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4th order polynomial coefficients for pressure at temperature readings, Help!!!

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Nominal Animal:

--- Quote from: The Electrician on August 27, 2020, 12:40:01 pm ---I don't find the maximum absolute error for a least squares solution to be "around .02", but rather I get .005115, which is only about twice the GDO error, not 10 times.  This problem has a condition number of about 2.646E9, which is why I used rational arithmetic to solve it rather than floating point, otherwise errors in results may be larger than expected at first glance.
--- End quote ---
Right; I apologize.  I'm on a different machine (about 1Mm away from my normal machine), and remembered the difference wrong.  In any case, halving the error is in my opinion still significant enough to consider.

Of course, it all depends on the calibration method.  For the GDO to make sense, you need 40-50 calibration samples at minimum, preferably more – I'd prefer around a hundred, with more unique temperatures.  This is why I suggested research into continuous calibration, i.e. measuring a test environment whose pressure and temperature can be varied, but not controller, with at least two other measurement units, and doing GDO dynamically during measurement.  (There is no need for the temperature counts and actual pressures to form a regular grid; the only critical thing is the sample density in the parameter space.  I'm thinking of a small pressure vessel in a thermal bath, with more than one set of sensors, so that both pressure and temperature can be varied (but not precisely), and measured very precisely.)


--- Quote from: The Electrician on August 27, 2020, 12:40:01 pm ---This is often a choice to be made as to the "best" method; minimize absolute error, or residual.

--- End quote ---
Yes, exactly: choosing what to minimize.  Least squares fitting is reliable and easy to document; gradient descent optimization to minimize error at calibration points is harder, but potentially more useful for end users.  I mean, if you can calibrate a device to higher precision than guaranteed, should you or should you not?  What is the physical accuracy of the sensors themselves, and how fast do they drift?  Would the "extra precision" lead users astray?

We do not know how repeatable the calibration samples are, nor how much physical error there is in the pressure and temperature readings.

For example, if for each calibration sample you had error bars, you could treat both the calibration measurements and the device coefficients as a statistical problem instead.  (Then, you wouldn't have anything exact to fit to, but would need to treat the five coefficients as parameters choosing the most likely approximation.)

I do know that calibrating PT100 and PT1000 temperature sensors can give very, very precise results, because they are inherently quite stable devices at these approx. room temperatures.  (Self-heating due to the measurement current passing through the device can be a problem, but in my experience, it only tends to be an issue with small amounts of material to measure, and/or much lower temperatures.)

As to pressure sensors, there are many different types, and we don't know which kind it is.  Most of them are quite stable (strain gauge based ones for example), so I would guess that their time-dependent variation is small, and precise calibration can actually give better readings in the long term (as opposed to just at the moment of calibration).

It is definitely an interesting problem in my opinion, and something I'd like to see put some thought into, as an end user myself.  (Not OP's device, but pressure and PT100 temperature sensors in general, in a fusor project I'm helping with for example.)

b_force:
Calibrating a sensor is one thing, but ever thought about the practical variables that influence the accuracy?
Especially measuring very accurate temperatures is extremely difficult, sensor NOT included.

Move the sensor a couple of millimetres and you're readings WILL be off.
Changes the airflow a teeny tiny bit and you'r readings WILL vary.
Same goes for humidity, type of mounting, mounting torque, use of thermal paste, type of thermal paste.
The list goes on and on.

You will be easily of ±0.5 degrees, which is even pretty generous.
I have seen differences over ±2 degrees in a professional mixing device, stabilized for 2 hours.

When working with samples, you should use the standard deviation with the amount of sigma you need (for accurate measurements, that's usually either 3 or 4 sigma), in combination with the standard error.
Depending if you're interested in the spread or how accurate your expected average will be.

I also agree to some extend with @Doctorandus_P .
You should focus on what you would expect, not if your mathematical equation is going to fit to a million decimals.
The default curve of a PT100 an alike isn't complicated at all, so it's not difficult to fit a regression line in there.

The rest of the systems accuracy you need to calculate with the relative error formula.

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