Discrete Iterative Curve Alignment

Transcription

1 Discrete Iterative Curve Alignment A Tool for Optimal Time Domain Calibration By Dr. Tim Butters Data Assimilation & Numerical Analysis Specialist tim.butters@sabisu.co

2 Contents 1 Introduction 2 2 Challenge 2 3 Solution Transformations in Tim Transformations in Time and Magnitude Accuracy Curve Interpolation Examples Equipment Failure Rate Plant Production Comparison Conclusions 6 Appendices 8 A Normal Equations 8 B Barycentric Rational Interpolation 9 List of Figures 1 DICA Method DICA Example DICA Example with Magnitude Scaling Sabisu Ltd. 1 V1.0

3 1 Introduction Curve alignment is a technique that can be used to find the optimal linear transformation between two datasets. This can provide information about the differences/similarities in the datasets by calculating the time shift and stretch between them. Comparisons of this type are often used to compare performance between different periods in time (e.g. between different quarters or different years), or to compare different facilities. The time shift and stretch quantify the difference in commencement time and rate between the datasets respectively. Discrete Iterative Curve Alignment (DICA) not only aligns curves of the same length (a relatively trivial task), but also curves of different lengths. This allows smaller sections of data, e.g. one month s worth, to be compared to a much larger set, e.g. the previous years data. DICA will find the optimal fit between the two sets, giving information about whether or not the current months data fits with the recordings from the same period last year. If not, the shift reported by DICA will give the period from last years data that it fits most closely to, and the stretch quantifies the difference in rate between the two periods. 2 Challenge Although fitting curves of the same length is trivial, fitting curves of different lengths that span different time periods is more complicated. An iterative approach is needed to find the best fit between the shorter curve and different sections of the longer curve. As well as transformations in time, there are applications for which a transformation in magnitude is also required. This could be, for example, to compare two plants of different sizes that perform similar tasks. In this case the production rates would not be expected to have the same absolute values, however, it is possible that the relative variation through the year due to seasonal conditions would be the same. To determine this a curve alignment solution is required that can account for the differences in magnitude between the data from each plant Sabisu Ltd. 2 V1.0

4 3 Solution 3.1 Transformations in Tim Discrete Iterative Curve Alignment provides an efficient and effective method to solve this problem by first discretizing the longer (reference) curve into n sections. The shorter curve is then shifted and stretched to fit between each curve section, or consecutive set of curve sections, as shown in figure 1. For each of these transformations the goodness-of-fit is measured. The optimal fit for this iteration is then simply the transformation that yields the best fit. The algorithm then takes the section of the reference curve to which the smaller curve was mapped and discretizes this into a further n sections. The algorithm is then repeated in a recursive manner a pre-defined number of times to achieve the desired accuracy. 3.2 Transformations in Time and Magnitude To apply transformations to both the time and value data the same algorithm is used with the addition of an extra step at each iteration. This extra step solves the normal equations for linear fitting (given in appendix A) to determine the optimal shift and stretch in the y direction to account for any differences in magnitude. This is possible as with the above algorithm the start and end points in the x direction are explicitly defined. 3.3 Accuracy Although there is a theoretical minimum precision with this recursive method, the expansion of the matched section before discretization can lead to the algorithm repeating iterative runs if the smaller curve matches the reference curve best when transformed to the full length of the reference curve (or the length of the current iterative section of the reference curve), or one step in from this, at both ends. 3.4 Curve Interpolation To speed up DICA, and to allow it to align very large datasets, barycentric rational interpolation (BRI) is utilised to smooth the curves being processed. This uniformly reduces the number of points in large datasets and reduces 2014 Sabisu Ltd. 3 V1.0

5 Figure 1: Selected iterative steps from the Dynamic Iterative Curve Alignment of two datasets. Panels A and B show the first and third iterative steps with the left hand edges of the two curves aligned. Panels C and D show the first and third iterative steps with the left hand side of the smaller curve aligned with the second section of the reference curve Sabisu Ltd. 4 V1.0

