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How to find the inital formula behind engineering test data ?
2

How to find the inital formula behind engineering test data ?

How to find the inital formula behind engineering test data ?

(OP)
Recently, my job seems to work on a mathematic problem.

We did many efforts on engineer test and get a perfect graph curve by discrete data (100 points on EXCEL).  
But my customer interest also the inital Formula behind those data. Perhaps it’s a function or calculus.....not clear.

Engineers and physicists distinguish between continuous data and discrete data.

Anyone leads us first step of this cross?
Books, software, method?
 
 
 

RE: How to find the inital formula behind engineering test data ?

SOMEONE must know what you were measuring.  Data does not randomly create itself.  

The physical process that you measure should have some sort of model behavior.  That is the equation.  You can then run least squares or other regression of the data against that equation to show correlation to the model.

TTFN



RE: How to find the inital formula behind engineering test data ?

I'm not sure I completely understand the question, but perhaps it amounts to communicating to the customer the logic behind the spreadsheet.  By the nature of spreadsheets, that can often be difficult.  It is helped if the spreadsheet it well documented.

Some tools for documenting spreadsheets and making them easier to use:
Label the frequently-used variables with a descriptive name instead of using cell reference. Often it is handy to put the name of the variabel immediately adjacent to it and create name using insert/name function/create / top-row or left-column.

If you don't have a lot of named-variables, then if you do a printout of the spreadsheet you absolutely have to set it up so it prints the row and column headers so readers can understand it.

One thing that can be very helpful in documenting results which won't take a lot of time is the getformula function described earlier in this forum. It will allow you to display the formula contents of a cell in a separate cell for documntation purposes.  

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RE: How to find the inital formula behind engineering test data ?

If all you have is a bunch of values (no calculations), then my previous comments won't help.

A clear description of where the data came from is best as suggested by IRStuff

Also excel has a number of curve-fitting functions which may or may not be relevant to satisfying the customer's request.

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RE: How to find the inital formula behind engineering test data ?

(OP)
Thanks for comments at the moment.

I try to make the problem clear, but I believe that many engineers have the same problem as I had.
It exists on all engineering to physical world.

I assume, physical model is Load to Stress, or Speed to Torque, or Current to Voltage...whatever
( My customer will not pay me, if detail and physical model is open from my side.)

Below is the procedure.

1. We did many jobs by our test rig.
2. Then we have reliable test data.
3. We analyzed those data, with as much as knowledge, we have at the moment.
4. At this stage, we filter some parameters and relation by physical and logical think.
5. Now we have a curve, Let’s call it coefficient curve on EXCEL spreadsheet.
This curve composed by 100 points on X-Y coordinates.
6. This curve can be trust and smoothly, But it is discrete data. Not precise enough for 0.000 digital.
7. My customer needs a precise continuous data or a continuous curve or a reliable formulate so that he can make programme to any points he likes. Perhaps some new relation or theory behind the curve…. I don’t know.
8. What to do next step…..

This is same thing for Newton on hundreds years before. G=9.8 X M. ( F=a X m)
He saw, apples drop from tree. Then he did many tests, and find this links to apple mass.
So he divide G/ M and found it is a constant G/ M = 9.8
Modern people laugh the simple equation. But it’s great development on history.

My problem is no more a simple constant. It is a smoothly curve.
I try on EXECL. Not powerful. Perhaps I will try other math software.
One of my mathematic friends told me, it could be thousand function or equations match with only one curve.
I don’t expect Nobel’s price.
I only need to convert discrete data to continuous curve, so my customer can programmed later base on it.
A function formulate with some reasonable tolerance will be perfect.
Or some math company knows?

RE: How to find the inital formula behind engineering test data ?

Then, you've asked for two different things.  

1>  If you are simply collecting data for a client, then the model is the province of the client.  Who else whould know about the mathematical abstraction of his physical system?

2>  If all the client cares about is a continous function, then feed him the simplest spline fit you can find, like a cubic spline.  Anything beyond that requires his internal knowledge of the system under test.

Personally, I can't see why he's asking you to provide this for him.  It's his design; whatever idiosyncrasies there are in the data should be something that he models, not you.

TTFN



RE: How to find the inital formula behind engineering test data ?

(OP)
Thanks TTFN

The kernel physical model belongs to nobody at the moment.
Because nobody understand it very clear.
Who knows it better, who has it.

