/usr/share/octave/packages/optim-1.3.0/doc-cache is in octave-optim 1.3.0-1.
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# name: cache
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LinearRegression
# name: <cell-element>
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-- Function File: [P,E_VAR,R,P_VAR,Y_VAR] = LinearRegression (F,Y)
-- Function File: [P,E_VAR,R,P_VAR,Y_VAR] = LinearRegression (F,Y,W)
general linear regression
determine the parameters p_j (j=1,2,...,m) such that the function
f(x) = sum_(i=1,...,m) p_j*f_j(x) is the best fit to the given
values y_i = f(x_i)
parameters:
* F is an n*m matrix with the values of the basis functions at
the support points. In column j give the values of f_j at the
points x_i (i=1,2,...,n)
* Y is a column vector of length n with the given values
* W is n column vector of of length n vector with the weights of
data points
return values:
* P is the vector of length m with the estimated values of the
parameters
* E_VAR is the estimated variance of the difference between
fitted and measured values
* R is the weighted norm of the residual
* P_VAR is the estimated variance of the parameters p_j
* Y_VAR is the estimated variance of the dependend variables
Caution: do NOT request Y_VAR for large data sets, as a n by n
matrix is generated
See also: regress,leasqr,nonlin_curvefit,polyfit,wpolyfit,expfit.
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general linear regression
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# elements: 1
# length: 6
adsmax
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# length: 1909
ADSMAX Alternating directions method for direct search optimization.
[x, fmax, nf] = ADSMAX(FUN, x0, STOPIT, SAVIT, P) attempts to
maximize the function FUN, using the starting vector x0.
The alternating directions direct search method is used.
Output arguments:
x = vector yielding largest function value found,
fmax = function value at x,
nf = number of function evaluations.
The iteration is terminated when either
- the relative increase in function value between successive
iterations is <= STOPIT(1) (default 1e-3),
- STOPIT(2) function evaluations have been performed
(default inf, i.e., no limit), or
- a function value equals or exceeds STOPIT(3)
(default inf, i.e., no test on function values).
Progress of the iteration is not shown if STOPIT(5) = 0 (default 1).
If a non-empty fourth parameter string SAVIT is present, then
`SAVE SAVIT x fmax nf' is executed after each inner iteration.
By default, the search directions are the co-ordinate directions.
The columns of a fifth parameter matrix P specify alternative search
directions (P = EYE is the default).
NB: x0 can be a matrix. In the output argument, in SAVIT saves,
and in function calls, x has the same shape as x0.
ADSMAX(fun, x0, STOPIT, SAVIT, P, P1, P2,...) allows additional
arguments to be passed to fun, via feval(fun,x,P1,P2,...).
Reference:
N. J. Higham, Optimization by direct search in matrix computations,
SIAM J. Matrix Anal. Appl, 14(2): 317-333, 1993.
N. J. Higham, Accuracy and Stability of Numerical Algorithms,
Second edition, Society for Industrial and Applied Mathematics,
Philadelphia, PA, 2002; sec. 20.5.
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ADSMAX Alternating directions method for direct search optimization.
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 7
battery
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# type: sq_string
# elements: 1
# length: 474
battery.m: repeatedly call bfgs using a battery of
start values, to attempt to find global min
of a nonconvex function
INPUTS:
func: function to mimimize
args: args of function
minarg: argument to minimize w.r.t. (usually = 1)
startvals: kxp matrix of values to try for sure (don't include all zeros, that's automatic)
max iters per start value
number of additional random start values to try
OUTPUT: theta - the best value found - NOT iterated to convergence
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battery.m: repeatedly call bfgs using a battery of
start values, to attempt t
# name: <cell-element>
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# length: 7
bfgsmin
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bfgsmin: bfgs or limited memory bfgs minimization of function
Usage: [x, obj_value, convergence, iters] = bfgsmin(f, args, control)
The function must be of the form
[value, return_2,..., return_m] = f(arg_1, arg_2,..., arg_n)
By default, minimization is w.r.t. arg_1, but it can be done
w.r.t. any argument that is a vector. Numeric derivatives are
used unless analytic derivatives are supplied. See bfgsmin_example.m
for methods.
Arguments:
* f: name of function to minimize (string)
* args: a cell array that holds all arguments of the function
The argument with respect to which minimization is done
MUST be a vector
* control: an optional cell array of 1-8 elements. If a cell
array shorter than 8 elements is provided, the trailing elements
are provided with default values.
* elem 1: maximum iterations (positive integer, or -1 or Inf for unlimited (default))
* elem 2: verbosity
0 = no screen output (default)
1 = only final results
2 = summary every iteration
3 = detailed information
* elem 3: convergence criterion
1 = strict (function, gradient and param change) (default)
0 = weak - only function convergence required
* elem 4: arg in f_args with respect to which minimization is done (default is first)
* elem 5: (optional) Memory limit for lbfgs. If it's a positive integer
then lbfgs will be use. Otherwise ordinary bfgs is used
* elem 6: function change tolerance, default 1e-12
* elem 7: parameter change tolerance, default 1e-6
* elem 8: gradient tolerance, default 1e-5
Returns:
* x: the minimizer
* obj_value: the value of f() at x
* convergence: 1 if normal conv, other values if not
* iters: number of iterations performed
Example: see bfgsmin_example.m
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bfgsmin: bfgs or limited memory bfgs minimization of function
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bfgsmin_example
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initial values
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initial values
# name: <cell-element>
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brent_line_min
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-- Function File: [S,V,N] brent_line_min ( F,DF,ARGS,CTL )
Line minimization of f along df
Finds minimum of f on line x0 + dx*w | a < w < b by bracketing. a
and b are passed through argument ctl.
Arguments
--------------
* F : string : Name of function. Must return a real value
* ARGS : cell : Arguments passed to f or RxC : f's only
argument. x0 must be at ARGS{ CTL(2) }
* CTL : 5 : (optional) Control variables, described below.
Returned values
--------------------
* S : 1 : Minimum is at x0 + s*dx
* V : 1 : Value of f at x0 + s*dx
* NEV : 1 : Number of function evaluations
Control Variables
----------------------
* CTL(1) : Upper bound for error on s Default=sqrt(eps)
* CTL(2) : Position of minimized argument in args Default= 1
* CTL(3) : Maximum number of function evaluations Default= inf
* CTL(4) : a Default=-inf
* CTL(5) : b Default= inf
Default values will be used if ctl is not passed or if nan values
are given.
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Line minimization of f along df
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cauchy
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-- Function File: cauchy (N, R, X, F )
Return the Taylor coefficients and numerical differentiation of a
function F for the first N-1 coefficients or derivatives using the
fft. N is the number of points to evaluate, R is the radius of
convergence, needs to be chosen less then the smallest singularity,
X is point to evaluate the Taylor expansion or differentiation.
