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Deconvolved Moments

In the Deconvolved Moments section of the Numeric Summary, measured (integrated) moments are given for the various deconvolution levels that arise from the fitting. Note that the second moment is listed as the standard deviation, the square root of the variance or second moment.

In general, for the maximum in analytic accuracy of moments, you will want to use information from a deconvolution where the theoretical peak is recovered and both the instrumental distortion (IRF) and intrinsic non-ideality (ZDD) effects are removed. The accuracy should be appreciably greater. The following example illustrates this for one of the two main models in the program, the GenNLC.

GenNLC Example

We will start with two fits, the first of data containing three peaks which are fronted and thus enable an easy determination of the IRF. Here we fit the data peaks to a GenNLC<ge>, a composite model containing the peak and instrumental distortion.

Fitted Parameters

r^2 Coef Det      DF Adj r^2      Fit Std Err      F-value        ppm uVar

0.99999626        0.99999625      0.00737447       69,224,300   3.73855756

Peak      Type            a0              a1              a2              a3              a4             a5              a6              a7

1         GenNLC<ge>      4.15166233      3.21604491      0.00027284      -0.0062463      1.1400634      0.00568777      0.04577913      0.65908137

2         GenNLC<ge>      1.18456089      5.16110351      0.00032154      -0.0011089      1.1400634      0.00568777      0.04577913      0.65908137

3         GenNLC<ge>      1.3883167       6.19097353      0.00032485      -0.0006517      1.1400634      0.00568777      0.04577913      0.65908137

Peak      Type            Area            % Area          Mean            StdDev          Skewness        Kurtosis

1         GenNLC<ge>      4.15171638      61.7393391      3.34062754      0.06878558      -0.0363613      3.83804159

2         GenNLC<ge>      1.18455943      17.6153451      5.20747294      0.06763697      0.28609566      3.96583262

3         GenNLC<ge>      1.38831249      20.6453158      6.22747272      0.072091        0.26971205      3.74802768

All       Total           6.72458831      100

In the second fit, we fit these three peaks after first performing a <ge> IRF deconvolution using the parameters [0.00641, 0.04724, 0.6426] as the average expected instrumental IRF. Note that these IRF parameters do not exactly match the fitted <ge> in the shared a5-a7 above.

Fitted Parameters

r^2 Coef Det      DF Adj r^2      Fit Std Err      F-value        ppm uVar

0.99999079        0.99999076      0.01322582       30,090,980   9.20810079

Peak      Type            a0              a1              a2              a3             a4

1         GenNLC          4.15792306      3.21576056      0.00027022      -0.0062455      1.16623974

2         GenNLC          1.18532879      5.16075375      0.00031975      -0.0011165      1.16623974

3         GenNLC          1.38923105      6.19059088      0.00032333      -0.0006602      1.16623974

Peak      Type            Area            % Area          Mean            StdDev          Skewness        Kurtosis

1         GenNLC          4.15792306      61.7591329      3.32199812      0.06017143      -0.6417655      2.87813101

2         GenNLC          1.18532879      17.6061166      5.18878808      0.05877753      -0.1969076      2.99103142

3         GenNLC          1.38923105      20.6347505      6.20876832      0.06384297      -0.1061161      2.99302052

All       Total           6.7324829       100

In looking at the first five parameters of the peaks in the two fits, you see very good agreement. The Deconvolved Moments section of the Numeric Summary uses these fitted parameters to furnish integrated moments which are likely to be more useful than the actual moments of the peaks in the data. In this example, we know from the fits' a3 that the three peaks are fronted. The skewness of the fitted GenNLC<ge> peaks, which include the IRF distortion, hardly reflect such. If we look at the actual moments of the fits of the pre-deconvolved data, the third and fourth moments of the data with the IRF pre-deconvolved track much more closely with the fitted chromatographic distortion.

We can now look at the Deconvolved Moments section for the GenNLC<ge> fit:

Deconvolved Moments

Peak      Type        Area            Mean            StdDev          Skewness        Kurtosis

1         GenNLC      4.15166233      3.3823982       0.07747355      -0.6753206      2.77493617

2         GenNLC      1.18456089      5.21478532      0.06160322      -0.3999334      3.02166993

3         GenNLC      1.3883167       6.22632043      0.06506786      -0.2519095      3.00812973

Peak      Type        Area            Mean            StdDev          Skewness        Kurtosis

1         NLC         4.15166233      3.3823982       0.07747355      -0.6753206      2.77493617

2         NLC         1.18456089      5.21478532      0.06160322      -0.3999334      3.02166993

3         NLC         1.3883167       6.22632043      0.06506786      -0.2519095      3.00812973

Peak      Type        Area            Mean            StdDev          Skewness        Kurtosis

1         Giddings  4.15166233      3.21604491      0.04189202      0.01953892      3.00050903

2         Giddings  1.18456089      5.16110351      0.0576108       0.01674375      3.0003738

3         Giddings  1.3883167       6.19097353      0.06342177      0.01536635      3.00031483

We can also look at the Deconvolved Moments for the GenNLC fit with the IRF pre-deconvolved:

Deconvolved Moments

Peak      Type        Area            Mean            StdDev          Skewness        Kurtosis

1         NLC         4.15792306      3.38228533      0.07739938      -0.6750866      2.77293732

2         NLC         1.18532879      5.21489917      0.06151559      -0.4037511      3.0220643

3         NLC         1.38923105      6.2264643       0.06496737      -0.256191       3.00854726

Peak      Type        Area            Mean            StdDev          Skewness        Kurtosis

1         Giddings  4.15792306      3.21576056      0.04168863      0.01944577      3.00050418

2         Giddings  1.18532879      5.16075375      0.05744806      0.01669758      3.00037175

3         Giddings  1.38923105      6.19059088      0.06327111      0.01533079      3.00031338

Note first that the GenNLC (the peak absent the instrument effects) varies considerably in the two higher moments with the two different approaches for managing the IRF. The GenNLC is a peak that will be free of the instrumental distortions, but not the intrinsic deviations from the theoretical model.

The NLC is the pure theoretical model (with the chromatographic distortion) and the Giddings is the pure zero-distortion theoretical density (no chromatographic distortion). Note the consistency in the higher moments of the NLC and Giddings models across the two approaches.

If the object of obtaining the moments is to quantify rather than discard the instrumental effects and intrinsic non-idealities, as in determining instrument variations or degradation of column function, the reverse applies: the moments of the distorted peaks will be target metrics.

A GenHVL example would look almost the same as the GenNLC above except that the HVL is the theoretical final model for the moments and the Gaussian is the ZDD.