We are responsible engineering and physical world.
My client is responsible at software, and converts the knowledge to market product.

Does any powerful math software assistant for such analyze to find a spline?
MathCAD or Mathematica ? What do you recommend?
Experience or Procedure can share?

RE: How to find the inital formula behind engineering test data ?

Every time I have seen a spline, it amounts to an exact fit to each point. I don't think that's what you're looking for. You are looking for a curve fit.

Here is a very simple curve fit procedure in excel.

Guess the functional form of the function.
It might be any of the following or many others
y = k0 + k1x + k2x^2 +k3 x^3 etc
y = k0 + k1*cos(k3*x)
y = k0 + k1 * tan(k2*x)
etc (you get the idea

Pick one form. P the k_i's into cells and make an initial guess at their values.  For each x_i in your data set, compute y_hat_i which is your estimated function.

Compute sum of residuals squared sum(y_hat_i-yi)^2

Use the solver tool to minimize the sum of residuals squared by varying the k0, k1, k2 etc.

If you haven't used solver, let us know. We can walk you through it.

There are other curve fitting tools in excel as well but this seems like a good simple approach to me and extremely flexible in terms of the functional form of the solution (pick any form you want).

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RE: How to find the inital formula behind engineering test data ?

"(pick any form you want)"

I didn't mean it was a random choice. I meant that if your eyeball gives you a clue of what type of function or the theory gives you some type of clue of what type of function, you can make that function your guess.

Also note that this approach is not limited to one input variable. You can compute your y from multiple inputs.  You could also have multiple outputs in which case you would have to sum the residuals from estimates of all outputs into a single cell.

Another form often used is
y = k0 + k1 * x^k2

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RE: How to find the inital formula behind engineering test data ?

A correction to the trigonmetric functions. Should allow for a phase shift as follows:
y = k0 + k1*cos(k3*x+k4)
y = k0 + k1 * tan(k2*x+k4)

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RE: How to find the inital formula behind engineering test data ?

I still don't follow.  Your X,Y data points must mean something.  If X is force, and Y is displacement, then (assuming a linear relationship)):

Y=mX+b => Displacement = m*Force + b

If you have 3 variables, then you will have to graph in 3D.  Unfortunately, I don't think Excel can regress a 3 variable equation.

As far a digital versus continuous data, you will NEVER have truly continuous data.  You will always work from a finite data set and use some sort of regression to define the relationship between the variables.

RE: How to find the inital formula behind engineering test data ?

What is the desired end result; what is your client attempting to do?

If you need a best fit curve, there is no shortage of methods.  If the curve is very complex, then as your mathmatician friend says, you may need to superimpose a lot of functions.

If you need a function that has to return each and every coordinate pair exaclty, and interpolate inbetween, than you need a lookup table and a method of interpolation.

RE: How to find the inital formula behind engineering test data ?

It would be helpful if you listed your type
of input data and what your output data is
for anyone to even guess at what you are
trying to ascertain.
What does your curve look like?  Is it always
sloping up, always sloping down, or combination
of slopes?
I am assuming that your input data is constantly
increaseing by some increment?

RE: How to find the inital formula behind engineering test data ?

melone says there is no capability to handle multiple input variables. I think with solver it can be done.

For multiple input/single-output problem with many sets of data i=1..N

yhat_i = k0 + k11*x1_i + k12*x1_i^2 + k13*x1_i^3
            + k21*x2_i + k22*x2_i^2 + k23*x2_i^3
            + k31*x3_i + k32*x3_i^2 + k33*x3_i^3
            etc
For the i'th set of data (x1_i, x2_i, x3_i, y_i) compute the residual
r_i = y_hat_i((x1_i, x2_i, x3_i))- y_i

Compute the sum of squares of r_i (1..N) in a target cell.

Use solver to find value of k's that minimize the value of the target cell.

If several outputs, as a first simpler attempt you could add all the sum of the squares together in a single target cell and try to solve all at once. But that would complicate the problem in terms of complexity and the need for a good initial guess. Probably better to split into several multiple-input/single-output problems and solve each individually as shown above.

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RE: How to find the inital formula behind engineering test data ?

