Thermocouple and RTD linearisation question

[Peter]

I am building a board with an ADC on it which is to measure these two
sensor types.

The tricky bit is the linearisation.

All the equations I can find come in multiple parts.

For example the Callendar–Van Dusen equation has a different set of
coefficients below 0C and above 0C. But you don't know the temperature
until you have evaluated the equation! So you could end up in a
situation where the algorithm oscillates between the two equations.

Same with thermocouples, e.g. here
http://www.ti.com/lit/ug/tidua11a/tidua11a.pdf
although to make life even harder that one gives you the voltage in
terms of temperature. I can see one could solve it iteratively. The
ARM CPU I have has hardware floats and runs at 150MHz.

I have found some thermocouple equations which cover the whole range
with one polynomial e.g. the NIST Polynomial Coefficients for
Temperature-to-Voltage Conversion. However I have found these for only
some of the eight thermocouple types. I am trying to support B E J K N
R S T types.

How do instrument manufacturers deal with this? Do they store a lookup
table and interpolate it?

I can see that with some effort one could write software (running on a
PC) which takes in these equations and generates lookup tables for the
whole lot.

Is there some easier way?

Maths is not my strong point although I can write C code to evaluate a
polynomial.

Very many thanks for any tips.

 

[Jan Panteltje]

On a sunny day (Wed, 02 Oct 2019 14:45:49 +0100) it happened Peter
<nospam@nospam9876.com> wrote in <qn29mg$108$1@dont-email.me>:

I am building a board with an ADC on it which is to measure these two
sensor types.

The tricky bit is the linearisation.

http://srdata.nist.gov/its90/download/type_e.tab
http://srdata.nist.gov/its90/download/type_j.tab
http://srdata.nist.gov/its90/download/type_k.tab
http://srdata.nist.gov/its90/download/type_r.tab
http://srdata.nist.gov/its90/download/type_s.tab
http://srdata.nist.gov/its90/download/type_t.tab


My C code:
http://panteltje.com/panteltje/newsflex/download.html#th
top link th-0.4.tgz th-0.4.lsm

Microchip PIC hardware implementaton:
http://panteltje.com/panteltje/pic/th_pic/

Works for me, but no other thing to compare at extremes.

 

[Peter]

jlarkin@highlandsniptechnology.com wrote:

The big error source is usually the reference junction. That's tricky
to measure right, and you need a reverse set of lookup tables to map
Tj into the proper offset correction. Cheap instruments apply a linear
correction at the ADC input, which isn't really right.

I have been reading about this, here

https://paginas.fe.up.pt/saic/Docencia/ci_lem/Ficheiros/an043.pdf

They arrive at a 0.45C error between the two methods of doing CJC.

AFAICS the problem with the simple method gets worse the further the
cold junction is from 0C, no?

It also ought to be possible to have an equation which takes in the
thermocouple output voltage and the cold junction temperature and it
gives you "exact" hot junction temperature. I guess this is the same
thing as calculating the two polynomials; first you do the CJC one
(temperature to equivalent voltage) and then you measure the
thermocouple voltage, add the CJC voltage to it, and do the
thermocouple polynomial on the result.

So I do need polynomials both ways, for all eight types...

 

On Wed, 02 Oct 2019 14:45:49 +0100, Peter <nospam@nospam9876.com>
wrote:

I am building a board with an ADC on it which is to measure these two
sensor types.

The tricky bit is the linearisation.

All the equations I can find come in multiple parts.

For example the Callendar–Van Dusen equation has a different set of
coefficients below 0C and above 0C. But you don't know the temperature
until you have evaluated the equation! So you could end up in a
situation where the algorithm oscillates between the two equations.

Same with thermocouples, e.g. here
http://www.ti.com/lit/ug/tidua11a/tidua11a.pdf
although to make life even harder that one gives you the voltage in
terms of temperature. I can see one could solve it iteratively. The
ARM CPU I have has hardware floats and runs at 150MHz.

I have found some thermocouple equations which cover the whole range
with one polynomial e.g. the NIST Polynomial Coefficients for
Temperature-to-Voltage Conversion. However I have found these for only
some of the eight thermocouple types. I am trying to support B E J K N
R S T types.

How do instrument manufacturers deal with this? Do they store a lookup
table and interpolate it?

I have generally use the NIST polynomials to generate lookup tables,
which I stored in the product. Runtime is lookup+interpolation. With a
modern ARM cpu with hardware float, you might consider executing the
polynomials in real time.