For example,
If X is a scalar, the function F is evaluated in a row vector of
length N. If X is a column vector, F is evaluated in a matrix of
length(x)-by-N elements and must return a matrix of the same size.
d = cauchy(16, 1.5, 0, @(x) exp(x));
=> d(2) = 1.0000 # first (2-1) derivative of function f (index starts from zero)
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Return the Taylor coefficients and numerical differentiation of a
function F for
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cdiff
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c = cdiff (func,wrt,N,dfunc,stack,dx) - Code for num. differentiation
= "function df = dfunc (var1,..,dvar,..,varN) .. endfunction
Returns a string of octave code that defines a function 'dfunc' that
returns the derivative of 'func' with respect to it's 'wrt'th
argument.
The derivatives are obtained by symmetric finite difference.
dfunc()'s return value is in the same format as that of ndiff()
func : string : name of the function to differentiate
wrt : int : position, in argument list, of the differentiation
variable. Default:1
N : int : total number of arguments taken by 'func'.
If N=inf, dfunc will take variable argument list.
Default:wrt
dfunc : string : Name of the octave function that returns the
derivatives. Default:['d',func]
stack : string : Indicates whether 'func' accepts vertically
(stack="rstack") or horizontally (stack="cstack")
arguments. Any other string indicates that 'func'
does not allow stacking. Default:''
dx : real : Step used in the symmetric difference scheme.
Default:10*sqrt(eps)
See also : ndiff, eval, todisk
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c = cdiff (func,wrt,N,dfunc,stack,dx) - Code for num.
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cg_min
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-- Function File: [X0,V,NEV] cg_min ( F,DF,ARGS,CTL )
NonLinear Conjugate Gradient method to minimize function F.
Arguments
--------------
* F : string : Name of function. Return a real value
* DF : string : Name of f's derivative. Returns a (R*C) x 1
vector
* ARGS: cell : Arguments passed to f.
* CTL : 5-vec : (Optional) Control variables, described below
Returned values
--------------------
* X0 : matrix : Local minimum of f
* V : real : Value of f in x0
* NEV : 1 x 2 : Number of evaluations of f and of df
Control Variables
----------------------
* CTL(1) : 1 or 2 : Select stopping criterion amongst :
* CTL(1)==0 : Default value
* CTL(1)==1 : Stopping criterion : Stop search when value
doesn't improve, as tested by ctl(2) > Deltaf/max(|f(x)|,1)
where Deltaf is the decrease in f observed in the last
iteration (each iteration consists R*C line searches).
* CTL(1)==2 : Stopping criterion : Stop search when updates are
small, as tested by ctl(2) > max { dx(i)/max(|x(i)|,1) | i in
1..N } where dx is the change in the x that occured in the
last iteration.
* CTL(2) : Threshold used in stopping tests. Default=10*eps
* CTL(2)==0 : Default value
* CTL(3) : Position of the minimized argument in args Default=1
* CTL(3)==0 : Default value
* CTL(4) : Maximum number of function evaluations Default=inf
* CTL(4)==0 : Default value
* CTL(5) : Type of optimization:
* CTL(5)==1 : "Fletcher-Reves" method
* CTL(5)==2 : "Polak-Ribiere" (Default)
* CTL(5)==3 : "Hestenes-Stiefel" method
CTL may have length smaller than 4. Default values will be used if
ctl is not passed or if nan values are given.
Example:
-------------
function r=df( l ) b=[1;0;-1]; r = -( 2*l{1} - 2*b +
rand(size(l{1}))); endfunction
function r=ff( l ) b=[1;0;-1]; r = (l{1}-b)' * (l{1}-b);
endfunction
ll = { [10; 2; 3] };
ctl(5) = 3;
[x0,v,nev]=cg_min( "ff", "df", ll, ctl )
Comment: In general, BFGS method seems to be better performin in
many cases but requires more computation per iteration
See also: bfgsmin,
http://en.wikipedia.org/wiki/Nonlinear_conjugate_gradient .
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NonLinear Conjugate Gradient method to minimize function F.
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cpiv_bard
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[lb, idx, ridx, mv] = cpiv_bard (v, m[, incl])
v: column vector; m: matrix; incl (optional): index. length (v)
must equal rows (m). Finds column vectors w and l with w == v + m *
l, w >= 0, l >= 0, l.' * w == 0. Chooses idx, w, and l so that
l(~idx) == 0, l(idx) == -inv (m(idx, idx)) * v(idx), w(idx) roughly
== 0, and w(~idx) == v(~idx) + m(idx, ~idx).' * l(idx). idx indexes
at least everything indexed by incl, but l(incl) may be < 0. lb:
l(idx) (column vector); idx: logical index, defined above; ridx:
~idx & w roughly == 0; mv: [m, v] after performing a Gauss-Jordan
'sweep' (with gjp.m) on each diagonal element indexed by idx.
Except the handling of incl (which enables handling of equality
constraints in the calling code), this is called solving the
'complementary pivot problem' (Cottle, R. W. and Dantzig, G. B.,
'Complementary pivot theory of mathematical programming', Linear
Algebra and Appl. 1, 102--125. References for the current
algorithm: Bard, Y.: Nonlinear Parameter Estimation, p. 147--149,
Academic Press, New York and London 1974; Bard, Y., 'An eclectic
approach to nonlinear programming', Proc. ANU Sem. Optimization,
Canberra, Austral. Nat. Univ.).
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[lb, idx, ridx, mv] = cpiv_bard (v, m[, incl])
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curvefit_stat
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-- Function File: INFO = curvefit_stat (F, P, X, Y, SETTINGS)
Frontend for computation of statistics for fitting of values,
computed by a model function, to observed values.
Please refer to the description of 'residmin_stat'. The only
differences to 'residmin_stat' are the additional arguments X
(independent values) and Y (observations), that the model function
F, if provided, has a second obligatory argument which will be set
to X and is supposed to return guesses for the observations (with
the same dimensions), and that the possibly user-supplied function
for the jacobian of the model function has also a second obligatory
argument which will be set to X.
See also: residmin_stat.
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Frontend for computation of statistics for fitting of values, computed
by a mode
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d2_min
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[x,v,nev,h,args] = d2_min(f,d2f,args,ctl,code) - Newton-like minimization
Minimize f(x) using 1st and 2nd derivatives. Any function w/ second
derivatives can be minimized, as in Newton. f(x) decreases at each
iteration, as in Levenberg-Marquardt. This function is inspired from the
Levenberg-Marquardt algorithm found in the book "Numerical Recipes".
ARGUMENTS :
f : string : Cost function's name
d2f : string : Name of function returning the cost (1x1), its
differential (1xN) and its second differential or it's
pseudo-inverse (NxN) (see ctl(5) below) :
[v,dv,d2v] = d2f (x).
args : list : f and d2f's arguments. By default, minimize the 1st
or matrix : argument.
ctl : vector : Control arguments (see below)
or struct
code : string : code will be evaluated after each outer loop that
produced some (any) improvement. Variables visible from
"code" include "x", the best parameter found, "v" the
best value and "args", the list of all arguments. All can
be modified. This option can be used to re-parameterize
the argument space during optimization
CONTROL VARIABLE ctl : (optional). May be a struct or a vector of length
---------------------- 5 or less where NaNs are ignored. Default values
are written <value>.