Needless to say, after you run solver you have to inspect the solution to see if you are satisfied with it. One quick easy way is to look at the computed residuals. If they are all below 0.0001 and you're only looking for 0.001 accuracy, you probably have a good solution.  If residuals are not good, you may not have picked a suitable guess at the functional form, the data may not have been well-behaved, or some combination. Also for non-linear minimization problems there is more than one minimum and excel might not find the best one.  That situation might be resolved by chaning/improving your initial guess.

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RE: How to find the inital formula behind engineering test data ?

Correction:
"Also for non-linear minimization problems there may be more than one minimum..."

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Eng-tips forums: The best place on the web for engineering discussions.

RE: How to find the inital formula behind engineering test data ?

Last correction:
"Also for non-linear minimization problems there may be more than one local minimum..."

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RE: How to find the inital formula behind engineering test data ?

A random attempt at curve fitting usually generatespoor results.  As an example, polynomial fits improve with the order, but the result begins to look like a spline fit, since each additional degree of freedom allows the curve to more closely fit each datapoint exactly, which is what a spline fit does.  Without an a priori equation, there's no way to determine the correct order of a polynomial fit.

Without the actual physical model equation, there is no plausible approach to figure out the correct equation.

TTFN



RE: How to find the inital formula behind engineering test data ?

You can curve fit various curves, then look at the statistical deviations to determine which is best.  However, you will likely never get to the original equation...

RE: How to find the inital formula behind engineering test data ?

"Without the actual physical model equation, there is no plausible approach to figure out the correct equation."

So if you were asked to fit a set of measured data without having complete access the underlying physical model,  you would give up and go home?  Most people are a little more resourceful than that.  They would use a review of the data itself (along with any limited physical knowledge of the problem they do have) to formulate some educated guesses at the most likely functional forms.  

If the data itself and physical knowledge don't give any clues, then there is no harm to try the multi-input/single output polynomial form I suggested above....  When you're all done, assuming the data is not sparse (100 points!) and encompasses the range of interest, the final success of this approach can be reasonably evaluated by examination of the residuals (again without requiring exact knowledge of the underlying phenomena).  If residuals are not acceptable than obviously more work is required.

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RE: How to find the inital formula behind engineering test data ?

I must admit, running around waving my arms in the air didn't strike me as a natural response either. If you truly do not know what the underlying physical mechanism is then you start with astraight line, then a log log graph, and then just suck it and see.

If you think it is a polynomial, then very few processes are above eighth order.

Oh, and of course a dimensional analysis might well give you a clue for the likely relationship.





Cheers

Greg Locock

Please see FAQ731-376 for tips on how to make the best use of Eng-Tips.

RE: How to find the inital formula behind engineering test data ?

Before anyone else says it, I will point out a serious weakness of the model I mentioned above (yhat_i = k0 + k11*x1_i + k12*x1_i^2 + k13*x1_i^3 + k21*x2_i + k22*x2_i^2 + k23*x2_i^3 + k31*x3_i + k32*x3_i^2 + k33*x3_i^3)

It does not allow for interaction among terms. For example a dependence upon x1*x2 would never be identified.  So clearly it will not work for all cases.

The more inputs you have, the tougher the problem becomes.  Any knowledge about the system that can be modeled into the estimate y_hat will help the problem solving process. On that point I agree with IRStuff and the others.

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RE: How to find the inital formula behind engineering test data ?

(OP)
Thanks all for comments and suggestion, I appreciate that.
Thanks for IFRs, that could be pretty close to what I desired….
Perhaps I am looking for a statistical analyze software.

My little experience with Mathematica a few years before...
If your input is math, the output is also logical math, with nice graphic.
Dose latest Math software, including such statistical Feature also?

electricpete
I do try with some basic function, it doesn’t fit exactly.

IRstuff
No more solid approved physical knowledge are available for me,
therefore we call it coefficient- curve.
Co-efficient means un-known.

RE: How to find the inital formula behind engineering test data ?

You have 100 points on x-y plot i.e. single input/single output?

I would think it should not be too tough to fit in excel using solver

Can you paste the x/y pairs here?

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RE: How to find the inital formula behind engineering test data ?

