Casestudie Breakdown prediction Contell PILOT - Transumo

More documents

Recommendations

Info

MTBF Availability = MTBF + MTTR Formula 5-10: The Definition of Availability [Masing88] The general idea, to calculate the estimated availability during a specified time, seems to be promising. But this method of failure- and availability ratios is faced with a major problem, when applying it to the setting of sensor based temperature monitoring. Most manufacturers of cooling devices do not offer ratios like MTTF [Nijmegen06]. As cooling devices are long-life products, a determination of these measures is also impossible. Hence, the appliance of failure- and availability ratios is not applicable within the setting of sensor based temperature monitoring. 5.7 Markov Chains Another approach of predicting breakdowns is the usage of Markov chains. These chains are simple time-discrete stochastic processes ( n ) n N0 X ∈ with a countable state space I that comply with the following Formula 5-11 for all points in time n∈ N 0 and all states i ,K, i , i i ∈ I : ([Waldmann04], p. 11) 0 n−1 n, n+ 1 P X i | X = i , , X = i , X = i ) = P( X = i | X = i ( n+ 1 = n+ 1 0 0 K n−1 n−1 n n n+ 1 n+ 1 n n Formula 5-11: The Markov Property ) This Markov property is the specific characteristic of Markov chains. It says that the probability for changing to another state is only influenced by the last observed state and not by prior ones. Hence, the probability that X n+ 1 takes the value i n+ 1 is only influenced by i n ∈ I and not by i ,K in ∈ I . ([Waldmann04], p. 11) 0 , −1 The conditional probability P ( X n + 1 = in + 1 | X n = in ) is called the processes’ transition probability. If this transition probability is independent from the point in time n , the Markov chain is called homogeneous. Otherwise it is called inhomogeneous ([Waldmann04], p. 11). In the following, this thesis will first of all focus on homogeneous Markov chains. To improve the readability, they will just be named Markov chains. In the majority of cases the transition probability is written as a matrix P . It contains the probabilities p ij of all possible changes between old state i and new state j as 68
pictured in Formula 5-12. Beside a change in state it is also possible that the state remains the same for another time interval. This probability is given by each column. ([Beichelt97], p. 146) p ii within ⎛ p ⎜ ⎜ p P = ⎜ ⎜ M ⎜ ⎝ p 00 10 i0 p p M p 01 11 i1 L p L p L M L p 0 j 1 j ij ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ Formula 5-12: Transition Probability Matrix As every p represents a probability, they all have to comply with 0 ≤ p ≤1. ij Moreover, every state must have a succeeding state. Hence, the probability to take one of the available countable states as the next one has to be one hundred percent. This leads to the conditions pictured in Formula 5-13 for the transition probability matrix. ([Jondral02], p. 186-187) ij 0 ≤ p ij ≤1 ∀i, j and N ∑ j= 1 p ij = 1 ∀i Formula 5-13: Conditions for the Transition Probability Matrix As the sum of every row within that Matrix has to be 1, p = 0 entries can be left out to offer a better overview. To achieve an even better overview, Markov chains are often visualized as a graph. Every node of that graph represents a possible state and every arrow a possible transition with a positive probability. ([Waldmann04], p. 17) ij A Markov chain can be used, for instance, to describe the following gamble between two people: A coin is thrown. Depending on which side is faced up, one of the two players wins the coin. Player one starts with four coins, player two with two coins. The game ends as soon as one of the players owns all six coins. This leads to seven possible states because a player can own every number of coins between zero and six. Provided that every coin has the same winning probability p the transition probability matrix would look like the one pictured in Formula 5-14. As described above, this Markov chain can be visualized as a graph to allow a better overview of the described process. A comparison of Formula 5-14 and the corresponding Figure 5-5 shows this improvement. 53 53 Example taken from ([Waldmann04], Chapter 2) 69
Page 1 and 2:
Technische Universität Braunschwei
Page 3 and 4:
4.4 Review of Current State of Rese
Page 5 and 6:
Index of Tables Table 2-1: Error of
Page 7 and 8:
1 Introduction 1.1 Initial Position
Page 9 and 10:
2 Sensor Based Temperature Monitori
Page 11 and 12:
compressor. They are served by a ce
Page 13 and 14:
the specified value, it works very
Page 15 and 16:
of this, over a hundred irregular p
Page 17 and 18:
a centralized one. An isolated appl
Page 19 and 20:
Figure 2-8: Lack of Information Pro
Page 21 and 22:
Moreover, the fridge’s filling le
Page 23 and 24: store this kind of data but only of
Page 25 and 26: 3 Current Monitoring Systems The la
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
Page 65 and 66: The mode is the most frequent value
Page 67 and 68: Figure 5-1: Two Samples of Regressi
Page 69 and 70: 5.4.2 The Major Problems of Regress
Page 71 and 72: The next section will introduce tim
Page 73: explained in section 5.4. If such a
Page 77 and 78: P ( m) = P ( r ) ⋅ P ( m−r ) wi
Page 79 and 80: But the greatest problem is again t
Page 81 and 82: clustering as well as an analysis o
Page 83 and 84: The general idea of supervised lear
Page 85 and 86: sometimes data of a door opening se
Page 87 and 88: Beside the introduced notification
Page 89 and 90: A classification could be achieved
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
Page 97 and 98: Figure 6-3: Maximum Values at Dayti
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
Page 109 and 110: Interviews with several employees f
Page 111 and 112: 7 Summary Cooling devices within me
Page 113 and 114: Bibliography Books and Articles: [B
Page 115 and 116: [Wittenberg98] Reinhard Wittenberg,
Page 117 and 118: Appendix 1 - Implementation of Inte
Page 119 and 120: %Name and location of the source fi
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
Page 127 and 128:
xlswrite(strcat(path, 'Excel\', fil
Page 129 and 130:
print('-dtiff', strcat(path, 'Graph
Page 131 and 132:
ar([dailydate(1):dailydate(length(s
Page 133 and 134:
Appendix 3 - Implementation of Data
Page 135 and 136:
Duration = 0; maxtemp = 0; elseif (
Page 137 and 138:
%Contains [Duration, Number of Occu
Page 139 and 140:
disp(strcat('Dooropening
show all

Casestudie Breakdown prediction Contell PILOT - Transumo

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?