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I have multiple sensors of the same kind that measure tempearture. The temperature it returns is actually in a digital (byte) format. The manufacturer have provided us with a linear equation (with coefficients m and b) to convert the digital bytes sent by the sensor back into a real temperature value.

The coefficients m and b are provided by the manufacturer but we are allowed to adjust them to improve accuracy. We see that there is some error in the sensor measurement which is slightly different in each sample of the sensor. We have the real temperature value in the room with an expensive temperature sensor that we know is 100% correct.

Is there any easy and quick (to understand) optimisation technique that we can apply (preferably in Excel) where I can give it the error we see in each sample and it can provide me the best values of m and b that will minimise the error across all sensor samples?

The value of m and b has to be same for all samples, so we need a optimisation technique that can minimise the error across all samples with a single value for m and a single value for b.

If anyone could point me in the right direction on what optimisation technique I should investigate? We only need to run it once, so execution time isn't a problem.

muosac
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1 Answers1

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Calling $\theta_k$ the number (byte), this number according to the sensor manufacturer means a temperature $T_k = b + m \theta_k$. Now if we know the real room temperature as $R_k$ indicating an error $e_k = R_k-T_k = R_k-b-m\theta_k$ we can recalculate $b, m$ by minimizing

$$ \sum_k e_k^2 = \sum_k \left(R_k-b-m\theta_k\right)^2 $$

so for $n$ measures we have to solve

$$ \left(\matrix{n & \sum_k\theta_k\\ \sum_k\theta_k & \sum_k\theta^2_k}\right)\left(\matrix{b^*\\ m^*}\right) = \left(\matrix{\sum_k R_k\\ \sum_k R_k\theta_k}\right) $$

with $b^*, m^*$ as the new calibrated values.

Cesareo
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