(OP)
Here is X

0.0000
0.0500
0.1000
0.1500
0.2000
0.2500
0.3000
0.3500
0.4000
0.4500
0.5000
0.5500
0.6000
0.6500
0.7000
0.7500
0.8000
0.8100
0.8200
0.8300
0.8400
0.8500
0.8550
0.8600
0.8650
0.8700
0.8750
0.8800
0.8850
0.8900
0.8950
0.9000
0.9020
0.9040
0.9060
0.9080
0.9100
0.9120
0.9140
0.9160
0.9180
0.9200
0.9220
0.9240
0.9260
0.9280
0.9300
0.9320
0.9340
0.9360
0.9380
0.9400
0.9420
0.9440
0.9460
0.9480
0.9500
0.9510
0.9520
0.9530
0.9540
0.9550
0.9560
0.9570
0.9580
0.9590
0.9600
0.9610
0.9620
0.9630
0.9640
0.9650
0.9660
0.9670
0.9680
0.9690
0.9700
0.9705
0.9710
0.9715
0.9720
0.9725
0.9730
0.9735
0.9740
0.9745
0.9750
0.9755
0.9760
0.9765
0.9770
0.9775
0.9780
0.9785
0.9790
0.9795
0.9800
0.9805
0.9810
0.9815
0.9820
0.9825
0.9830
0.9835
0.9840
0.9845
0.9850
0.9855
0.9860
0.9865
0.9870
0.9875
0.9880
0.9885
0.9890
0.9895
0.9900
0.9905
0.9910
0.9915
0.9920
0.9925
0.9930
0.9935
0.9940
0.9945
0.9950
0.9955
0.9960
0.9965
0.9970
0.9975
0.9980
0.9985
0.9990
0.9995
1.0000

Here is Y

0.02363
0.02434
0.02528
0.02623
0.02717
0.02812
0.02930
0.03048
0.03190
0.03332
0.03497
0.03710
0.03923
0.04183
0.04513
0.04891
0.05435
0.05553
0.05671
0.05813
0.05978
0.06144
0.06238
0.06333
0.06427
0.06546
0.06664
0.06758
0.06900
0.07018
0.07160
0.07302
0.07373
0.07443
0.07491
0.07562
0.07632
0.07727
0.07798
0.07869
0.07940
0.08034
0.08105
0.08200
0.08294
0.08389
0.08483
0.08578
0.08696
0.08814
0.08932
0.09050
0.09168
0.09310
0.09428
0.09570
0.09736
0.09806
0.09901
0.09972
0.10066
0.10161
0.10255
0.10350
0.10444
0.10563
0.10657
0.10775
0.10893
0.10988
0.11106
0.11248
0.11366
0.11484
0.11626
0.11768
0.11933
0.12028
0.12122
0.12193
0.12288
0.12382
0.12477
0.12571
0.12642
0.12737
0.12855
0.12973
0.13067
0.13186
0.13304
0.13398
0.13516
0.13658
0.13776
0.13918
0.14036
0.14178
0.14320
0.14462
0.14627
0.14792
0.14958
0.15123
0.15289
0.15478
0.15690
0.15927
0.16163
0.16376
0.16588
0.16848
0.17132
0.17415
0.17699
0.18006
0.18337
0.18739
0.19140
0.19542
0.20015
0.20511
0.21078
0.21669
0.22354
0.23087
0.23984
0.24882
0.26040
0.27364
0.28970
0.31073
0.33673
0.37265
0.43786
0.56594
?

First Point : ( X= 0.0000, Y= 0.02363 )
Last pointt : ( X= 1.0000, Y= ?       )

RE: How to find the inital formula behind engineering test data ?

(OP)
? =infinity

RE: How to find the inital formula behind engineering test data ?

(OP)
Our thought is like this.

We don't know A, we don't know B
We assume C ( specification )
We calucate to have D  ( Physical + Math )
.....
.....
Then we test to have G ( Engineering Test data )
we analyse to have F   ( Co-efficient curve)

Can we go from F to E ? How ?

RE: How to find the inital formula behind engineering test data ?

A scatter diagram of your data presents a graph that can be easily fitted by a rational function R(x)=P(x)/Q(x) where   P(x) is a polinomial of a degree n and Q(x) is a polynomial of degree n+1.
Of course this fit (or any other) does not tell us anything about the physical background of your system.
m777182

RE: How to find the inital formula behind engineering test data ?

m777182, I'm sure you are right, what is the method that that is called?

Meanwhile

Y=1/($E$1*SQRT($E$2-x)+$E$3+$E$4*SQRT(SQRT($E$5-x)))

where e1 thru e5 are

39.26003141
1
1.240802562
-0.121279377
1.142702314

seems to fit pretty well. I got that using inspection and a variation on electricpete's method.

