This invention relates generally to a method for computing correlation anomaly scores and, more specifically, to a method for computing correlation anomaly scores using change analysis.
Change detection is an unsupervised learning task that decides whether a data generating mechanism behind data has been changed or not When considering multivariate systems, however, more advanced tasks involving anomaly or change detection are also relevant. After detecting a change, it is desirable to determine which variables (or, degrees of freedom) are responsible for the change. Determining which variables are responsible for a change is called change analysis. Change analysis can be achieved by comparing a target data set with some reference data based on predefined criteria. If the degrees of freedom are independent of each other, change analysis amounts to change detection of individual degrees of freedom. However, degrees of freedom are not usually independent in general, and the dependence makes change analysis challenging.
It is desirable for a method of computing correlation anomaly scores to be effective in multivariate systems having highly correlated degrees of freedom.
The shortcomings of the prior art are overcome and additional advantages are achieved through an exemplary method for computing correlation anomaly scores, the method including, defining a first similarity matrix for a target run of data, wherein the target run of data includes an N number of sensors, defining a second similarity matrix for a reference run of data wherein the target run of data includes the N number of sensors, developing a k-neighborhood graph Ni of the i-th node for the target run of data, wherein the k-neighborhood graph of the i-th node is defined as a graph comprising the i-th node and its k-nearest neighbors (NN), developing a k-neighborhood graph
Additional features and advantages are realized through the techniques of the present invention. Other embodiments and aspects of the invention are described in detail herein and are considered a part of the claimed invention. For a better understanding of the invention with advantages and features, refer to the description and to the drawings.
The subject matter that is regarded as the invention is particularly pointed out and distinctly claimed in the claims at the conclusion of the specification. The foregoing and other aspects, features, and advantages of the invention are apparent from the following detailed description taken in conjunction with the accompanying drawings in which:
The detailed description explains the preferred embodiments of the invention, together with advantages and features, by way of example with reference to the drawings.
Systems and methods involving computing correlation anomaly scores are provided. Several exemplary methods are described.
In this regard, multivariate systems having highly correlated degrees of freedom can be naturally represented as graphs. Computing correlation anomaly scores may be described as: given two different graphs with nodes, providing the score of each node based on how much the node is responsible for the difference between the graphs.
An exemplary application addresses cases where the nodes are given by time-series data from physical sensors. An adjacency matrix (or the similarity matrix between sensors) can be changed over time, and the comparison is made for graphs at different time points.
One exemplary application of computing correlation anomaly scores is in sensor validation, where sensor signals are inspected for detecting proper operation.
To compute correlation anomaly scores, an unknown structure hidden deep inside of graphs should be determined, and the correlation between the nodes should be unfolded. An exemplary method for computing correlation anomaly scores between sensor signals is based on a notion of neighborhood preservation: under normnal system operation, a neighborhood graph of each node is almost invariant against the fluctuations of experimental conditions. The exemplary method computes anomaly scores by evaluating how much this assumption is broken. The score can be obtained as a probability value by considering a canonical distribution on the neighborhood graph.
Referring to
In a single run, let xi(t) be the observation of the i-th sensor (i=1, 2, . . . , N) at a time index t (t=1, 2, . . . , T). The given data of a target run, which is expected to have some anomalies is shown in
There are at least two considerations regarding comparing the target run with the reference run. First, in highly dynamic systems such as automobiles and computer systems, direct comparison by overlapping between individual target and reference signals is generally meaningless.
The second consideration regarding comparing the target run with the reference run is that most of the nontrivial anomalies are related to the correlations between signals. Generally, errors that occur in individual sensors are not difficult to detect. For example, a dead sensor can be easily detected by using a threshold value. However, if, for example, two working sensors have been exchanged by some human error in wiring, the error would appear primarily as a correlation anomaly. Since the anomaly score is computed for individual sensors, unknown correlations between signals are “unfolded”.
