Extraction of nonstationarity information from correlation noise

Analytica Chimica Acta, 177 (1986) 15-22 Elsevier Science Publishers B.V., Amsterdam -Printed

EXTRACTION OF NONSTATIONARITY CORRELATION NOISE

SCOTT FRAZER Department

in The Netherlands

INFORMATION

FROM

and MICHAEL F. BURKE*

of Chemistry,

University of Arizona, Tucson, AZ 85721

(U.S.A.)

(Received 24th June 1986)

SUMMARY A gas chromatograph is used as a model of a monitored process stream, and a method is developed to detect fixed-amplitude changes either in component concentration or system response to a perturbation. The multiple injection input and its advantages are based on the principles of correlation chromatography. The deconvolution of the aperiodic input pattern from the output results in a signal peak representing average system response and correlation “noise” which, when calibrated, gives the direction and magnitude of the change which has occurred. Because of the signal-averaging effect of the multiple inputs and deconvolution process, the effects of random fluctuations and noise are reduced.

In most analyses, the assumption is made that certain factors, such as sample concentrations, are at a steady state and will remain constant during data collection. However, in many continuous analytical procedures, including kinetic studies and process stream analysis, this assumption is not valid and, in fact, it is the extent of this nonstationarity that is of interest. If random fluctuations are of the same magnitude as the fixed-amplitude change that is occurring, it becomes difficult to evaluate accurately the direction and magnitude of the nonstationarity by simple observation of the output. To develop a data evaluation method for monitoring concentrations or establishing process dynamics, a gas chromatograph was used as a small model of a process stream. Single-component samples were injected to simulate, in effect, the input of a tracer, voltage or temperature pulse, etc. to characterize a system by determining its transient response [l].The input, or “forcing”, imposed upon the chromatograph and the advantages that result from it are based on correlation chromatography, a method which uses multiple inputs and a correlation operation. Specifically, a unique input pattern, which may be a stream-switching type [2] or, as in this case, a set of multiple injections [3] is continuously fed into a gas chromatograph at a rate that will generally overlap the fronts or peaks in the output to some extent. When the input pattern representation is deconvolved from this output, the result, or correlogram, will represent the average of all the peaks

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Cc)

(b)

Fig. 1. Typical (a) input pattern, x(t), in correlation chromatography.

(b) output, y(t),

and (c) correlation

function h(t)

present in the output file and will appear as a single injection chromatogram (Fig. 1). The inherent averaging effect of this correlation process reduces the amplitude of the random noise in the raw data to give a correlogram with an improved S/N ratio. Because one does not wait for one injection to elute before making another, this S/N enhancement occurs without requiring the time commitment of ensemble averaging. Considering these properties, it is not surprising that correlation chromatography has found application in trace analysis [4, 51. However, just as the technique might be used quickly to detect small signal peaks buried in noise, it can also be used to detect small changes in that signal that occur during the course of the injection sequence. Because the multiple injection process is continuous, the possibility of using it to monitor process streams has been suggested before [5] . The problem encountered in attempting this application is that the deconvolution operation assumes a stationary system and any nonstationarity, such as a fluctuation in input concentration or system response during data collection, causes the appearance of correlation noise in the final results [6, 71. This noise decreases the reliability of quantitative information and, if severe enough, can lead to a complete breakdown of the correlogram [8] . In this study, it was found that this correlation “noise” contains information on the type and magnitude of changes in system response and can be used to monitor these changes. Random fluctuations and noise are diminished through the same processes which allow S/N enhancement and an accurate evaluation of the constant concentration or response changes that occur during data collection can be made. In addition, however, by finding the

17

magnitude of the noise under the principal peaks in the correlogram and removing it, reliable quantitation of the average input concentration or system response can also be achieved. As stated earlier, this study was done with single-component samples to simulate a flowing system which is monitored without a chromatographic separation. The multiple signal or input aspect of the technique may be realized by truly injecting sample into the system (e.g., as in flow injection analysis) or in process dynamics studied by simply taking multiple readings from the monitoring device (e.g., a spectrometer) at set times. Multiplecomponent samples requiring a separation will show the same type of correlation noise behavior though, as the number of sample components that have changing concentrations increases, so must the sophistication of the procedure for the data analysis. THEORY

As mentioned before, a correlogram (i.e., the information of interest, h(r)) can be calculated by deconvolving the input pattern representation x(t) from the output data, y(t). When done in the Fourier domain, this deconvolution is a simple division of the two files [9],

4~) = IFT [W/W)1

(1)

where IFT indicates the inverse Fourier transform. The appearance of a system nonstationarity (e.g., a continual concen tration increase) may be thought of as a function NS(t) which is added upon y(t) to give a new output (Fig. 2a). Because of the linear addition property of the Fourier transform [9], Eqn. 1 becomes 47)

= IFT [Y(f)lX(f)

