| Re: Finding coefficients in multiple nonlinear equations using MPFIT [message #71070] |
Fri, 28 May 2010 23:33  |
Craig Markwardt
Messages: 1869
Registered: November 1996
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Senior Member |
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Greetings--
On May 29, 1:54 am, Junum <junshi...@gmail.com> wrote:
> Hi all,
>
> I got a problem that I have spent several days, but did not solve.
> I have two nonlinear equations.
> Equation 1 has 6 coefficients (c1 through c6) and Eq. 2 has 5
> equations (d1 through d5).
> Eq. 1 : c1 * A * B^c2 + c3*C^c4 + c5*exp(D^c6) = 0
> Eq. 2 : d1*(A/B)^d2 + d3*(1/C)^d4 = 0
> Actually, original equations are much complex.
>
> From lab experiments I have data sets of A, B, C, and D, more than
> 40000.
> My goal is to find optimum coefficients using these large data sets.
> I've tried to use MPFIT, but it seems that MPFIT has limit to define a
> function.
> What I did is
>
> expr = '[ '
> st = 31
> en = 44
> FOR j=st,en DO BEGIN
> s_L = STRING(FORMAT='( D9.3)',L[j]) + 'D'
> s_Lapp = STRING(FORMAT='( D9.3)',Lapp[j]) + 'D'
> s_Z = STRING(FORMAT='(D11.6)',Z[j]) + 'D'
> s_foc = STRING(FORMAT='( D8.3)',foc[j]) + 'D'
> s_rnd = STRING(FORMAT='( D8.3)',round[j]) + 'D'
> s_bg = STRING(FORMAT='(D11.5)',bg[j]) + 'D'
>
> exp1 = '( -1.*' + s_foc + ' )'
> exp2 = '( (ABS(' + s_L + '/' + s_rnd + '))^P[5] )'
> exp3 = '(' + s_bg + ')'
> exp4 = '( -1.*ABS(P[1]*(' + exp3 + '^P[3] )) )'
> exp5 = '( ABS(' + s_Z + ' + ABS(' + s_Z + ')*P[2] )^P[4] )'
> exp6 = '( EXP(' + exp4 + ' * ' + exp5 + ') )'
>
> expr1 = exp1 + ' + ' + exp2 + ' * ' + exp3 + ' * ' +
> exp6 + ' * P[0]'
>
> exp7 = '( -1.*(' + s_Lapp + '/' + s_L + ') )'
> exp8 = '(' + s_Z + '/((' + s_L + '*'+ s_rnd + ')^(2.)) )'
> exp9 = '(ABS(' + exp8 + '))'
> exp10 = '( ' + exp8 + ' * P[7] )'
> exp11 = '( (' + exp9 + ' + ' + exp10 + ')^P[8] )'
> exp12 = '( ((' + s_L + ')^P[10])*P[9] )'
> exp13 = '( ' + exp11 + ' * ' + exp3 + ' )'
>
> expr2 = exp7 + ' + ' + exp13 + ' * P[6] + ' + exp12
>
> IF ( j eq st ) THEN BEGIN
> expr = expr + expr1 + ', ' + expr2
> ENDIF ELSE BEGIN
> expr = expr + ', ' + expr1 + ', ' + expr2
> ENDELSE
>
> ENDFOR
>
> expr = expr + ' ]'
>
> fit_status = 0
> num_iter = 0
> resid = 0.D
>
> st_gs = [-10., -10., -10., -10., -10., -10., -10., -10., -10.,
> -10., -10. ]
> P = mpfitexpr(expr, 1, 1, 0.1, st_gs, MAXITER=900)
>
> 14 (so 28 elements on defined function) is ok, but 15 (so 30 elements
> on defined function) does not work.
>
> How can I solve this problem?
> Can any one help me?
You don't say how the code "does not work" so it is difficult to
comment. The example is not complete, so I can't test it.
I think you need to define better what you are trying to do. I guess
that you are trying to solve your two equations in a least squares
sense using all the data jointly, but that is not clear.
Your equation has 11 coefficients, so I don't understand why 14 or 15
are important. If I misinterpreted your intent, and you want to solve
each equation separately, then you can just run MPFIT*() 40000 times
and achieve your goal. I don't know why you need to join them
together into an expression. I doubt any IDL program will be able to
solve an equation with 40000 parameters, if that is really what you
are attempting.
