In higher order Kalman filtering applications the analyst often has very little insight into the nature of the observability of the system. For example, there are situations where the filter may be estimating certain linear combinations of state variables quite well, but this is not apparent from a glance at the error covariance matrix. It is shown here that the eigenvalues and eigenvectors of the error covariance matrix, when properly normalized, can provide useful information about the observability of the system. In the two decades since R.E. Kalman [1] introduced the idea of least square recursive filtering, there have been fantastic advances in computer technology. This has made it feasible to implement considerably more complicated filters than one would have dreamed possible 20 years ago. For example, in the global positioning system (GPS) ground control segment, the state vector being estimated will contain over 200 elements when the system becomes fully operational [2].In complex estimation applications, it becomes most difficult for the analyst to maintain good physical insight into the problem. The difficulty with most computer analyses of higher order systems is that the analyst is provided with an utter deluge of information without, at the same time, being provided with means of sorting the relatively important information from that which is not. The discussion that follows is intended to help the analyst with this sorting process and to help provide additional insight into complicated estimation problems.It is certainly apparent to those familiar with estimation and control theory that there is a close connection between observability and estimation. Observability in a deterministic sense simply means that an observation of the output over the time span (0,t) provides sufficient information to determine the initial state of the system [3]. This, along with knowledge of the system driving function, uniquely specifies the state at any other time within the interval. However, there are problems in applying the observability concept to complicated situations. The difficulties are essentially twofold: (1) The test for observability is not easy to apply in higher order systems. In a fixed-parameter system it requires determination of the rank of a large rectangular matrix; in a time-variable system it suffices to say that the usual test is even more difficult to apply [3]. (2) Even when one can apply the test of observability successfully, it only provides a yes-no type answer. It tells one nothing about the degree of observability. One might think that these difficulties are solved by doing off-line Kalman filter analysis, and, in a sense, this is true. However, as mentioned before, the problem here is that the computer tells the analyst more than he wants to know. In a few minutes a computer can output more numbers than one can intelligently digest in months of study! Allowing for symmetry, there are n(n + 1)/2 terms to examine in the error covariance matrix. If n = 200, the analyst must at least glan...

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We have investigated the QRS complex, extracted from electrocardiogram (ECG) data, using fuzzy adaptive resonance theory mapping (ARTMAP) to classify cardiac arrhythmias. Two different conditions have been analyzed: normal and abnormal premature ventricular contraction (PVC). Based on MIT/BIH database annotations, cardiac beats for normal and abnormal QRS complexes were extracted from this database, scaled, and Hamming windowed, after bandpass filtering, to yield a sequence of 100 samples for each ORS segment. From each of these sequences, two linear predictive coding (LPC) coefficients were generated using Burg's maximum entropy method. The two LPC coefficients, along with the mean-square value of the QRS complex segment, were utilized as features for each condition to train and test a fuzzy ARTMAP neural network for classification of normal and abnormal PVC conditions. The test results show that the fuzzy ARTMAP neural network can classify cardiac arrhythmias with greater than 99% specificity and 97% sensitivity.

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We have investigated the use of a time-domain optimal filtering method to simultaneously minimize both the baseline variation and high-frequency noise in near-infrared (NIR) spectrophotometric absorption data of glucose dissolved in a simple aqueous (deionized water) matrix. By coupling a third-order (six-pole) digital Butterworth bandpass filter with partial least-squares (PLS) regression modeling, glucose concentrations were determined for a set of test data with a standard error of prediction (SEP) of 10.53 mg/dl (mean percent error: 4.24%) using seven PLS factors. Compared to the unfiltered test data for six PLS factors and a SEP = 17.00 (mean percent error: 7.38%) this results shows more than a 38% decrease in the error. The glucose concentrations ranged from 51 mg/dl to 493 mg/dl, and the NIR spectral region between 2088 nm and 2354 nm (4789 cm-1 and 4248 cm-1) was used to develop the optimal PLS model. The optimal PLS model was determined from a sequence of three-dimensional performance response maps for different numbers of PLS factors (2-10). A total of 99 NIR spectra were generated for glucose dissolved in deionized water using a NIRsystems 5000 dispersive spectrophotometer. Nine of these spectra were generated for only water, which were averaged and subtracted from the remaining 90 spectra to generate the training and test data sets, thereby, removing the intrinsic high background absorption due to the water. The training set consisted of 57 spectra and associated glucose concentration target values, and the test set was comprised of the remaining 33 spectra and target values. Performance results were compared for three different digital Butterworth bandpass filters (four-poles, six-poles, and eight-poles), and a digital Gaussian filter design approach (i.e., Fourier filtering).