hypothesis about the original event sequence. If the values are far from zero, this is a sign that we need to revisit our assumptions and possibly discard the i.i.d. We are going to estimate the expectations 𝔼 M(Y) and use the estimates as scores: Values close to zero can be interpreted as evidence speaking for geometric distribution with parameter p. This relation can be used to test whether a given sample of waiting times belongs to the geometric distribution with parameter p. Holds, if and only if Y∈ Geom( p) and k>0. To formally define the waiting time sequence y = (y ,…,y ) based on x, consider the sequence of indices I=(i ,i ,…), such that for each i∈ I we have x =1. Note that the waiting time calculation discards the initial and trailing zeros in the event sequence x. Yields the following sequence of waiting times. To rigorously investigate the question of serial dependence in the sequence X based on the observation x, we can look at the distribution of waiting times between the events. Typical examples are the “winning or lucky streaks” experienced by gamblers in casinos. We often tend to see serial dependence in situations where there is none. How can we check whether our events occur independently of each other and gather evidence that there is no serial dependence between the event times? We say that “an event” occurred at time i, if x = 1. To set an appropriate context for the serial dependence detection problem, let me remark that random binary sequences x are often associated with discreet observations of a system with two states. This is because the expectation of X is 𝔼 X = p. If we assume that the elements of x are independent samples from a fixed Bernoulli random variable X, then we can estimate the probability p by calculating the average of x, x, …, x. The random variables X are independent.The random variables X have the distribution Ber( p) (Bernoulli distribution with the success probability p).This is a very difficult and deep problem, and in this blog post I am going to focus on collecting evidence to support or reject the following two basic hypotheses: a finite sequence of zeros and ones (x, x., x). This investigation should be based on a sample x drawn from X, i.e. Given a sequence of random variables X = (X, X, …, X) taking values in the set and a probability p∈(0,1), I would like to investigate the serial dependencies between the elements of X.
How to test dependency scoring methods using simple Monte-Carlo experiments involving Markov Chains with a known dependency structure.How to calculate the Meixner Dependency Score for a waiting times sequence to quantify the dependence strength.How to derive the Meixner polynomials from the Geometric distribution.How the problem of serial dependency measurement can be formulated in statistical terms.The Meixner Dependency Score is easy to implement and is based on the orthogonal polynomials associated with Geometric distribution.Īfter reading through this blog post, you will know: In this blog post, I am going to develop and test a scoring method for quantifying the dependence strength in random binary sequences. For that, we need to take a close look at the dependence structure of the binary sequence under consideration. For example, the occurrence of an incident in a production system should not make the system more incident prone. In many of those situations, it is necessary to ensure that the tracked events occur independently of each other. Value-at-Risk exceedance indicators in financial market risk models.Binary sequences often encode the occurrence of random events: Primary outputs generated by most random number generators are binary. Random binary sequences are sequences of zeros and ones generated by random processes. In this blog post, I am going to investigate the serial (aka temporal) dependence phenomenon in random binary sequences.
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