An RTD is fairly linear to start, so a low order poly fixes that. Not
CvanD.

This is all lookup tables and interpolation:

http://www.highlandtechnology.com/DSS/V450DS.shtml

I programmed that in 68K assembly, all fixed point math, no floats.

I can see that with some effort one could write software (running on a
PC) which takes in these equations and generates lookup tables for the
whole lot.

I did that in Power Basic.

You can also just look up the numbers in an Omega handbook to make the
runtime lookup tables. A point every 10 deg C is probably good enough.

You could just brute-force the voltage-to-temp polynomials in double
floats in c. That would run pretty fast on a decent uP. In real life,
you don't need the sixteen or whatever terms that NIST uses; you can't
measure voltage or resistance well enough to justify that.

$10 DVMs do it.

The big error source is usually the reference junction. That's tricky
to measure right, and you need a reverse set of lookup tables to map
Tj into the proper offset correction. Cheap instruments apply a linear
correction at the ADC input, which isn't really right.



--

John Larkin Highland Technology, Inc

lunatic fringe electronics

 

[Martin Brown]

On 02/10/2019 14:45, Peter wrote:

I am building a board with an ADC on it which is to measure these two
sensor types.

The tricky bit is the linearisation.

All the equations I can find come in multiple parts.

For example the Callendar–Van Dusen equation has a different set of
coefficients below 0C and above 0C. But you don't know the temperature
until you have evaluated the equation! So you could end up in a
situation where the algorithm oscillates between the two equations.

Same with thermocouples, e.g. here
http://www.ti.com/lit/ug/tidua11a/tidua11a.pdf
although to make life even harder that one gives you the voltage in
terms of temperature. I can see one could solve it iteratively. The
ARM CPU I have has hardware floats and runs at 150MHz.

I have found some thermocouple equations which cover the whole range
with one polynomial e.g. the NIST Polynomial Coefficients for
Temperature-to-Voltage Conversion. However I have found these for only
some of the eight thermocouple types. I am trying to support B E J K N
R S T types.

How do instrument manufacturers deal with this? Do they store a lookup
table and interpolate it?

Most likely these days just store the coefficients.

I can see that with some effort one could write software (running on a
PC) which takes in these equations and generates lookup tables for the
whole lot.

Is there some easier way?

Maths is not my strong point although I can write C code to evaluate a
polynomial.

There is a fairly simple trick to it in that most important calibration
problems are fundamentally either linear or quadratic with a smallish
zero offset and some extra nuisance non-linearity thrown in.

Depending on the range of temperatures your sensor is expected to
encounter then you can choose the right coefficients. How many you need
depends on how accurate you want the calibration to be.

You obtain a crude estimate of the right answer from the leading terms
up to quadratic and then iterate using Newton-Raphson or if the
non-linearity is quite small and you need to be very fast another
polynomial recast in the form that takes a voltage in and returns a
temperature.

Linear first guess would be T = (E-d0)/d1

Quadratic first guess would be T = 2(E-d0)/(d1+d1*sqrt(1-4*d2*(E-d0)))

* be careful solving the quadratic or you end up with the tiny
difference of two very large numbers and loss of precision.

** subject to typos

Very many thanks for any tips.

--
Regards,
Martin Brown

 

[Peter]

Martin Brown <'''newspam'''@nezumi.demon.co.uk> wrote:

>Most likely these days just store the coefficients.

I can find the polynomial coefficients for EJKRST here

https://paginas.fe.up.pt/saic/Docencia/ci_lem/Ficheiros/an043.pdf

for both directions. I just need them for B and N.

Depending on the range of temperatures your sensor is expected to
encounter then you can choose the right coefficients. How many you need
depends on how accurate you want the calibration to be.

I am trying to support the full documented temp range for each type.

However I have had no luck yet finding a *single* resistance to
temperature equation for the RTD.

Tables can be found for all these so one could generate a lookup
table.

I know that in principle one can generate a polynomial for almost any
curve, and these curves being monotonic, it is even easier. If you
want to fit 10 points exactly, you need a polynomial with 10 (11?)
terms. How to do this, I don't know, but clearly it is well known.

 

[Peter]

Jan Panteltje <pNaOnStPeAlMtje@yahoo.com> wrote:

On a sunny day (Wed, 02 Oct 2019 14:45:49 +0100) it happened Peter
nospam@nospam9876.com> wrote in <qn29mg$108$1@dont-email.me>:

I am building a board with an ADC on it which is to measure these two
sensor types.