FIELD VECTOR
NAME POS
ftol, f N/A : Stop search when value doesn't improve, as tested by
f > Deltaf/max(|f(x)|,1)
where Deltaf is the decrease in f observed in the last
iteration. <10*sqrt(eps)>
utol, u N/A : Stop search when updates are small, as tested by
u > max { dx(i)/max(|x(i)|,1) | i in 1..N }
where dx is the change in the x that occured in the last
iteration. <NaN>
dtol, d N/A : Stop search when derivative is small, as tested by
d > norm (dv) <eps>
crit, c ctl(1) : Set one stopping criterion, 'ftol' (c=1), 'utol' (c=2)
or 'dtol' (c=3) to the value of by the 'tol' option. <1>
tol, t ctl(2) : Threshold in termination test chosen by 'crit' <10*eps>
narg, n ctl(3) : Position of the minimized argument in args <1>
maxev,m ctl(4) : Maximum number of function evaluations <inf>
maxout,m : Maximum number of outer loops <inf>
id2f, i ctl(5) : 0 if d2f returns the 2nd derivatives, 1 if <0>
it returns its pseudo-inverse.
verbose, v N/A : Be more or less verbose (quiet=0) <0>
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[x,v,nev,h,args] = d2_min(f,d2f,args,ctl,code) - Newton-like minimization
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dcdp
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function prt = dcdp (f, p, dp, func[, bounds])
This is an interface to __dfdp__.m, similar to dfdp.m, but for
functions only of parameters 'p', not of independents 'x'. See
dfdp.m.
dfpdp is more general and is meant to be used instead of dcdp in
optimization.
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function prt = dcdp (f, p, dp, func[, bounds])
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de_min
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de_min: global optimisation using differential evolution
Usage: [x, obj_value, nfeval, convergence] = de_min(fcn, control)
minimization of a user-supplied function with respect to x(1:D),
using the differential evolution (DE) method based on an algorithm
by Rainer Storn (http://www.icsi.berkeley.edu/~storn/code.html)
See: http://www.softcomputing.net/tevc2009_1.pdf
Arguments:
---------------
fcn string : Name of function. Must return a real value
control vector : (Optional) Control variables, described below
or struct
Returned values:
----------------
x vector : parameter vector of best solution
obj_value scalar : objective function value of best solution
nfeval scalar : number of function evaluations
convergence : 1 = best below value to reach (VTR)
0 = population has reached defined quality (tol)
-1 = some values are close to constraints/boundaries
-2 = max number of iterations reached (maxiter)
-3 = max number of functions evaluations reached (maxnfe)
Control variable: (optional) may be named arguments (i.e. "name",value
---------------- pairs), a struct, or a vector, where
NaN's are ignored.
XVmin : vector of lower bounds of initial population
*** note: by default these are no constraints ***
XVmax : vector of upper bounds of initial population
constr : 1 -> enforce the bounds not just for the initial population
const : data vector (remains fixed during the minimization)
NP : number of population members
F : difference factor from interval [0, 2]
CR : crossover probability constant from interval [0, 1]
strategy : 1 --> DE/best/1/exp 7 --> DE/best/1/bin
2 --> DE/rand/1/exp 8 --> DE/rand/1/bin
3 --> DE/target-to-best/1/exp 9 --> DE/target-to-best/1/bin
4 --> DE/best/2/exp 10--> DE/best/2/bin
5 --> DE/rand/2/exp 11--> DE/rand/2/bin
6 --> DEGL/SAW/exp else DEGL/SAW/bin
refresh : intermediate output will be produced after "refresh"
iterations. No intermediate output will be produced
if refresh is < 1
VTR : Stopping criterion: "Value To Reach"
de_min will stop when obj_value <= VTR.
Use this if you know which value you expect.
tol : Stopping criterion: "tolerance"
stops if (best-worst)/max(1,worst) < tol
This stops basically if the whole population is "good".
maxnfe : maximum number of function evaluations
maxiter : maximum number of iterations (generations)
The algorithm seems to work well only if [XVmin,XVmax] covers the
region where the global minimum is expected.
DE is also somewhat sensitive to the choice of the
difference factor F. A good initial guess is to choose F from
interval [0.5, 1], e.g. 0.8.
CR, the crossover probability constant from interval [0, 1]
helps to maintain the diversity of the population and is
rather uncritical but affects strongly the convergence speed.
If the parameters are correlated, high values of CR work better.
The reverse is true for no correlation.
Experiments suggest that /bin likes to have a slightly
larger CR than /exp.
The number of population members NP is also not very critical. A
good initial guess is 10*D. Depending on the difficulty of the
problem NP can be lower than 10*D or must be higher than 10*D
to achieve convergence.
Default Values:
---------------
XVmin = [-2];
XVmax = [ 2];
constr= 0;
const = [];
NP = 10 *D
F = 0.8;
CR = 0.9;
strategy = 12;
refresh = 0;
VTR = -Inf;
tol = 1.e-3;
maxnfe = 1e6;
maxiter = 1000;
Example to find the minimum of the Rosenbrock saddle:
----------------------------------------------------
Define f as:
function result = f(x);
result = 100 * (x(2) - x(1)^2)^2 + (1 - x(1))^2;
end
Then type:
ctl.XVmin = [-2 -2];
ctl.XVmax = [ 2 2];
[x, obj_value, nfeval, convergence] = de_min (@f, ctl);
Keywords: global-optimisation optimisation minimisation
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de_min: global optimisation using differential evolution
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deriv
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-- Function File: DX = deriv (F, X0)
-- Function File: DX = deriv (F, X0, H)
-- Function File: DX = deriv (F, X0, H, O)
-- Function File: DX = deriv (F, X0, H, O, N)
Calculate derivate of function F.
F must be a function handle or the name of a function that takes X0
and returns a variable of equal length and orientation. X0 must be
a numeric vector or scalar.
H defines the step taken for the derivative calculation. Defaults
to 1e-7.
O defines the order of the calculation. Supported values are 2
(h^2 order) or 4 (h^4 order). Defaults to 2.
N defines the derivative order. Defaults to the 1st derivative of
the function. Can be up to the 4th derivative.
Reference: Numerical Methods for Mathematics, Science, and
Engineering by John H. Mathews.
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Calculate derivate of function F.
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dfdp
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function prt = dfdp (x, f, p, dp, func[, bounds])
numerical partial derivatives (Jacobian) df/dp for use with leasqr
--------INPUT VARIABLES---------
x=vec or matrix of indep var(used as arg to func) x=[x0 x1 ....]
f=func(x,p) vector initialsed by user before each call to dfdp
p= vec of current parameter values
dp= fractional increment of p for numerical derivatives
dp(j)>0 central differences calculated
dp(j)<0 one sided differences calculated
dp(j)=0 sets corresponding partials to zero; i.e. holds p(j) fixed
func=function (string or handle) to calculate the Jacobian for,
e.g. to calc Jacobian for function expsum prt=dfdp(x,f,p,dp,'expsum')
bounds=two-column-matrix of lower and upper bounds for parameters
If no 'bounds' options is specified to leasqr, it will call
dfdp without the 'bounds' argument.
