PFChrom v5 Documentation Contents AIST Software Home AIST Software Support

IRF Deconvolution and Fitting (Tutorial)

In the second tutorial, we estimated a <ge> system IRF by fitting 16 different data sets with different concentrations of standards and differing amounts of additive. We will use those estimates to deconvolve the IRF prior to fitting a closed form model.

We will also explore independent IRF estimates using a genetic algorithm for IRF Deconvolution optimization.

Importing the Data

Click the **Open** button, the first button in the program's main toolbar. Select the file **CationTutorial2.pfd**
from the program's installed default data directory (\ProgramData\PFChromv5\Data).

Deconvolving a Fitted IRF

When a standard is accurately fitted with an IRF, those parameters can be used in the IRF Deconvolution procedure to remove that instrument/system response prior to fitting a closed form model. Although this is a two step process, it can be quite fast and once an IRF is known, it can be removed from any applicable data set. To effectively fit an IRF model, the peaks need to be baseline resolved and a convolution integral is fitted, which can be appreciably slower.

Click on the **IRF Deconvolution** button in the main window.

if you did the second tutorial, click the **Import from Fit** button. We will import averaged values
from the fits saved in the second tutorial.

Select the **<ge>** fits in the **Average Fits Across Multiple Data Sets** dropdown. Click
**OK**.

If you did not do the second tutorial, you will need to manually enter the three IRF parameter values as listed below.

Be sure the following values are set in the IRF Deconvolution dialog:

Response Fn

<ge> AreaSum g,e

Deconvolve Right

IRF Parameters

0.0073714

0.0429846

0.6343680

Genetic Alg Optimization

These settings will not be used

Fourier Filter

D2 Automatic (the FFT threshold will be determined
using a smoothed 2nd derivative)

8 (8% smoothing for the D2 algorithm)

Post-Processing

Zero Baseline Points - unchecked (baseline will be
actual subtracted values)

Zero Negative Points - unchecked (negative
baseline points are permitted)

**
**Click the **Zoom-In Applies to All Graphs - X Scaling Only **button in the toolbar of the graphs**.**

**Use the mouse to zoom-in the first three peaks** in the first graph.

Viewed in this way, the averaging appears adequate. The deconvolutions look reasonable.

Click **OK** to exit the IRF Deconvolution procedure. Click **OK** to accept the default titles
for this new data level in the data file containing this deconvolved data. The deconvolved data set is
now selected in the main window.

Fitting the Data with the IRF Removed

We used Fourier deconvolution to remove the IRF from the data. We can now fit the fast closed form models to this deconvolved data. We assume the instrumental/system distortions have been mathematically removed.

Fitting the GenNLC Closed Form Model

Click the **Local Maxima Peaks** button in the main window to open the peak placement screen. Ensure
the following settings are selected:

Peak Detection

Set Sm
n(1) to 25

Peak Type

Select Chromatography
in the first dropdown

Select GenNLC as the model in the second dropdown

Scan

Set the Amp %
threshold to 1.5 %

Leave Use Baseline Segments unchecked

Be sure Use IRF,ZDD is checked

Vary

Leave width a2
and shape a3 checked, all other unchecked

Click the **Peak Fit** button in the lower left of the dialog to open the fit
strategy dialog. Select the **Fit with Reduced Data Prefit, Cycle Peaks, 2 Pass, Lock Shared Parameters
on Pass 1**. Leave the **Fit using Sequential Constraints** box checked. Click **OK**.

As the fitting proceeds, you may wish to uncheck the **Iteration Update**. This updates each fit's
graph with each successive improved iteration of that fit. Fits are faster without this graphical update.

Although we are not fitting an IRF, we are still fitting a higher order ZDD
parameter. The **2-Pass** option first estimates the main peak parameters while the starting estimate
for the ZDD asymmetry is held constant. When multiple peaks are present, especially with a significant
variation in amplitude, the **Cycle Peaks** option will individually fit peaks within the overall fit,
enabling even small amplitude peaks to be accurately fitted.

The sixteen fits should require just a few seconds. All of the threads/cores of your CPU are simultaneously used to make this fitting as fast as possible.

Click **Review Fit**.

The GenNLC Fits with the IRF Pre-deconvolved Data

Click **Numeric**, and in the **Options** menu choose **Select Only Fitted Parameters** and check
**Average Multiple Fits**. Scroll to the bottom of the Numeric
Summary.

