Derivative-Fourier transforms-polynomial fit: a coupling of interest in common non-ideal cases arises during trace metal analysis using graphite furnace atomic absorption spectrometry

Abstract

Chemometric treatment was proposed for handling graphite furnace atomic absorption (GFAAS) data. It aims at correcting the errors encountered in such a technique due to improper method parameter optimization such as: temperature of pyrolysis, absence of a chemical (matrix) modifier, slit width, concentration working range and cleaning of the graphite tube between injections. These common examples of improper method parameters optimization lead to non-ideal cases of linearity, at which the calibration curves suffer from a pronounced downward curvature. This will eventually cause a complete loss of accuracy and precision. The suggested chemometric handling of data involves the application of a derivative method to GFAA peaks followed by their convolution using 8 points sin x_i polynomials (discrete Fourier functions). The calibration data points before and after the chemometric treatment were fitted using different polynomial orders to investigate the efficiency of the proposed chemometric. Also, assessment of precision and accuracy was done to test the efficiency of the method and to check the absence of polynomial over-fitting. For comparison, the analysis was done using optimized method parameters, ideal case of linearity. For the non-ideal cases, t and F tests were applied to test the presence of significance differences in accuracy and precision results obtained before the data treatment and after the data treatment using different chemometric methods along with the different order polynomials used to fit the different calibration data.

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1. Introduction

One of the most important analytical techniques used for trace analysis of metals in a variety of matrices is graphite furnace atomic absorption spectrometry (GFAAS). The technique is sensitive, selective and allows the injection of samples without or with minimum treatment before the analysis.^1,2 However, the GFAAS technique exhibits many limitations and is subject to different types of interferences, so a thorough optimization of the analysis parameters is necessary.^1,2

The chemical and matrix interferences are the most common in GFAAS. They mainly affect the atomization efficiency of the analyte.^1,2 This cause the analyte signal obtained from the sample to be either enhanced or suppressed compared to the signal obtained from the standard. Serial dilutions are necessary to confirm the absence of interference. This could be done by successive dilution and reanalyzing the sample to determine whether the interference can be eliminated.^1,2 However, it is time consuming and may result in increasing the risk of samples contamination. Generally, the analysis should be performed using standard addition method. These types of interferences usually affect the slopes of the calibration graphs and cause a downward curvature in them.³ It is common practice that chemical (matrix) modifiers are added to all samples.

Modifiers affect the thermal processes in the atomizer in order to decrease the analyte loss during pyrolysis and to facilitate the removal of interfering species in the matrix.³ In order to attain this, some type of modifiers modify the sample matrix while others stabilizes the analyte to prevent its loss.

The graphite furnace has a greater temperature variation than flame. An optimum temperature programming is necessary to achieve the maximum analyte absorbance but with the minimum loss of analyte and minimum background interference during atomization.^1,2 Optimum temperature programming is essential for a reproducible signal (in terms of height and shape of the peak) of the analyte to be recorded.

Another limitation of the GFAAS is the limiting concentration working range. This happens at high concentrations, at which the intensity of the resonance line is small compared with the intensity of the non-absorbed line. This effect is displayed as a severe downward curvature of the calibration graph.^4,5

By decreasing the slit width of the monochromator, the light from the non-absorbed line is blocked. As a result, the graph becomes less curved.^4,5 However, decreasing the slit cause a decrease in the amount of light from the resonance line itself. This mainly causes an increase in the baseline noise which affects the detection and quantitation limits of the analyte. An optimization of the slit width should be done to get the minimum calibration graph curvature and the least baseline noise level.^4,5

The previously stated parameters are only examples of many other parameters that should be optimized when using GFAAS in trace analysis, as the cleaning out procedure of the graphite tube itself between injections. The failure to optimize and adjust these parameters will eventually lead to deviation from linearity of calibration graphs and a subsequent loss of accuracy and precision of the method.⁶

There are various approaches to overcome the above mentioned inconvenience. In the presented work, chemometric⁷ and mathematical⁷ handling of the GFAAS data are being investigated. Few attempts were found in literature trying to solve the problems of GFAAS technique using chemometric.^8–11