The tricky bit is the linearisation.

http://srdata.nist.gov/its90/download/type_e.tab
http://srdata.nist.gov/its90/download/type_j.tab
http://srdata.nist.gov/its90/download/type_k.tab
http://srdata.nist.gov/its90/download/type_r.tab
http://srdata.nist.gov/its90/download/type_s.tab
http://srdata.nist.gov/its90/download/type_t.tab


My C code:
http://panteltje.com/panteltje/newsflex/download.html#th
top link th-0.4.tgz th-0.4.lsm

Microchip PIC hardware implementaton:
http://panteltje.com/panteltje/pic/th_pic/

Works for me, but no other thing to compare at extremes.

This is brilliant stuff - thank you!

Are there tables for types B and N?

 

[Martin Brown]

On 02/10/2019 17:04, Peter wrote:

Martin Brown <'''newspam'''@nezumi.demon.co.uk> wrote:

Most likely these days just store the coefficients.

I can find the polynomial coefficients for EJKRST here

https://paginas.fe.up.pt/saic/Docencia/ci_lem/Ficheiros/an043.pdf

for both directions. I just need them for B and N.

Depending on the range of temperatures your sensor is expected to
encounter then you can choose the right coefficients. How many you need
depends on how accurate you want the calibration to be.

I am trying to support the full documented temp range for each type.

However I have had no luck yet finding a *single* resistance to
temperature equation for the RTD.

It won't hurt much if you use the high range polynomial for temperatures
a little below zero or the low range one for room temperatures. It
really matters which you use when you get to very hot or very cold. The
two range calibration methods should agree pretty well in their overlap.

Providing a bit of hysteresis so you only swap to high range at +30C and
to low range at -10C would be a reasonable compromise. I haven't checked
what maximum error that would incur (you ought to though).

Tables can be found for all these so one could generate a lookup
table.

I know that in principle one can generate a polynomial for almost any
curve, and these curves being monotonic, it is even easier. If you
want to fit 10 points exactly, you need a polynomial with 10 (11?)
terms. How to do this, I don't know, but clearly it is well known.

That is an unfortunate tendency of engineers to overfit their
calibration data and get a polynomial that fits all their calibration
points exactly and oscillates wildly at all points in between.

--
Regards,
Martin Brown

 

On Wed, 02 Oct 2019 17:04:19 +0100, Peter <nospam@nospam9876.com>
wrote:

Martin Brown <'''newspam'''@nezumi.demon.co.uk> wrote:

Most likely these days just store the coefficients.

I can find the polynomial coefficients for EJKRST here

https://paginas.fe.up.pt/saic/Docencia/ci_lem/Ficheiros/an043.pdf

for both directions. I just need them for B and N.

Depending on the range of temperatures your sensor is expected to
encounter then you can choose the right coefficients. How many you need
depends on how accurate you want the calibration to be.

I am trying to support the full documented temp range for each type.

However I have had no luck yet finding a *single* resistance to
temperature equation for the RTD.

Tables can be found for all these so one could generate a lookup
table.

I know that in principle one can generate a polynomial for almost any
curve, and these curves being monotonic, it is even easier. If you
want to fit 10 points exactly, you need a polynomial with 10 (11?)
terms. How to do this, I don't know, but clearly it is well known.

To curve fit, it's generally better to have a bunch more points than
the degree of the polynomial. Given N points, you can draw a roller
coaster curve that still hits them all.

Some polynomial regression algorithms produce radically dumb curves.

In real life, 3rd order mostly works.




--

John Larkin Highland Technology, Inc

lunatic fringe electronics

 

[Jan Panteltje]

On a sunny day (Wed, 02 Oct 2019 16:57:57 +0100) it happened Peter
<nospam@nospam9876.com> wrote in <qn2he8$flp$1@dont-email.me>:

Jan Panteltje <pNaOnStPeAlMtje@yahoo.com> wrote:

On a sunny day (Wed, 02 Oct 2019 14:45:49 +0100) it happened Peter
nospam@nospam9876.com> wrote in <qn29mg$108$1@dont-email.me>:

I am building a board with an ADC on it which is to measure these two
sensor types.

