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FIGURE 2 Early detection by moving column c by1 row
0 0 0 0 0 0 0 0 0 1 0 0 0
0 0 0 0 0 0 0 0 0 0 1 0 0
where st , j ′ and st , j ′′ represent the first and second derivatives of s t , j , respectively. The first derivative is related to a gradual change in time series which may not be sensitive to a sudden machine breakdown, while the second derivative is more useful to detect sharp changes that appeared in the streaming signals. For the rest of this article, we use the second derivative to seize precursors of immediate failure. In this project, we aim to detect the failure earlier before it occurs. One simple way to achieve this goal is to use the class label of k time points ahead for the current instance’s class label so that classifiers are able to learn to predict c t + k the system condition at k time units ahead. 2
c t ← c t + k , where k = 1 , 2 , … .
(4)
We set k = 1, which implies that we build a model to detect a failure 2 minutes earlier than its occurrence. Figure 2 depicts this process. In classification problems with streaming data, temporal sequence data can normally secure more information com- pared with the data point sampled at a single time step. 33 Accordingly, we extract small fragments of sequences by conducting, namely, time window processing. For a given window size m , a window instance consists of the last m sen- sor measurements up to time t which corresponds to the rows of MSTS data given in Equation (1) with time indices t − m + 1, … , t . The class label of the window instance is given as c t so that it represents the system condition at the last time point of the window. These window instances provide features to be used in a machine learning algorithm. In addi- tion, we address the problem of severely imbalanced class labels of the original MSTS data while constructing the window instances by making a balance between two labels to some extent. That is, for time window processing we select all the time points t where c t = 1 and only randomly select t where c t = 0 such that it makes difference between the number of class labels not too large. The constructed window instances and those class labels are given as the following form.
⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ,
⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝
w ′ w ′
1 , 1 w ′ 2 , 1 w ′
… w ′ … w ′
1 , p y 1
1 , 2
2 , 2 2 , p y 2 ⋮ ⋮ ⋮ ⋮ ⋮ w ′ n , 1 w ′ n , 2 … w ′ n , p y n
(5)
( Wy ) =
where each row represents each window instance. That is, w i , j is the sequence of length m which is the second derivatives of the j th sensor signals, and y i is the class label of the i th window instance. Note that the row index i = 1, … , n merely distinguishes each window instance, not necessarily implies the time point. The time window processed training dataset ( W train y train ) and test dataset ( W test ) are used as input of Algorithms 1 and 2, respectively. 3.4 1-NN for time series classification In the field of data mining and machine learning, one of the most frequently studied problems is classification. 34 Theclas- sification process is to evaluate the similarities in a dataset to classify them into designated classes. One of the differences
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