Get equation that describes set of measured values

[mr02077]

Hello.

A whole decade passed since I graduated mathematics and shifted to other profession, so my knowledge is very rusty.

There is an important problem for a scientific work that I need help for.

Let's say factor t is being calculated from factors x, y and z, all some parameters from living environment.
There is a bunch of experimental measures where x, y and t were measured out precisely, but not z. z is a value that this work is about and is not measurable anyhow in reality.

The question is how to determine a math equation that gives dependency between the 4 values :
t = f(x,y,z)

I understand standard interpolation requires known values for all variables in interpolation points, so I am not even sure the formulation of the problem is ok
I do not know if it would be possible to do this even if z is known in all points, since this is a function from three variables.

So my first question would be if it is possible to get this equation at all.

If it is, I would be very glad that someone directs me to some interpolation method I could use to resolve this.I have tried with polynomial approximation:
a_n*x^n + ... + a_1*x + a_0
+ b_m*y^m + ... + b_1*y + b_0
+ c_p*z^p + ... + c_1*z + c_0
= t

When you substitute values from a very broad set of measured tests, this becomes a system of equations per following variables:
a_n, ..., a_0, b_m, ..., b_0, c_p, ..., c_0, z
where for z this goes to the power of p.
This is obviously not solvable.

So, there might be numerical analysis that could help, but I do not have knowledge to do this

Thank you very much!

 

[Stephen Tashi]

but not z. z is a value that this work is about and is not measurable anyhow in reality.

The question is how to determine a math equation that gives dependency between the 4 values :
t = f(x,y,z)

You must clarify your question.

I assume you have a reasonable problem in mind, but it seems crazy to ask how to express t = f(x,y,z) if you have no data about the values of z. If you have no data about z, and somebody said "Use f(x,y,z) = 3x + 0.2y + 3z^2" then how would you determine if that function worked or didn't work?

 

[mr02077]

Yes, that was part of my question, if this problem is formulated ok at all.

It is just that some effort realized depends on two components that are measurable and one that is certainly not.
So I was wondering is it possible to get math equation between the four, if we have multiple points where three values are measured (of which one is result).

Somehow that formulation of the problem seemed logical, but it seems it is not good..

 

[Nidum]

Is this a purely numerical problem or do x , y , t and z have dimensions ? A dimensional analysis may be possible which could at least show you the general form of the linking equation .

 

[mr02077]

I am not sure what you are asking exactly.
What I have is a big set of point where for some x and y predetermined and set value t has been experimentally measured.
And I want to show it depends from z too and to get the equation.

 

[BvU]

Without any information on z at all, it is difficult to establish the dependency of t on z. Can you be more concrete with a comparable example ?

 

[Stephen Tashi]

I am not sure what you are asking exactly.
.

(In regard to dimensions) - for example, is "x" measured in units of length (e.g. meters) ? units of currency (e.g. US dollars)? Is "t" measured in units of time?

You will probably get better advice if you describe the problem completely. If your math is rusty, you aren't likely to succeed in the job of taking a real world problem and trying to state it as an abstract mathematical problem. Let people who know math assist you in stating the real world problem as a mathematical problem.

 

[mr02077]

This is what the scientific work deals with:
t is success rate for a type of specific very precise machines that is easy to calculate
x is skillfulness of worker, which is also able to be determined in terms of tests he does and certificates he already has.
y is readiness of machine, which is also able to be determined in terms of regular repairs it passes and how used-up it currently is.
z is something very important about this machines - daily psychological mood of worker.
z of course cannot ever be calculated
The goal was to find equation that gives t from x, y and z.
Thank you for sharing opinions

 

[BrettNortje]

This is what the scientific work deals with:
t is success rate for a type of specific very precise machines that is easy to calculate
x is skillfulness of worker, which is also able to be determined in terms of tests he does and certificates he already has.
y is readiness of machine, which is also able to be determined in terms of regular repairs it passes and how used-up it currently is.
z is something very important about this machines - daily psychological mood of worker.
z of course cannot ever be calculated
The goal was to find equation that gives t from x, y and z.
Thank you for sharing opinions

This may be calculated with biorithms and heat sensors, visualizing the heat of the worker into something that can be measured, due to the color of the worker's heat signature - red usually implies stress, blue complete calm and yellow means they are dreaming - not too sure about the dreaming. this is the three primary colors.

 

[jack action]

First, You do a set of tests where $y$ and $z$ are constant. Your data will fit $t = f(x)$, where $f(x)$ is some function. I will assume it is a polynomial of the second order, $f_1(x) = a_1x^2+b_1x+c_1$.