The tricky bit is the linearisation.

http://srdata.nist.gov/its90/download/type_e.tab
http://srdata.nist.gov/its90/download/type_j.tab
http://srdata.nist.gov/its90/download/type_k.tab
http://srdata.nist.gov/its90/download/type_r.tab
http://srdata.nist.gov/its90/download/type_s.tab
http://srdata.nist.gov/its90/download/type_t.tab


My C code:
http://panteltje.com/panteltje/newsflex/download.html#th
top link th-0.4.tgz th-0.4.lsm

Microchip PIC hardware implementaton:
http://panteltje.com/panteltje/pic/th_pic/

Works for me, but no other thing to compare at extremes.

This is brilliant stuff - thank you!

Are there tables for types B and N?

Inferred from the above URL where only one character changes:
https://srdata.nist.gov/its90/download/type_b.tab
https://srdata.nist.gov/its90/download/type_n.tab
---------------------------------------------^

 

On Wed, 02 Oct 2019 16:23:15 +0100, Peter <nospam@nospam9876.com>
wrote:

jlarkin@highlandsniptechnology.com wrote:

The big error source is usually the reference junction. That's tricky
to measure right, and you need a reverse set of lookup tables to map
Tj into the proper offset correction. Cheap instruments apply a linear
correction at the ADC input, which isn't really right.

I have been reading about this, here

https://paginas.fe.up.pt/saic/Docencia/ci_lem/Ficheiros/an043.pdf

They arrive at a 0.45C error between the two methods of doing CJC.

AFAICS the problem with the simple method gets worse the further the
cold junction is from 0C, no?

Generally so. The usual assumption is to cancel the slope of the tc
voltage around room temperature. Since thermocouples are nonlinear,
the correction gets worse further away from room temp.

Plus, it's just hard to measure the reference junction temp
accurately. We like a platinum RTD in an isothermal box.

It also ought to be possible to have an equation which takes in the
thermocouple output voltage and the cold junction temperature and it
gives you "exact" hot junction temperature.

The t/c voltage is not a function of the temp difference between the
hot and cold junctions. So computing an equivalent ref junction temp
doesn't help. You need to inverse compute the equivalent offset
voltage of the t/c, add it to the measured t/c voltage, then do the
nonlinear lookup.

Really, it's not bad.


I guess this is the same

thing as calculating the two polynomials; first you do the CJC one
(temperature to equivalent voltage) and then you measure the
thermocouple voltage, add the CJC voltage to it, and do the
thermocouple polynomial on the result.

Yes, that's right.

So I do need polynomials both ways, for all eight types...

Yup!

Or lookup tables.






--

John Larkin Highland Technology, Inc

lunatic fringe electronics

 

[Martin Brown]

On 02/10/2019 17:49, jlarkin@highlandsniptechnology.com wrote:

On Wed, 02 Oct 2019 17:04:19 +0100, Peter <nospam@nospam9876.com
wrote:


Martin Brown <'''newspam'''@nezumi.demon.co.uk> wrote:

Most likely these days just store the coefficients.

I can find the polynomial coefficients for EJKRST here

https://paginas.fe.up.pt/saic/Docencia/ci_lem/Ficheiros/an043.pdf

for both directions. I just need them for B and N.

Depending on the range of temperatures your sensor is expected to
encounter then you can choose the right coefficients. How many you need
depends on how accurate you want the calibration to be.

I am trying to support the full documented temp range for each type.

However I have had no luck yet finding a *single* resistance to
temperature equation for the RTD.

If there was a way to do it that was adequate they would have done it.

There do tend to be two sorts of users. Those doing precision cryogenics
and those doing furnace control - there isn't a lot of overlap needed.

Tables can be found for all these so one could generate a lookup
table.

I know that in principle one can generate a polynomial for almost any
curve, and these curves being monotonic, it is even easier. If you
want to fit 10 points exactly, you need a polynomial with 10 (11?)
terms. How to do this, I don't know, but clearly it is well known.

To curve fit, it's generally better to have a bunch more points than
the degree of the polynomial. Given N points, you can draw a roller
coaster curve that still hits them all.

It is essential to have plenty more points than free parameters!

> Some polynomial regression algorithms produce radically dumb curves.

You can generally resolve it by rescaling the problem so that the
polynomial variable x is in the range -1 to 1. Otherwise the matrix you
have to invert gets a terrible condition number and crazy results.

For equally spaced tabular data Chebeshev polynomials are handy for a
quick and dirty fit. I do wonder how they arrived at some of the spline
coefficients for that particular curve since it is not continuous at
T90=0C ( d0 + b0.exp(b1*126.9686^2) != 0

> In real life, 3rd order mostly works.

