How to calculate a correction factor for two sensors

Question

Suppose one has two sensors placed in the same environment (thus should be reading the same environmental data). But not being perfect instruments, there is a discrepancy between the two sets of numbers.

Sensor 1   Sensor 2
1         1
2         1.9
3         2.7
2         2
2         1.9
3         3
2         1.8
1         1
2         2
3         2.8

How does one calculate a correction factor between these two sets of numbers. The hope being that this first set of data could be a calibration set. Then (knowing the sensors are off) one could then move the sensors to different environments and record a new set of data.

S1      S2
1   10
2   12
3   13
4   15
5   15
6   16
7   17
8   19
9   19
10  20

and then apply a correction factor (based on the initial calibration) to the new data to account for the discrepancy between instruments.

Answer 1

If you have some reason to believe there is a simple relationship between the reading on one sensor and on the other, you can use linear regression to estimate the parameters of that relationship based on a calibration measurement like you suggest.

For example, if are measuring a parameter V, using sensors "1" and "2", and you think the only difference between these sensors is that signal conditioning circuits in these sensors have different gain and offset terms, you would guess at a model like

\$V_2 = a + bV_1\$.

Or, if the first sensor is a carefully calibrated instrument and sensor 2 is something you've built yourself, you might also think there's some nonlinearity in the response of sensor 2. Then you might guess a model like

\$V_2 = a + bV_1 + cV_1^2\$.

Notice that linear regression doesn't require the relationship between the two V's to be linear --- it only requires that one V be expressed as a linear combination of other observable values (in this case, each of these values being some function of \$V_1\$).

You can find formulas for linear regression at Wikipedia or any first-year statistics textbook (see the Wikipedia article on Simple Linear Regression to get the formulas written out rather than in matrix notation). These sources also clarify some additional requirements for your calibration measurement that are necessary for the standard linear regression formulas to provide an optimal least squares estimate of the true relationship.

For the simple case of a straight-line fit, you may get a result like this:

This chart shows the observations (points), the fit line, and 95% confidence intervals (dashed lines). The confidence intervals indicate the region where, if you were to repeat the calibration measurement multiple times, the calculated fitting line would lie 95% of the time.

Notice that the confidence intervals diverge from the fitting line on either end, outside the range of the observations. Because of this divergence, its much preferable for the calibration measurement to cover the same range as the actual measurements.

Edit

1. I should point out that one of the limitations of regression analysis is that the \$V_1\$ values in your calibration measurement should be noise-free. If sensor 1 is a more precise sensor than sensor 2, or if it can be operated in a low-noise mode (maybe with a slower input filter, or averaging multiple readings) this might be a reasonable assumption.

   If it's not reasonable to assume sensor 1 is noise-free, then linear regression will still very likely get you a reasonable result. But if you want to check against a statistically more correct method, you'll need to look in to total least squares regression.

2. Geometrikal has it right in the comments. Having done your calibration, when you later make a measurement with sensor 2 and you want to know what sensor 1 would have measured, you have to invert your model. For the simple linear model you have

   \$ V_1 = (V_2 - a) / b \$.