Casestudie Breakdown prediction Contell PILOT - Transumo

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⎛ 1 ⎜ ⎜1− p ⎜ 0 P = ⎜ ⎜ 0 ⎜ ⎜ 0 ⎝ 0 0 0 1− p 0 0 0 0 p 0 1− p 0 0 0 0 p 0 1− p 0 0 0 0 p 0 0 0 0 0 0 p 1 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ Formula 5-14: Sample Transition Probability Matrix Figure 5-5: Sample Transition Probability Graph Up to now, the transition probability matrix only made it possible to obtain the probability for a single change. But also important are probabilities of several changes in a row, as pictured in Formula 5-15. ([Beichelt97], p. 147) p ( m) ij = P( X + = j | X = i) m = 1,2,... n m n Formula 5-15: Transition Probabilities of Several Changes in a Row (m) p ij symbolizes the probability that state i will change to state j after m steps. Apparently, p = is complied. The calculation of m > 1 can be done by using the (1) ij p ij formula of Chapman-Kolmogorov, which is pictured in Formula 5-16. ([Beichelt97], p. 147) m ( r ) ( m−r ) pij = ∑ pik pkj r = 1,2, K , m −1 k∈I Formula 5-16: Formula of Chapman-Kolmogorov Using the knowledge that one state of the countable state space has to be taken after r steps in combination with the knowledge of the total probability and the Markov property leads to an easy argumentation. 54 As a result the transition probability matrix for r steps is determined by multiplying r times the matrix by itself. This enables a simplified version of Formula 5-16. ([Beichelt97], p. 148) 54 See ([Beichelt97], p. 147) for details 70
P ( m) = P ( r ) ⋅ P ( m−r ) with P ( m) = ( m) ( p ) m = 1,2, K ij Formula 5-17: Formula of Chapman-Kolmogorov (Simplified Version) This simplified Formula 5-17 allows an argumentation that shows that every Markov chain can be described completely by just giving a starting distribution at step 0 and a transition matrix. 55 As mentioned above, a Markov chain is often used to predict breakdowns. Therefore, the existing states have to be classified as critical and uncritical ones. In general, states i ∈ I with p = 1 are critical ones. They are called absorbing states. Figure 5-5 ii contains two absorbing states because after taking state 0 or 6 all following states will remain the same. Markov chains can now be used to determine the probability a critical state is taken. If an absorbing state is taken with a probability of one hundred percent, the mean number of steps can also be determined after which an absorbing state is taken. ([Waldmann04], p. 18) This determination can be done by calculating ( ) P m with m = 1,2, K, ∞ . Markov chains often converge to a stationary distribution, so that the probability for an absorbing state can be given for an infinite number of state changes. Formula 5-18 introduces a counter-example that does not converge. Hence, the probability an absorbing state is taken during the whole processing time of the Markov chain cannot be obtained in any case but in many cases. ([Waldmann04], p. 40) ⎛0 P = ⎜ ⎝1 1⎞ ⎟ 0⎠ Formula 5-18: Identity Matrix as an Example of a non Converging Markov Chain The Markov property can also be transferred to the setting of time-continuous stochastic processes. The result is called Markov process. The biggest difference to the Markov chains is the non-applicability of the above described state probability calculations. The results can no longer be determined by just multiplying matrices but by solving differential equations. To ease calculations the underlying process is often 55 See ([Beichelt97], p. 148) for details 71
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Technische Universität Braunschwei
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4.4 Review of Current State of Rese
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Index of Tables Table 2-1: Error of
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1 Introduction 1.1 Initial Position
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2 Sensor Based Temperature Monitori
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compressor. They are served by a ce
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the specified value, it works very
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of this, over a hundred irregular p
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a centralized one. An isolated appl
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Figure 2-8: Lack of Information Pro
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Moreover, the fridge’s filling le
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store this kind of data but only of
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Page 27 and 28: device’s condition. Especially th
Page 29 and 30: a converter that is also available
Page 31 and 32: To indicate important events within
Page 33 and 34: 1. Display stored data in table for
Page 35 and 36: This way of visualizing data is the
Page 37 and 38: free text. Moreover it is possible
Page 39 and 40: dependent limit and delay settings,
Page 41 and 42: Aside from these functions, the kin
Page 43 and 44: 1. Temperature verification in retr
Page 45 and 46: alarms by the use of additionally a
Page 47 and 48: The third mentioned Centron Environ
Page 49 and 50: 4 Current State of Research As alre
Page 51 and 52: even this improvement is still face
Page 53 and 54: Another current approach is regress
Page 55 and 56: S o n 2 = ∑ Yi −Yi n −1 i= 2
Page 57 and 58: This algorithm returns an upper and
Page 59 and 60: • Critical values • Pre-warning
Page 61 and 62: 5 Possible and Promising Ways of Da
Page 63 and 64: could be: “The mean increase of a
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Page 67 and 68: Figure 5-1: Two Samples of Regressi
Page 69 and 70: 5.4.2 The Major Problems of Regress
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Page 73 and 74: explained in section 5.4. If such a
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Page 91 and 92: Table 5-1: Estimated Improvements A
Page 93 and 94: As all these software products fail
Page 95 and 96: 6.2 Case Study The UMC St. Radboud
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Page 99 and 100: Figure 6-5: Minimum Values at Dayti
Page 101 and 102: Figure 6-9: Standard Deviation at D
Page 103 and 104: Table 6-3: Reported Notifications (
Page 105 and 106: Important for the determination of
Page 107 and 108: Remarkable is a comparison to the a
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Page 111 and 112: 7 Summary Cooling devices within me
Page 113 and 114: Bibliography Books and Articles: [B
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Page 117 and 118: Appendix 1 - Implementation of Inte
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Page 121 and 122: Appendix 2 - Implementation of Stat
Page 123 and 124: dailydate = [dailydate; floor(date(
Page 125 and 126: %Count Dooropenings per Day dailydo
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xlswrite(strcat(path, 'Excel\', fil
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print('-dtiff', strcat(path, 'Graph
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ar([dailydate(1):dailydate(length(s
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Appendix 3 - Implementation of Data
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Duration = 0; maxtemp = 0; elseif (
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%Contains [Duration, Number of Occu
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disp(strcat('Dooropening
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Casestudie Breakdown prediction Contell PILOT - Transumo

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