The polynomial fit in Excel charts is remarkably good (apart from when
they broke it in XL2007 briefly to make it agree with MATLAB). You just
have to force the display equation format to show you enough digits!

Their other polynomial fit is dodgy. It is a particularly bad problem if
you feed x values with a large DC offset into some fitting routines.

--
Regards,
Martin Brown

 

[Peter]

Martin Brown <'''newspam'''@nezumi.demon.co.uk> wrote

If there was a way to do it that was adequate they would have done it.

There do tend to be two sorts of users. Those doing precision cryogenics
and those doing furnace control - there isn't a lot of overlap needed.

It sounds like that for both RTDs and TCs I should use just the tables
and interpolate them. Then you can have the whole range covered.

The max error is also easy to quantify.

It doesn't make sense to use polynomials for 6 of the TCs and table
interpolation for the other 2.

Is there a way to generate a polynomial from one of these tables?
Surely this must be a well worn challenge.

 

[Peter]

Martin Brown <'''newspam'''@nezumi.demon.co.uk> wrote

>If there was a way to do it that was adequate they would have done it.

https://srdata.nist.gov/its90/type_k/kcoefficients_inverse.html

but they have created polynomials for different bits of the
temperature ranges

 

[whit3rd]

On Wednesday, October 2, 2019 at 7:55:06 AM UTC-7, jla...@highlandsniptechnology.com wrote:

On Wed, 02 Oct 2019 14:45:49 +0100, Peter <nospam@nospam9876.com
wrote:

I am building a board with an ADC on it which is to measure these two
sensor types.

The tricky bit is the linearisation.

The big error source is usually the reference junction.

It doesn't need to be. If your case is always cool, the
reference junction can be an insulated glob with a heater
that always stays at one setpoint. That cuts out half the calculation
overhead, and all it takes is a thermostat (very simple electronics).

It isn't a surface-mount off-the-shelf jellybean, though. And it takes a few seconds
to come on-line.

 

[John Larkin]

On Wed, 2 Oct 2019 13:36:09 -0700 (PDT), whit3rd <whit3rd@gmail.com>
wrote:

On Wednesday, October 2, 2019 at 7:55:06 AM UTC-7, jla...@highlandsniptechnology.com wrote:
On Wed, 02 Oct 2019 14:45:49 +0100, Peter <nospam@nospam9876.com
wrote:

I am building a board with an ADC on it which is to measure these two
sensor types.

The tricky bit is the linearisation.

The big error source is usually the reference junction.

It doesn't need to be. If your case is always cool, the
reference junction can be an insulated glob with a heater
that always stays at one setpoint. That cuts out half the calculation
overhead, and all it takes is a thermostat (very simple electronics).

It isn't a surface-mount off-the-shelf jellybean, though. And it takes a few seconds
to come on-line.

If you heat the reference junction, all the incoming thermocouple pair
transitions to copper must be at the same temp. The heater creates
thermal gradients.

 

[Clifford Heath]

On 3/10/19 3:31 am, Martin Brown wrote:

On 02/10/2019 17:49, jlarkin@highlandsniptechnology.com wrote:
On Wed, 02 Oct 2019 17:04:19 +0100, Peter <nospam@nospam9876.com
wrote:


Martin Brown <'''newspam'''@nezumi.demon.co.uk> wrote:

Most likely these days just store the coefficients.

I can find the polynomial coefficients for EJKRST here

https://paginas.fe.up.pt/saic/Docencia/ci_lem/Ficheiros/an043.pdf

for both directions. I just need them for B and N.

Depending on the range of temperatures your sensor is expected to
encounter then you can choose the right coefficients. How many you need
depends on how accurate you want the calibration to be.

I am trying to support the full documented temp range for each type.

However I have had no luck yet finding a *single* resistance to
temperature equation for the RTD.

If there was a way to do it that was adequate they would have done it.

There do tend to be two sorts of users. Those doing precision cryogenics
and those doing furnace control - there isn't a lot of overlap needed.

Tables can be found for all these so one could generate a lookup
table.

I know that in principle one can generate a polynomial for almost any
curve, and these curves being monotonic, it is even easier. If you
want to fit 10 points exactly, you need a polynomial with 10 (11?)
terms. How to do this, I don't know, but clearly it is well known.

To curve fit, it's generally better to have a bunch more points than
the degree of the polynomial. Given N points, you can draw a roller
coaster curve that still hits them all.

It is essential to have plenty more points than free parameters!

