Pulsed NMR techniques are used in instruments for the measurement of the type, property and quantity of lattice bound and free, magnetically active nuclei within a sample. Some of the substances and properties that have been measured by NMR techniques are: moisture, polymers and copolymers, oils, fats, crystalline materials, density and melt indices.
Pulsed NMR uses a burst or pulse of energy that is designed to excite the nuclei of a particular nuclear species of a sample being measured (the protons, or the like, of such sample having first been precessed in an essentially static magnetic field); in other words the precession is modified by the pulse. After the application of the pulse there occurs a free induction decay (FID) of the magnetization associated with the excited nuclei. That is, the transverse magnetization associated with the excited nuclei relaxes back to its equilibrium value of zero. This relaxation produces a changing magnetic field which is measured in adjacent pickup coils. A representation of this relaxation is the FID curve.
The general problem in data analysis of the FID curves and model making in NMR systems is to determine an acceptable modeling equation to predict the desired dependent variable (moisture and oils in foodstuffs or density, MI (melt index) or other such parameters in polyolefins) from the set of explanatory data derived from the Marquardt-Levenberg (M-L) analysis of the FIDs obtained from the NMR measurements. Present theory can suggest that some of the explanatory data will be important (crystalline/amorphous ratios for density, for example), but except for some very simple cases, is unable yet to specify an actual functional relationship among the dependent variables (designated in equations as the `y`s) and the explanatory data (designated the `x`s)
Generally the position taken is that there exists a relationship between the `x` and `y` data; but it is unknown. If a relationship does exist, it can probably be represented as a power series expansion of the `x` data. These are definitely assumptions and may not be true in particular cases, but so far they have led to acceptable results.
The analysis method described in the above related applications is to decompose the FID waveform into a sum of separate time function equations. The coefficients of these equations are derived from the FID by use of a Marquardt-Levenberg (M-L) iterative approximation that minimizes the Chi-squared function--a technique well known in the art. Some of the time function equations found useful are: Gaussians, exponentials, Abragams (Gaussian)*(sin(t))*(1/t), modified Gaussian (Gaussian)*(cos(sqrt(t))) and trigonometric. From these time functions all possible ratios of the amplitude parameters (including Hahn echoes, if present) were formed. These ratios, together with the various waveform time decays (T2's) and the reciprocals of all these (which together form the `x` data set) were fed into a stepwise statistical modeling program to select those parameters which could best forecast the dependent (the `y`) variable (density, melt index, and the like), and produce a general regression equation to predict the desired parameters. It is not unusual for such an `x` data set to contain 40 or more parameters. This approach generally gave good results with moisture, oils and density, but had limitations when applied to melt index or flow rate ratios in polyolefins.
Explanatory data used in the above referenced patent applications were essentially first order terms (amplitude ratios, T2's and their reciprocals), and there were indications that modeling the more difficult `y` data (e.g., MI) required more flexibility in the `x` data set. To this end, the `x` data set was expanded to include, in addition to the above, all meaningful cross products of the `x` data (terms such as x/x=1 are excluded, but second order terms such as x*x (x times x) are included). This process can expand the presumed relationship between the "x" and "y" data to the second order, and has been found generally adequate to meet the prediction accuracies required. It is to be understood that expansion beyond the second order may be required in some instances. This technique generates great numbers of potential `x` explanatory terms (in the order of 1000), and generally requires relatively large `y` data sets before any meaningful models can be obtained. Fortunately, the generally available statistical techniques can be coerced into dealing effectively with data sets of this size, and much improved results began to appear with the more difficult `y` terms.
There has been one known problem which was not serious with the relatively small `x` data sets previously used. This is the problem of high internal correlations among the `x` data--called multicollinearity. Multicollinearity is a natural feature of the analysis of FID data since, for example, a high density sample of polyethylene has a large crystalline component of the FID, with the result that the amorphous region must be small. Similar correlations exist among most of the other parameters, such as the T2's--the time decays of the FID component curves.
When performing regressions with correlated explanatory variables, the danger exists that the various matrices used in such regression analysis may be nearly singular; the more correlated the `x` data are, the more nearly singular are the matrices (effectively working with, say, ten variables may yield results equivalent to having only 1 or 2 independent variables). This condition can cause the resulting models to be unstable in the sense that relatively small changes in the `x` data set can cause extremely large variations in the model coefficients or even in the actual explanatory (`x`) terms chosen for the model.
With large numbers of the `x` parameters associated with the inclusion of second order terms, the problem of multicollinearity can not be ignored. Many of the added second order terms have even higher internal correlations than the original data. Removal of the highly correlated (say correlation coefficients above 0.95) is helpful but the problem still remains. That problem is that while prediction may be acceptable for the calibration data set, the on-line predictions are poor because the model is unstable when used with `x` data that were not part of the calibration data.
In addition to this problem, we have previously noted that the iterative curve fitting M-L techniques may go awry and produce meaningless results. There is a need to find an efficient test for this condition. In the above referenced patent applications, a Marquardt Reference Ratio (MRR) has had some success in finding these M-L failures. MRR makes use of the fact that there are high correlations among the `x` data.
It is a principal object of this invention to find an improved test to indicate when the M-L iterative technique has produced an erroneous result.
It is a principal object of the present invention to apply statistical processes to alleviate the instability of prediction model equations due to multicollinearity.
It is a principal object of the present invention to obtain flow rates for plastics, (melt index, melt flow and flow rate ratios for polyethylene and polypropylene) via NMR techniques.
It is yet another object of this invention to relate the type, property and quantity of target nuclei of interest accurately and precisely.