----------OUTPUT VARIABLES-------
prt= Jacobian Matrix prt(i,j)=df(i)/dp(j)
================================
dfxpdp is more general and is meant to be used instead of dfdp in
optimization.
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function prt = dfdp (x, f, p, dp, func[, bounds])
numerical partial derivative
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dfpdp
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function jac = dfpdp (p, func[, hook])
Returns Jacobian of func (p) with respect to p with finite
differencing. The optional argument hook is a structure which can
contain the following fields at the moment:
hook.f: value of func(p) for p as given in the arguments
hook.diffp: positive vector of fractional steps from given p in
finite differencing (actual steps may be smaller if bounds are
given). The default is .001 * ones (size (p)).
hook.diff_onesided: logical vector, indexing elements of p for
which only one-sided differences should be computed (faster); even
if not one-sided, differences might not be exactly central if
bounds are given. The default is false (size (p)).
hook.fixed: logical vector, indexing elements of p for which zero
should be returned instead of the guessed partial derivatives
(useful in optimization if some parameters are not optimized, but
are 'fixed').
hook.lbound, hook.ubound: vectors of lower and upper parameter
bounds (or -Inf or +Inf, respectively) to be respected in finite
differencing. The consistency of bounds is not checked.
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function jac = dfpdp (p, func[, hook])
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dfxpdp
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function jac = dfxpdp (x, p, func[, hook])
Returns Jacobian of func (p, x) with respect to p with finite
differencing. The optional argument hook is a structure which can
contain the following fields at the moment:
hook.f: value of func(p, x) for p and x as given in the arguments
hook.diffp: positive vector of fractional steps from given p in
finite differencing (actual steps may be smaller if bounds are
given). The default is .001 * ones (size (p));
hook.diff_onesided: logical vector, indexing elements of p for
which only one-sided differences should be computed (faster); even
if not one-sided, differences might not be exactly central if
bounds are given. The default is false (size (p)).
hook.fixed: logical vector, indexing elements of p for which zero
should be returned instead of the guessed partial derivatives
(useful in optimization if some parameters are not optimized, but
are 'fixed').
hook.lbound, hook.ubound: vectors of lower and upper parameter
bounds (or -Inf or +Inf, respectively) to be respected in finite
differencing. The consistency of bounds is not checked.
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function jac = dfxpdp (x, p, func[, hook])
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expfit
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USAGE [alpha,c,rms] = expfit( deg, x1, h, y )
Prony's method for non-linear exponential fitting
Fit function: \sum_1^{deg} c(i)*exp(alpha(i)*x)
Elements of data vector y must correspond to
equidistant x-values starting at x1 with stepsize h
The method is fully compatible with complex linear
coefficients c, complex nonlinear coefficients alpha
and complex input arguments y, x1, non-zero h .
Fit-order deg must be a real positive integer.
Returns linear coefficients c, nonlinear coefficients
alpha and root mean square error rms. This method is
known to be more stable than 'brute-force' non-linear
least squares fitting.
Example
x0 = 0; step = 0.05; xend = 5; x = x0:step:xend;
y = 2*exp(1.3*x)-0.5*exp(2*x);
error = (rand(1,length(y))-0.5)*1e-4;
[alpha,c,rms] = expfit(2,x0,step,y+error)
alpha =
2.0000
1.3000
c =
-0.50000
2.00000
rms = 0.00028461
The fit is very sensitive to the number of data points.
It doesn't perform very well for small data sets.
Theoretically, you need at least 2*deg data points, but
if there are errors on the data, you certainly need more.
Be aware that this is a very (very,very) ill-posed problem.
By the way, this algorithm relies heavily on computing the
roots of a polynomial. I used 'roots.m', if there is
something better please use that code.
Demo for a complex fit-function:
deg= 2; N= 20; x1= -(1+i), x= linspace(x1,1+i/2,N).';
h = x(2) - x(1)
y= (2+i)*exp( (-1-2i)*x ) + (-1+3i)*exp( (2+3i)*x );
A= 5e-2; y+= A*(randn(N,1)+randn(N,1)*i); % add complex noise
[alpha,c,rms]= expfit( deg, x1, h, y )
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USAGE [alpha,c,rms] = expfit( deg, x1, h, y )
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fmin
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alias for fminbnd
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alias for fminbnd
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fmins
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-- Function File: [X] = fmins( F,X0,OPTIONS,GRAD,P1,P2, ...)
Find the minimum of a funtion of several variables. By default the
method used is the Nelder&Mead Simplex algorithm
Example usage: fmins(inline('(x(1)-5).^2+(x(2)-8).^4'),[0;0])
*Inputs*
F
A string containing the name of the function to minimize
X0
A vector of initial parameters fo the function F.
OPTIONS
Vector with control parameters (not all parameters are used)
options(1) - Show progress (if 1, default is 0, no progress)
options(2) - Relative size of simplex (default 1e-3)
options(6) - Optimization algorithm
if options(6)==0 - Nelder & Mead simplex (default)
if options(6)==1 - Multidirectional search Method
if options(6)==2 - Alternating Directions search
options(5)
if options(6)==0 && options(5)==0 - regular simplex
if options(6)==0 && options(5)==1 - right-angled simplex
Comment: the default is set to "right-angled simplex".
this works better for me on a broad range of problems,
although the default in nmsmax is "regular simplex"
options(10) - Maximum number of function evaluations
GRAD
Unused (For compatibility with Matlab)
P1,P2, ...
Optional parameters for function F
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Find the minimum of a funtion of several variables.
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gjp
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m = gjp (m, k[, l])
m: matrix; k, l: row- and column-index of pivot, l defaults to k.
Gauss-Jordon pivot as defined in Bard, Y.: Nonlinear Parameter
Estimation, p. 296, Academic Press, New York and London 1974. In
the pivot column, this seems not quite the same as the usual
Gauss-Jordan(-Clasen) pivot. Bard gives Beaton, A. E., 'The use of
special matrix operators in statistical calculus' Research Bulletin
RB-64-51 (1964), Educational Testing Service, Princeton, New Jersey
as a reference, but this article is not easily accessible. Another
reference, whose definition of gjp differs from Bards by some
signs, is Clarke, R. B., 'Algorithm AS 178: The Gauss-Jordan sweep
operator with detection of collinearity', Journal of the Royal
Statistical Society, Series C (Applied Statistics) (1982), 31(2),
166--168.
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m = gjp (m, k[, l])
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jacobs
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-- Function File: Df = jacobs (X, F)
-- Function File: Df = jacobs (X, F, HOOK)
Calculate the jacobian of a function using the complex step method.
Let F be a user-supplied function. Given a point X at which we
seek for the Jacobian, the function 'jacobs' returns the Jacobian
matrix 'd(f(1), ..., df(end))/d(x(1), ..., x(n))'. The function
uses the complex step method and thus can be applied to real
analytic functions.
The optional argument HOOK is a structure with additional options.
