it's been a while since my last math class
Bobb Fwed
Posts: 1,119
I am trying to get an equation that I can use in the propeller float math objects. I have three data points to create the function. Everything will be positive inputs, and result in a positive value. I am pretty sure it can simplify into the ax^2 + bx + c but it has been so long since math class, I don't remember how to fill in the a, b, and c. Any help would be appreciated.
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April, 2008: when I discovered the answers to all my micro-computational-botherations!
Some of my objects:
MCP3X0X ADC Driver - Programmable Schmitt inputs, frequency reading, and more!
Simple Propeller-based Database - Making life easier and more readable for all your EEPROM storage needs.
String Manipulation Library - Don't make strings the bane of the Propeller, bend them to your will!
▔▔▔▔▔▔▔▔▔▔▔▔▔▔▔▔▔▔▔▔▔▔▔▔
April, 2008: when I discovered the answers to all my micro-computational-botherations!
Some of my objects:
MCP3X0X ADC Driver - Programmable Schmitt inputs, frequency reading, and more!
Simple Propeller-based Database - Making life easier and more readable for all your EEPROM storage needs.
String Manipulation Library - Don't make strings the bane of the Propeller, bend them to your will!
Comments
Leon
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Amateur radio callsign: G1HSM
Post Edited (Leon) : 11/9/2009 7:53:56 PM GMT
Not tested by me.
John Abshier
with equation calculated by excel
Just enter your X-Y-Values into the green cells and the aprox-function will be calculated by excel.
best regards
Stefan
If you are trying to fit 3 (x,y) data points to a Quadratic polynomial then what
you are trying to do is easy.
What you want is the inverse of a 3x3 matrix and multiply it by a 3x1 vector
the elements of the matrix are as follows
n is the number of points you are trying to fit to the parabola (3 in your case)
sum(Xi) means sum of all values X1, X2, X3
sum(Xi^2) means the sum of X1^2 and X2^2 and X3^2
sum(XiYi) means the sum of X1*Y1 and X2*Y2 and X3*Y3
etc.
_·········· n······· ···· sum(Xi)······· sum(Xi^2)
A··=·· sum(Xi)······· sum(Xi^2)··· Sum(Xi^3)
· ····· Sum(Xi^2)··· sum(Xi^3)··· Sum(Xi^4)
The 3x1 vector is
_··············· Sum(Yi)
V·=··········· Sum(XiYi)
·············· Sum(Xi^2 Yi)
So now you can do the matrix inversion to get·inverse of A·call it A'·· and then
multiply A' by V which will result in a 3x1 vector call it C
_······· c
C·=···· b
··········a
where the quadratic polynomial is aX^2+bX+c
Now if you know that you will ALWAYS have three data points
you can develop ALGEBRAIC formulas involving X1..3· and Y1..3
and get the elements of the A matrix and the V vector as algebraic
equations.
Also since A is only a 3x3 you can develop algebraic formulas for
the elements of A' (the inverse of A)
and thus you end up with algebraic formulas for the coefficients a,b,c.
With these algebraic formulas you can just plug in X1..3 and Y1..3 and obtain a,b,c.
·
Here is a RobotBASIC program that does the above but for ANY number of data points and
ANY polynomial.
I just wrote this program and it is very SIMPLE....you can always change it to make it more
user friendly and utilize RobotBASIC's GUI ability·to have the user enter the data and display
the results. BUT...I take it you are a programmer and you can do all this by yourself.....this
is just for you to see how it is done.
You can get RobotBASIC (interpreter/compiler) from www.RobotBASIC.com
Edit
I forgot to mention that the program makes use of RB's matrix math commands that
make it easy to multiply and invert matrices with just one command....you may want
to develop your own matrix inversion algorithm on the Propeller.....but it should not be
necessary if all you have is a 3x3 matrix since this can be a simple ALGEBRAIC formula.
//------Enter your data points here as shown data x;-4.5 ,-3.2 ,-1.4 ,0.8 ,2.5 ,4.1 data y; 0.7 ,2.3 ,3.8 ,5.0 ,5.5 ,5.6 //-----set m to the power of the polynomial you desire m = 2 //2 for a quadratic 3 for a cubic 4 for a quartic etc. //------------------------------------- //the following code does the work....do not change this unless you know what you are doing //------------------------------------- n = maxdim(x,1) //---calc sum of powers of X dim xp[noparse][[/noparse]2*m+1] mconstant xp,0 for i=0 to 2*m for j=0 to n-1 xp[noparse][[/noparse]i] = xp[noparse][[/noparse]i]+x[noparse][[/noparse]j]^i next next //----calc xy calc sum of powers of X multiplied by Y dim xy[noparse][[/noparse]m+1,1] mconstant xy,0 for i=0 to m for j=0 to n-1 xy[noparse][[/noparse]i,0] = xy[noparse][[/noparse]i,0]+(x[noparse][[/noparse]j]^i)*y[noparse][[/noparse]j] next next //---Calculate matrix dim P[noparse][[/noparse]m+1,m+1] P[noparse][[/noparse]0,0] = n for i = 1 to m P[noparse][[/noparse]0,i] = xp[noparse][[/noparse]i] next for i=1 to m for j=0 to m P[noparse][[/noparse]i,j]=xp[noparse][[/noparse]j+i] next next mInvert P,P1,d //invert the matrix mmultiply P1,xy,a //multiply the inverse by the vector //print out the coefficients of the powers of x....remember x^0 means 1 for i=0 to m print "Coefficient for X^",i," is ",a[noparse][[/noparse]i,0] next·
Sam·
·
Post Edited (SamMishal) : 11/9/2009 8:06:57 PM GMT
That's the same as my suggestion. It will work.