Average for 16 Fits

Fitted Parameters

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

0.99995804
0.99995795
0.02531326
24,665,342
__41.9592113__

Peak
Type
a_{0}
a_{1}
a_{2}
a_{3}
a_{4}

1 GenNLC 8.07479019 2.30934258 0.00027187 -0.0099375 1.04831845

2 GenNLC 2.25529475 3.77250138 0.00029195 -0.0016416 1.04831845

3 GenNLC 2.61961968 4.54344339 0.00027784 -0.0008759 1.04831845

4 GenNLC 1.18644131 7.22104750 0.00039602 0.00189999 1.04831845

5 GenNLC 2.52708365 12.3209593 0.00086309 0.01873860 1.04831845

6 GenNLC 3.58674858 15.2890117 0.00099722 0.02611136 1.04831845

Note that the a_{0} area and a_{3} chromatographic distortion parameter averages mean
little since these parameters are concentration dependent, and the data samples vary a full order of magnitude
in concentration.

The goodness of fit values range from 9.80 - 131 ppm.

The second tutorial fitted these same sixteen data sets to a GenNLC<ge> model with an average goodness of fit of 10.3 ppm unaccounted variance, varying from 5.0 - 20.6 ppm:

Average for 16 Fits

Fitted Parameters

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

0.99998974
0.99998972
0.01494899
58,763,437
__10.2583889__

Peak
Type
a_{0}
a_{1}
a_{2}
a_{3}
a_{4}
a_{5}
a_{6}
a_{7}

1 GenNLC<ge> 8.07933699 2.30784582 0.00026474 -0.0099790 1.11887780 0.00737138 0.04298459 0.63436801

2 GenNLC<ge> 2.25247000 3.77100711 0.00029000 -0.0016692 1.11887780 0.00737138 0.04298459 0.63436801

3 GenNLC<ge> 2.61798274 4.54185798 0.00027606 -0.0009040 1.11887780 0.00737138 0.04298459 0.63436801

4 GenNLC<ge> 1.18633675 7.21994862 0.00039513 0.00189375 1.11887780 0.00737138 0.04298459 0.63436801

5 GenNLC<ge> 2.52764231 12.3193707 0.00086418 0.01870488 1.11887780 0.00737138 0.04298459 0.63436801

6 GenNLC<ge> 3.58747556 15.2877892 0.00099840 0.02611155 1.11887780 0.00737138 0.04298459 0.63436801

By preprocessing the data with the average IRF from the 16 fits, removing this average IRF from all 16 data sets using Fourier deconvolution, we see see an average error of 42 ppm. Since one is deconvolving an average IRF estimate on all data sets, where such was seen as having a some variability, and since deconvolution adds noise, we expect a weaker goodness of fit.

Click **OK** to close the Review window. Check **Save updated information to the current data file
when adding fits** and click **OK**, accepting the default name for the fit. Click **OK** to confirm
and click **OK** one last time to leave the placement screen and return to main screen.

An Alternative to IRF Determination by Convolution Fitting and Averaging

Averaging a number of IRF estimates from standards which are truly representative of the production samples will serve you well, provided you do not vary formula or settings which cause the IRF to vary from expectation, and assuming column health remains good.

PFChrom offers an alternative to convolution integral modeling for determining the IRF of real-world data for those instances where the peaks are baseline resolved, and the baseline is cleanly subtracted. In this instance, a genetic optimization is used to estimate the values of the IRF in an approach wholly distinct from the nonlinear convolution fitting.

This is especially attractive for the <pe> IRF, which is harder to fit in convolution models because the long tails of the 1.5 order kinetic component are more easily correlated with the slower exponential component at certain parameter guesses in the iterative fitting.

Change the **Data Level** to the entry that begins with **'BL'**, the baseline correction that was
fitted. It will be second in the dropdown.

With the exception of the Section procedure, which does not actually modify the data (only the active and inactive state of the points), the different procedures in the program that alter the data values always process the uppermost level of data. To create an alternative convolution, we must not use the uppermost "DC" level we just created since this consists of data that has already seen this deconvolution. We must first create a copy of the baseline-corrected data we used for this first deconvolution.