MPFITEXPR() is not the ideal solution for your problem; it is better
for quick jobs. MPFITEXPR has limitations because it needs to parse
the user expression many times. IDL cannot parse or evaluate
infinitely long expressions. For solving equations, MPFIT() is the
desirable method, especially since you have already expressed your
equations as (function) = 0. That is exactly what MPFIT() solves.
If my initial suspicion was right, it should be as simple as something
like this (untested),
function myfunc, p, a=a, b=b, c=c, d=d
return, [p[0]* a * b^p[1] + p[2]*c^p[3] + p[4]*exp(d^p[5]), $
p[6] * (a/b)^p[7] + p[8]*(1/c)^p[9] ]
end
The [ ... ] notation will glue the two residual arrays together.
MPFIT() will try to solve for parameters which drive all equations to
zero.
p0 = [ ... initial guess of parameters ... ]
p = mpfit('myfunc', p0, functargs={a:a, b:b, c:c, d:d})
Your equations look a little troublesome. You may have to worry about
whether they are linearly independent, and whether certain parameters
will be strongly correlated. You also don't define whether the
equations should be weighted equally or not. The above formulation
gives equal weight.
Craig
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| Re: Finding coefficients in multiple nonlinear equations using MPFIT [message #71167 is a reply to message #71070] |
Sat, 29 May 2010 12:32  |
Junum
Messages: 13
Registered: May 2010
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Junior Member |
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On May 29, 1:33 am, Craig Markwardt <craig.markwa...@gmail.com> wrote:
> Greetings--
>
> On May 29, 1:54 am, Junum <junshi...@gmail.com> wrote:
>
>
>
>
>
>> Hi all,
>
>> I got a problem that I have spent several days, but did not solve.
>> I have two nonlinear equations.
>> Equation 1 has 6 coefficients (c1 through c6) and Eq. 2 has 5
>> equations (d1 through d5).
>> Eq. 1 : c1 * A * B^c2 + c3*C^c4 + c5*exp(D^c6) = 0
>> Eq. 2 : d1*(A/B)^d2 + d3*(1/C)^d4 = 0
>> Actually, original equations are much complex.
>
>> From lab experiments I have data sets of A, B, C, and D, more than
>> 40000.
>> My goal is to find optimum coefficients using these large data sets.
>> I've tried to use MPFIT, but it seems that MPFIT has limit to define a
>> function.
>> What I did is
>
>> expr = '[ '
>> st = 31
>> en = 44
>> FOR j=st,en DO BEGIN
>> s_L = STRING(FORMAT='( D9.3)',L[j]) + 'D'
>> s_Lapp = STRING(FORMAT='( D9.3)',Lapp[j]) + 'D'
>> s_Z = STRING(FORMAT='(D11.6)',Z[j]) + 'D'
>> s_foc = STRING(FORMAT='( D8.3)',foc[j]) + 'D'
>> s_rnd = STRING(FORMAT='( D8.3)',round[j]) + 'D'
>> s_bg = STRING(FORMAT='(D11.5)',bg[j]) + 'D'
>
>> exp1 = '( -1.*' + s_foc + ' )'
>> exp2 = '( (ABS(' + s_L + '/' + s_rnd + '))^P[5] )'
>> exp3 = '(' + s_bg + ')'
>> exp4 = '( -1.*ABS(P[1]*(' + exp3 + '^P[3] )) )'
>> exp5 = '( ABS(' + s_Z + ' + ABS(' + s_Z + ')*P[2] )^P[4] )'
>> exp6 = '( EXP(' + exp4 + ' * ' + exp5 + ') )'
>
>> expr1 = exp1 + ' + ' + exp2 + ' * ' + exp3 + ' * ' +
>> exp6 + ' * P[0]'
>
>> exp7 = '( -1.*(' + s_Lapp + '/' + s_L + ') )'
>> exp8 = '(' + s_Z + '/((' + s_L + '*'+ s_rnd + ')^(2.)) )'
>> exp9 = '(ABS(' + exp8 + '))'
>> exp10 = '( ' + exp8 + ' * P[7] )'
>> exp11 = '( (' + exp9 + ' + ' + exp10 + ')^P[8] )'
>> exp12 = '( ((' + s_L + ')^P[10])*P[9] )'
>> exp13 = '( ' + exp11 + ' * ' + exp3 + ' )'
>
>> expr2 = exp7 + ' + ' + exp13 + ' * P[6] + ' + exp12
>
>> IF ( j eq st ) THEN BEGIN
>> expr = expr + expr1 + ', ' + expr2
>> ENDIF ELSE BEGIN
>> expr = expr + ', ' + expr1 + ', ' + expr2
>> ENDELSE
>
>> ENDFOR
>
>> expr = expr + ' ]'
>
>> fit_status = 0
>> num_iter = 0
>> resid = 0.D
>
>> st_gs = [-10., -10., -10., -10., -10., -10., -10., -10., -10.,
>> -10., -10. ]
>> P = mpfitexpr(expr, 1, 1, 0.1, st_gs, MAXITER=900)
>
>> 14 (so 28 elements on defined function) is ok, but 15 (so 30 elements
>> on defined function) does not work.