In the present work, the use of numerical differentiation of signals with respect to time is used to correct for constant and linear interferences in GFAAS measurements by using first and second derivative methods, respectively.¹²

Moreover, other types of interferences that may be present in the measurement are corrected by the convolution of the resulted first and second derivative curves using 8-points sin x_i polynomials (discrete Fourier functions).^13–17

If the calibration data points cannot be represented by a straight line, they could be fit well to a curved algebraic function. The simplest way to obtain such a fit is to use the polynomial functions.³ However, polynomials should be carefully used to prevent the problem of over-fitting. The over-fitting takes place upon using high order polynomials. It is characterized by obtaining excessively complex curve; with pronounce inflections, which may fit to noise and errors.³ The line will fit the data points well but fails to predict new data (loss of precision and accuracy). Consequently, extrapolation from polynomials is misleading.

Using the GFAAS derivative curves instead the original curves with their subsequent convolution using 8-points sin x_i polynomials (discrete Fourier functions) will eventually eliminate different types of interferences.¹⁸ This will produce highly purified analytical peaks for the quantitative analysis using GFAAS.¹⁹

Polynomial functions were used to fit the calibration data before and after the chemometric treatment. Low order polynomials were suitable to fit the calibration data after the proposed chemometric method compared to the data before the chemometric treatment.

To emphasis the effect of the proposed chemometric methods,^20–22 different non-ideal cases were intentionally created during the analysis of different metals using GFAAS. Data resulted from the application of non-ideal cases led to non-linear calibration graphs with complete loss of accuracy and precision. Chromium will be taken as an example during the present study and the non-ideal cases are illustrated in Table 1.

Table 1 Summary of the atomic absorption graphite furnace analysis parameters for chromium in different cases

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The purpose of the present work is to evaluate the efficiency of differentiating the GFAA peaks followed by their convolution using 8-points sin x_i polynomials (discrete Fourier functions) for treating the non-linear calibration graphs in GFAAS, aiming at solving the most common non-ideal cases that could appear in such technique. The data after chemometric treatment were fitted with different order polynomials to test the efficiency of the chemometric treatment. Complete validation of the method was done. Accuracy and precision were important to be validated to test the efficiency of the proposed method and to detect the over-fitting problem. In the same time, the analysis of the selected metal (chromium) was also performed using the optimized parameters (ideal case). A comparison, using t and F tests, was done between the different chemometric methods in the non-ideal cases along with the different orders polynomials used to fit the different calibration data obtained and the direct measurement using the linear fit of regression points.

2. Experimental

2.1. Instruments and tools

An atomic absorption with graphite furnace and auto sampler unit VARIAN-AA240FS Fast Sequential Atomic Absorption Spectrometer, serial no. AA1007M053, GTA120 (Graphite Tube Atomizer) (GFAAS) and Smart water Chiller H150-1500, serial no. 1003118945 (Lab Tech – Italy).

2.2. Materials

Pure reagents were used as: nitric acid 65% pure (POCH – Poland) and hydrochloric acid 37.0% w/w, (Sigma-Aldrich). The water used through the study was bi-distilled water. Ascorbic acid (made in Sun Tin Co. – China) was used as the second modifier. The metal standard solution (1000 ppm) used was of analytical grade chromium, (MERCK – Germany as nitrate in HNO₃ 0.5 mol L⁻¹).

2.3. Procedure

2.3.1. Preparation of chromium working standard and serial diluted solutions. The working standard solution of chromium (250 ppb) was prepared in 0.1 M HCl from the corresponding stock (1000 ppm) solution (Merck). Serial diluted standard solutions in the ideal and the non-ideal linearity ranges mentioned in Table 1 were prepared by auto dilution with 0.1 M HCl solution. This was done through auto sampler procedure.

2.3.2. Procedures for chromium analysis in the GFAAS.
2.3.2.1. For the ideal case of linearity. A volume of 25 μL of each of the finally prepared solutions was auto injected into GFAAS. This was done using 0.1 M HCl solution as a blank after adjusting the parameters of analysis as described under ideal case in Table 1, Fig. 1a.

Fig. 1 Signal graphics of chromium metal, (a) is for the ideal case at 40 ppb conc. level and (b–e) are for the different non ideal cases at the same conc. level (40 ppb) while (f) is for the non ideal case of improper working range selection at 100 ppb conc. level.