Leon
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Amateur radio callsign: G1HSM
My method is a LEAST-SQUARES method for curve fitting and is not EXACT it is a
numerical analysis approximation.
So you may be better off with the method in the PDF rather than mine.
For higher polynomials though my method works great since a simple
algebraic formula is not EASY to get or even to write down.
Sam
Post Edited (SamMishal) : 11/9/2009 8:17:31 PM GMT
data x;-4.5·· ,-3.2·· ,-1.4··
data y; 0.7·· , 2.3··· , 3.8
The results from the LEAST-Squares method are
······· Coefficient for X^0· is· 4.39230769230769
······· Coefficient for X^1· is· 0.243589743589763
······· Coefficient for X^2· is· -0.128205128205126
I also plugged the same data in the formulas given in the above PDF and here are the results
······· C·· =··· 4.39230769230769
······· B·· =··· 0.243589743589743
········A·· =·· -0.128205128205128
Amazingly close..............I was not even expecting this level of closeness.......so the
Least squares method works GREAT ........but you must of course have ONLY THREE
(x,y) points...........
If you have more than three points the PDF's method will not work as a BEST FIT....
while the LEAST squares will give you a BEST FIT............
Samuel
·
Leon
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Amateur radio callsign: G1HSM
In spin:
In PHP:
Neither of them yield correct results. What am I doing wrong?
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April, 2008: when I discovered the answers to all my micro-computational-botherations!
Some of my objects:
MCP3X0X ADC Driver - Programmable Schmitt inputs, frequency reading, and more!
Simple Propeller-based Database - Making life easier and more readable for all your EEPROM storage needs.
String Manipulation Library - Don't make strings the bane of the Propeller, bend them to your will!
I cut and pasted your PHP code into RobotBASIC. Of course I removed all the $ and ;
And it ran OK....
I added a few lines to test the other points and print the results. The output was
76.0000000000001
20
156
As it should be .....so I do not know why you are·having trouble........
Sam
Post Edited (SamMishal) : 11/10/2009 5:58:42 AM GMT
I did a graphical analyse and then you can see that it does not fit very well for values between the three given points.
The R^2=1 indicates the given points match 100%. But the deviatation for other values is big.
What I have learned about math is: never thrust a result until you have proven it with another method or a lot of experience.
If you add two more points the aproximation gets better.
You get a really good aproximation with a logarithmic function.
And what is really astonishing: I estimated two points inbetween
with the logarithmic aprox-function the three-point-version is closer than the five-point.
I think this is because my estimation set the point as it would be a linear function between each of the two points
The R^2 gives an information how good the aproximation is. A higher R^2-Value means a better aproximation.
The Float32-object provides logarithmic calculations
See attached pictures.
best regards
Stefan
Post Edited (StefanL38) : 11/10/2009 10:12:23 AM GMT
I looked over the Spin code and I am not sure it should EVEN COMPILE
1- You are not using an Obj section and thus all the floating point calls are not going to compile
2- You have the BOLDED variables (see below) but you have not defined them anywhere as global
··· or local variables (A,B,C,x,val)
I did not check the nesting if it is·OK...it gave me a headache
But I am sure you can just scan it and make sure the nesting of all the functions and () are ok.
I suggest you do only a couple of add/multiplies at a time and print out the results (use FullDuplexSerial and PST)
and·test that it is giving the right answer then add more levels and test and so on.
The help I wanted was with the mess of nested float commands. But no matter now (read below).
@StefanL38:
That last graph is what I was looking for all along, I guess those quadratic equations (I think that what they are called) don't cut it. The log-based equation looks like the way I have to go. I tested it out, and it looks very close to the values I was expecting. What program did you use to get those equations, graphs, and values? If I wanted to get new equations with new numbers (I expect I will need it a couple more times soon, and a few more times throughout my life -- that is an understatement), where could I get it? There is a lot less float computations that need to be made which makes it more readable and faster.
Next thing I am looking for is float log function that doesn't use PASM (no need to waste the cog for a 4 float computations every second). Log is the only one that I need that is not in the FloatMath object. Any ideas where I could get that? I guess I could port it, but that seems like a major hassle.
