Constitución del Ecuador

So far I have assumed that the DCEG models a dynamic process where measure- ments are taken at regular intervals, such as daily or monthly. For example in the DCEG of the flu example in Figure 5.3, the individual is, every month, at risk of catching flu: If he catches flu, he traverses through the rest of the DCEG ending up either atw₁ or back atw0; if not he loops back directly to w0. In this section I

will extend the above methodology so that time spent until an event occurs can be modelled directly.

Going back to the tree representation of a problem call the time an individual stays in a situationsitheholding timeHsi associated with this situation. Further,

let theconditional holding timeassociated with each edge esik, k= 1, . . . , mi in

the tree be denoted by Hsik. This describes the time an individual stays at a

situation si given that he moves along the edge esik next. Analogously to this,

holding times on the positions of the associated DCEG_Dcan be defined as follows: LetHw be the random variable describing the holding time at position w2W(D)

in the DCEG andHwk, k= 1, .., mw the random variable describing the conditional

holding time onwgiven the individual moves along the edge ewk next.

In a DCEG the time an individual stays in a particular position w, with a loop into itself, simply follows a geometric distribution. So, if we assume that thekth edge ofwloops back intow, then the probability that an individual stays in position

wfortmonths is equal to⇡t

wk⇥(1 ⇡wk), whereewk=e(w, w). Further, it has been

assumed that once an individual catches flu, only the events of taking treatment, recovering, and receiving a vaccine are recorded and not the time until these events occur. These could, for example, be recorded retrospectively when measurements are taken a month later. The holding time distributions on a position without a loop into itself are therefore degenerate.

As in the flu example, the processes to be modelled are often event driven and these are well represented within a tree and hence a DCEG: When moving from one position to another the individual transitions away from a particular state into a di↵erent state associated with a new probability distribution of what will happen next. In these scenarios, interest commonly lies not only in the transition probabilities through the graph but also in the amount of time spent at each position. Hence, rather than measurements being taken at regular time-steps it is more natural to think of the measurements being taken when an event happens, where the time until the event happens is recorded. For example, the individual may not record whether he catches flu or not every month but instead monitor the time spent atw0

not catching flu, until one day he falls ill. Similarly, the time until seeing the doctor for treatment or the time until recovery may be of di↵erent lengths and so he spends di↵erent amounts of time at each position in the DCEG. In order to incorporate this into the graph conditional holding time distributions can be attached to each edge in the DCEG.

By the definition of a DCEG, two situations are in the same stage whenever their emanating edges have the same probability. Similarly, it is assumed that, the conditional holding time depends only on the current stage and the next edge the individual moves along but not on the previous path up to reaching the current stage.

Definition 26. A DCEG is time-homogeneous whenever two situations that are in the same stage also have the same conditional holding time distributions on their edges, i.e. the holding times are independent of the path taken through which the stage is reached. Denote the random variable of the conditional holding time asso- ciated with each stage by Huk, k= 1, . . . , mu.

implies that when two positions are in the same stage u then their conditional holding time distributions are also the same. Note that an individual may spend a certain amount of time in position wi 2 u before moving along the kth edge to a

positionwj which is in the same stage. So an individual may make a transition into

a di↵erent position but arrive at the same stage.

I further assume throughout that the conditional probabilities of going along a particular edge after reaching a stage, do not vary with previous holding times. In the flu example this would mean that the time until catching flu does not ef- fect the probability of taking treatment and the probability of recovery without treatment. Similarly, the holding times are assumed to be independent of previous holding times. So, for example, the time until recovery is independent of the time to catching flu. Contexts where the holding time distribution may a↵ect the transition probabilities and future holding times can provide an interesting extension to the DCEG, which, however, will not be covered in this thesis. Under these assumptions a time-homogeneous DCEG with holding times can therefore be defined as follows:

Definition 27. A DCEG with holding times, D = (V(D), E(D)) is a DCEG with no loops from a position into itself and with conditional holding time distri- butions conditioned on the current stage, u, and the next edge, euk, to be passed

through:

Fuk(h) =P(Huk h|u, euk), h 0, u2U, k= 1, . . . mu.

Hence Fuk(h) describes the time an individual stays in a position w 2 u before

moving along the kth edge, ewk. A frame around a position in D indicates that

holding time distributions have been attached to its associated edges.

Consequently, given a position w ₂ W(_D) is reached, the joint probability of staying at this position for a time less than or equal tohand then moving along thekth edge is

P(Hwk h, ewk|w) =P(Hwk h|w, ewk)P(ewk|w) =Fuk(h)⇡uk, w2u. (5.1)

Finally, the joint density ofewk and h is

p(ewk, h|w) =⇡ukfuk(h),

wherefukis the pdf or pmf of the holding time at stageugoing along edgeewk, w2u

next. A time-homogeneous DCEG _D with stage partition U(_D) is therefore fully specified by its set of conditional holding time distributions

process of a DCEG could be extended to include the elicitation of holding time dis- tributions for each position and its associated edges. The conditional holding times could in general take any distribution. For example, an exponential holding time distribution may be plausible if it is assumed that the event will occur at a constant rate. Other plausible distributions would be alternative survival distributions such as the Weibull distribution, when the occurrence of the event is expected to increase or decrease with time, or a log-normal distribution or a log-logistic distribution, when a unimodal event rate is appropriate. I will postpone further discussion of the holding time distribution for this example to the end of this chapter when looking at the learning of the parameters of the DCEG.

Example 15. Consider the following variant of the flu example represented by the infinite tree, T⇤_{, in Figure 5.4. Instead of measuring every month whether the} individual catches flu, the individual will spend a certain amount of time ats0 before

moving along the tree. Hence the second edge emanating from s0 in Figure 5.2 and

its entire subtree have been removed. As before, it is assumed that the probability of

s0 s1 s3 l9 No surv_ival s6 l8 Get flu vaccine s7 Catch flu s11... Resume normal li fe Recover y No tre_at m ent s2 l5 Get_flu vac cine s4 Catch flu s10... Resume normal life Treat men t Catch flu

Figure 5.4: The beginning of the infinite tree,T⇤_{, for the flu example where catching} flu is represented by the time spent at the root vertex

catching flu and the decision to take treatment does not depend on whether the flu has been caught before. Also, recovery with or without treatment is assumed not to a↵ect the probability of receiving a vaccine. The corresponding DCEG is given in

Figure 5.5 with the stages and positions given by

w0 =u0 ={s0, s4, s7, . . .}, w1 =u1 ={s1, s10, s11, . . .},

w2 =u2={s2, s6, . . .}, w3 =u3 ={s3, . . .}, w1={l5, l8, l9, . . .}. (5.2)

In comparison to Figure 5.3 the loop from w0 into itself has been removed. Instead

w0 Catch flu / /w1 Treatment / / No treatment $ $ w2 Resume normal life

v v Get vaccine ) )_w 1 w3 Recovery O O No survival 5 5

Figure 5.5: The DCEG with holding times for the flu example, where catching flu is described by the time spent at the root vertex

the time spent at w0 is described by the holding time at position w0. Similarly, the

time until treatment is taken or not, the time until recovery or death and the time to receiving the flu vaccine or not are of interest and holding time distributions can be defined on these. Hence, visually the only di↵erence between Figures 5.3 and 5.5 is that the positions have a frame around them to illustrate that the conditional holding times are of interest andw0 no longer contains a loop into itself.