Click the **Copy** button in the Data Levels portion of the dialog. Click **Yes** to confirm the
copy of the 'BL' baseline-corrected level to the uppermost level and **OK** to accept the default data
level name and the data titles for this level. These will be updated when this copied level becomes a
second deconvolved data level.

Click on the **IRF Deconvolution** button in the main window. The values should be those we just entered
so that we can visually explore this average across the different data sets. Here we will also specify
the generic algorithm settings:

Response Fn

<ge> AreaSum g,e

Deconvolve Right

IRF Parameters

0.0073714

0.0429846

0.6343680

Genetic Alg Optimization

90% (Parameter
1) 0.0073714 ± 90%

20% (Parameter 2) 0.0429846 ± 20%

50% (Parameter 3) 0.6343680 ± 50%

BsLnZero (we want to maximize the baseline
points)

0.50 % (the baseline region is defined as
±0.5% of the amplitude)

Fourier Filter

D2 Automatic (the FFT threshold will be determined
using a smoothed 2nd derivative)

8 (8% smoothing for the D2 algorithm)

Post-Processing

Zero Baseline Points - unchecked (baseline will be
actual subtracted values)

Zero Negative Points - unchecked (negative
baseline points are permitted)

**
**The **Zoom-In Applies to All Graphs **option in the toolbar of the graphs will be selected upon
entry into the procedure.** **We will use this zoom-in to see a fixed zone of baseline in each graph.

**Use the mouse to zoom-in the first three peaks** in the first graph. **Additionally zoom-in just
the baseline region**.

When the deconvolution using the average IRF from the convolution fits is strongly zoomed-in, it is apparent this averaged IRF is not at an optimum for most of the data sets. The concentration increases by rows: 5 ppm in the first row, 10 ppm in the second, 25 ppm in the third, and 50 ppm in the fourth. The additive increases by columns: 0x, 1x, 1.5x, and 2x. Looking at the additive trends across the four rows of graphs, we see we tend to be too strong in the deconvolution with no additive and too weak with 2x additive. With increasing concentration, we see mostly the scaling to a fixed y range. If you look very closely, you will probably find something to dislike visually with each of the 16 deconvolutions.

Click the **Generic Optimize** button.

The procedure is mathematically intensive, and will require perhaps minute to optimize all sixteen data sets. All cores and threads are used in the genetic algorithm, so the more capable your CPU, the faster the optimizations will be. These do require a random progression (a genetic differential evolution) and for each set of trial estimates a Fourier deconvolution must be performed and analyzed. When an optimization is complete, that data set will be graphed with the optimized deconvolution and the background color will change to indicate a custom parameter set (differing from the values set in the dialog).

The GA optimization is based on the premise that the settings which maximize the baseline represent the optimum deconvolution.

When the optimizations are complete, click the **List Parameter Estimates** button.

IRF Parm1 Parm2 Parm3 Data

<ge> 0.0007371 0.0443169 0.5909612 CationStd-5ppm-0x

<ge> 0.0007371 0.0458846 0.5739107 CationStd-5ppm-1x

<ge> 0.0007371 0.0465199 0.5716800 CationStd-5ppm-1.5x

<ge> 0.0007371 0.0462845 0.5623338 CationStd-5ppm-2x

<ge> 0.0007371 0.0428577 0.5727528 CationStd-10ppm-0x

<ge> 0.0007371 0.0452781 0.5869663 CationStd-10ppm-1x

<ge> 0.0007371 0.0452583 0.5698625 CationStd-10ppm-1.5x

<ge> 0.0007371 0.0465040 0.5797757 CationStd-10ppm-2x

<ge> 0.0007371 0.0420473 0.5726280 CationStd-25ppm-0x

<ge> 0.0007372 0.0436653 0.5739285 CationStd-25ppm-1x

<ge> 0.0011291 0.0447747 0.6163836 CationStd-25ppm-1.5x

<ge> 0.0007371 0.0450597 0.5760548 CationStd-25ppm-2x

<ge> 0.0007371 0.0399514 0.5532984 CationStd-50ppm-0x

<ge> 0.0009950 0.0422471 0.5861017 CationStd-50ppm-1x

<ge> 0.0008931 0.0445735 0.6028895 CationStd-50ppm-1.5x

<ge> 0.0007411 0.0445609 0.5772500 CationStd-50ppm-2x

Note that a genetic algorithm uses random values to arrive at its solution, and you may see results which slightly differ from the above IRF estimates. This kind of Fourier optimization problem, plagued by local minima, is ideally managed by a fast genetic algorithm.

