Download Slides 17: Waiting for Disaster (PDF, 135 KB)

Stochastic Models
Waiting for Disaster
Walt Pohl
Universität Zürich
Department of Business Administration
May 16, 2013
Survival Analysis
We will use a few concepts from survival analysis.
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Stochastic Models
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Survival Analysis
We will use a few concepts from survival analysis.
Survival analysis models waiting for an event to happen.
It was first developed in medicine, where it was used to
model how long terminally ill patients had to live. The
associated terminology is grim.
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Stochastic Models
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Survival Analysis
We will use a few concepts from survival analysis.
Survival analysis models waiting for an event to happen.
It was first developed in medicine, where it was used to
model how long terminally ill patients had to live. The
associated terminology is grim.
We will use it to model how long companies have to live,
so for us the terminology fits.
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Stochastic Models
May 16, 2013
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Survival
Let T be a random variable that represents the amount
of time until an event occurs (death, for example), and
let
Walt Pohl (UZH QBA)
Stochastic Models
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Survival
Let T be a random variable that represents the amount
of time until an event occurs (death, for example), and
let
F (t) = P(T < t)
be its CDF.
Walt Pohl (UZH QBA)
Stochastic Models
May 16, 2013
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Survival
Let T be a random variable that represents the amount
of time until an event occurs (death, for example), and
let
F (t) = P(T < t)
be its CDF.
Then the probability that the event doesn’t happen
before time t, F̂ , is
Walt Pohl (UZH QBA)
Stochastic Models
May 16, 2013
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Survival
Let T be a random variable that represents the amount
of time until an event occurs (death, for example), and
let
F (t) = P(T < t)
be its CDF.
Then the probability that the event doesn’t happen
before time t, F̂ , is
F̂ (t) = P(T > t) = 1 − F (t).
Walt Pohl (UZH QBA)
Stochastic Models
May 16, 2013
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Survival
Let T be a random variable that represents the amount
of time until an event occurs (death, for example), and
let
F (t) = P(T < t)
be its CDF.
Then the probability that the event doesn’t happen
before time t, F̂ , is
F̂ (t) = P(T > t) = 1 − F (t).
F̂ is known as the survival function for T .
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Stochastic Models
May 16, 2013
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Hazard Rate
Before the event occurs, the only information we have is
how long we’ve already waited. This makes the
conditional probability distribution,
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Hazard Rate
Before the event occurs, the only information we have is
how long we’ve already waited. This makes the
conditional probability distribution,
P(T ≤ t 0 |T > t),
interesting.
Walt Pohl (UZH QBA)
Stochastic Models
May 16, 2013
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Hazard Rate
Before the event occurs, the only information we have is
how long we’ve already waited. This makes the
conditional probability distribution,
P(T ≤ t 0 |T > t),
interesting.
In discrete time, all we need to know is that probability
of the event happening at time t + 1, given that it hasn’t
happened at time t.
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Stochastic Models
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Hazard Rate, cont’d
The analoguous concept in continuous time is the hazard
rate, which is the rate of change in the probability of the
event happening in the next instant, given that it has not
yet happened.
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Stochastic Models
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Hazard Rate, cont’d
The analoguous concept in continuous time is the hazard
rate, which is the rate of change in the probability of the
event happening in the next instant, given that it has not
yet happened.
The hazard rate is defined as
P(t < T < t + ∆t|T = t)
f (t)
lim
=
,
∆t→0
∆t
1 − F (t)
where f is the pdf of T .
Walt Pohl (UZH QBA)
Stochastic Models
May 16, 2013
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Hazard Rate, cont’d
The analoguous concept in continuous time is the hazard
rate, which is the rate of change in the probability of the
event happening in the next instant, given that it has not
yet happened.
The hazard rate is defined as
P(t < T < t + ∆t|T = t)
f (t)
lim
=
,
∆t→0
∆t
1 − F (t)
where f is the pdf of T .
Given the hazard rate h(t), we can reconstruct the
original distribution,
F (t) = 1 − e −
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Rt
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0
h(s)ds
.
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Exponential Distribution
Of particular interest is the case of a constant hazard
rate, h(t) = λ, which describes the exponential
distribution,
P(t ≤ T ) = 1 − e −λt .
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Stochastic Models
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Exponential Distribution
Of particular interest is the case of a constant hazard
rate, h(t) = λ, which describes the exponential
distribution,
P(t ≤ T ) = 1 − e −λt .
Lots of things follow an exponential distribution. For
example, the waiting time for the first jump in a Poisson
(or compound Poisson) process is exponentially
distributed.
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Stochastic Models
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Estimation and Prediction
The main purpose of survival analysis is prediction:
to predict the probability of the event happening
within a prescribed period, given that it has not yet
happened.
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Estimation and Prediction
The main purpose of survival analysis is prediction:
to predict the probability of the event happening
within a prescribed period, given that it has not yet
happened.
If we have a large prior history, we can use that
history to fit a model for the event.
Walt Pohl (UZH QBA)
Stochastic Models
May 16, 2013
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Estimation and Prediction
The main purpose of survival analysis is prediction:
to predict the probability of the event happening
within a prescribed period, given that it has not yet
happened.
If we have a large prior history, we can use that
history to fit a model for the event.
This is how survival analysis is used to calculate actuarial
rates for life insurance, for example. But what if we
don’t have a long history of data?
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Stochastic Models
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