HOOK can have the following fields:
* 'h' - can be used to define the magnitude of the complex step
and defaults to 1e-20; steps larger than 1e-3 are not allowed.
* 'fixed' - is a logical vector internally usable by some
optimization functions; it indicates for which elements of X
no gradient should be computed, but zero should be returned.
For example:
f = @(x) [x(1)^2 + x(2); x(2)*exp(x(1))];
Df = jacobs ([1, 2], f)
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Calculate the jacobian of a function using the complex step method.
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leasqr
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function [f,p,cvg,iter,corp,covp,covr,stdresid,Z,r2]=
leasqr(x,y,pin,F,{stol,niter,wt,dp,dFdp,options})
Levenberg-Marquardt nonlinear regression of f(x,p) to y(x).
Version 3.beta
Optional parameters are in braces {}.
x = vector or matrix of independent variables.
y = vector or matrix of observed values.
wt = statistical weights (same dimensions as y). These should be
set to be proportional to (sqrt of var(y))^-1; (That is, the
covariance matrix of the data is assumed to be proportional to
diagonal with diagonal equal to (wt.^2)^-1. The constant of
proportionality will be estimated.); default = ones( size (y)).
pin = vec of initial parameters to be adjusted by leasqr.
dp = fractional increment of p for numerical partial derivatives;
default = .001*ones(size(pin))
dp(j) > 0 means central differences on j-th parameter p(j).
dp(j) < 0 means one-sided differences on j-th parameter p(j).
dp(j) = 0 holds p(j) fixed i.e. leasqr wont change initial guess: pin(j)
F = name of function in quotes or function handle; the function
shall be of the form y=f(x,p), with y, x, p of the form y, x, pin
as described above.
dFdp = name of partial derivative function in quotes or function
handle; default is 'dfdp', a slow but general partial derivatives
function; the function shall be of the form
prt=dfdp(x,f,p,dp,F[,bounds]). For backwards compatibility, the
function will only be called with an extra 'bounds' argument if the
'bounds' option is explicitely specified to leasqr (see dfdp.m).
stol = scalar tolerance on fractional improvement in scalar sum of
squares = sum((wt.*(y-f))^2); default stol = .0001;
niter = scalar maximum number of iterations; default = 20;
options = structure, currently recognized fields are 'fract_prec',
'max_fract_change', 'inequc', 'bounds', and 'equc'. For backwards
compatibility, 'options' can also be a matrix whose first and
second column contains the values of 'fract_prec' and
'max_fract_change', respectively.
Field 'options.fract_prec': column vector (same length as 'pin')
of desired fractional precisions in parameter estimates.
Iterations are terminated if change in parameter vector (chg)
relative to current parameter estimate is less than their
corresponding elements in 'options.fract_prec' [ie. all (abs
(chg) < abs (options.fract_prec .* current_parm_est))] on two
consecutive iterations, default = zeros().
Field 'options.max_fract_change': column vector (same length as
'pin) of maximum fractional step changes in parameter vector.
Fractional change in elements of parameter vector is constrained to
be at most 'options.max_fract_change' between sucessive iterations.
[ie. abs(chg(i))=abs(min([chg(i)
options.max_fract_change(i)*current param estimate])).], default =
Inf*ones().
Field 'options.inequc': cell-array containing up to four entries,
two entries for linear inequality constraints and/or one or two
entries for general inequality constraints. Initial parameters
must satisfy these constraints. Either linear or general
constraints may be the first entries, but the two entries for
linear constraints must be adjacent and, if two entries are given
for general constraints, they also must be adjacent. The two
entries for linear constraints are a matrix (say m) and a vector
(say v), specifying linear inequality constraints of the form
`m.' * parameters + v >= 0'. If the constraints are just bounds,
it is suggested to specify them in 'options.bounds' instead,
since then some sanity tests are performed, and since the
function 'dfdp.m' is guarantied not to violate constraints during
determination of the numeric gradient only for those constraints
specified as 'bounds' (possibly with violations due to a certain
inaccuracy, however, except if no constraints except bounds are
specified). The first entry for general constraints must be a
differentiable vector valued function (say h), specifying general
inequality constraints of the form `h (p[, idx]) >= 0'; p is the
column vector of optimized paraters and the optional argument idx
is a logical index. h has to return the values of all constraints
if idx is not given, and has to return only the indexed
constraints if idx is given (so computation of the other
constraints can be spared). If a second entry for general
constraints is given, it must be a function (say dh) which
returnes a matrix whos rows contain the gradients of the
constraint function h with respect to the optimized parameters.
It has the form jac_h = dh (vh, p, dp, h, idx[, bounds]); p is
the column vector of optimized parameters, and idx is a logical
index --- only the rows indexed by idx must be returned (so
computation of the others can be spared). The other arguments of
dh are for the case that dh computes numerical gradients: vh is
the column vector of the current values of the constraint
function h, with idx already applied. h is a function h (p) to
compute the values of the constraints for parameters p, it will
return only the values indexed by idx. dp is a suggestion for
relative step width, having the same value as the argument 'dp'
of leasqr above. If bounds were specified to leasqr, they are
provided in the argument bounds of dh, to enable their
consideration in determination of numerical gradients. If dh is
not specified to leasqr, numerical gradients are computed in the
same way as with 'dfdp.m' (see above). If some constraints are
linear, they should be specified as linear constraints (or
bounds, if applicable) for reasons of performance, even if
general constraints are also specified.
Field 'options.bounds': two-column-matrix, one row for each
parameter in 'pin'. Each row contains a minimal and maximal value
for each parameter. Default: [-Inf, Inf] in each row. If this
field is used with an existing user-side function for 'dFdp'
(see above) the functions interface might have to be changed.
Field 'options.equc': equality constraints, specified the same
way as inequality constraints (see field 'options.inequc').
Initial parameters must satisfy these constraints.
Note that there is possibly a certain inaccuracy in honoring
constraints, except if only bounds are specified.
_Warning_: If constraints (or bounds) are set, returned guesses
of corp, covp, and Z are generally invalid, even if no constraints
are active for the final parameters. If equality constraints are
specified, corp, covp, and Z are not guessed at all.
Field 'options.cpiv': Function for complementary pivot algorithm
for inequality constraints, default: cpiv_bard. No different
function is supplied.
OUTPUT VARIABLES
f = column vector of values computed: f = F(x,p).
p = column vector trial or final parameters. i.e, the solution.
cvg = scalar: = 1 if convergence, = 0 otherwise.
iter = scalar number of iterations used.
corp = correlation matrix for parameters.
covp = covariance matrix of the parameters.
covr = diag(covariance matrix of the residuals).
stdresid = standardized residuals.
Z = matrix that defines confidence region (see comments in the source).
r2 = coefficient of multiple determination, intercept form.
Not suitable for non-real residuals.
References:
Bard, Nonlinear Parameter Estimation, Academic Press, 1974.
Draper and Smith, Applied Regression Analysis, John Wiley and Sons, 1981.