Looks like a stiffness function.





Cheers

Greg Locock

Please see FAQ731-376 for tips on how to make the best use of Eng-Tips.

RE: How to find the inital formula behind engineering test data ?


First Point : ( X= 0.0000, Y= 0.02363 )
Last pointt : ( X= 1.0000, Y= 1.00000 )?

Are you saying that it cannot be measured?

RE: How to find the inital formula behind engineering test data ?

Are these clearances or deflection
values in a ball bearing similar to
your other questions in the bearing
forum?

RE: How to find the inital formula behind engineering test data ?

A plot of the residuals shows that GregLocock's formula has some issues.  Residuals should be randomly distributed about 0.  These residuals show a clear trend.  HOWEVER, it appears that GregLocock is on the right track with the equation.

RE: How to find the inital formula behind engineering test data ?

(OP)
Well done. GregLocock
Now i assume, we have the first equation.

If i give this to my boss. He will accept only e2. OK.
Then he ask me, “ Freddy, what do you name e1 e3 e4 and e5 ? ”
I say, "Of course, this is a Factor."
He say, “Factor ? Factor means you are un-known."
.........
(I do think about to kill him in first year.)

My dear friends,
My suggestion is?
Is that possible to have one more less Factor
Or have a real e = 2.718282 or pi= 3.1415926.....or SQRT(2)=1.414.....
Is that a way, we approach from engineering world to physical world?

To Diamondjim
This curve has nothing to do with ball bearing.
I am new for bearing, therefore i start for ball bearing only.
I see, you have a lot of knowledge and experience on that field.
We made good discussion. (I am new, but not too new.)
I am glad to consult to you later, if possible.

RE: How to find the inital formula behind engineering test data ?

Y=1/($E$1*SQRT($E$2-x)+$E$3+$E$4*SQRT(SQRT($E$5-x)))

You could say these constants came out of a least-squares fit program with an assumed solution including a term varying in x^1/2, a term varying in x^1/4, and a bias term.
E1 is magnitude scaling factor for the ½ power term (centered on E2)
E4 is the scaling factor for the ¼ power term (centered on E5).
E4 is a constant (bias) term.
Maybe Greg has a better way.  To get more help on physical interpretation from the folks on the forum I would say you have to give more details of the physical problem (maybe that's something that you don't want to do to protect business secrets?).

Unfortunately I went the opposite direction.  I spent (wasted?) some time messing around to add more terms to Greg’s model to reduce the residuals.  I did get good reduction in residuals at a cost of substantial increase in complexity.  This is probably not what you’re looking for but since I’ve done it, here it is:

The following columns are in my spreadsheet
X, Y = Original data
Ygreg = Greg’s fit * Note - Greg’s fit constants changed when I reoptimized with new terms
Rgreg = Greg’s residual = Y – Ygreg  *Same Note
Fpete = my fit to greg’s residual
Ypete = Ygreg +Fpete = estimate of Y
Rpete = Y - Ypete
Rpete/Y = Fractional residuals

I had some difficulties minimizing errors on both the top and bottom end.  What finally resolved the problem was to optimize sum of squares of FRACTIONAL residuals , rather than residuals themselves.  Probably if I tried this approach from the beginning I could have reduced the complexity. I didn’t go back and try to clean it up.  What is left is a complex formula which I’m sure could be simplified or done better. But this is what I have for now.

** The largest residual magnitude (Rpete) is 0.0075
** The largest fractional residual magnitude (Rpete/Y) is 1.7% over the entire range up to 0.9995.  That worst performance occurs at 0.999.   Below 0.9965 all fractional residuals are less than 1% of Y

If one wanted, one could start with my solution and look at the residuals and attempt to add terms to minimize those residuals (that’s what I did to Greg’s solution).  

