Do you know about operational equations?

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1. Oct 8, 2016

Schimmel

Many years ago, I encountered a problem involving four numerical data in a square array or a rectangular array. The standard method for interpolating that design is the bilinear equation.

For example, let the array be ACIG as below left. If A=1, C=3, G=7, I=9, then the bilinear equation yields z = (1+2x+6y) in the (x, y) = (0 .. 1, 0 ..1) coordinate system. However, if the data are the squares of
the cited data, the bilinear equation yields z = 35 at the center of the design. That is not a
G I good estimate of the true value z = 25. I sought help at universities but was rebuffed by
A C the remark that no other equation for the four-point rectangle is possible. A few years
later I happened on the relation (e^x)F(x) = F(x+h) where is 'e' is now an operator and F(x) is any function of (x).

This ultimately led to an interpolating equation that is exact on bilinear data and on the squares of bilinear data. That led to a nasty verbal interchange. It also resulted in two papers: "No Free Lunch: Comments on Silver" where the new equation was declared impossible. Read it and see: Quality Engineering 5(3) 369-373 (1993) by Norman R Draper and Dennis K J Lin. It also led to my rebuttal: Free Lunch, Bigger Menu, Better Food" by G. L. Silver in Quality Engineering 6(2) 307-310 (1994). Read it and see.

Draper and Lin were never heard from again. After I arrived at Los Alamos National Laboratory (New Mexico) I added to the altercation: "Operational equations for data in rectangular array" by G. L. Silver, Computational Statistics and Data Analysis 28 (1998) 211-215. That citation gives the complete equation in the -1 .. 1 coordinate system. It is exact on (positive) bilinear numbers and on the squares of such numbers.

For example, if A=1, C=3, G=7, I=9 then the equation is z = (5+x+3y) but if A=1, C=9, G=49, I=81 then z = (5+x+3y)^2. There are lots of operational equations. Most are "substitute and see" equations. They are probably pertinent to physics. You can join in the fun, too!

Last edited by a moderator: Oct 13, 2016
2. Oct 13, 2016

Stephen Tashi

If any of the papers you cite are available online, you should give links. Otherwise, it would be helpful if you stated the problem.

You mention "design" but it isn't clear whether you are talking about data involving "the design of experiments" or simply some arbitrary data without any specific interpretation.

You haven't made it clear why z = 25 is the "true value". What criteria is used to determine the true value at the center of the rectangle?

3. Oct 13, 2016

Schimmel

Reply to Stephen Tashi about Operational Equations from G. L. Silver. The four-point rectangle ACIG has A=1 at lower left corner, C=3 at lower right corner, G=7 at upper left corner, I=9 at upper right corner. These are bilinear numbers. Let (x) be the x-coordinate and (y) be the y-coordinate in the -1..1 coordinate system for both (x) and (y). Let (z) be the altitude over the x,y plane. The bilinear equation for this design is z = (A+C+G+I)/4 + (I+C-A-G)(x/4) + (I+G-A-C)(y/4) + (I+A-C-G)(xy/4). (The bilinear equation is not new. It is Eq. (8) in the literature citation below.) The bilinear equation for this four-point design is z = (5+x+3y). At the center point of the rectangle we have (x,y) = (0,0) so z = 5. Now suppose the data are squared so that A=1, C=9, G=49, I=81. That is, we apply the squaring function to the four-point design. At the center point of the rectangle, the bilinear equation yields z = (35+10x+30y+6xy) so that at (x,y)=(0,0) and we have z=35. That is a poor approximation to the true value z=25 at (0,0). This deficiency attracted my attention with respect to work in another field. I wanted something more accurate. By chance, one day I saw the relationship exp(x)F(x) = F(x+h). That was years later but it reminded me of the rectangle problem so I used exp(x)F(x)=F(x+h) to find a new interpolating equation for the four-point rectangular design. The explicit equation is Eq. (7) in Computational Statistics and Data Analysis 28(1998)211-215. If you put A=1, C=9, G=49, I=81 into the cited Eq. (7) the result is an interpolating equation that factors to z = (5+x+3y)^2. (That is the square of the bilinear equation.) In summary: the bilinear equation is exact on four bilinear numbers in a rectangular array. The operational equation is exact on four POSITIVE bilinear numbers in a rectangular array AND on the squares of those numbers. That was supposed to be impossible! Make no mistake: emotions ran very high! In summary, the bilinear equation is to four data in a rectangular array what the straight line is to two data. The operational equation is to four data in a rectangular what the parabola is to three equidistant curvilinear data. That analogy is good enough. The derivation of the operational equation is suggested in the 1985 reference found on page 215 of the cited Comp. Stat. Data Analysis paper. (Do not bother with that citation. It isn't pertinent here.) In summary, there are three kinds of calculus: the differential calculus, the integral calculus, and the operational calculus. I applied the last one to geometry. (That is where the term "operational equation" comes from.) That is the story. I hope it helps.

4. Oct 13, 2016

Stephen Tashi

What is a "bilinear number" ?

I have no idea why we would expect to get the same answer from a formula if we replace the original inputs with their squares.

I don't have access to the journals you mention, so unless you can describe the problem coherently, I must leave this thread to other forum numbers who have the appropriate subscriptions.

5. Oct 13, 2016

Schimmel

As a courtesy, I will try to type the operational equation (for the four-point rectangle) into this letter. It is a tedious job, so if it fails the problem is likely a typographical mistake.

z = ((C+G)(A-C+G-I)(A+C-G-I) - (C-G)^2(A-C-G+I)) / (2(A-C+G-I)(A+C-G-I)) + (I+C-A-G)(x/4) + (G+I-A-C)(y/4) + (A+I-C-G)(xy/4)
+ ((I+A-C-G)(I+C-A-G)x^2) / (8(G+I-A-C)) + ((I+A-C-G)(G+I-A-C)y^2) / (8(I+C-A-G))

Remember, the coordinate system is -1 .. +1 in BOTH the x- and y-directions. If you still have trouble, ask a colleague (or someone on the PhysicsForum) to help you. Surely the management at PhysicsForum will have someone to assist the members. Do those things FIRST. This is a matter of public-school algebra so there should be a nearby colleague or member of the Forum to help with such problems. --- Schimmel

6. Oct 14, 2016