When fitting a convolution model, each peak is a convolution integral and there must be limits on the lower bound of the half-Gaussian, the 'g' component in the <ge> IRF. On the other hand, when deconvolving the <ge> IRF, as is being done in this procedure, one is deconvolving the IRF from the entire data set, irrespective of the count of peaks present. For this GA optimized deconvolution, this narrow width lower bound in the IRF of convolution models is not imposed. You will thus note that every data set optimized to a very narrow half-Gaussian width (Parm1), some to the lower limit set in the GA % limits. This optimization confirms the narrow width component is significant, better than half the area of the IRF in every instance, but of so small a width it is almost an impulse. This is one explanation as to why the <ge>, <pe>, and <e2> IRFs can be used almost interchangeably. Note also the constancy in the higher width exponential (Parm 2) and the area fraction of the narrow component (Parm3).

**Close the List window. **

An Additional Optional Exercise

If you would like an interesting additional exploration at this point, select the **<e2>** or
**<pe>** IRF and click the **Genetic Optimize** button. Despite significant differences in
the narrow width component in the <ge>, <e2> and <pe> IRFs, the deconvolutions will
look remarkably similar. You might also find it interesting to explore the single exponential, the **<e>**
IRF, if you feel the narrow width component could perhaps be omitted. You will not be able to optimize
the <e> IRF-every value of an <e> parameter produces a poorer merit function than the non-deconvolved
data. You will have to manually enter a value similar to **0.03** for the <e> parameter to see
why this is so. It is a lovely way to visually confirm the necessity of this narrow width component in
a chromatographic IRF. If you explore any of these other IRFs, select and optimize the **<ge>**
once again before continuing with this tutorial.

The GenNLC Fits with the GA Optimized IRF Pre-deconvolved Data

Note that we did not use an average IRF; each of the data sets had its own estimated IRF in the deconvolution.

Click **OK** to exit the IRF
Deconvolution procedure, **OK** to acknowledge the local deconvolution notification (all are locally
set if GA estimated), and **OK** to accept the default titles for this second deconvolved data level
in the data file. This new deconvolved data level is now uppermost in the data file and it is also selected
in the main window.

Click the **Local Maxima Peaks** button in the main window to open the peak placement screen. The prior
settings are remembered. We will again be fitting the closed form GenNLC
model.

Click the **Peak Fit** button in the lower left of the dialog to open the fit
strategy dialog. Select the **Fit with Reduced Data Prefit, Cycle Peaks, 2 Pass, Lock Shared Parameters
on Pass 1**. Leave the **Fit using Sequential Constraints** box checked. Click **OK**.

Click **Review Fit**.

Click **Numeric**, and in the **Options** menu choose **Select Only Fitted Parameters** and check
**Average Multiple Fits**. Scroll to the bottom of the Numeric
Summary.

**Average for 16 Fits**

Fitted Parameters

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

0.99996768
0.99996761
0.02640061
23,900,180
__32.3182938__

Peak
Type
a_{0}
a_{1}
a_{2}
a_{3}
a_{4}

1 GenNLC 8.07888262 2.31690150 0.00029312 -0.0097789 0.98443932

2 GenNLC 2.25586357 3.77827862 0.00029543 -0.0016222 0.98443932

3 GenNLC 2.62037724 4.54892853 0.00028019 -0.0008688 0.98443932

4 GenNLC 1.18691783 7.22571489 0.00039726 0.00188481 0.98443932

5 GenNLC 2.52745307 12.3255234 0.00086439 0.01872755 0.98443932

6 GenNLC 3.58716871 15.2936918 0.00099916 0.02611948 0.98443932

By this GA optimization, we have reduced the average error across the 16 data sets from 42.0 ppm using an average of the IRFs from the convolution fits to 32.7 ppm. This is an improvement, but still nowhere close to the 10 ppm average error of the GenNLC<ge> convolution fits. The errors of fit across the 16 data sets range from 9.01 - 46.7 ppm, a significant improvement in the data sets least effectively fitted when using an average IRF from fitting convolution models.

Click **OK** to close the Review window. Click **OK**, accepting the default name for the fit. Click
**OK** to confirm and click **OK** one last time to leave the placement screen and return to main
screen.