2.3.2.2. For the non-ideal cases of linearity. The same procedures were repeated except that each time one parameter was altered to yield non linear calibration data points as presented under non-ideal cases in Table 1, Fig. 1b–f.

2.3.3. Treatment of GFAA data of chromium analysis in the ideal and non ideal cases of linearity.
2.3.3.1. Application of derivative technique (D method) to GFAAS response data. Derivative calculation was applied to GFAAS response data. When direct method (atomic absorbance method) exhibits interference, the derivative methods can be used instead. The direct method for chromium determination was done using the recorded absorbance values at 63.8 s. Constant and linear interferences could be eliminated by calculating the first (D₁) and second (D₂) derivative, respectively.^13–17 For the atomic absorbance curve of chromium at the non ideal cases of linearity, the absorbance (A) was recorded versus time at 0.1 s as time interval then the D₁ and D₂ curves were obtained using Excel software. The D₁ or D₂ values at peak to zero were selected and related to concentration for the ideal case (Fig. 2) and for the different non-ideal cases. For the ideal case, the selected points were 63.4 and 63.3 s for D₁ and D₂, respectively. The non-ideal case of improper slit width selection was taken as a representative example (Fig. 3). Its selected D₁ and D₂ values at peak to zero were 63.5 and 63.4 s, respectively.

Fig. 2 First (a) and second (b) derivative curves derived from the signal graphic in Fig. 1(a) of the ideal case and their convoluted Fourier curves (a′) and (b′), the arrows indicates the selected points used.

Fig. 3 First (a) and second (b) derivative curves derived from the signal graphic in Fig. 1c (40 ppb) of the non ideal case (improper slit width selection) and their convoluted Fourier curves (a′) and (b′), the arrows indicates the selected points used.

2.3.3.2. Application of Fourier functions to derivatives data (D/FF method). The first and the second derivative curves were convoluted using 8-points sin x_i polynomial at 0.1 s time interval according to the following equation:

t = (0)D′₀ + (+0.707)D′₁ + (+1)D′₂ + (+0.707)D′₃ + (0)D′₄ + (−0.707)D′₅ + (−1)D′₆ + (−0.707)D′₇/4

where D′₀ to D′₇ stand for the eight derivative values; at 0.1 time interval. The numbers in brackets are values of the selected Fourier function.

After that, the optimum D₁/FF, D₂/FF values at the selected points were related to concentration for the ideal case (Fig. 2) and for the different non-ideal cases. For the ideal case, the selected points were 63.3 and 63.4 s for D₁/FF and D₂/FF, respectively. The non-ideal case of improper slit width selection was taken as a representative example (Fig. 3). Its optimum D₁/FF and D₂/FF values at peak to zero were 63.3 and 63.5 s, respectively.

Since convolution using Fourier functions can eliminate many types of interferences except for linear interference, application of Fourier functions on derivatives data would finally lead to removal of linear and many non-linear types of interferences found in the original analytical signals. Data at the selected points were used for the assessment of linearity, precision and accuracy.^13–17

This was beneficial in the presented non ideal cases of linearity at which the accuracy and precision of the method were recovered, as will be discussed later in the section of accuracy and precision assessment. The selected points were based on that the background noise was neglected and the metal response at each point was maximum response.

2.3.3.3. Polynomial fitting of the calibration points data. A polynomial is a function showing how a variable y depends on a combination of increasing powers of another variable x. Polynomials are usually fitted to data containing error in y by an extension of the least-squares principle. If the first order line is far from the calibration points of the non ideal cases of linearity (Fig. 4) then, the second order curve is suitable to be used. The second order curve will provide a curve of a type that is often observed in analytical systems, a smoothly decreasing gradient curve with increasing concentration.³ As increase the order of polynomial, the closer is the curve to the data points but “over-fitting” may arise. The over-fitted curve shows pronounced inflections, which certainly do not exist in a true calibration function. In many cases, small extrapolations of polynomial fits above order 2 could lead to unreliable estimates of concentration.³ First, second and third order polynomials were tested to fit the calibration data points of the non-ideal cases of linearity after and before the proposed chemometric method. Over-fitting was carefully checked.