+

Wf)l-W=)l

(2)

The result of this operation is the sum of the deconvolution of x(t) from y(t) and x(t) from NS(t); the latter of these provides the correlation noise (Fig. 2b). With this explanation, the correlation noise in h(7) is expected to be consistent with the function NS(t). If the nonstationarity is increased or decreased by some factor (e.g., if the concentration increase is halved), the correlation noise peaks and valleys, while remaining in the same position, will increase or decrease in amplitude by that same factor (Fig. 3a). When the nonstationarity trend is reversed (e.g., if the solute concentration from file start to finish is allowed to decrease), the correlation noise follows the change and one sees the mirror image of the original noise as peaks become valleys and valleys become peaks (Fig. 3b). Both of these facts may be thought of as being due to the deconvolution process linearly converting nonstationarity changes in the output y(t) to correlation noise in the correlogram h(7).

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(0)

2

3

D8splocement

(b)

0

I

I

2 Displacement

3 (mln)

Fig. 2. Output and the correlation the same input of Fig. 1.

0

I

(rn~n)

(b)

2 Displacement

3 (mln)

function resulting from a nonstationary

response to

Fig. 3. Comparison of correlation noise patterns: (-) correlation functions of (a) one half the concentration increase of Fig. 2, and (b) a concentration decrease of the same magnitude as the increase of Fig. 2; (. . .) correlogram of Fig. 2 for comparison. EXPERIMENTAL

Very few modifications are necessary to convert a gas chromatograph to one with multiple injection capabilities. A Becker model 1452D gas chromatograph oven with a Varian flame ionization detector was used in this study. A computer-controlled autoinjector provided the multiple injections. Though other sampling systems have been suggested and used in multiple injection work [lo] , the Seiscor model VIII valve used in this study is quite reliable, readily available and has a switching time of only 10 ms. Injection control, data acquisition, and the deconvolution operation are all done on an IBM 9000 computer. The internal high-resolution timer which controls injection timing and data collection provides an accuracy of 30.5 MS. At set times, an injection pulse is provided by the computer via a single bit transfer through the parallel port in a nonstrobed mode to a

19

hardwired pulse-width controller. This, in turn, fires a Clippard Minimatic model EOV-3 nitrogen-powered solenoid which opens the Seiscor valve for the set length of time. The computer takes data from a Keithley model 417 high-speed picoammeter, using an internal A/D converter card. This A/D board allows sample rates up to 30 points per second and is set at +/- 10 V full scale. Because this was basically a data-processing investigation, columns, temperatures, flow rates and solutions were chosen to give certain retention times and peak widths. The principal solute used was heptane which had a retention time of 64 s at a temperature of 91°C and flow rate of 20 ml min-’ in a Durapak C8 column (l/8 in. X 24 in.). Calibration files can be simulated either experimentally or on the computer. To effect concentration changes experimentally, the resistance of the RC time constant which controlled pulse width was gradually increased or decreased allowing more or less solute to enter the chromatograph and thereby simulating a concentration change, Computer simulations of concentration changes were done by using an output data file with no concentration change and then modifying that data to simulate a nonstationarity effect. All programs were written in FORTRAN-77, using several library subprograms for input/output operations. RESULTS AND DISCUSSION

This experiment necessarily consists of first establishing a set of simulation files with known concentration changes. From the resulting correlograms, several points are chosen and calibrated for the effect that these concentration changes have had. The points chosen should be at the tops of noise peaks (or the bottoms of noise valleys) to allow better resolution and relative accuracy. As explained, a plot of concentration change vs. point value for each point should be linear. While it is possible to use only one point for calibration and monitoring nonstationary effects, it is probably best to average the results of a number of points to minimize random noise effects. Linear concentration changes are the easiest fluctuations to calibrate and detect (Fig. 4A). In this case, only the magnitude of the change will affect

Time

Fig. 4. Plot of possible input concentration

change. For explanation, see text.

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the correlation noise, so that only a few calibrations are needed. The evaluation process is quite straightforward. If concentration maxima or minima are present in the file (Fig. 4B), the calibration and evaluation become more complex as the position of a maximum, for example, will greatly affect the correlation noise pattern. In the calibration process, several files with a concentration maximum in various locations of the file will need to be simulated and calibrated. In the evaluation process of an unknown concentration change, the locations of any maxima must first be established. Then, from the appropriate calibration plot, the magnitude of the change can be calculated. To illustrate how this might be done, the data for a very simple evaluation process are given in Table 1. The calibration points used were chosen for the purpose of differentiating between files that have linear changes or a single concentration maximum (or minimum) and then quantifying that change. The first decision to be made is if there is a concentration maximum present and, if present, where in the data file it is located. This question was answered by noting that while both the position of a concentration maximum and its magnitude affect point values, only the maximum position (or, more generally, output pattern) governs where each calibration point from a group of simulation files crosses zero. For example, point 92 is near zero only when there is no concentration maximum or minimum in the file. Point 203 crosses zero when the turnaround point is at 50% of the file length, no matter what the magnitude of the concentration change at that time. By determining the near-zero values of a correlogram and comparing them to the calibration files, the position of a concentration turnaround can be approximated. Because the correlation noise changes only gradually with turnaround location, an approximation will still allow good accuracy though the higher the frequency change, the more accurate the approximation must be. The values of points 67, 352, and 462 would then be compared to the appropriate calibration to characterize fully the nonstationarity. The number of points to use and the sophistication of their analysis depends TABLE

