2.4 Resolución de ambigüedad de sentidos de palabras
2.4.3 El rol del contexto
First, we present the basics of order statistics, which are applied in many approaches of travel time modeling. As our central approach bases on stochastic processes, we subsequently explain the important concepts in this regard.
Order statistics
Order statistics are used to denote the it h smallest or it h largest instant within a sample. Suppose X is a continuous random variable with a cumu-lative distribution function (cfd) FX(x) and a probability density function (pdf ) fX(x) and the independent, identically distributed random samples X1, X2, ··· Xn. The realization of these variables x(i ), i = 1,··· ,n can be or-dered such that x(1)≤ x(2)≤ · · · ≤ x(n). The order statistics X(1), X(2), . . . , X(n)
are random variables of the ordered values. The it h entry of the ordered sample is referred to as the it horder statistics. Note that x(i )is the realiza-tion that of the it horder statistics X(i )but, in general, not the realization of random variable Xi.
For the cumulative distribution function of the first order statistics we know:
FX(1)(x) = P(X(1)≤ x) = 1 − P (X(1)> x). (3.1) For the smallest value of the sample, X(1)> x holds if and only if X(i )> x for all i = 1,2,··· ,n. As the individual X(i )are stochastically independent, the smallest order statistics is (Devore and Berk 2012, p. 272 f.):
FX(1)(x)= 1 − P (X(1)> x)
= 1 − P (X1> x)P (X2> x) · · · P (Xn> x)
= 1 − (1 − F (x))n.
(3.2)
Devore and Berk (2012, p. 276 f.) present a clear derivation of the it horder statistics and determine the pdf of the it horder statistics by:
fX(i )(x)= n f (x)
The cdf of the it horder statistic is (Zörnig 2016, p. 178):
FX(i )(x)=
The range of the sample, R, is the difference between the largest and the smallest value, i.e., R = X(n)−X(1). According to Guttman, Wilks and Hunter (1982) the cdf and pdf, H (r ) and h(r ), respectively, of the range are defined as:
Stochastic processes are used to describe real procedures or system behav-ior over time. Usually, all possible states and the transition from one state to another are analyzed. A stochastic process is a family of random vari-ables {Xt, t ∈ T } with the index set T. T usually is a set of points in time when the system is observed. For T = N0the stochastic process is said to be discrete-time process and for T = [0,∞) a continuous-time process. In
the context of this work, discrete-time processes are relevant which is why we refrain from going into detail for the second group. The possible states the random variables Xtcan take are represented by the state space S. This means, a stochastic process is a sequence of random variables X0, X1, ···
that take values from S and are observed at the points in time T . The dif-ference between a stochastic process and a random sample of X is that the sample values X1, ··· , Xn are independent of each other whereas the ran-dom variables of the stochastic process are not.
A common example is queuing of customers waiting to be served at a counter. In this example, a random variable (Xt) is used to describe the state of the queuing system, which is the number of customers waiting in the queue. All possible numbers of waiting customers that can be observed is S = N0. Now consider a point in time t, where three waiting customers are observed (Xt= 3). The next point in time, t + 1 (after a state transition), depends on the previous state: If one customer arrives without completing any of the already waiting customers in service, the new number of waiting customers is four (Xt= 4). On the other hand, if no customer has arrived and the ongoing service was completed, the number of customers waiting is reduced to Xt= 2. Obviously t + 1 is dependent on t .
If the state of a stochastic process, t , is only dependent on state t − 1 and not of previous ones, this is called Markov Property. A process with the Markov Property is also called memoryless.
Discrete Time Markov Chains
A stochastic process {Xt, t ∈ T } taking values in a countable state space S is called Markov chain, if for a all points in time t ∈ T and all states i0, ··· ,it −1, it, it +1∈ S the following is true (Waldmann and Stocker 2013, p. 11):
P (Xt +1= it +1|X0= i0, ··· , Xt −1= it −1, Xt= it)
= P (Xt +1= it +1|Xt= it) (3.7)
This represents the Markov Property and is expressed through the transi-tion probabilities:
The conditional probability P (Xt +1= it +1|Xt= it) is called transition prob-ability of the process and represents the probprob-ability that for a given state it
the following state it +1is realized. This means that the transition probabil-ity P (Xt +1= it +1|Xt= it) from state itinto state it +1is only dependent of it, but of no other state prior to it .
For transition probabilities independent of the time of the transition t , the Markov chain is said to be homogeneous. The evolution of a homogeneous Markov chain can be described by
1. Initial probabilityπi(0) := P(X0= i ), i ∈ S,
The transition matrix is a stochastic matrix that describes the transitions of a Markov chain. As the entries of the square matrix represent transition probabilities, all entries are greater or equal zero and and all rows sum up to one. tran-sition probability which denotes the probability of going from state i to state j in n transitions. πj(t ) is the marginal distribution over states at time n, the probability distribution of the random variable {Xt, t ∈ T } is described as:
p(t )i j is obtained by adding up the probabilities of all sequences of states pi ,i1...pit −1, j (i , i1...it −1, j ∈ S), beginning in i and ending up in j after t steps. As p(t )i j is a conditional probability, it is multiplied by the initial prob-ability of state i in equation 3.8. Summarizing all initial states i ∈ S, ac-cording to the law of total probability the unconditional probabilityπj(t )
can be calculated (Waldmann and Stocker 2013, p. 15f ). We can interpret the state probabilitiesπj(t ) for all j ∈ S as a row vector π(t) giving the state distribution at time t.
It is possible to analyze the evolution of Markov chains for t → ∞ to de-rive a stationary distribution. For this purpose, we want to discuss some properties of Markov chains:
A state i is said to communicate with state j , if they are accessible from each other. A Markov Chain is said to be irreducible, if all states commu-nicate, i.e., for every state i there is a positive probability of going into state j . If an irreducible Markov chain has a finite state space, it has a unique stationary distribution (Waldmann and Stocker 2013, p. 36). A state i of a Markov chain is recurrent, if it has a finite return time, so with a probabil-ity of one the chain returns to state i after a finite number of transitions. A state i is called aperiodic, if the transition to the same state has a non-zero probability, which is pi i> 0, i ∈ S. An irreducible Markov chain is aperiodic if it has at least one aperiodic state (Waldmann and Stocker 2013, p. 41).
Let {Xt, t ∈ T } be an irreducible, aperiodic Markov chain with a stationary distributionπ, then π(t) converges to the stationary distribution for t → ∞.
The stationary distribution is independent of the initial distribution of the process. This means the stationary distribution is reached regardless of the starting point. A distribution is said to be stationary, if
πj=X
i ∈I
πip(t )i j, j ∈ S, ∀t ∈ N. (3.9)
This could also be expressed as the convergence of the transition matrix in the following way:
t →∞limpi j(t )= πj> 0 for all j ∈ S (3.10)
Let ui ∈ [0, 1], i ∈ S be a probability distribution of Xt. π is given as the solution of the following system of linear equations
uj = X
i ∈I
uipi j, j ∈ S (3.11)
ui ≥ 0, i ∈ S (3.12)
X
i ∈S
ui = 1 (3.13)
πj is the probability that the system is in state j for t → ∞ and, as can be seen from equation 3.10, is independent of the initial state. After a sufficient period of time,πj can also be interpreted as the mean proportion of time the system is in state j .