Fig. 4 The three polynomial fit of the calibration points for the non ideal case of improper working range selection, (a) is for the direct measurement, (b) and (c) are for the first and second derivative data while (b′) and (c′) are for their convoluted Fourier data.

3. Results & discussion

3.1. Assessment of linearity using linear parametric regression method

For the ideal case, direct measurement was done for assessment of linearity. The linearity of chromium by GFAAS method was done by plotting the concentrations of the final diluted solutions against their peaks height at 63.8 s, Table 2. For the non-ideal cases, both direct measurement and proposed chemometric methods were used for the assessment of linearity. For the chemometric method, the graphs obtained by plotting derivative and convoluted derivative Fourier functions data at the selected points, versus concentration, show various degrees of linearity and were compared to the linearity obtained using direct measurement. The non-ideal case of improper working range selection is taken as a representative example, Table 3. Using the method of least squares (first order polynomial), different statistical parameters were calculated as stated in Table 3. Upon applying D methods then D/FF methods in the linear parametric regression for all the non-ideal cases, the values of coefficient of determination (R²) and F variance ratio were increased indicating the good linearity of the calibration graphs obtained after treatment of data.²³ The values of (b) slope increased while the values of (a) intercept decreased upon applying different chemometric methods relative to the direct measurement. Taking Table 3 as an example, the (b) value was increased from 0.0181 to 1.0281, upon applying direct method relative to D₂ method which indicates that the sensitivity of the applied chemometric method is higher than the direct measurements. The S_y/x value in non-ideal cases represented in Table 3 decreased after the chemometric treatment. For example, it decreased from 21.5 × 10⁻² to 0.4 × 10⁻², upon applying direct method relative to D₁ method. This indicates that after application of chemometric methods, the error in the y-direction is minimized and the calibration points become closer to the linear regression line. Regression lines with high F-values are much better than those with lower ones²⁴ and this was achieved when the data were handled with the proposed chemometric methods. As can be seen in Table 3, the F-value increased from 48 to 64 905, upon applying the direct measurement relative to the D₂/FF method.

Table 2 The statistical parameters for the ideal case of linearity based on the least squares parametric method of regression in the direct measurement

	^aR^2	a^b	b^c	Sx/y^d	Sa^e	Sb^f	F^g	LOD^h	LOQ^i
a Coefficient of determination.b Intercept.c Slope.d Standard deviation of residuals.e Standard deviation of intercept.f Standard deviation of slope.g Variance ratio, equals the mean of squares due to regression divided by the of squares about regression (due to residuals).h Limit of detection (ppb).i Limit of quantitation (ppb).
Direct	0.9956	0.0069	0.0209	3.09 × 10^−2	3.35 × 10^−2	0.80 × 10^−3	676	4.44	14.79
D1	0.9999	−0.0089	0.0955	0.53 × 10^−2	0.58 × 10^−2	0.14 × 10^−3	473 296	0.17	0.56
D1/FF	0.9992	0.0304	0.0748	4.74 × 10^−2	5.14 × 10^−2	1.23 × 10^−3	3680	1.90	6.34
D2	0.9999	−0.0924	0.9554	4.78 × 10^−2	5.18 × 10^−2	1.24 × 10^−3	592 271	0.15	0.50
D2/FF	0.9999	0.3328	0.4589	33.94 × 10^−2	36.80 × 10^−2	8.82 × 10^−3	2706	2.22	7.40

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Table 3 The statistical parameters for the non ideal of improper working range selection based on the least squares parametric method of regression in the proposed chemometric methods

	^aR^2	a^b	b^c	Sx/y^d	Sa^e	Sb^f	F^g	LOD^h	LOQ^i
a Coefficient of determination.b Intercept.c Slope.d Standard deviation of residuals.e Standard deviation of intercept.f Standard deviation of slope.g Variance ratio, equals the mean of squares due to regression divided by the of squares about regression (due to residuals).h Limit of detection (ppb).i Limit of quantitation (ppb).
Direct	0.8745	0.4343	0.0181	21.5 × 10^−2	16.8 × 10^−2	0.26 × 10^−2	48	3.448	11.49
D1	0.9999	0.0078	0.0998	0.4 × 10^−2	0.3 × 10^−2	0.54 × 10^−4	342 801	0.469	1.563
D1/FF	0.9990	0.0161	0.0295	2.9 × 10^−2	2.3 × 10^−2	0.03 × 10^−2	7087	1.823	6.076
D2	0.9999	0.0521	1.0281	5.4 × 10^−2	4.2 × 10^−2	0.06 × 10^−2	2 522 417	0.151	0.503
D2/FF	0.9999	−0.0471	0.4719	15.4 × 10^−2	12.0 × 10^−2	0.18 × 10^−2	64 905	0.153	0.511

