Xiao Liu PhD Thesis.pdf - Faculty of Information and Communication ...

k 1
∑ − Mean i + 1 − Mean i
Rank ( k)
= i = 1
k −1
Formula 4.1
The theoretical basis for MaxSDev is Tchebysheff’s theorem [87] which has
been widely used in the statistics theory. According to the theorem, given a number
2
d greater than or equal to 1 and a set of n samples, at least [ 1 ( 1 ) ]
− of the
d
samples will lie within d standard deviations of their mean, no matter what the
actual probability distribution is. For example, 88.89% of the samples will fall into
the interval of ( µ − 3 σ , µ + 3σ
)
. The value of µ and σ can be estimated by the
k
sample mean and sample standard deviation as µ = ∑ Xi
k and
i=1
2
⎛ k ⎞
σ = ⎜ ∑ ( Xi − µ ) ⎟ ( k −1)
respectively. If it happens to be a normal distribution,
i=
1
⎝
⎠
Tchebysheff’s theorem is turning to one of its special cases, i.e. the “3σ ” rule which
means with a probability of 99.73% that the sample is falling into the interval of
( µ 3 σ , µ + 3σ
)
− [49]. Therefore, if we control the deviation to be less than m% of
the mean, then 3σ ≤ m % × µ is ensured. We can thus specify MaxSDev by Formula
4.2:
m% MaxSDev = × µ
Formula 4.2
3
K-MaxSDev is the core algorithm for pattern recognition in our strategy, and it is
applied to discover the minimal set of potential duration-series patterns.
4.3.4 Forecasting Algorithms
As presented in Section 4.3.2, the interval forecasting strategy for workflow
activities in data/computation intensive scientific applications are composed of four
major steps: duration series building, duration pattern recognition, duration pattern
matching, and duration interval forecasting. In this section, we will propose the
detailed algorithms for each step. Note that since pattern matching and interval
forecasting are always performed together, we illustrate them within an integrated
process.
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