>
>> How can I solve this problem?
>> Can any one help me?
>
> You don't say how the code "does not work" so it is difficult to
> comment. The example is not complete, so I can't test it.
>
> I think you need to define better what you are trying to do. I guess
> that you are trying to solve your two equations in a least squares
> sense using all the data jointly, but that is not clear.
>
> Your equation has 11 coefficients, so I don't understand why 14 or 15
> are important. If I misinterpreted your intent, and you want to solve
> each equation separately, then you can just run MPFIT*() 40000 times
> and achieve your goal. I don't know why you need to join them
> together into an expression. I doubt any IDL program will be able to
> solve an equation with 40000 parameters, if that is really what you
> are attempting.
>
> MPFITEXPR() is not the ideal solution for your problem; it is better
> for quick jobs. MPFITEXPR has limitations because it needs to parse
> the user expression many times. IDL cannot parse or evaluate
> infinitely long expressions. For solving equations, MPFIT() is the
> desirable method, especially since you have already expressed your
> equations as (function) = 0. That is exactly what MPFIT() solves.
>
> If my initial suspicion was right, it should be as simple as something
> like this (untested),
> function myfunc, p, a=a, b=b, c=c, d=d
> return, [p[0]* a * b^p[1] + p[2]*c^p[3] + p[4]*exp(d^p[5]), $
> p[6] * (a/b)^p[7] + p[8]*(1/c)^p[9] ]
> end
> The [ ... ] notation will glue the two residual arrays together.
> MPFIT() will try to solve for parameters which drive all equations to
> zero.
>
> p0 = [ ... initial guess of parameters ... ]
> p = mpfit('myfunc', p0, functargs={a:a, b:b, c:c, d:d})
>
> Your equations look a little troublesome. You may have to worry about
> whether they are linearly independent, and whether certain parameters
> will be strongly correlated. You also don't define whether the
> equations should be weighted equally or not. The above formulation
> gives equal weight.
>
> Craig
Craig, thank you very much.
Let me explain my problem.
I have two equations, f, g.
These are function of 4 parameters (A, B, C, and D), so f(A,B,C,D) and
g(A,B,C,D).
I can guess how these equations should be.
For example,
Eq. 1 : c1 * A * B^c2 + c3*C^c4 + c5*exp(D^c6) = 0
Eq. 2 : d1*(A/B)^d2 + d3*(1/C)^d4 = 0
Here, I don't know constants (c1 through c6 and d1 through d4).
From lab experiments, I have measurements of A, B, C, and D.
Therefore, my goal is to find optimum coefficients (c1 through c6 and
d1 through d4) using measurements of A, B, C, and D.
I have approximately 40000 datasets of A, B, C, D and want to find
coefficients that satisfy eqs. 1 and 2 with minimum residual error.
To make a problem simple, I may combine two equations together, so it
becomes c1 * A * B^c2 + c3*C^c4 + c5*exp(D^c6) + d1*(A/B)^d2 + d3*(1/
C)^d4 = 0.
This is one equation with 11 unknowns (i.e., coefficient).
Theoretically, I need 10 more equation to solve them.
But, I have 40000 equations and want to find optimum coefficients that
satisfy 40000 equations with some residuals.
It is clear that there will be exact solution (coefficient) that
satisfy 40000 equations.
In summary, my goal is to find coefficients that make minimum residual
error across 40000 equations.
I need this because we need to find empirical relationship based on
experiment and more data give us more higher confidence level.
"Does not work" means that I got "% MPFITEXPR: check syntax and
parameter usage" message when I add more expressions on "my
function".
Since my function ('expr' in original posting) is defined as a string,
I think that this is problem of length of string, as you indicated.
Does your suggestion associated with MPFIT solve equations 40000
times?
Do you have any suggestion?
Please let me know any question or comment.
I really thank for your help.
Thank you very much again.
Jun
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