The Waiting Time Distribution of Competing Patterns in Markov-Dependent Bernoulli Trials
Abstract
:1. Introduction and Literature Review
2. Definitions and Preliminaries
Probability Generating Function
3. Competing Patterns in High-Order Markov-Dependent Bernoulli Trials
3.1. Step 1. Experimental Design and Settings
3.2. Step 2. Stopping Paths
- (i)
- The number of competing patterns in each path satisfies:
- (ii)
- The number of paths in the set is given by (combinatorial considerations):(Note that the summing is over all patterns excluding ).
- (iii)
- Since including distinct patterns, we have:
3.3. Step 3. The Waiting Time Distribution
3.3.1. The Functions
3.3.2. The Functions
4. Algorithm and Examples
4.1. The Algorithm
- The parameter the space state the -square transition matrix .
- The competing patterns and their appearances:
- Calculate the steady-state probability vector satisfying
- Set
- Build the state space define the absorbing states
- Calculate the vector and the matrices and Derive (the matrix can be helpful with the appropriate eliminations).
- Apply (G14) to obtain and
4.2. Example 2
- (i)
- Two occurrences of the set of patterns i.e., either 101 occurs twice, or 11 occurs twice, or 101 and then or vice versa, all occurrences are not necessarily consecutive.
- (ii)
- Two (not necessarily consecutive) occurrences of two consecutive .
4.3. Example 3
- (1)
- Two occurrences (not necessarily consecutive) of an abnormal rate after a normal or abnormal rate. An abnormal rate that appears twice (even after a normal one) may indicate a cardiac problem and is worth checking. According to the company’s experience, the need to record two consecutive outcomes helps determine whether the result is part of an ongoing trend or a single event.
- (2)
- Three (not necessarily consecutively) sequences of an abnormal rate followed by two normal results. This situation may be caused by a malfunction of the device, or as a result of other factors that caused a positive change in the heart rate.
- (i)
- Two occurrences of 11, or two occurrences of or first 01 and then or vice versa—all occurrences are not necessarily consecutive.
- (ii)
- Two (not necessarily consecutive) occurrences of the pattern .
5. Summary and Future Research
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
Appendix A
- 1.1.
- Define: r be the Markov-order chain.
- 1.2.
- Generate the set of states ().
- 1.3.
- Define transition probabilities, , Build the matrix .
- 1.4.
- Define the probability vector Obtain by solving
- 1.5.
- For each competing pattern :
- Define
- Let
- Define -the number of appearances needed.
- 1.6.
- Let
- 2.1
- For
- Define to be the set of all paths that terminate the experiment via
- Calculate using
- For each generate the series of patterns .
- 2.2
- Build where ().
- 2.3
- Obtain distinct sets of absorbing states with regard to .
- 2.4
- Obtain the transient set of states,
- 2.5
- Derive —the transition probability matrix among the states in
- 2.6
- Derive —the absorbing probability matrix into states in
- 2.7
- Construct the Markov probability matrix as follows (Figure A1):Figure A1. The transition probability matrix.( is the identity matrix, and is the zero matrix, all with the appropriate dimensions).
- 3.1
- For :
- Derive and using a probability product of maximum length b (a probability tree diagram may be useful).
- Derive the vector (the matrix may be helpful).