If you're using a polynomial regression (approximate fit) then that's
true. If using interpolation (line goes through all data points) then it
can get dumb.

>> Some polynomial regression algorithms produce radically dumb curves.

Interpolation works well if your data is not noisy (i.e. when you're
trying to simplify a complex curve to a similar simpler one). I've
implemented interpolated cubic splines and they're very effective.
<https://en.wikipedia.org/wiki/Spline_interpolation>

B-splines get used a lot in computer graphics because the derivatives
are continuous (though the curves don't pass through the points so are
less easy to predictably manipulate than interpolated splines). The
functions yield curves that are very quick and easy for 3D graphic
hardware to reproduce. Basically all modern car bodies are recognisable
as modern precisely because they *all* use b-splines.

Either kind of spline will be continuous at/near all control points, but
interpolated splines of order N have non-continuous N-1 derivatives (I
think I stated that correctly).

They're both easier to fit and to evaluate than polynomials.

Clifford Heath.

 

[John Larkin]

On Wed, 2 Oct 2019 18:31:20 +0100, Martin Brown
<'''newspam'''@nezumi.demon.co.uk> wrote:

On 02/10/2019 17:49, jlarkin@highlandsniptechnology.com wrote:
On Wed, 02 Oct 2019 17:04:19 +0100, Peter <nospam@nospam9876.com
wrote:


Martin Brown <'''newspam'''@nezumi.demon.co.uk> wrote:

Most likely these days just store the coefficients.

I can find the polynomial coefficients for EJKRST here

https://paginas.fe.up.pt/saic/Docencia/ci_lem/Ficheiros/an043.pdf

for both directions. I just need them for B and N.

Depending on the range of temperatures your sensor is expected to
encounter then you can choose the right coefficients. How many you need
depends on how accurate you want the calibration to be.

I am trying to support the full documented temp range for each type.

However I have had no luck yet finding a *single* resistance to
temperature equation for the RTD.

If there was a way to do it that was adequate they would have done it.

It's a smooth physical process, and there's nothing fundamental about
0 degrees C. One polynomial can cover the whole range.

 

[Bill Sloman]

On Thursday, October 3, 2019 at 2:50:07 AM UTC+10, jla...@highlandsniptechnology.com wrote:

On Wed, 02 Oct 2019 17:04:19 +0100, Peter <nospam@nospam9876.com
wrote:


Martin Brown <'''newspam'''@nezumi.demon.co.uk> wrote:

Most likely these days just store the coefficients.

I can find the polynomial coefficients for EJKRST here

https://paginas.fe.up.pt/saic/Docencia/ci_lem/Ficheiros/an043.pdf

for both directions. I just need them for B and N.

Depending on the range of temperatures your sensor is expected to
encounter then you can choose the right coefficients. How many you need
depends on how accurate you want the calibration to be.

I am trying to support the full documented temp range for each type.

However I have had no luck yet finding a *single* resistance to
temperature equation for the RTD.

Tables can be found for all these so one could generate a lookup
table.

I know that in principle one can generate a polynomial for almost any
curve, and these curves being monotonic, it is even easier. If you
want to fit 10 points exactly, you need a polynomial with 10 (11?)
terms. How to do this, I don't know, but clearly it is well known.

To curve fit, it's generally better to have a bunch more points than
the degree of the polynomial. Given N points, you can draw a roller
coaster curve that still hits them all.

It's absolutely essential to have more points than parameters.

Since your N-points all have an additional bit of random error, it pays to have a lot more points than parameters, so the random errors have a chance to cancel out.

For my Ph.D. work, where I had to do non-linear multi-parameter curve fitting, the curve-fitting routine worked by generating a matrix of partial derivatives of the least square error between the curve and the data for each parameter, which it used to improve the parameter estimates from one pass through the data to the next, and you could also use that matrix to work out how closely the data defined the best-fit parameters after you had got a fit.

If you tried to fit too many parameters (four in my case, rather than three) the confidence limits got a whole lot worse on all of the parameters.

> Some polynomial regression algorithms produce radically dumb curves.

Some people who use curve fitting don't know what they are doing.

> In real life, 3rd order mostly works.

In John Larkin's experience, this may be true. It probably doesn't generalise.

--
Bill Sloman, Sydney

 

[Rick C]

On Wednesday, October 2, 2019 at 8:50:55 PM UTC-4, John Larkin wrote:

It's a smooth physical process, and there's nothing fundamental about
0 degrees C.

There is if you're a fish.

--

Rick C.

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