The spreadsheet is here:
http://home.houston.rr.com/electricpete/Fit2.xls
(scroll to bottom of sheet 1 to see summary statistics)

The formulas are
Ygreg=1/($B$3*SQRT($B$4-C3)+$B$5+$B$6*SQRT(SQRT($B$7-C3)))

Ypete = Ygreg+$B$8/($B$11+(C3-0.995)^2)  +  $B$9*(1-C3)*EXP($B$10*(1-C3))  +  $B$12*(0.5-C3)^2*(ATAN(1000*(C3-0.5))+PI()/2)+$B$13*EXP($B$15*C3)

where cells are as follows:
C3     is     x  (copy down as relative cell reference)
B3    50.63746184  (remaining items used as absolute reference)
B4    1.000239763
B5    0.823257421
B6    -0.006540995
B7    3.654165646
B8    0.015665365
B9    772.441016
B10    -2859.484503
B11    1.23573751
B12    0.008220401
B13    -0.002908095
B14    -10

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RE: How to find the inital formula behind engineering test data ?

Why should he accept e2 any more than any other number?

Your suggestion, unfortunately, is naive.  There should be no expectation of seeing e, or pi, or whatever, in real data fits.

You have yet to even describe what the data comes from.  Without that critical information, there can be no meaningful discussion about the meaning and validity of the fitting coefficients.

TTFN



RE: How to find the inital formula behind engineering test data ?

"centered" on E5 and E2 was not good terminology. Those functions (X^1/2 and X^1/4) are shifted along the x coordinate by E5 and E2.

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RE: How to find the inital formula behind engineering test data ?

Looking at the original data, I think I could come up with a much simpler solution than I gave above starting again from scratch with simple functions:
x^n   (n=>1)
(x+p)^n  
(1-x)^m  (m<=0)
a^(b*x)
Maybe log-log plot would give some clues.  I'll give it a try when I have some time.

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RE: How to find the inital formula behind engineering test data ?

Nope, I was wrong. Starting from scratch I couldn't find any simple way to fit the data.

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RE: How to find the inital formula behind engineering test data ?

I took the inverse of the y values to give me a less alarming function. It looked like a parabola. It didn't quite fit so I chucked in another term. Then I used solver to 'minimise' the errors in the coefficients. Brutal, but quick. As melone pointed out, not a mathematically 'good' result.

Now, /if/ this is a Hertzian contact stress problem (which it intuitively looked like to me) then there are a wealth of papers on this stuff, including a good derivation in Timoshenko. I suspect that one problem is that we are looking at (say) force vs deflection, the mathematical derivation is more likely to be based on rate (stiffness) versus axial displacement. So, the data we are fitting is the derivative of a rapidly changing function. Anyway, if you understand what the problem is then you will be able to find some meaning for the terms, and maybe even justify my admittedly arbitrary choice of exponents.

Cheers

Greg Locock

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RE: How to find the inital formula behind engineering test data ?

OK.  You've got me intrigued.

Data appears to have two dominating characteristics:  for small x it is close to a straight line with a positive intercept and a shallow slope;  as x approaches 1 we have a vertical asymptote.

A model that satisfies these two is
Y = a + b*X +c*[sec(pi*X/2)]^d
for which Excel's Solver, when fed all the points, homes in on
a = -0.017917
b = 0.0083520
c = 0.042542
d = -0.36422
For these values, the maximum "error" over the full set of given points is 0.0032 in absolute terms or 4% in relative terms (at different points).

The maximum errors are larger that ElectricPete is citing above, but an argument can be mounted that my approach honours the asymptote (and the asymptote is a given, based on FreddyMusic's data).

And, I've even got a pi in there.

RE: How to find the inital formula behind engineering test data ?

A plot of the raw data looks like a plot of the Tangent function.....  it appears to be asymtotic.

RE: How to find the inital formula behind engineering test data ?

to GregLocock

A rational function R(x)is a simple method to fit an asymptotic function.You start with a low degree n and can get even better fit by increasing it. With m-n>1 you can fit even multi modal functions easily.
No background, experience only.
m777182

RE: How to find the inital formula behind engineering test data ?

Yes, but you mentioned "R(x)=P(x)/Q(x) where   P(x) is a polinomial of a degree n and Q(x) is a polynomial of degree n+1"

How did you know about the n+1 bit?

Cheers

Greg Locock

Please see FAQ731-376 for tips on how to make the best use of Eng-Tips.

RE: How to find the inital formula behind engineering test data ?

(OP)
Don't look at me. I am also thinking.

To Denial

There is no argument.
To approach vertical asymptote X=1 is important.
Meanwhile, middle section of curve is the best place, which we can measure precisely in available test environment.

Is that possible to shoot one more co-efficiency off ?

RE: How to find the inital formula behind engineering test data ?

