Based on your comments, what you want to do is a calibration, which you also want to validate:
- reference measurements of a temperature (thermometer A), and
- measurements of instrument B which is not a thermometer yet, as you do not get response of the physical quantity temperatures but of a physical quantity like e.g. electrons/s.
Camera readout is not the same physical quantity as a temperature.
So in fact your task is to find the conversion between electrons/s and temperature, i.e. to calibrate your camera output to temperatures.
I'm chemometrician, I do calibrations to relate instrument readout to chemical quantities. There are whole books written on the subject of how to obtain a good calibration model (your question 2) and then how to validate this method (your question 1).
Question 1: how to compute the parameter $k$?
This is called fitting the calibration model.
And this part actually starts with deciding what kind of model is appropriate.
This is what your assumption (multiplicative) is.
In chemometrics, sometimes the terms soft and hard models are sometimes used to distinguish:
- hard models: deriving the ansatz for the model from first (global) principles,
e.g. describing camera readout as function of temperature (e.g. black body radiation, quantum efficiency of the camera at different wavelengths, ...) and then solving for temperature and simplifying as much as possible by merging as many parameters as possible into fewer parameters that need to be determined experimentally.
- soft models: modelling the calibration function by approximations that are independent of the exact physical connection.
E.g. you may assume that if your temperature range is narrow enough, you can approximate the unknown hard ansatz by a linear model. If that isn't enough, quadratic may be appropriate etc. Or, you may expect a sigmoid behaviour etc.
Recommendation 1: do a bit of thinking and decide roughly what type of relationship you expect.
Soft modelling is a valid and widely used option, but you should be able to give reasoning why multiplicative relationship is sensible compared to other families of functions like sigmoid or exponential or logarithmic.
Question 3: What to do with more $L$s?
I'm not sure whether I understand correctly what the different $L$s are.
if they are measurements of parts with other temperature, you are going to need them as Peter Flom and gung already said.
Usually, extrapolating outside the calibrated range (i.e. the temperature range spanned by your model fitting data) is not considered valid. You may argue for an exception if you validate (see below) the method for a wider range; but if you can get a wide range of validation data, there is no reason why you couldn't get training data for that range as well.
if you refer to the camera having many pixels: it will depend on the properties of the camera whether you can reasonably assume that all pixels follow the same calibration or whether you need to calibrate each pixel.
Question 1: How to know whether multiplicative relationship is appropriate? Part I
In chemometrics, multiplicative without intercept is not even done in situations where the hard model suggests multiplicative-only relationship (e.g. Beer-Lambert-law) as there are usually many things in the construction of instruments that lead to an intercept.
My experience suggests multiplicative relationship without an intercept term is hardly ever appropriate for camera readout.
E.g. all camera readout I've worked with so far had a bias or dark current which would be an intercept in the model.
Recommendation 2: if you decide for a multiplicative model without intercept, you should be able to give very good reasons why no intercept can possibly occur. This may be easier the other way round: try to invent situations that would lead to an intercept for the camera readout. If you can come up with an intercept, you should include one into the model.
The so called regression diagnostics for linear models will tell you if the intercept cannot be distinguished from zero. That would be evidence that allows you to fit a model without intercept. Likewise, you can fit a quadratic model and see whether the quadratic term can be distinguished from zero.
Question 1: How to know whether multiplicative relationship is appropriate? Part II
While you can spot certain things going wrong within the set of measurements used for building the calibration model, "valid" means more than that. Usually, it means demonstrating that your calibration can be successfully applied to camera readout of completely unknown samples (possibly measured some time after the calibration was done). Again there is a whole body of literature to validation, and depending on what your exact field is, there are also norms that you should follow.
Briefly, for validation you need a second set of measurements that was not involved in any way in building the calibration. You then compare the reference instrument's output to the predictions of the calibration. Looking at the deviations, you can assess several aspects of correctness of your calibration:
- bias (i.e. your model has a systematic deviation)
- variance (random uncertainty)
- drift (i.e. $k$ changes over time; requires appropriate planning of measurements)
- IUPAC Recommendations: Guidelines for calibration in analytical chemistry. Part I. Fundamentals and single component calibration
This is like a norm.
- American Laboratory has a series called "Statistics in Analytical Chemistry"
lots of stuff here, including case studies
Richard G. Brereton: Introduction to multivariate calibration in analyticalchemistry, Analyst, 2000,125, 2125-2154.
seems to cover univariate calibration as well.
- Esbensen, K. H. & Geladi, P. Principles of Proper Validation: use and abuse of re-sampling for validation J. Chemometrics, John Wiley & Sons, Ltd., 2010, 24, 168-187
gives a good discussion what you need to keep in mind when choosing validation samples.