1

Simulation Cont. change

(%)”

-25 +25 +50 +lO,-40 + 25, -25 t40, -10 aPercent

results

to illustrate

Cont. maximum locationb

0.25 0.50 0.75 of largest

evaluation Point 67 1364 -1355 -2710 1491 -760 -2245

peak in file. b Fraction

procedure values 92 1 -2 -3 -269 -997 -670

203

352

1803 -1802 -3605 2417

1371 -1370 -2740 2355 525 -1594

6 -2358

of file length.

462 -879 880 1760 -840 1198 1485

803 -1668 1668 3335 -2147 -139 2266

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on the accuracy desired and the random noise present, and is limited only by the number of points in the file. Fortunately, once a calibration is complete for a certain input pattern, it should require few modifications. If there is more than one maximum or minimum present in the file (Fig. 4C), very extensive calibration becomes necessary. To avoid this, files of shorter length, though still long enough to include the longest retained eluent, can be collected. Once the nonstationarity has been identified, the noise directly under the principal peaks can be evaluated and removed. The calibration of this “hidden” noise was accomplished by affecting two simulations of output data, one possessing a certain concentration change and the other a change of equal magnitude but opposite direction. If negligible random noise can be assumed, then, subtracting one of these files from the other leaves only twice the noise of the first file as the principal peaks subtract out but the noise contributions add together (see Fig. 5). When the concentration change has been calibrated, the noise under the principal peaks can be calculated and removed. If one is concerned only with peak height, this is a one-point process; to integrate the peak, noise must be removed over the entire peak width. The adjusted peak should then represent the average peak height or area of the peaks present in the output file. Ultimately, the resolution and accuracy of this analysis is limited by the random noise (including random nonstationarity) of the system. To establish the effects of this randomness on the technique, the simple evaluation process described was applied to data files which showed various levels of random deviation from linear concentration changes. The results are shown in Table 2. Obviously, random deviations will increase error, but because these fluctuations are averaged in the correlation process, their effect is diminished.

I

D~splocement

2 (mini

Fig. 5. Correlation noise isolated from peak by subtracting correlograms concentration changes. Dotted line represents the former peak position.

of opposing

22 TABLE 2 Effect on results of random deviations from linear changes (Values are given as percent of initial concentration and those in parentheses are expected averages) Cont. change (%) -5

-25

-50

Random deviation (%) 0 1 2 0 2 5 0 2 5

Calculated cont. change (So) -5.7 -5.2 -4.7 -25.3 -26.5 -23.6 -50.4 -51.6 -52.1

Calculated cont. average (%) 97.5 (97.5) 97.4 96.8 87.3 (87.6) 87.3 88.2 74.8 (75.0) 74.3 73.6

Thus, small changes in system response to a given input which might be difficult to detect and quantify because of random fluctuations and noise, can be evaluated by using multiple inputs and a deconvolution process which effectively averages the results of the multiple signals. Points in a correlation noise pattern directly quantify changes that occur over the length of an entire data file, making the change much more amenable to computer analysis. Finally, the analysis is very fast and continuous, constantly updating old data and giving new values for response changes and averages. It is a data-handling method applicable to a variety of signals and its application to process-stream control, flow-injection analysis and other continuous monitoring devices is direct and viable. REFERENCES 1 J. 0. Hougen, Measurement and Control Applications, 2nd edn., Instrument Society of America, Pittsburgh, 1979. 2 H. C. Smit, Trends Anal. Chem., 2(l) (1983) 1. 3 J. B. Phillips and M. F. Burke, J. Chromatogr. Sci., 14 (1976) 495. 4 J. B. Phillips, Anal. Chem., 52 (1980) 468A. 5 R. Annino and E. J. Grushka, Chromatogr. Sci., 14 (1976) 265. 6 R. Annino and L. E. Bullock, Anal. Chem., 45 (1983) 1221. 7 T. T. Lub and H. C. Smit, Anal. Chim. Acta, 112 (1979) 341. 8 K. R. Godfrey and M. Devenish, Meas. Control, 2 (1969) 228. 9 E. 0. Brigham, The Fast Fourier Transform, Prentice-Hall, 1974. 10 R. Annino, M. F. Gonnard and G. Guoichon, Anal. Chem., 51 (1979) 379.