Calculate a polynomial function from other polynomial functions

[drpratai]

Background information:

I have come up against a mathematical question which I as somebody with relatively limited exposure to maths can not seem to answer. I am a student working on a thesis dealing with near-infrared spectroscopy (NIRS). The NIR scanner is able to measure the moisture content in different materials. The scanner outputs two values: Absorbance and moisture content in percent. Currently, the scanner is not calibrated and the moisture content in percent is therefore inaccurate. My objective is to calibrate the scanner with regards to the moisture content for different materials as well as for different mixtures of these materials. To do this I am plotting calibration curves by taking a material (e.g. wood) and preparing several samples with different moisture contents between 0 and 40 %. I then scan the samples and record the absorbance values. Parallel to this I determine the moisture contents of the samples in the laboratory using drying ovens.

With these values I can now plot my calibration curve; the values for absorbance are entered into the y-axis and the moisture contents in percent are entered into the x-axis. I can now best describe my correlation with a polynomial of 2nd degree (y = ax^2 + bx +c). Now when I analyse a new wood sample with an unknown moisture content I can insert the value for absorbance as x into the function and calculate the moisture content in percent.

I have already plotted calibration graphs for the following materials:

Wood y = -6,0066x2 + 118,25x + 5,6884
Textiles y = 388,75x2 + 131,85x + 9,0221
PVC y = 85,467x2 + 18,372x + 0,1075Mathematical query:

I would now like to analyse different mixtures of these materials. My approach to plotting the correlation graphs will be the same as with the homogeneous samples. Additionally, I wish to test if it is possible to calculate a new calibration graph and compare that to the one I will plot as described above. An example of a heterogeneous sample would be a mixture of wood, textiles and PVC with the following composition:

Wood: 10 %
Textiles: 10 %
PVC: 80 %

I then would calculate the function which should belong to this mixed materal sample in the following way. I newly calculate the polynomial coefficients according to the percentages of the components (i.e. wood: 10 %, textiles: 10 % and PVC: 80 %):

a = (0,1 * -6,0066) + (0,1 * 388,75) + (0,8 * 85,467) = 1,0664794
b = (0,1 * 118,25) + (0,1 * 131,85) + (0,8 * 18,372) = 0,397076
c = (0,1 * 5,6884) + (0,1 * 9,0221) + (0,8 * 0,1075) = 0,0155705

The new polynomial function which describes the correlation of the heterogeneous sample is then:

y = 1,0665x^2 + 0,3971x + 0,0156

I have tried this in excel and it looks feasible from a graphical point of view, the slope of the calibration curve changes just as I thought it should etc..

My question here is if anybody can advise me whether this is mathematically correct to calculate a polynomial function from the three functions and the percentages? Is there anywhere a law or calculation rules regarding this what I have done (calculation of polynomial functions)? I would like to be able to to cite source from a book to back up my calculation.

Sorry for the long-winded question!
Kind regards

 

[jackmell]

Background information:

I have come up against a mathematical question which I as somebody with relatively limited exposure to maths can not seem to answer. I am a student working on a thesis dealing with near-infrared spectroscopy (NIRS). The NIR scanner is able to measure the moisture content in different materials. The scanner outputs two values: Absorbance and moisture content in percent. Currently, the scanner is not calibrated and the moisture content in percent is therefore inaccurate. My objective is to calibrate the scanner with regards to the moisture content for different materials as well as for different mixtures of these materials. To do this I am plotting calibration curves by taking a material (e.g. wood) and preparing several samples with different moisture contents between 0 and 40 %. I then scan the samples and record the absorbance values. Parallel to this I determine the moisture contents of the samples in the laboratory using drying ovens.

With these values I can now plot my calibration curve; the values for absorbance are entered into the y-axis and the moisture contents in percent are entered into the x-axis. I can now best describe my correlation with a polynomial of 2nd degree (y = ax^2 + bx +c). Now when I analyse a new wood sample with an unknown moisture content I can insert the value for absorbance as x into the function and calculate the moisture content in percent.

I have already plotted calibration graphs for the following materials:

Wood y = -6,0066x2 + 118,25x + 5,6884
Textiles y = 388,75x2 + 131,85x + 9,0221
PVC y = 85,467x2 + 18,372x + 0,1075


Mathematical query:

I would now like to analyse different mixtures of these materials. My approach to plotting the correlation graphs will be the same as with the homogeneous samples. Additionally, I wish to test if it is possible to calculate a new calibration graph and compare that to the one I will plot as described above. An example of a heterogeneous sample would be a mixture of wood, textiles and PVC with the following composition:

Wood: 10 %
Textiles: 10 %
PVC: 80 %

I then would calculate the function which should belong to this mixed materal sample in the following way. I newly calculate the polynomial coefficients according to the percentages of the components (i.e. wood: 10 %, textiles: 10 % and PVC: 80 %):

a = (0,1 * -6,0066) + (0,1 * 388,75) + (0,8 * 85,467) = 1,0664794
b = (0,1 * 118,25) + (0,1 * 131,85) + (0,8 * 18,372) = 0,397076
c = (0,1 * 5,6884) + (0,1 * 9,0221) + (0,8 * 0,1075) = 0,0155705

The new polynomial function which describes the correlation of the heterogeneous sample is then:

y = 1,0665x^2 + 0,3971x + 0,0156

I have tried this in excel and it looks feasible from a graphical point of view, the slope of the calibration curve changes just as I thought it should etc..

My question here is if anybody can advise me whether this is mathematically correct to calculate a polynomial function from the three functions and the percentages? Is there anywhere a law or calculation rules regarding this what I have done (calculation of polynomial functions)? I would like to be able to to cite source from a book to back up my calculation.

Sorry for the long-winded question!
Kind regards

I don't think you need a reference for that. It's just straight percentages. You have for the mixture:

$$m=0.1 w+0.1 t+0.8 p$$

with:

$$w=a_1 x^2+a_2 x+a_3$$
$$t=b_1 x^2+b_2 x+b_3$$
$$p=c_1 x^2+c_2 x+c_3$$

plug all that in and you get the same thing you wrote above.

 

[Robert1986]

Well, you have the absorbance and you want the moisture content, right? So, let's say for each material (wood, textiles, pvc, etc) there is a function $f_{m}(x) = y$ where $x$ is the absorbance and $y$ is the moisture content and $m$ is the material type. Then, your polynomials $p_{m}$ are approximations to the actual functions. That is $f_m(x)\simeq p_m(x)$. Now, say you have a material that is 10% wood, 10% textiles, 80% PVC. If it is correct that the moisture content of this sample is given by $.1f_{wood}(x) + .2f_{textiles}(x) + .8f_{pvc}(x)$ then the polynomial apporximation that you have given is correct. But, I think it would be best if you were able to calibrate the equipment with mixtures of materials. That is, instead of being a function that takes absorbance and spits out moisture content, make a function that takes absorbance, %wood, %textiles, %pvc and spits out moisture content. I'm away from my normal computer so I don't have excel, but I think this is something that can be done in excel.

 

[drpratai]

Thanks for the feedback!

@Robert1986, I do intend on calibrating the scanner for a typical mixture of
materials. I will then compare the results with a calculated calibration. If it
works well I plan to write a small program that allows the operator to enter the
composition of the sample so that a calibration is calculated in real time which
wold be advantageous as the calibration process can take a week or more due to
the necessity of drying samples etc.