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Deming regression - eliminate one solution by sign of 2nd derivative at root of 1st
08-04-2019, 09:37 PM (This post was last modified: 08-05-2019 08:40 PM by Hans Wurst.)
Post: #1
Deming regression - eliminate one solution by sign of 2nd derivative at root of 1st
To be more precise, it's only about one special case of Deming regression, the orthogonal R. This is a spin-off from this thread to fokus on the second issue with several problems (for me).

What I did so far: start HP Prime Virtual Calculator, Menu/Calculator/Reset... and hit Yes to confirm 'clear all memory'. Hit CAS.

i) Save the formula "Sum sqared orthogonal distances from observation data to best fit straight line y=mx+b" for later use:
a=sum((y(k)-(m*x(k)+b))^2,k,1,n)/(1+m^2)

ii) Find b by solving da/db=0:
\(\textbf{zero}\left (\frac{\partial a}{\partial b}, b \right )\)
Expected result: \(b=\bar{y}-m*\bar{x}\) with overbar indicating 'arithmetic average'.
HP Prime shows: ["Invalid function x(k) perhaps a missing * for multiplication Error: Bad Argument Value"]
Question: what do I wrong?

iii) Substitute b (found elsewhere) in a:
subst(a,b=((sum(y(k),k,1,n)-m*sum(x(k),k,1,n))/n))
Note: I do not know (yet) if HP Prime "knows" overbar so I spelled out the arithmetic means of x and y.
Result: sum((-m*x(k)-(-m*sum(x(k),k,1,n)+sum(y(k),k,1,n))/n+y(k))^2,k,1,n)/(m^2+1)
Note: I did not check if this is what I expected because next step fails anyway.

iv) As this formula looks awful (or awesome?) I tried to simplify it, inspired by the formulas given in HP-IOC Owner's Handbook, Appendix B, p. 103. Alas, subst() does just the reverse what I'd llike to do. Instead of replacing a single variable by the content of the stored one I want it to find the stored parts of a variable and replace those therms by the letter of the variable. Example: if n*sum(x(k)^2,k,1,n)-sum(x(k),k,1,n)^2 is found in the formula it should be replaced by u.
Question: Is the CAS of HP Prime able to do so? I assume yes, I only could not find it in the manual.
Expected result would be: \(\displaystyle \frac{m^2*u-2*m*w+v}{n*(1+m^2)}\) (found elsewhere). Saved it in a.

v) Find m by da/dm=0:
\(\textbf{simplify}\left (\textbf{zero}\left (\frac{\partial a}{\partial m}, m \right ) \right )\)
Result: [(-u+v+√(u^2-2*u*v+v^2+4*w^2))/(2*w),(-u+v-√(u^2-2*u*v+v^2+4*w^2))/(2*w)] Save it as r (like roots). Another simplification would be nice, like in step iv I do not know how to let {v-u=>p,2*w=>q};

vi) Find sign of 2nd derivative at first root of 1st derivative (positive indicates a minimum).
subst(diff(a,m,2),m=r(1))
Result: have fun fiddling out the sign of this 'conglomeration' Wink I see no chance without a. m. simplifications and eliminate all not changing the sign of the formula.

I hope it is obvious what I miss: the canonical way to define items \(x_i\) and \(y_i\) in sums to avoid "Invalid function x(k)..." error and how to do a "reverse substitute", for example replace all occurences of u-v in a formula by variable p.

TIA
H.

Edit: 100 viewes, 0 replies. No answer is also an answer. I may conclude either i) Prime may do a lot but not what I try, or ii) false, Prime may do it, but nobody knows how, or iii) false too, alas the experts keep the secret.

It is what it is. I quit.
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08-06-2019, 02:05 PM
Post: #2
RE: Deming regression - eliminate one solution by sign of 2nd derivative at root of 1st
The problem is (may be) the use of round brackets for indices, here squared brackets may help.
The other problem: 100 views, no reply. Recently it has become quiet here, so I come here once a week to see what has happened, others perhaps do the same or come less often.
Arno
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08-06-2019, 04:11 PM
Post: #3
RE: Deming regression
Lots of people are on Holiday Wink
Best,

Aries Smile
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08-06-2019, 05:31 PM (This post was last modified: 08-06-2019 08:30 PM by StephenG1CMZ.)
Post: #4
RE: Deming regression - eliminate one solution by sign of 2nd derivative at root of 1st
Deleted: Correction: I misread the sum

Stephen Lewkowicz (G1CMZ)
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