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3.2. Assessment of linearity using polynomial fitting

For the non-ideal cases, the calibration data points before and after the chemometric treatment were fit using polynomial functions. Fig. 4 and Table 4 represent the non-ideal case of improper working range selection as representative example. For the direct measurement, the polynomial fit alone didn't cause a remarkable enhance in the regression parameters. On the contrary, the pronounced enhancement in the linearity parameters was achieved upon the polynomial fit of calibration data points after chemometric treatment. For example, the S_y/x value of the direct measurement decreased from 21 × 10⁻² to 0.4 × 10⁻² after the D₁ treatment in the linear fit (Table 3) while it decreased to 6.40 × 10⁻² and 10.03 × 10⁻² upon using the quadratic and the cubic polynomial fit with no chemometric treatment (Table 4). Using the same example of D₁ method, a less remarkable change was recorded for the S_y/x value. Its value changed from 0.4 × 10⁻² in the linear fit to 0.58 × 10⁻² and 0.81 × 10⁻² in the quadratic and the cubic fit, respectively. As mentioned in Section 3.1, the F value of the direct measurement increased from 48 to 64 905 upon applying the D₂/FF method in the linear fit (Table 3). In the same time, this value was changed to 28 877 and 17 470 after fitting the D₂/FF data points using quadratic and cubic polynomial fit (Table 4). The pronounced effect of the chemometric treatment alone compared to the effect of the polynomial fit indicates that the chemometric treatment is useful in enhancing the results by eliminating interferences and thus produces purified analytical peaks. Moreover the chemometric treatment is not subject to the problem of over-fitting as does the polynomial fit.

Table 4 The regression parameters for the non ideal of improper working range selection based on the quadratic and cubic fit in the proposed chemometric methods

		^aR^2	a^b	b^c	Sx/y^d	Sa^e	Sb^f	F^g
a Coefficient of determination.b Intercept.c Slope (two obtained variables in quadratic fit and three in the cubic fit).d Standard deviation of residuals.e Standard deviation of intercept.f Standard deviation of slope.g Variance ratio, equals the mean of squares due to regression divided by the of squares about regression (due to residuals).
Direct	Quad	0.9945	−0.152	X2 = −0.3 × 10^−3	6.40 × 10^−2	X2 = 2.15 × 10^−5	4.84 × 10^−2	544
				X = 4.57 × 10^−2		X = 0.25 × 10^−2
	Cubic	0.9949	−0.104	X3 = −6 × 10^−7	10.03 × 10^−2	X3 = 9.32 × 10^−7	5.10 × 10^−2	327
				X2 = −1.5 × 10^−4		X2 = 1.6 × 10^−4
				X = 4.13 × 10^−2		X = 0.74 × 10^−2
D1	Quad	0.9999	0.0028	X2 = −2.1 × 10^−6	0.58 × 10^−2	X2 = 1.97 × 10^−6	0.44 × 10^−2	1 743 136
				X = 100.1 × 10^−3		X = 0.23 × 10^−3
	Cubic	0.9999	−0.0056	X3 = 1.05 × 10^−7	0.81 × 10^−2	X3 = 7.54 × 10^−8	0.41 × 10^−2	1 345 528
				X2 = −1.9 × 10^−5		X2 = 1.25 × 10^−5
				X = 100.9 × 10^−3		X = 0.6 × 10^−3
D1/FF	Quad	0.9990	0.0058	X2 = −4.3 × 10^−6	4.09 × 10^−2	X2 = 1.38 × 10^−5	3.10 × 10^−2	3086
				X = 29.9 × 10^−3		X = 1.6 × 10^−3
	Cubic	0.9990	−0.0029	X3 = 1.09 × 10^−7	6.68 × 10^−2	X3 = 6.21 × 10^−7	3.39 × 10^−2	1725
				X2 = −2.2 × 10^−5		X2 = 1.03 × 10^−4
				X = 30.75 × 10^−3		X = 0.50 × 10^−2
D2	Quad	0.9999	0.0612	X2 = 4.14 × 10^−6	7.62 × 10^−2	X2 = 2.57 × 10^−5	5.80 × 10^−2	1 085 712
				X = 102.76 × 10^−2		X = 0.297 × 10^−2
	Cubic	0.9999	−0.0969	X3 = 1.98 × 10^−6	8.10 × 10^−2	X3 = 7.5 × 10^−7	4.10 × 10^−2	1 444 187
				X2 = −0.32 × 10^−3		X2 = 0.12 × 10^−3
				X = 104.24 × 10^−2		X = 0.598 × 10^−2
D2/FF	Quad	0.9999	0.035	X2 = 3.46 × 10^−5	21.43 × 10^−2	X2 = 7.24 × 10^−5	16.30 × 10^−2	28 877
				X = 46.8 × 10^−2		X = 0.84 × 10^−2
	Cubic	0.9999	0.2021	X3 = −2.1 × 10^−6	33.65 × 10^−2	X3 = 3.13 × 10^−6	0.1711	17 470
				X2 = 0.38 × 10^−3		X2 = 0.52 × 10^−3
				X = 45.24 × 10^−2		X = 2.494 × 10^−2

