Triangular episodes

Triangular Episode A Geometric Language to Describe Trends... 

Completeness. Every trend can be represented by a legal sequence of triangular episodes (Fig. 4b). 

Correctness. The recursive refinement of triangular episodes allows the description of a trend at any level of detail, converging to the realvalued description of a signal. 

Fig. 4. (a) The seven primitives of the triangular description of trends (b) sequence of triangular episodes describing a specific trend of a signal. 

Robustness. The relative ordering of the triangular episodes in a trend is invariant to scaling of both the time axis and the function value. It is also invariant to any linear transformation (e.g., rotation, translation). Finally it is quite robust to uncertainties in the real value of the signal (e.g., noise), provided that the extent of a maximal episode is much larger than the period of noise. 

From Fig. 5 we conclude that the original function can be represented by six (6) distinct trends (Fig. 5a, b, d, f, h, j), each with its own sequence of triangular episodes. Each successive trend of Fig. 5 contains information at a coarser resolution (scale). The differences among two successive trends are shown in Figs. 5c, 5e, 5g, 5i, and 5k, assuming for presentation purposes perfectly local filters. 

Scale-space filtering provides a multiscale description of a signal s trends in terms of its inflexion points (second-order zero crossings). The only legal sequences of triangles between two adjacent inflexion points are (in terms of triangular episodes) ... 

If a signal is represented by a sequence of triangular episodes, scale-space filtering manipulates the sequences of triangular episodes with very concrete mechanisms. Here is the complete list of syntactic manipulations carried out by scale-space filtering ... 

Once the stable reconstruction of a signal has been accomplished, its subsequent representation can be made at any level of detail, i.e. qualitative, semi-quantitative, or fully real-valued quantitative. The triangular episodes (described in Section I, A) can be constructed to offer an explicit, declarative description of process trends. 

Consider a measured operating variable, xit), and its M distinct measurement records, [)], / = 1,2,..., A/ over the same range of time. Using the multiscale decomposition of measured variables, discussed in Section III, we can represent each measurement record, [x(t)], / = 1,2,..., M by a finite state of trends, where each trend is a pattern of triangular episodes ... 

Two patterns are qualitatively equivalent if and only if their corresponding sequences of triangular episodes are qualitatively equivalent episode by episode i.e., the condition... 

The procedure for generating generalized trends was described earlier. The features of the generalized trends are given by the triangular episodes at any level of required detail, i.e., qualitative, semiquantitative, realvalued analytic. 

Fault Diagnosis Using Triangular Episodes and HMMs... 

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