Temporal Point Processes the Conditional Intensity Function

Transcription

1 Temporal Point Processes the Conditional Intensity Function Jakob G. Rasmussen Department of Mathematics Aalborg University Denmark February 8, /10

2 Temporal point processes A temporal point process is a point process defined on the time line. Examples of point (= events) patterns: Earthquakes or other disasters Visits at a server Defective products made at a factory Accidents at a road junction Distinguishing features of temporal point processes: S = R the events have the natural order of time Temporal point processes can also have marks: Magnitudes or positions of earthquakes Time spent by a visitor at a server Type or number of defects on a product Severity or cost of an accident 2/10

3 Definition of an unmarked temporal point process Definitions: 1. a random sequence T on R 2. a random sequence of intervals T on [0, ) 3. a random counting measure N on R Interpretations: 1. Event times: T = (...,t 1,t 2,...) is a sequence of event times. 2. Interevent times: T = (...,τ 1,τ 2,...) is a sequence of non-negative random variables interevent times. 3. Number of events: N(A) counts the number of events falling in any Borel set A R. All three definitions are equivalent. Simple (or orderly) point process: We only consider cases where no events occur at the same time with probability one. In practice we often consider a point process defined on an interval of finite length. 3/10

4 Evolutionarity, history and interevent times Evolutionarity: what happens in the present depends only on the past, not the future. History: H t = (...,t 1, t 2,...,t n ) is a vector of all past events t i < t. We can define a point process by the distribution of all interevent times, τ n = t n t n 1, with density function f (t) = f (t H t ) The is a sloppy (but convenient) notation for conditioning on the history. Likelihood function: f (...,t 1, t 2,...) = i f (t i H ti ) = i f (t i ) 4/10

5 A simple examples using interevent time distributions Renewal process: A point process where all the interevent times have i.i.d. distributions. Homogeneous Poisson process: Special case of renewal process with exponential distributed interevent time. Wold process: generalization of renewal process where the interevent times is a homogeneous Markov chain. Note: here f (t) only depends on the last event before time t (renewal/poisson process) or two last events (Wold process). However, more complicated processes are not handled well by specifying the interevent time distribution. 5/10

6 The conditional intensity function Conditional intensity (/risk/rate/hazard) function: λ (t) = f (t) 1 F (t) f (t) is the density function and F (t) is the distribution function of the interevent times. Interpretation: λ (t)dt E[N(dt) H t ] Integrated conditional intensity function: Λ (t) = t 0 λ (s)ds The conditional intensity function for a renewal or Wold process can easily be found using the above equation. (inhomogeneous) Poisson process: if λ (t) = λ(t) is independent of H t we get the Poisson process with intensity λ(t) 6/10

7 Example: Hawkes process Hawkes process: λ (t) = µ + t i <t γ(t t i ) where γ(t) 0, e.g. γ(t) = α exp( βt) This produces clustered point patterns. This model can also be defined as a model with Immigrants: T0 Poisson(µ, R) 1. order offspring: Each immigrant ti T 0 gives birth to T i,1 Poisson(αγ( t i ), R), and T 1 = i T i,1 2. order offspring: Each immigrant tj T 1 gives birth to T j,2 Poisson(αγ( t j ), R), and T 2 = j T j,2 And so on... µ is called immigration rate, γ is called offspring rate This is a model for reproducing populations, e.g. earthquakes, plants or viruses. 7/10

8 Can we choose any function as conditional intensity? Proposition: A conditional intensity function λ (t) uniquely defines a point process if it satisfies the following conditions for all t and all possible point patterns before t: 1. λ (t) is well-defined and non-negative, 2. the integral t t n λ (s)ds is well-defined, 3. t t n λ (s)ds for t. Lemma: ( t ) f (t) = λ (t)exp ( F (t) = 1 exp λ (s)ds t n ) λ (s)ds t n For finite point processes (i.e. a finite number of points with probability one), we drop item 3. in the proposition. t 8/10

9 The marked case History: In the marked case the marks are also included in the history H t = (...,(t 1, κ 1 ), (t 2, κ 2 ),...,(t n, κ n )). Definition: Conditional intensity function: Thus λ (t, κ) = λ (t)f (κ t) λ (t, κ) = λ (t)f (κ t) = f (t)f (κ t) 1 F (t) = f (t, κ) 1 F (t) Note: F (t) is only the distribution function for time (not mark), but may still depend on previous marks. Unpredictable marks: f (κ t) does not depend on the past. Independent marks: f (κ t) does not depend on the past or future (= future does not depend on marks). 9/10

10 Example: ETAS model The ETAS model is a marked Hawkes process with conditional intensity function ( λ (t, κ) = µ + α ) e βκ i e γ(t t i) δe δκ. t i <t Or equivalently, with ground intensity λ (t) = µ + α t i <t e βκ i e γ(t t i), and (conditional) mark density f (κ t) = δe δκ. 10/10