Then you redo the tests with a different $y$ but the same $z$. You should end up with, $f_2(x) = a_2x^2+b_2x+c_2$ (The function should have the same form as the first set). Redo it for different value of $y$. For each value of $y$ (from value $1$ through $n$), you will find a function $f_y(x) = a_yx^2+b_yx+c_y$.

Then you can find $a(y)$, $b(y)$ and $c(y)$ which are also some functions. Say $a(y) = A_1\sin(B_1y)$, $b(y) = C_1$ and $c = D_1\ln(E_1y)$; Where $A_1$, $B_1$, $C_1$, $D_1$ and $E_1$ are constants.

Now your equation is $t = A_1\sin(B_1y)x^2 + C_1x + D_1\ln(E_1y)$

Then you redo the tests with a different $z$ for different $y$. Similarly, you will find functions of $z$ for the constants $A$, $B$, $C$, $D$ and $E$.

Your final equation will be:

$t = A(z)\sin(B(z)y)x^2 + C(z)x + D(z)\ln(E(z)y)$

 

[mr02077]

Jack, thank you for this interesting answer. I thought about it for a while.

But does not your step 3 assume I can have numeric value of z, in order to substitute it in functions obtained from interpolation, and get A(z), B(z), C(z), D(z), E(z) ?
I can get z to be the same for some set of tests (all tests done in the same day time, when workers abilities are the same), but I cannot obtain numeric z.

Is my observation right?

 

[Svein]

You can create a formula that matches your data points exactly - but I do not think you want that. If your data points are the result of a set of measurements, you need to consider:

• What is the precision of the data points?
• How much "noise" is implicit in the data?
• What kind of complexity do you want to implement?

When these points are clear, you can decide what kind of formula you want and how to get about finding it.

Hints:

• Least squares approximation (linear, quadratic or exponential)
• Curve fitting using splines
• Approximation using orthogonal polynomials (Legendre, Chebyshev,...)

 

[jack action]

but I cannot obtain numeric z.

This is when you will have to create your own scale. Maybe there is already one that exists.

For example, say you have these 2 workers: one that is always upbeat, always smiling, whistling while working. The other one is always complaining, going through a messy divorce, always eating alone in the office. You might set those 2 workers their $z$ values to be $100$ and $0$. You have just created a scale.

This is how temperature scales were made. Why does water boils at 100°C and and freezes at 0°C? Because someone put mercury in a tube of a certain dimension, draw a line at the level when water was freezing and another at the level when it was boiling and finally arbitrarily put the number 0 and 100 beside the lines.

How do you define «daily psychological mood of worker»? This is not my area of expertise, but I'm sure some studies were done to put some value according to certain criteria.

Worst case scenario, you will end up with a set of equations, specifying something like:

  mood           equation
excellent       t = f(x,y)
  good          t = g(x,y)
   OK           t = h(x,y)
   bad          t = i(x,y)
  worst         t = j(x,y)

No matter what, you will have to define «daily psychological mood of worker» somehow.

 

[Nidum]

Having though about the problem some more I think that a simple point scoring system might do all that you want .

Point scoring systems might in any case be the only thing that will work with such limited data available .

On each test each of your variables is given a points score and the sum of scores is the measure of 'goodness' of the system .

Scores for each variable can be equally weighted to begin with but they can be given more/less weight to tune the system to give more accurate results as test experience increases .

 

[mr02077]

Hello.

I need some advice from you guys.

I have a table of 92 rows with 4 values for variables x,y,z,t in each row.
These are all experimentally measured out values for function t = f(x,y,z)

My goal is to get the equations, most likely with some form of interpolation.

Is there any software I could use to get this?
Is there any numerical analysis method that I could use to get the equation and could you give me some link so I read about it? It's been long time since I have graduated mathematics, but if I read enough on the method, I think I could do it.

Thank you very much!

 

[BvU]

You could try to explore any a priori knowledge you have about this relationship and fit a model in the usual way (least-squares fitting -- however, for a 3D model 92 observations is often meager statistics.).
Or you could attempt a brute-force method using a product like http://www.sigmaplot.co.uk/products/tablecurve3d/tablecurve3d.php

I sure hope this is not https://www.physicsforums.com/threads/get-equation-that-describes-set-of-measured-values.897728/ because then you don't really have what you claim to have.

Oh, and a very late , mr020 !

 

[Mark44]

I sure hope this is not https://www.physicsforums.com/threads/get-equation-that-describes-set-of-measured-values.897728/ because then you don't really have what you claim to have.

I'd be willing to bet that it is.