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function [f,p,cvg,iter,corp,covp,covr,stdresid,Z,r2]=
leasqr(
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line_min
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[a,fx,nev] = line_min (f, dx, args, narg, h, nev_max) - Minimize f() along dx
INPUT ----------
f : string : Name of minimized function
dx : matrix : Direction along which f() is minimized
args : cell : Arguments of f
narg : integer : Position of minimized variable in args. Default=1
h : scalar : Step size to use for centered finite difference
approximation of first and second derivatives. Default=1E-3.
nev_max : integer : Maximum number of function evaluations. Default=30
OUTPUT ---------
a : scalar : Value for which f(x+a*dx) is a minimum (*)
fx : scalar : Value of f(x+a*dx) at minimum (*)
nev : integer : Number of function evaluations
(*) The notation f(x+a*dx) assumes that args == {x}.
Reference: David G Luenberger's Linear and Nonlinear Programming
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[a,fx,nev] = line_min (f, dx, args, narg, h, nev_max) - Minimize f() along dx
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linprog
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-- Function File: X = linprog (F, A, B)
-- Function File: X = linprog (F, A, B, AEQ, BEQ)
-- Function File: X = linprog (F, A, B, AEQ, BEQ, LB, UB)
-- Function File: [X, FVAL] = linprog (...)
Solve a linear problem.
Finds
min (f' * x)
(both f and x are column vectors) subject to
A * x <= b
Aeq * x = beq
lb <= x <= ub
If not specified, AEQ and BEQ default to empty matrices.
If not specified, the lower bound LB defaults to minus infinite and
the upper bound UB defaults to infinite.
See also: glpk.
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Solve a linear problem.
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mdsmax
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MDSMAX Multidirectional search method for direct search optimization.
[x, fmax, nf] = MDSMAX(FUN, x0, STOPIT, SAVIT) attempts to
maximize the function FUN, using the starting vector x0.
The method of multidirectional search is used.
Output arguments:
x = vector yielding largest function value found,
fmax = function value at x,
nf = number of function evaluations.
The iteration is terminated when either
- the relative size of the simplex is <= STOPIT(1)
(default 1e-3),
- STOPIT(2) function evaluations have been performed
(default inf, i.e., no limit), or
- a function value equals or exceeds STOPIT(3)
(default inf, i.e., no test on function values).
The form of the initial simplex is determined by STOPIT(4):
STOPIT(4) = 0: regular simplex (sides of equal length, the default),
STOPIT(4) = 1: right-angled simplex.
Progress of the iteration is not shown if STOPIT(5) = 0 (default 1).
If a non-empty fourth parameter string SAVIT is present, then
`SAVE SAVIT x fmax nf' is executed after each inner iteration.
NB: x0 can be a matrix. In the output argument, in SAVIT saves,
and in function calls, x has the same shape as x0.
MDSMAX(fun, x0, STOPIT, SAVIT, P1, P2,...) allows additional
arguments to be passed to fun, via feval(fun,x,P1,P2,...).
This implementation uses 2n^2 elements of storage (two simplices), where x0
is an n-vector. It is based on the algorithm statement in [2, sec.3],
modified so as to halve the storage (with a slight loss in readability).
References:
[1] V. J. Torczon, Multi-directional search: A direct search algorithm for
parallel machines, Ph.D. Thesis, Rice University, Houston, Texas, 1989.
[2] V. J. Torczon, On the convergence of the multidirectional search
algorithm, SIAM J. Optimization, 1 (1991), pp. 123-145.
[3] N. J. Higham, Optimization by direct search in matrix computations,
SIAM J. Matrix Anal. Appl, 14(2): 317-333, 1993.
[4] N. J. Higham, Accuracy and Stability of Numerical Algorithms,
Second edition, Society for Industrial and Applied Mathematics,
Philadelphia, PA, 2002; sec. 20.5.
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MDSMAX Multidirectional search method for direct search optimization.
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minimize
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[x,v,nev,...] = minimize (f,args,...) - Minimize f
ARGUMENTS
f : string : Name of function. Must return a real value
args : list or : List of arguments to f (by default, minimize the first)
matrix : f's only argument
RETURNED VALUES
x : matrix : Local minimum of f. Let's suppose x is M-by-N.
v : real : Value of f in x0
nev : integer : Number of function evaluations
or 1 x 2 : Number of function and derivative evaluations (if
derivatives are used)
Extra arguments are either a succession of option-value pairs or a single
list or struct of option-value pairs (for unary options, the value in the
struct is ignored).
OPTIONS : DERIVATIVES Derivatives may be used if one of these options
--------------------- uesd. Otherwise, the Nelder-Mean (see
nelder_mead_min) method is used.
'd2f', d2f : Name of a function that returns the value of f, of its
1st and 2nd derivatives : [fx,dfx,d2fx] = feval (d2f, x)
where fx is a real number, dfx is 1x(M*N) and d2fx is
(M*N)x(M*N). A Newton-like method (d2_min) will be used.
'hess' : Use [fx,dfx,d2fx] = leval (f, args) to compute 1st and
2nd derivatives, and use a Newton-like method (d2_min).
'd2i', d2i : Name of a function that returns the value of f, of its
1st and pseudo-inverse of second derivatives :
[fx,dfx,id2fx] = feval (d2i, x) where fx is a real
number, dfx is 1x(M*N) and d2ix is (M*N)x(M*N).
A Newton-like method will be used (see d2_min).
'ihess' : Use [fx,dfx,id2fx] = leval (f, args) to compute 1st
derivative and the pseudo-inverse of 2nd derivatives,
and use a Newton-like method (d2_min).
NOTE : df, d2f or d2i take the same arguments as f.
'order', n : Use derivatives of order n. If the n'th order derivative
is not specified by 'df', 'd2f' or 'd2i', it will be
computed numerically. Currently, only order 1 works.
'ndiff' : Use a variable metric method (bfgs) using numerical
differentiation.
OPTIONS : STOPPING CRITERIA Default is to use 'tol'
---------------------------
'ftol', ftol : Stop search when value doesn't improve, as tested by
ftol > Deltaf/max(|f(x)|,1)
where Deltaf is the decrease in f observed in the last
iteration. Default=10*eps
'utol', utol : Stop search when updates are small, as tested by
tol > max { dx(i)/max(|x(i)|,1) | i in 1..N }
where dx is the change in the x that occured in the last
iteration.
'dtol',dtol : Stop search when derivatives are small, as tested by
dtol > max { df(i)*max(|x(i)|,1)/max(v,1) | i in 1..N }
where x is the current minimum, v is func(x) and df is
the derivative of f in x. This option is ignored if
derivatives are not used in optimization.
MISC. OPTIONS
-------------
'maxev', m : Maximum number of function evaluations <inf>
'narg' , narg : Position of the minimized argument in args <1>
'isz' , step : Initial step size (only for 0 and 1st order method) <1>
Should correspond to expected distance to minimum
'verbose' : Display messages during execution
'backend' : Instead of performing the minimization itself, return
[backend, control], the name and control argument of the
backend used by minimize(). Minimimzation can then be
obtained without the overhead of minimize by calling, if
a 0 or 1st order method is used :
[x,v,nev] = feval (backend, args, control)
or, if a 2nd order method is used :
[x,v,nev] = feval (backend, control.d2f, args, control)
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[x,v,nev,...] = minimize (f,args,...) - Minimize f
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nelder_mead_min
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[x0,v,nev] = nelder_mead_min (f,args,ctl) - Nelder-Mead minimization
Minimize 'f' using the Nelder-Mead algorithm. This function is inspired
from the that found in the book "Numerical Recipes".