- Compute the vector (a probability tree diagram may be useful)
- Derive and by:
- Use the law of total expectation to obtain:
- 3.2
- The final is obtained by
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Px,y | ΩY | ||||||
---|---|---|---|---|---|---|---|
010 | 011 | 101 | 110 | α1 | α2 | ||
ΩY | 010 | 0 | 0 | p4 | 0 | p3 | 0 |
011 | 0 | 0 | 0 | p7 | 0 | p8 | |
101 | p5 | p6 | 0 | 0 | 0 | 0 | |
110 | 0 | 0 | p4 | 0 | p3 | 0 | |
α1 | 0 | 0 | 0 | 0 | 1 | 0 | |
α2 | 0 | 0 | 0 | 0 | 0 | 1 |
ΩY | |||||||||
---|---|---|---|---|---|---|---|---|---|
000 | 001 | 010 | 011 | 100 | 101 | 110 | 111 | ||
2-state trials | 00 | p1 | p2 | ||||||
01 | p5 | p6 | |||||||
10 | p3 | p4 | |||||||
11 | p7 | p8 |
ΩY | |||||||||
---|---|---|---|---|---|---|---|---|---|
000 | 001 | 010 | 011 | 100 | 101 | 110 | 111 | ||
ΩY | 000 | p1 | p2 | ||||||
001 | p5 | p6 | |||||||
010 | p3 | p4 | |||||||
011 | p7 | p8 | |||||||
100 | p1 | p2 | |||||||
101 | p5 | p6 | |||||||
110 | p3 | p4 | |||||||
111 | p7 | p8 |
ΩY | |||||||||
---|---|---|---|---|---|---|---|---|---|
000 | 001 | 010 | 011 | 100 | 101 | 110 | 111 | ||
2-state trials | 00 | (p1)3 | (p1)2p2 | p1p2p5 | p1p2p6 | p2p5p3 | p2p5p4 | p2p6p7 | p2p6p8 |
01 | p5p3p1 | p2p5p3 | (p5)2p4 | p5p4p6 | p6p7p3 | p6p7p4 | p6p8p7 | p6(p8)2 | |
10 | p3(p1)2 | p3p1p2 | p2p5p3 | p3p2p6 | p5p4p3 | p5(p4)2 | p6p7p4 | p4p6p8 | |
11 | p7p3p1 | p7p3p2 | p7p4p5 | p7p4p6 | p8p7p3 | p8p7p4 | (p8)2p7 | (p8)3 |
ΩY | |||||||||
---|---|---|---|---|---|---|---|---|---|
000 | 001 | 010 | 011 | 100 | 101 | 110 | 111 | ||
2-state trials | 00 | (p1)3 | (p1)2p2 | p1p2p5 | p1p2p6 | p2p5p3 | p2p5p4 | p2p6p7 | p2p6p8 |
01 | p5p3p1 | p2p5p3 | (p5)2p4 | p5p4p6 | p6p7p3 | p6p7p4 | p6p8p7 | p6(p8)2 | |
10 | p3(p1)2 | p3p1p2 | p2p5p3 | p3p2p6 | p5p4p3 | p5(p4)2 | p6p7p4 | p4p6p8 | |
11 | p7p3p1 | p7p3p2 | p7p4p5 | p7p4p6 | p8p7p3 | p8p7p4 | (p8)2p7 | (p8)3 |
ΩY | |||||||||
---|---|---|---|---|---|---|---|---|---|
000 | 001 | 010 | 011 | 100 | 101 | 110 | 111 | ||
2-state trials | 00 | (p1)3 | (p1)2p2 | p1p2p5 | p1p2p6 | p2p5p3 | p2p5p4 | p2p6p7 | p2p6p8 |
01 | p5p3p1 | p2p5p3 | (p5)2p4 | p5p4p6 | p6p7p3 | p6p7p4 | p6p8p7 | p6(p8)2 | |
10 | p3(p1)2 | p3p1p2 | p2p5p3 | p3p2p6 | p5p4p3 | p5(p4)2 | p6p7p4 | p4p6p8 | |
11 | p7p3p1 | p7p3p2 | p7p4p5 | p7p4p6 | p8p7p3 | p8p7p4 | (p8)2p7 | (p8)3 |
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Moshkovitz, I.; Barron, Y. The Waiting Time Distribution of Competing Patterns in Markov-Dependent Bernoulli Trials. Axioms 2025, 14, 221. https://doi.org/10.3390/axioms14030221
Moshkovitz I, Barron Y. The Waiting Time Distribution of Competing Patterns in Markov-Dependent Bernoulli Trials. Axioms. 2025; 14(3):221. https://doi.org/10.3390/axioms14030221
Chicago/Turabian StyleMoshkovitz, Itzhak, and Yonit Barron. 2025. "The Waiting Time Distribution of Competing Patterns in Markov-Dependent Bernoulli Trials" Axioms 14, no. 3: 221. https://doi.org/10.3390/axioms14030221
APA StyleMoshkovitz, I., & Barron, Y. (2025). The Waiting Time Distribution of Competing Patterns in Markov-Dependent Bernoulli Trials. Axioms, 14(3), 221. https://doi.org/10.3390/axioms14030221