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3.3. Limit of detection and limit of quantitation (LOD & LOQ)

The LOD and LOQ after the chemometric treatment of the data are much less than those obtained in the direct measurements before chemometric treatment of data. As seen in Table 3, the LOD and LOQ were decreased from 3.448 and 11.49 ppb to 0.151 and 0.503 ppb upon applying the direct method relative to D₂ method respectively. This indicates that the proposed chemometric treatment of data could expand the linearity range of GFAAS technique.

3.4. Accuracy and precision

In order to test the accuracy and the precision of the presented method, different concentration levels within the linearity range were used to calculate the recovery% ± RSD%.This was done either using direct method (peak height method) or the chemometric methods (derivative and convoluted derivative using Fourier function) in the linear parametric and different polynomial fit methods of analysis. The non-ideal case of improper temperature program adjustment is taken as a representative example, Table 5. Using the method of least squares (first order polynomial), the values of recovery% of each individual concentration level were calculated as stated in Table 5. Upon applying D methods then D/FF methods in the linear parametric regression for all the non-ideal cases, the values of recovery% were in the accepted ranges (98–102%) indicating the good accuracy of the method obtained after treatment of data. For example in Table 5, the values of recovery% of concentration levels 10 ppb and 50 ppb decreased from 130.31 and 113.26% to 100.07 and 100.15% upon applying the direct measurement relative to the D₂/FF method.

Table 5 The accuracy and precision of the direct and the proposed chemometric methods for the non ideal case of improper temperature program adjustment using the linear parametric method of regression

Labeled conc. (ppb)	Direct	D1	D1/FF	D2	D2/FF
a F – critical value is 5.05.b t – critical values for one tail test is 2.02 and two tail test is 2.57.
10	130.31	101.57	99.70	99.66	100.07
25	84.74	103.41	100.13	100.07	100.07
30	97.11	99.25	100.04	100.13	100.07
45	113.26	98.87	99.89	99.99	100.07
50	113.21	100.90	99.70	100.08	100.15
65	99.79	100.26	99.65	99.99	100.07
Recovery% mean	106.4	100.71	99.85	99.99	100.08
SD	15.90	1.66	0.20	0.17	0.03
RSD	14.94	1.65	0.20	0.17	0.03
F-test^a		91.52	6287	8845	237 051
t-test^b		0.8724	1.009	0.9884	0.9735

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In the same time, the RSD% for the different concentration levels used in the study showed a great decrease in their values upon applying the different chemometric methods relative to the direct method using the linear polynomial fit. For example in Table 5, the value of RSD% decrease from 14.94 to 0.03 upon applying direct method relative to D₂/FF method. This indicates that the proposed chemometric methods are valid for the estimation of the unknown metal concentrations in the whole linearity range with acceptable accuracy and precision.

