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Deming regression - eliminate one solution by sign of 2nd derivative at root of 1st - Hans Wurst - 08-04-2019 09:37 PM
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'
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.
RE: Deming regression - eliminate one solution by sign of 2nd derivative at root of 1st - Arno K - 08-06-2019 02:05 PM
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
RE: Deming regression - Aries - 08-06-2019 04:11 PM
Lots of people are on Holiday
Best,
Aries
RE: Deming regression - eliminate one solution by sign of 2nd derivative at root of 1st - StephenG1CMZ - 08-06-2019 05:31 PM
Deleted: Correction: I misread the sum