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April, 2008: when I discovered the answers to all my micro-computational-botherations!
Some of my objects:
MCP3X0X ADC Driver - Programmable Schmitt inputs, frequency reading, and more!
Simple Propeller-based Database - Making life easier and more readable for all your EEPROM storage needs.
String Manipulation Library - Don't allow strings to be the bane of the Propeller, bend them to your will!
You are fitting the points to a Logarithmic (natural log) curve
y = a*Ln(x)+b
This is a Line formula similar to· Y = aX+b
So what you need is to fit your points using Linear Least Square Line Fit. But you are not fitting the
(x,y)…..rather you are fitting the (Ln(x), y)
Once you do the fit you will obtain the values of a and b
So now you know a and b and thus the formula y = a*ln(x)+b is defined.
·
Now remember when you do the Regression Analysis you must use Ln(x) and y [noparse][[/noparse]not x,y]
The formulas for the linear regression are
a = (sum(Xi)*sum(Yi) - n*sum(XiYi))/(sum(Xi)^2 - n*sum(Xi^2))
b·= (sum(Xi)*sum(XiYi) - sum(Xi^2)*sum(Yi))/(sum(Xi)^2-n*sum(Xi^2))
Remember the Xi in the above is actually Ln(xi)
So· the process is
1-····· Plug the values of Ln(X) and Y in the formulas above
2-····· Step 1 will give you a and b.
3-····· You now have the curve y = a*Ln(x)+b
·
Here is a program in RobotBASIC that does the above work and ALSO...it plots the data.
Again...just an Engineer's program...no Frills and Whistles.....you can do that yourself if you need to.
//-----enter your data here data x; 23553.0, 4660.0 , 680.0 data y; 20.0 , 76.0 , 156.0 n = maxdim(y,1) //----do sums SumX = 0 \ SumY = 0 \ SumXY = 0 \ SumX2=0 For i = 0 to n-1 SumX = SumX+NLog(x[noparse][[/noparse]i]) SumX2 = SumX2 + NLog(x[noparse][[/noparse]i])^2 SumY = SumY + y[noparse][[/noparse]i] SumXY = SumXY+NLog(x[noparse][[/noparse]i])*y[noparse][[/noparse]i] next a = (SumX*SumY - n*SumXY)/(SumX^2 - n*SumX2) b = (SumX*SumXY - SumX2*SumY)/(SumX^2-n*SumX2) //----print out the data print "a = ",a;"b = ",b print "X";"Y";"Calc Y" print sRepeat("-",50) for i=0 to n-1 data yy;a*nLog(x[noparse][[/noparse]i])+b print x[noparse][[/noparse]i];y[noparse][[/noparse]i];yy[noparse][[/noparse]i] next //-------generate graphs original data but also a smooth claculated curve i = min(x) \ h = (max(x)-min(x))/30 while i <= max(x) data xx;i data yyy;a*nlog(i)+b i = i+h wend data GPSpec;100,130,500,400 data GSpec;100,130,500,400 mGraphPaper GPSpec mPlotXY GSpec,x,y GSpec[noparse][[/noparse]4] = 4 \ GSpec[noparse][[/noparse]5] = 2 mPlotXY GSpec,xx,yyyHere is the printed output from the program....the graph is shown in the Jpg attached
a = -38.4640530675537······ b = 404.993703132938
X······ ·· Y······ Calc Y
23553·· 20····· 17.7757547318452
4660··· 76····· 80.0966656355471
680···· 156···· 154.127579632612
Post Edited (SamMishal) : 11/10/2009 8:37:31 PM GMT
big smile on my face: I already mentioned with which program I did that.
Even bigger smiling: And I bet you already have it.
Almost nobody uses Microsoft Excel or OpenOffice Calc that way
This is simply a X-Y-Diagram created with Excel (OpenOffice Calc will do the same)
with a added trendline with activated option show trendline equation
see attached picture. As I'm from germany I don't have an eglish-speaking Excel handy
but I think the symbol will guide you how to do it
Sam shows how to do it in Robotbasic
best regards
Stefan
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April, 2008: when I discovered the answers to all my micro-computational-botherations!
Some of my objects:
MCP3X0X ADC Driver - Programmable Schmitt inputs, frequency reading, and more!
Simple Propeller-based Database - Making life easier and more readable for all your EEPROM storage needs.
String Manipulation Library - Don't allow strings to be the bane of the Propeller, bend them to your will!
(by the way: to all the helpfile-authors:
screenshots showing the COMPLETE clickway is what is missing in 99% of all help-files
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April, 2008: when I discovered the answers to all my micro-computational-botherations!
Some of my objects:
MCP3X0X ADC Driver - Programmable Schmitt inputs, frequency reading, and more!
Simple Propeller-based Database - Making life easier and more readable for all your EEPROM storage needs.
String Manipulation Library - Don't allow strings to be the bane of the Propeller, bend them to your will!