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.7. Outlier removal and principal components estimates of the samples.

progress of the batch in real time is an active research and development area [607, 647].

A different paradigm is the use of smart sensors that include in the sensor system functionality to detect outliers, reduce noise and conduct self diagnosis of operation status. This will shift the burden of information reliability assessment from the data collection and process operations computer to the individual sensors. In coming years, cost reductions and improvements in reliability would increase the feasibility of this option.

3.5 Data Pretreatment: Signal Noise Reduction

Sensors and transmission lines can pick up noise from the environment and report compromised readings. Various signal processing and filtering paradigms have been proposed to reduce signal noise. Some simple remedies can be developed by boosting the signal strength or selecting a variable that is less susceptible to noise. An example of the former is noise reduction in thermocouple signals that are at millivolt level. If the signal is amplified to volt level before transmitting, the signal to noise ratio during transmission is significantly improved. Converting a voltage signal to current that is less noise prone is an example of the latter remedy.

If several measurements are made for the same variable at a specific sampling time, measurement error is reduced by averaging these. This is the approach used in traditional statistical quality control in discrete manufacturing processes. It may also be used for quality variables at the end of a batch if several measurements can be made easily and at low cost. The problem is more challenging if a single measurement is made for each variable at each sampling instant. The paradigms used in this case include averaging over time, decomposing the signal into low and high frequency components, and use of multivariate data analysis techniques such as PCA to separate signal information from noise. A general framework for data averaging for signal filtering is presented and the use of PCA for noise reduction is illustrated in Section 3.5.1. The PCA method is discussed in detail in Section 4.1. Signal decomposition can be implemented by various techniques described in the signal processing literature [45, 207, 271]. Most of these techniques are available in commercial software such as Matlab Signal Processing Toolbox [373]. A particular signal decomposition approach that has captured attention in recent years is the wavelet decomposition, which can implement time-frequency decomposition simultaneously. Wavelets and their use in noise reduction are discussed in Section 3.5.2. Multivariate data analysis techniques use a different premise: If a coordinate transformation can be made to explain the major variation in data, what has not been extracted from the measurements would be mostly random noise. If the signals are reconstructed by using only the information retained in the new coordinate system, then the noise will be filtered out.

3.5.1 Signal Noise Reduction Using Statistical Techniques

Simple noise filtering tools can be developed by using time series model representation (Section 4.3.1) where the filtered signal y(n) at sampling time n is related to the signal x(n)\

where n is the current sampling time and na and rit, are the lengths of the past sampling time windows for y and x signals, respectively. This is the standard time-domain representation of a digital filter. Assuming zero initial conditions and starting with y( 1), the progression of this representation is:

y{3) = 6n(3) + b2x(2) + bsx(l) - aay(2) - a3y(l) •

For example, if rib = 2 and na = 2, then y(6) is computed by writing Eq. 3.22 for this set of index values:

2/(6) = bnr(6) + b2x(5) + 63*(4) - a2y(b) - a3y(4) . (3.24)

To compute the estimated value y(6) for sensor reading x(6), the weighted sum of the three most recent sensor readings and two most recent estimated (filtered) values are used. Two limiting cases may be considered to generate the estimates by using a time series model: the use of sensor readings only (all at — 0) and the use of previous estimates only (all bl =0). The former is called moving average (MA) since all readings included in a sliding time window (the window width is determined by the value of n;, + 1) are used to estimate the filtered signal. One option is to assign all values equal weights (all bi are equal). Another option is to assign them different weights, perhaps to give a higher emphasis to more recent readings. If only past estimated values are used, then the current value is regressed over the previous estimates, yielding an auto regressive (AR) model. If an MA model is used, the last few readings are averaged to eliminate random noise. This is reasonable because the noise is "random" and consequently sometimes it will be positive and at other times it will be negative with a mean value that is zero in theory. By averaging a few readings, it is hoped that the noise components in each measurement cancel out in the averaging process. A pure AR model is not appealing because the estimation is based only on past estimated values and the actual measurements are ignored. The filter is called autoregressive moving average (ARMA) when both AR and MA terms are included. The reduction of noise by using ARMA and MA filters is illustrated in the following example.

Example Consider a low frequency, high amplitude sinusoidal signal containing high frequency, low amplitude noise (Figure 3.8(a)). Moving average (MA) and autoregressive moving average (ARMA) filtering are applied to denoise this signal. Filter design and fine tuning the parameters are important issues that are influential as concerns the result (Figure 3.8). After filtering (especially with good performance filters), increase in the signal-to-noise ratio is obvious. □

(a) Noisy raw signal.

(b) Filtered signal with a poor MA filter.

(c) Filtered signal with a good MA filter.

(d) Filtered signal with a poor ARMA (e) Filtered signal with a good ARMA filter. filter.

Figure 3.8. Noise reduction using MA and ARMA filters.

Noise Reduction by1 PCA. Multivariate statistical analysis tools process the entire data set and make use of the relations between all measured variables. Consequently, the noise reduction philosophy is different than the filtering techniques discussed earlier where each measured variable is treated separately. The multivariable statistical technique is utilized to separate process information from the noise by decomposing relevant process information from random variations. This is followed by reconstruction of the signals using only the process information. PCA determines the new coordinates and extracts the essential information from a data set. Principal Components (PC) are a new set of coordinates that are orthogonal to each other. The first PC indicates the direction of largest variation in data, the second PC indicates the largest variation not explained by the first PC in a direction orthogonal to the first PC (Fig. 4.1). Consequently, the first few PCs describe mostly the actual variations in data while the portion of the variation not explained by these PCs contains most of the random measurement noise. By decomposing the data to their PCs and reconstructing the measurement information by using only the PCs that contain process information, measurement noise can be filtered. The methods for computing PCs and determining the number of PCs that contain significant process information are described in Section 4.1. The random noise filtering by PCA is illustrated in the following example.

Example An MPCA model with 4 principal components is developed using a reference data set containing 60 batches, 14 variables and 2000 samples of each variable over the batches. Scores of the first two principal components are plotted in Figure 3.9, for the reference data set representing normal operating conditions. As a result of PCA modeling, noise level is decreased as shown in Figure 3.10. This is expected since a PC model extracts the most relevant correlation information among the variables, unmodeled part of the data being mostly noise. □

3.5.2 Wavelets and Signal Noise Reduction

Wavelets were developed as an alternative to Short Time Fourier Transform (STFT) for characterizing non-stationary signals. Wavelets provide an opportunity to localize events in both time and frequency by using windows of different lengths while in STFT the window length is fixed. A wavelet transform can be represented as

where represents the mother wavelet, x(t) is the original signal, a and b are scale and translation parameters, respectively, and the factor l/^/faf

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