FreddyMusic,

I take it you are asking whether the fit could be improved by adding another term into the equation.  The answer is almost certainly "yes", but there does come a point when you begin to kid yourself (statistically speaking).  Your data is based on physical experiments, and so it will inevitably contain "noise":  at some point you begin to overly accommodate this noise.

As someone indirectly pointed out above, if you want an EXACT fit to your 99 data points you could always resort to a polynomial of order 98, but I don't think you'd be any the wiser from the exercise.  A statistician would express this in terms of degrees of freedom, and tell you that your 98-order polynomial had zero predictive power because the number of degrees of freedom in your model was zero.

Occam's Razor is never better applied than in the area of curve fitting.

Having said all that, if you really want to "improve" the fit by adding one further term then you have to decide what sort of term to add.  Since we already have what I would regard as a good fit, you will have to resort to good old trial&error, guided slightly by your mathematical instincts, and perhaps by any knowledge you (but not the rest of us) might have of the underlying phyical phenomenon.

The first trial I would undertake (since you are insisting on going further) would be to add a quadratic term to the model I used above.  Thus:
Y = a + b*X + c*X^2 + d*[sec(pi*X/2)]^e

Happy hunting.

RE: How to find the inital formula behind engineering test data ?

(OP)
Denial,

No need to go further. Something is wrong in the beginning.

Now I found the correct direction.

Thank you all of you.

Best regards,

FreddyMusic

RE: How to find the inital formula behind engineering test data ?

So what is the correct direction?

RE: How to find the inital formula behind engineering test data ?

(OP)
Melone
Perhaps this link interested you. I get from my Math friend
http://www.statsoft.com/textbook/stathome.html

Seems like some company do study for that. They call it DOE (design of Experiment)
Still is not clear, what they can do exactly? How much they want to charge me for software?


Denial

I have a kind of feeling (only feeling), what Greg Locock found e2=1 is a good number.
That seems like predict the vertical asymptote X=1.
Perhaps we can modify the math model to something like
Y = 1/ (X-1) * ………….

Of course, I like Pi. But this physical model doesn’t seem like a period’s function or a period's curve.
Or is sec(x) only a kind “noise "? I don’t know.

I understand, this is big-time-consume job.
Only thinking or searching, No Working !!!
I am great appreciated.

FreddyMusic

RE: How to find the inital formula behind engineering test data ?

(OP)
To Occam's Razor

Philosopher always speaks paradox, therefore they are always Right
Unfortunately, engineer always works like trial & error, therefore they are always Wrong.

FreddyMusic

RE: How to find the inital formula behind engineering test data ?

(OP)
Denial,

You curve seems up to down, or perhaps my eyes goes wrong?lol



FreddyMusic

RE: How to find the inital formula behind engineering test data ?

DoE won't help, if you have described the problem you have, correctly.

Cheers

Greg Locock

Please see FAQ731-376 for tips on how to make the best use of Eng-Tips.

RE: How to find the inital formula behind engineering test data ?

(OP)
Greglock

I assume, you has understand the "problem" clearly.
I feel, I have no more correctly input....

FreddyMusic

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RE: How to find the inital formula behind engineering test data ?

FreddyMusic,

You are correct.  My value for "d" should be positive, not negative.  Apologies.  [This happened because Excel does not have a sec() function, so I used cos() instead.  Then forgot to change the sign when transcribing results into my post.  Other values are unaffected.]

RE: How to find the inital formula behind engineering test data ?

(OP)
Congratulation Denial,

That looks pretty nice.

Your friend


FreddyMusic

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RE: How to find the inital formula behind engineering test data ?

(OP)
Congratulation to GregLocock.
Congratulation to electricpete also.

With your guys help, I understand the "problem" and "curve" much better in the last two weeks.

It has been great pleasure for me to meet you all in this forum.





FreddyMusic

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Readme FAQ731-376

RE: How to find the inital formula behind engineering test data ?

to greglock
yes, you are right. The Q(x) should be in general of degree m and m>n. In the most simple case m=n+1. M and n is got by nonlinear best fit (Solver).
m777182

RE: How to find the inital formula behind engineering test data ?

Fit your data with TableCcurve 2D

www.systat.com/products/TableCurve2D

It will give you the best fit, the statics for the fit, residuals, prediction interals and fit intervals with a select confidence level.

Then put the formula into a spread sheet

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