Locking the Average IRF Values for Subsequent Fits

We will explore one further option in this tutorial. We know IRF deconvolution adds noise. An option to remove this impact is to fit the convolution models with the IRF parameters locked at the previously estimated averages. The fits will not be closed form, and thus slower because of the convolution integrals still being fitted, but the noise arising from the deconvolution will be absent.

Change the **Data Level** to the entry that begins with **'BL'**, the baseline correction that was
fitted.

Click the **Local Maxima Peaks** button in the main window to open the peak placement screen. The prior
settings are remembered. We will again be fitting the closed form GenNLC
model.

Click the **IRF** button and enter **0.0073714**, **0.0429846**, **0.6343680**, our averages
for the <ge> parameters, in the **<ge>** section in this dialog (upper left). Also click
the **Lock** buttons on all three parameters. Click **OK**.

Select the **GenNLC<ge>** model. Click the **Peak Fit** button in the lower left of the dialog
to open the fit
strategy dialog. Select the **Fit with Reduced Data Prefit, Cycle Peaks, 2 Pass, Lock Shared Parameters
on Pass 1**. Leave the **Fit using Sequential Constraints** box checked. Click **OK**.

After the fitting is complete. click **Review Fit**.

Click **Numeric**, and in the **Options** menu choose **Select Only Fitted Parameters** and check
**Average Multiple Fits**. Scroll to the bottom of the Numeric
Summary.

Average for 16 Fits

Fitted Parameters

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

0.99997735
0.99997730
0.01898578
37,488,984
__22.6508064__

Peak
Type
a_{0}
a_{1}
a_{2}
a_{3}
a_{4}
a_{5}
a_{6}
a_{7}

1 GenNLC<ge> 8.08594987 2.30807271 0.00026604 -0.0099521 1.09407482 0.00737140 0.04298460 0.63436800

2 GenNLC<ge> 2.25705595 3.77110554 0.00028976 -0.0016624 1.09407482 0.00737140 0.04298460 0.63436800

3 GenNLC<ge> 2.62101056 4.54191884 0.00027588 -0.0008993 1.09407482 0.00737140 0.04298460 0.63436800

4 GenNLC<ge> 1.18734441 7.22006712 0.00039529 0.00189655 1.09407482 0.00737140 0.04298460 0.63436800

5 GenNLC<ge> 2.52765222 12.3199427 0.00086380 0.01875207 1.09407482 0.00737140 0.04298460 0.63436800

6 GenNLC<ge> 3.58746198 15.2882163 0.00099839 0.02616161 1.09407482 0.00737140 0.04298460 0.63436800

We now have a basis for estimating the analytical cost of using Fourier deconvolution in a preprocessing
step followed by a very fast closed-form fitting. The error with locked estimates of the IRF on the convolution
models averages 22.6 ppm across the sixteen data sets. The error with the Fourier deconvolution and closed
form fits was 41.9 ppm. If we compare the a_{2}-a_{4} higher moment parameter averages,
the differences are modest. If you want the minimum of noise in your fits, and fitting time is not an
issue for you, this approach of locking the known IRF in convolution model fits is recommended. Here we
repeat the fit averages for the deconvolved data.

To summarize the average goodness of fit for the sixteen data sets:

Full GenNLC<ge> convolution model = 10.25 ppm

GenNLC<ge> with locked <ge> average IRF = 22.65 ppm

Fourier deconvolution using individual estimates of IRF, fitted by GenNLC
= 32.32 ppm

Fourier deconvolution using fitted IRF average, fitted by GenNLC = 41.96
ppm

We can assume the additional unaccounted variance arising from using an averaged IRF is about a doubling, and that the noise introduced in the Fourier deconvolution adds approximately another doubling.

Click **OK** to close the Review window. Click **OK**, accepting the default name for the fit. Click
**OK** to confirm.

Click the **IRF** button and remove the **Lock** on the three **<ge>** parameters previously
set. The IRF and ZDD customizations are global and remain in place until changed. Click **OK**.

Click **OK** one last time to leave the placement screen and return to main screen.