6 noise. These interpolated curves are then aligned using DICA, and the optimal parameters are applied to the original data to produce the correct results. The mathematical details of LBRI are given in appendix B. 4 Examples 4.1 Equipment Failure Rate Figure 2A shows two time-dependent curves with different numbers of datapoints. The black reference curve could be thought of as a manufacturer s failure curve for some specific piece of equipment, with the smaller red curve being the measurements taken from the plant. As shown in panel B, DICA finds the optimal fit between the two curves, and although not obvious from panel A, the curves clearly align well. DICA also outputs the stretch and shift applied, with the values for this example being 0.37 and 34.5 respectively. In this particular example of comparison with failure data, the shift represents a delay in the commencement of recordings on the plant. The stretch provides the most useful result in this case, as it shows whether or not the equipment is degrading at the rate predicted by the manufacturer (stretch= 1), faster than expected (> 1), or slower than expected (< 1). The value of 0.37 indicates that the equipment on the plant is degrading at only 37% of the rate predicted by the manufacturer. 4.2 Plant Production Comparison When comparing two plants of different sizes magnitude scaling is required to account for the different production characteristics of each installation. Figure 3 shows the application of DICA to a pair of curves on significantly different scales. The reference curve (shown in panel A) varies between 0 and 450, whereas the smaller curve (panel B) lies within the range Panel C shows the two curves plotted together, due to the large difference in scales it is not possible to pick out any features of the red smaller curve. Clearly magnitude scaling and shifting are required if these datasets are to be aligned. DICA finds the optimal transformation in both time and magnitude that aligns the two curves, and as shown in panel D, the two curves are comparable. Were these real plant production curves this would indicate that the variation in production over time varies in a similar manner but 2014 Sabisu Ltd. 5 V1.0

7 Figure 2: Example time dependent data before (A) and after (B) Discrete Iterative Curve Alignment. From panel A alone it is not clear that the two curves coincide. with some offset and stretch. The time shift and stretch for this transformation are 38.9 and 0.29 respectively. This means that the smaller plant has a lower rate of production, at 29% of the rate of the larger plant. The time shift could correspond to a difference in the commencement of recordings in the production cycle between the two plants. The shift in magnitude is 11.3, which could relate to a difference in starting value, e.g. stock remaining from the last period for the larger plant. The stretch in magnitude is 24.0, which relates to the difference in size between the two installations. 5 Conclusions DICA provides a fast and effective method for aligning time-series data, the results of which can be used to quantitatively compare the datasets. These comparisons provide important information such as the difference in rates between the sets, which can be used to asses performance. As well as this, any absolute shift between the data sets is reported, which could correspond to variation in the operational cycle or other key differences such as initial levels of product remaining from the previous period Sabisu Ltd. 6 V1.0

8 Figure 3: Discrete Iterative Curve Alignment applied to two datasets of different magnitudes. Panels (A) and (B) show the reference curve and smaller curve respectively. When plotted on the same axis (C) the smaller curve appears as a flat line. After alignment (D) it can be seen that the curves are comparable Sabisu Ltd. 7 V1.0

9 The ability to also scale the data in magnitude as well as time allows curves of various scales to be compared easily. This provides a large amount of flexibility with many industrial uses. One such use would be to find the similarities and differences between operations of different sizes to assess the economic feasibility of up-scaling production. DICA is a highly effective analysis tool that provides fast methods to maximise the information extracted from many different kinds of data. Appendices A Normal Equations The normal equations provide a closed-form analytical solution to the general minimisation problem Ax = b. The linear form required for this application is: 1 a 1 1 a 2 [ ] α A =.., x =, (1) β 1 a n Ax = b, (2) A T Ax = A T b, (3) x = ( A T A ) 1 A T b. (4) The matrix A is the affine matrix of the parameters to be scaled and shifted, in the application of DICA this would be the magnitude values of the smaller curve. The vector x contains the fitting parameters for shift (α) and stretch (β) that are being calculated, and b is a vector of the magnitude values for the section of the reference curve that overlaps with the smaller curve. This provides the optimal parameters α and β to fit the smaller curve to the reference curve in the y-direction Sabisu Ltd. 8 V1.0

10 B Barycentric Rational Interpolation For this application the linear form of barycentric rational interpolation is used. r n (x) = n i=0 w i,n x x i,n f i,n n i=0, (5) w i,n w i,n = ( 1) i. (6) Where r n (x) is the resulting interpolant. In cases where the dataset needed to be reduced, averaging was performed on sets of consecutive points before interpolation. The number of points averaged in each set depends on the reduction required, for example, to reduce a dataset to a third of its original size points would be averaged in sets of three. This was found to produce a good approximation to the original curve Sabisu Ltd. 9 V1.0