1
1∫DK(s, si) dsi
K(s, si).
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Inhomogeneous Poisson Processes
Final Issues
• How do you simulate from an inhomogeneous point process?
• Also possible to model λ(s) parametrically, e.g.,
log λ(s) = x(s)Tβ
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Second-Order Properties
Where are we?
1 Last Words about the Frequency Domain
2 Point Processes in Time and Space
3 Inhomogeneous Poisson Processes
4 Second-Order Properties
5 Wrapping Up
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Second-Order Properties
Why Second-Order Properties?
Processes may still exhibit clustering or inhibition.
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Second-Order Properties
Examples of Clustering and Inhibition
redwood
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Second-Order Properties
Second-Order Intensity
λ(s) = lim|ds|→0
E(N(ds))
|ds|µ(s) = E(y(s))
λ2(s, s′) = lim
|ds|→0
E(N(ds)N(ds′))
|ds||ds′|Σ(s, s′) = E(y(s)y(s′))− µ(s)µ(s′)
λ2 is called the second-order intensity.
A process is stationary if λ(s) ≡ λ and λ2(s, s′) = λ2(s− s′).
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Second-Order Properties
Ripley’s K-function
• λ2 is hard to estimate.• For stationary and isotropic processes, K-function is used instead:
K(r) =1
λE#{events within distance r of a randomly chosen event}
=1
λE
1
N(D)
N(D)∑i=1
∑j 6=i
1{d(si, sj) ≤ r}
• This has a natural estimator:
K(r) =1
λ
#{(i, j) : d(si, sj) ≤ r}N(D)
• General strategy for fitting models:
minimizeθ
∫ r0
0w(r)(K(r)−Kθ(r))
2 dr
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Second-Order Properties
Handling Inhomogeneity
• What if λ(s) 6≡ λ but is known?
• Solution: Re-weight distances by chance of observing event.
KI(r) = E
1
|D|
N(D)∑i=1
∑j 6=i
1{d(si, sj) ≤ r}λ(si)λ(sj)
KI(r) =
1
|D|
N(D)∑i=1
∑j 6=i
1{d(si, sj) ≤ r}λ(si)λ(sj)
• All computations proceed with KI and KI instead of K and K.
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Second-Order Properties
Modeling Clustering: Neyman-Scott Process
• Parent events are generated from Poisson process with intensity ρ(·).
• Each parent has S offspring, where Siid∼ pS .
• Offspring are located i.i.d. around their parent, according to somedensity function f(·).
• The process is the resulting offspring.
[Check: If ρ(·) ≡ ρ and f(s) = f(||s||), then stationary and isotropic.]
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Second-Order Properties
Modeling Clustering: Neyman-Scott Process
redwood
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> cauchy.estK(redwood)
Fits Neyman-Scott process with Cauchy ker-nel:
• parents from Poisson process withintensity κ
• S ∼ Pois(µ)
• f(s) = 12πω2
(1 + ||s||2
ω2
)−3/2
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Second-Order Properties
Modeling Clustering: Neyman-Scott Process
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Minimum contrast fit (object of class minconfit)
Model: Cauchy process
Fitted by matching theoretical K function to Kest(redwood)
Parameters fitted by minimum contrast ($par):
kappa eta2
12.446917419 0.008454113
Derived parameters of Cauchy process ($modelpar):
kappa omega mu
12.44691742 0.04597312 4.98115300
Converged successfully after 259 iterations
Starting values of parameters:
kappa eta2
1 1
Domain of integration: [ 0 , 0.25 ]
Exponents: p= 2, q= 0.25
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Second-Order Properties
Modeling Inhibition: Strauss Process
p(s1, ..., sn) =βnγ#{(i,j): d(si,sj)≤δ}
α(β, γ)
γ ∈ [0, 1], so last term penalizes processes that have points close together.
• γ = 1: no inhibition
• γ = 0: no event can be within δ of another (“hard-core” model)
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Second-Order Properties
Modeling Inhibition: Strauss Process
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> ppm(Q = cells, trend = ~ 1,
interaction = Strauss(0.1))
Fits Strauss model with δ = 0.1.
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Second-Order Properties
Modeling Inhibition: Strauss Process
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Stationary Strauss process
First order term:
beta
563.0123
Interaction: Strauss process
interaction distance: 0.1
Fitted interaction parameter gamma: 0.0389
Relevant coefficients:
Interaction
-3.247058
For standard errors, type coef(summary(x))
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Wrapping Up
Where are we?
1 Last Words about the Frequency Domain
2 Point Processes in Time and Space
3 Inhomogeneous Poisson Processes
4 Second-Order Properties
5 Wrapping Up
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Wrapping Up
Summary
• Point processes are distinguished by the randomness of the locations.
• Poisson processes are the simplest model for point processes.
• More complex models (Neyman-Scott, Strauss) can capturesecond-order interactions.
• Use the spatstat package in R!
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Wrapping Up
Homeworks
• Homework 3 deadline extended to Friday.
• Winners of prediction competition will be announced next Monday.
• Homework 4a and b: students doing a project need only complete oneof these.
• Homework 4a will be posted Wednesday, 4b by the end of the week.
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Wrapping Up
Projects
• Please submit your project proposal if you haven’t done so already!
• I will start responding to them today.
Dennis Sun Stats 253 – Lecture 9 July 21, 2014 Vskip0pt