Robert Israel wrote:
> Luna Moon <lunamoonmoon@[EMAIL PROTECTED]
> writes:
>=20
>> Hi all,
>>=20
>> How do I introduce correlation to two otherwise standard Poisson
>> processes?
>=20
> Here's one way. Note that the sum of two independent Poisson
> processes is a Poisson process.
> If X(t), Y(t) and Z(t) are independent Poisson processes with rates
> lambda_x, lambda_y and lambda_z respectively, then X(t) + Y(t) and
> X(t) + Z(t) are Poisson processes with rates lambda_x + lambda_y and
> lambda_x + lambda_z, and
> Cov(X(t)+Y(t), X(t)+Z(t)) =3D Var(X(t)) =3D lambda_x
> so the correlation coefficient of X(t)+Y(t) and X(t)+Z(t) is
> lambda_x/sqrt((lambda_x+lambda_y)(lambda_x+lambda_z)).
Three other ways:
(i) complimentary to the above ... take an underlyiing Poisson process =
.... create a first "observed" series by accepting in point in the =
underlying process at random (ie with a given probability) ... creare a =
second "observed" series by repeating the random deletion =
(independently) on the underlying process;
(ii) if you are not being strict about there term "Poisson process", so =
that variable rates are allowed, you could use doubly stochastic models =
where you allow the rate-processes to be correlated;
(iii) You could start from "waiting-time for next event" representation, =
and generate to sequences of exponential rv's, which are pairwise =
dependent across the different sequences.
David Jones
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