Correlation anomaly analysis is defined as: given a target graph having A and a reference graph having Ā, provide a score of each node which accounts for the difference between the graphs. Wherein, A and
The dissimilarity between graph nodes may be computed from the similarity matrix A. A k-neighborhood graph of the i-th node is defined as a graph comprising the i-th node and its k-NNs, with edges between the i-th node and its neighbors. Wherein, dij is the dissimilarity between the i-th and the j-th nodes. A node j is the nearest neighbor (NN) to another node i if j satisfies j=argminj′(≠i) dij′. Similarly, the k-th NN of the i-th node is defined as the one which has the k-th smallest dissimilarity to i (except for i itself). The i-th node is the central node in the k-neighborhood graph of the i-th node.
Most of the unimportant fluctuations in a highly dynamic system are due to weakly correlated pairs of sensors. Conversely, highly correlated pairs of sensors tend to be hardly affected by the change in experimental conditions, under normal system operation. These observations lead to the neighborhood preservation principle for the correlation anomaly analysis. The neighborhood preservation principle recites: if the system is working normally, the neighborhood graph of each node is almost invariant against the fluctuations of experimental conditions.
The dissimilarity dij between the i-th and j-th node is defined so as to satisfy the following conditions. (1) di,i=0 for ∀i, (2) dij ≈0 for highly-correlated pairs, and (3) dij→∞ for almost uncorrelated pairs. Regarding the difference between the second and the third conditions, a large correlation coefficient may be considered as a representation of the internal structure of the system. On the other hand, a value of small correlation coefficients is considered to carry no useful information of the system. Mathematically, the dissimilarity is defined by
dij2=φ(aij)
Where aij represents the (i,j) element of A, and φ(x) is a monotonically decreasing positive function satisfying
φ(x)→0, |x|→1, and
φ(x)→∞, |x|→0
For example. if aij is the correlation coefficient between the i-th and the j-th signals which satisfies |aij|≦1, φ(x)=log|x| satisfies these conditions.
In block 203 neighborhood graphs are developed. Two examples of neighborhood graphs target run graph 206 and reference run set of data graph 208. As shown in
where Ni+ defines the set of Ni∪i. Similarly,
By definition of coupling probability between nodes, p(j|i) should take a large value in a node with a small dissimilarity dij. On the other hand, the probability distribution over j should be as general as possible in terms of entropy. Thus, the distribution p(i↑i) is defined as the solution of the following optimization problem.
where Hi is the entropy of the probability distribution p(j|i), and is defined by
The symbol di2 represents the expected squared dissimilarity around the i-th node, which is defined by
By introducing Lagrange's multipliers 2σi2 and μi, the optimization problem is reduced to that of
Inserting the definition of Hi, and differentiating with respect to p(j|i), results in
where μi is removed by introducing the partition function Zi, which is defined by
The multiplier σi is to be determined by the condition about di2. This can be treated as an input parameter, which is independent of i. Also, by modifying the definition of φ accordingly, the relation may be fixed as 2σi2=1. This relation is used hereafter.
Equation (7), may be further explained by considering what happens if a node is almost uncorrelated with other nodes. In this case, since p(i|i)>>p(j|i) for j≠i, the probability weight concentrates at the central node i. Thus, p(j|i) is capable of naturally discounting loosely correlated neighbors. The self-coupling probabilities in the normalization equation (4). As a result, they tend to give unstable results since Zi can be very small for isolated nodes.
According to the neighborhood preservation principle, the difference between the following quantities should be small if the system is working correctly.
Equations (9) and (10) represent the tightness of the i-th neighborhood graph. Thus, ei measures the tightness of the coupling between the central node and its neighbors in a target run in terms of probability. Also, ēi(Ni) measures the tightness of the coupling around the i-th node in a reference run using the k-NN set defined in the target set (i.e. not using the NN set of the reference data). For defining ēi(Ni) (as well as ei(
Similarly, by replacing Ni with
The same holds for ēi. The minimum is obtained when the i-th node is totally uncorrelated to others, and the maximum is obtained when the nodes are perfectly correlated.
Using the tightnesses, an anomaly score of the i-th node is defined as
max{|ei(Ni)−ēi(Ni)|,|ei(
The anomaly score is determined after the coupling probabilities in block 205 of
While the preferred embodiment to the invention has been described, it will be understood that those skilled in the art, both now and in the future, may make various improvements and enhancements which fall within the scope of the claims which follow. These claims should be construed to maintain the proper protection for the invention first described.
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