Fitting an IRF Convolution Model vs Deconvolving the IRF and Fitting a Closed-Form Model

Fitting an IRF-bearing convolution model, such as the GenNLC<ge> (as we fit above with locked parameters) is not the same analytical process as employing a Fourier deconvolution to remove the <ge> IRF and then subsequently fitting a closed form GenNLC model. For one, only convolution occurs in the first instance, and only deconvolution in the second. The convolution process smoothes data, the deconvolution process adds noise. That noise is usually filtered within the Fourier domain during the inverse which follows the deconvolution, but that noise filtering may add its own bias. The results of both approaches will be quite close, but there will be differences, especially in the higher moments.

Convolution vs. Deconvolution

If you need the most accurate a_{3} chromatographic distortion and a_{4} asymmetry parameters
possible, you will probably want to fit the convolution integral directly. If you do not have baseline
resolved peaks with at least one of the peaks intrinsically fronted, you may need lock an average IRF
as we did in the last example. Otherwise, you should be able to use either of the approaches illustrated
in this tutorial for a Fourier deconvolution preprocessing step followed by a very fast and efficient
closed form model fit. We observed that the two-step process, despite using exactly the same IRF parameter
values, adds the noise from the Fourier deconvolution that was not filtered, as well as any bias arising
from this signal filtration.

In discrete Fourier processing, convolution and deconvolution will have subtle differences. When we fit
a convolution integral, such as the GenNLC<ge>, we fit a convolution. The different tiers of deconvolution
are parametric and derived from the fitting; one can generate an IRF-free GenNLC by using the a_{0}-a_{4}
parameters of the GenNLC<ge> fit, or an IRF-free and ZDD-free NLC by using the a_{0}-a_{3}
parameters, or even an infinite dilution Giddings peak using the a_{0}-a_{2} parameters.
In an <irf> fit, there is no Fourier deconvolution, only Fourier convolution. All deconvolution
is parametric, and noise free.** **

However elegant and efficient from a goodness of fit perspective, there are significant drawbacks to a
convolution model fit where the IRF is estimated. The data must be clean enough to expose the IRF for
such estimation, meaning at least certain peaks must be baseline resolved and ideally with one or more
intrinsically fronted (-a_{3}) peaks to help statistically separate a_{3} from a_{4}.
Because of multiple parameters which can compete in addressing tailing, the S/N must be strong. The fits
may not be possible if only intrinsically tailed peaks are present in the data. Also, convolution models
are integrals computed in Fourier domain convolutions, and thus much slower to fit than closed form models.

The main benefit of a convolution model fit is that every fit will contain parameters that give you estimates of system stability and column health to the extent the IRF parameters are accurately estimated. The goodness of fit is also potentially higher.

Deconvolution vs Convolution

Deconvolution is fast, efficient, and once an IRF for a given instrument, column, and prep is quantified, it can be used to render any data set amenable to closed form model fits. Data which have overlapping and hidden peaks aren't an issue, and with an automated filter, such as the D2 filter we used in this tutorial, varying S/N can often be transparently managed. This two-step approach works whether or not the peaks are baseline resolved, no matter how messy the data happens to be. Closed-form fits are also far less susceptible to failed fits arising from correlated parameters since no IRF is fitted. They are also exceptionally fast, usually mere seconds or less for multiple high sampling-density data sets.

There are drawbacks to deconvolution followed by a closed-form model fit. The IRF must first be predetermined by some method, either fitting a convolution model or by a genetic optimization of the deconvolution. The goodness of fit will definitely be weaker as a consequence of the noise introduced by discrete deconvolution, and unless each data set is independently optimized with the genetic algorithm, there will be a small diminishing of the goodness of fit by the imposition of an averaged IRF estimate.

In Perspective

To put these differences in perspective, the best fit realizable in prior releases of the technology, the NLC, generated an 3470 ppm average error for these sixteen fits. In the above discussion, we address average errors which range from about 10 - 40 ppm. The reason we give such attention to this refinement in fitting is that the higher moments and the IRF parameters are important in measuring system stability and performance, and in any process optimization. If you wish to know if small changes in formulations produce better separations, you need to know if a change increases or diminishes the IRF distortions and the deviations from ideality represented in the higher moment parameters.

For most analytical work, identifying the IRF for a given column's general procedure and using IRF Deconvolution followed by closed-form GenNLC or GenHVL fits may be all you ever need. As a crosscheck on system stability, however, we would recommend routine convolution fits for the standards to ensure there have been no changes in the system or column health.