For the different polynomial fit as can be seen in Table 6, the different polynomial fit of the calibration points in the direct measurement showed great variation in the recovery% of the different concentration levels. The mean recovery% ± RSD% went from 106.4 ± 14.94 to 105.94 ± 14.93 and 34.58 ± 111.8 upon applying the linear fit relative to the quadratic and cubic polynomial fit in the direct measurement. The cubic fit bad results suggest an over-fitting may arise especially upon assessing the accuracy and precision using concentration levels different than those used in the linearity assessment. As a result the quadratic fit was suitable and enough to fit the calibration points after the chemometric treatment of the data. The polynomial fit of the calibration points after the chemometric treatment yielded nearly the same good results as the chemometric handling of the data in the linear fit. This indicated that the effect of chemometric treatment in producing pure analytical signals is more pronounced than simply fitting the calibration points.

Table 6 The accuracy and precision of the direct and the proposed chemometric methods for the non ideal case of improper temperature program adjustment using the quadratic and cubic polynomial fit of regression data

Labeled conc. (ppb)	Direct		D1		D1/FF		D2		D2/FF
	Quad	Cubic	Quad	Cubic	Quad	Cubic	Quad	Cubic	Quad	Cubic
a F – critical value is 5.05.b t – critical values for one tail test is 2.02 and two tail test is 2.57.
10	129.79	110.97	98.57	99.30	100.38	100.24	99.03	100.04	100.09	100.05
25	84.44	1.77	104.01	103.51	100.51	100.53	99.66	100.01	100.03	100.15
30	96.77	15.86	99.96	99.53	100.40	100.38	99.76	100.05	100.03	100.15
45	112.74	27.25	99.51	100.89	100.25	100.12	99.76	99.95	100.03	100.15
50	112.65	27.48	101.39	100.89	100.08	99.94	99.88	100.05	100.11	100.24
65	99.24	24.17	100.21	99.00	100.08	99.99	99.92	100.03	100.04	100.19
Recovery% mean	105.94	34.58	100.70	100.41	100.28	100.20	99.73	100.04	100.05	100.16
SD	15.82	38.66	1.71	1.51	0.18	0.23	0.30	0.05	0.03	0.05
RSD	14.93	111.8	1.70	1.51	0.18	0.23	0.30	0.05	0.03	0.05
F-test^a			69.05	534.19	7913	35 176	2349	1 026 298	197 057	382 346
t-test^b			0.8194	4.1733	0.8756	4.157	0.971	4.1456	0.911	4.154

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For comparison, the F-test and t-test were conducted to test whether a significance difference was achieved upon applying the different chemometric methods and the different polynomial fit relative to the direct measurement in the linear fit. As can be seen in Tables 5 & 6, comparing each chemometric method in the different polynomial fit with the direct measurement using the linear fit indicates that a significance difference was achieved in the precision of the method. This was confirmed by calculating F-values exceeding the theoretical one. Moreover, the calculated t-values of the cubic fit only after the chemometric treatment exceeded the critical ones. This may confirm the presence of over-fitting problem in the higher orders. The results indicates that the proposed chemometric method with or without the polynomial fit was superior compared to the direct measurement either using the linear fit or the polynomial of regression points.

4. Conclusion

Determination of trace levels of several heavy metals in common non ideal cases in GFAAS was successfully done taking chromium as an example. High degree purity of the GFAAS signals was attained by convoluting their derivative curves. Different orders polynomial fit of the calibration points either before or after the chemometric treatment was done. The linearity parameters were enhanced permitting the estimation of different metal concentration levels with good accuracy and precision. Lower LOD and LOQ were attained after the chemometric treatment of data permitting the analysis and detection of minor metal concentrations and increasing the concentration working range. The effect of chemometric treatment has a pronounced effect than the polynomial fit alone. This indicates that convolution of derivative GFAAS curves in the non-ideal cases is a powerful tool in handling such cases and is a promising technique in the field of trace metals analysis.

Acknowledgements

Authors are thankful to Pharco-B International pharmaceutical company for providing metals standards, reagents and instruments used through the study.

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