ARGUMENTS
---------
f : string : Name of function. Must return a real value
args : list : Arguments passed to f.
or matrix : f's only argument
ctl : vector : (Optional) Control variables, described below
or struct
RETURNED VALUES
---------------
x0 : matrix : Local minimum of f
v : real : Value of f in x0
nev : number : Number of function evaluations
CONTROL VARIABLE : (optional) may be named arguments (i.e. "name",value
------------------ pairs), a struct, or a vector of length <= 6, where
NaN's are ignored. Default values are written <value>.
OPT. VECTOR
NAME POS
ftol,f N/A : Stopping criterion : stop search when values at simplex
vertices are all alike, as tested by
f > (max_i (f_i) - min_i (f_i)) /max(max(|f_i|),1)
where f_i are the values of f at the vertices. <10*eps>
rtol,r N/A : Stop search when biggest radius of simplex, using
infinity-norm, is small, as tested by :
ctl(2) > Radius <10*eps>
vtol,v N/A : Stop search when volume of simplex is small, tested by
ctl(2) > Vol
crit,c ctl(1) : Set one stopping criterion, 'ftol' (c=1), 'rtol' (c=2)
or 'vtol' (c=3) to the value of the 'tol' option. <1>
tol, t ctl(2) : Threshold in termination test chosen by 'crit' <10*eps>
narg ctl(3) : Position of the minimized argument in args <1>
maxev ctl(4) : Maximum number of function evaluations. This number <inf>
may be slightly exceeded.
isz ctl(5) : Size of initial simplex, which is : <1>
{ x + e_i | i in 0..N }
Where x == args{narg} is the initial value
e_0 == zeros (size (x)),
e_i(j) == 0 if j != i and e_i(i) == ctl(5)
e_i has same size as x
Set ctl(5) to the distance you expect between the starting
point and the minimum.
rst ctl(6) : When a minimum is found the algorithm restarts next to
it until the minimum does not improve anymore. ctl(6) is
the maximum number of restarts. Set ctl(6) to zero if
you know the function is well-behaved or if you don't
mind not getting a true minimum. <0>
verbose, v Be more or less verbose (quiet=0) <0>
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[x0,v,nev] = nelder_mead_min (f,args,ctl) - Nelder-Mead minimization
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nmsmax
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NMSMAX Nelder-Mead simplex method for direct search optimization.
[x, fmax, nf] = NMSMAX(FUN, x0, STOPIT, SAVIT) attempts to
maximize the function FUN, using the starting vector x0.
The Nelder-Mead direct search method is used.
Output arguments:
x = vector yielding largest function value found,
fmax = function value at x,
nf = number of function evaluations.
The iteration is terminated when either
- the relative size of the simplex is <= STOPIT(1)
(default 1e-3),
- STOPIT(2) function evaluations have been performed
(default inf, i.e., no limit), or
- a function value equals or exceeds STOPIT(3)
(default inf, i.e., no test on function values).
The form of the initial simplex is determined by STOPIT(4):
STOPIT(4) = 0: regular simplex (sides of equal length, the default)
STOPIT(4) = 1: right-angled simplex.
Progress of the iteration is not shown if STOPIT(5) = 0 (default 1).
STOPIT(6) indicates the direction (ie. minimization or
maximization.) Default is 1, maximization.
set STOPIT(6)=-1 for minimization
If a non-empty fourth parameter string SAVIT is present, then
`SAVE SAVIT x fmax nf' is executed after each inner iteration.
NB: x0 can be a matrix. In the output argument, in SAVIT saves,
and in function calls, x has the same shape as x0.
NMSMAX(fun, x0, STOPIT, SAVIT, P1, P2,...) allows additional
arguments to be passed to fun, via feval(fun,x,P1,P2,...).
References:
N. J. Higham, Optimization by direct search in matrix computations,
SIAM J. Matrix Anal. Appl, 14(2): 317-333, 1993.
C. T. Kelley, Iterative Methods for Optimization, Society for Industrial
and Applied Mathematics, Philadelphia, PA, 1999.
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NMSMAX Nelder-Mead simplex method for direct search optimization.
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nonlin_curvefit
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-- Function File: [P, FY, CVG, OUTP] = nonlin_curvefit (F, PIN, X, Y)
-- Function File: [P, FY, CVG, OUTP] = nonlin_curvefit (F, PIN, X, Y,
SETTINGS)
Frontend for nonlinear fitting of values, computed by a model
function, to observed values.
Please refer to the description of 'nonlin_residmin'. The only
differences to 'nonlin_residmin' are the additional arguments X
(independent values, mostly, but not necessarily, an array of the
same dimensions or the same number of rows as Y) and Y (array of
observations), the returned value FY (final guess for observed
values) instead of RESID, that the model function has a second
obligatory argument which will be set to X and is supposed to
return guesses for the observations (with the same dimensions), and
that the possibly user-supplied function for the jacobian of the
model function has also a second obligatory argument which will be
set to X.
See also: nonlin_residmin.
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Frontend for nonlinear fitting of values, computed by a model function,
to obser
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nonlin_min
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-- Function File: [P, OBJF, CVG, OUTP] = nonlin_min (F, PIN)
-- Function File: [P, OBJF, CVG, OUTP] = nonlin_min (F, PIN, SETTINGS)
Frontend for constrained nonlinear minimization of a scalar
objective function. The functions supplied by the user have a
minimal interface; any additionally needed constants can be
supplied by wrapping the user functions into anonymous functions.
The following description applies to usage with vector-based
parameter handling. Differences in usage for structure-based
parameter handling will be explained in a separate section below.
F: objective function. It gets a column vector of real parameters
as argument. In gradient determination, this function may be
called with an informational second argument, whose content depends
on the function for gradient determination.
PIN: real column vector of initial parameters.
SETTINGS: structure whose fields stand for optional settings
referred to below. The fields can be set by 'optimset()' with
Octave versions 3.3.55 or greater; with older Octave versions, the
fields must be set directly as structure-fields in the correct
case.
The returned values are the column vector of final parameters P,
the final value of the objective function OBJF, an integer CVG
indicating if and how optimization succeeded or failed, and a
structure OUTP with additional information, curently with only one
field: NITER, the number of iterations. CVG is greater than zero
for success and less than or equal to zero for failure; its
possible values depend on the used backend and currently can be '0'
(maximum number of iterations exceeded), '1' (fixed number of
iterations completed, e.g. in stochastic optimizers), '2'
(parameter change less than specified precision in two consecutive
iterations), '3' (improvement in objective function less than
specified), or '-4' (algorithm got stuck).
SETTINGS:
'Algorithm': String specifying the backend. Currently available
are '"lm_feasible"' (default) and '"siman"'. They are described in
separate sections below.
'objf_grad': Function computing the gradient of the objective
function with respect to the parameters. Will be called with the
column vector of parameters and an informational structure as
arguments. The structure has the fields 'f': value of objective
function for current parameters, 'fixed': logical vector indicating
which parameters are not optimized, so these partial derivatives
need not be computed and can be set to zero, 'diffp',
'diff_onesided', 'lbound', 'ubound': identical to the user settings
of this name, 'plabels': 1-dimensional cell-array of
column-cell-arrays, each column with labels for all parameters, the
first column contains the numerical indices of the parameters. The
default gradient function will call the objective function with the
second argument set with fields 'f': as the 'f' passed to the
gradient function, 'plabels': cell-array of 1x1 cell-arrays with
the entries of the column-cell-arrays of 'plabels' as passed to the
jacobian function corresponding to current parameter, 'side': '0'
for one-sided interval, '1' or '2', respectively, for the sides of
a two-sided interval, and 'parallel': logical scalar indicating
parallel computation of partial derivatives.
'objf_hessian': Function computing the Hessian of the objective
function with respect to the parameters. The default is backend
specific. Will be called with the column vector of parameters as
argument.
'diffp': column vector of fractional intervals (doubled for central
intervals) supposed to be used by gradient functions performing
finite differencing. Default: '.001 * ones (size (parameters))'.
The default gradient function will use these as absolute intervals
for parameters with value zero.
'diff_onesided': logical column vector indicating that one-sided
intervals should be used by gradient functions performing finite
differencing. Default: 'false (size (parameters))'.
'complex_step_derivative_objf', 'complex_step_derivative_inequc',
'complex_step_derivative_equc': logical scalars, default: false.
Estimate gradient of objective function, general inequality
constraints, and general equality constraints, respectively, with
complex step derivative approximation. Use only if you know that
your objective function, function of general inequality
constraints, or function of general equality constraints,
respectively, is suitable for this. No user function for the
respective gradient must be specified.
'cstep': scalar step size for complex step derivative
approximation. Default: 1e-20.
'parallel_local': logical scalar, default: false. Estimate
gradient of objective function and of constraints in parallel
processes. Works for default finite difference gradient function
and for complex step derivatives. Due to overhead, a speed
advantage can only be expected if objective function or constraint
functions are time consuming enough.
'fixed': logical column vector indicating which parameters should
not be optimized, but kept to their inital value. Fixing is done
independently of the backend, but the backend may choose to fix
additional parameters under certain conditions.
'lbound', 'ubound': column vectors of lower and upper bounds for
parameters. Default: '-Inf' and '+Inf', respectively. The bounds
are non-strict, i.e. parameters are allowed to be exactly equal to
a bound. The default gradient function will respect bounds (but no
further inequality constraints) in finite differencing.
'inequc': Further inequality constraints. Cell-array containing up
to four entries, two entries for linear inequality constraints
and/or one or two entries for general inequality constraints.
Either linear or general constraints may be the first entries, but
the two entries for linear constraints must be adjacent and, if two
entries are given for general constraints, they also must be
adjacent. The two entries for linear constraints are a matrix (say
'm') and a vector (say 'v'), specifying linear inequality
constraints of the form 'm.' * parameters + v >= 0'. The first
entry for general constraints must be a differentiable
column-vector valued function (say 'h'), specifying general
inequality constraints of the form 'h (p[, idx]) >= 0'; 'p' is the
column vector of optimized parameters and the optional argument
'idx' is a logical index. 'h' has to return the values of all
constraints if 'idx' is not given. It may choose to return only
the indexed constraints if 'idx' is given (so computation of the
other constraints can be spared); in this case, the additional
setting 'f_inequc_idx' has to be set to 'true'. In gradient
determination, this function may be called with an informational
third argument, whose content depends on the function for gradient
determination. If a second entry for general inequality
constraints is given, it must be a function computing the jacobian
of the constraints with respect to the parameters. For this
function, the description of 'dfdp' above applies, with 2
exceptions: 1) it is called with 3 arguments since it has an
additional argument 'idx', a logical index, at second position,
indicating which rows of the jacobian must be returned (if the
function chooses to return only indexed rows, the additional
setting 'df_inequc_idx' has to be set to 'true'). 2) the default
jacobian function calls 'h' with 3 arguments, since the argument
'idx' is also supplied. Note that specifying linear constraints as
general constraints will generally waste performance, even if
further, non-linear, general constraints are also specified.
'equc': Equality constraints. Specified the same way as inequality
constraints (see 'inequc'). The respective additional settings are
named 'f_equc_idx' and 'df_equc_idx'.
'cpiv': Function for complementary pivoting, usable in algorithms
for constraints. Default: cpiv_bard. Only the default function
is supplied with the package.
'TolFun': Minimum fractional improvement in objective function in
an iteration (termination criterium). Default: .0001.
'MaxIter': Maximum number of iterations (termination criterium).
Default: backend-specific.
'fract_prec': Column Vector, minimum fractional change of
parameters in an iteration (termination criterium if violated in
two consecutive iterations). Default: backend-specific.
'max_fract_change': Column Vector, enforced maximum fractional
change in parameters in an iteration. Default: backend-specific.
'Display': String indicating the degree of verbosity. Default:
'"off"'. Possible values are currently '"off"' (no messages) and
'"iter"' (some messages after each iteration). Support of this
setting and its exact interpretation are backend-specific.
'debug': Logical scalar, default: 'false'. Will be passed to the
backend, which might print debugging information if true.
Structure-based parameter handling
The setting 'param_order' is a cell-array with names of the
optimized parameters. If not given, and initial parameters are a
structure, all parameters in the structure are optimized. If
initial parameters are a structure, it is an error if 'param_order'
is not given and there are any non-structure-based configuration
items or functions.
The initial parameters PIN can be given as a structure containing
at least all fields named in 'param_order'. In this case the
returned parameters P will also be a structure.
Each user-supplied function can be called with the argument
containing the current parameters being a structure instead of a
column vector. For this, a corresponding setting must be set to
'true': 'objf_pstruct' (objective function), 'grad_objf_pstruct'
(gradient of objective function), 'hessian_objf_pstruct' (hessian
of objective function), 'f_inequc_pstruct' (general inequality
constraints), 'df_inequc_pstruct' (jacobian of general inequality
constraints), 'f_equc_pstruct' (general equality constraints), and
'df_equc_pstruct' (jacobian of general equality constraints). If a
gradient (jacobian) function is configured in such a way, it must
return the entries (columns) of the gradient (jacobian) as fields
of a structure under the respective parameter names. If the
hessian function is configured in such a way, it must return a
structure (say 'h') with fields e.g. as 'h.a.b = value' for
'value' being the 2nd partial derivative with respect to 'a' and
'b'. There is no need to also specify the field 'h.b.a' in this
example.
Similarly, for specifying linear constraints, instead of the matrix
(called 'm' above), a structure containing the rows of the matrix
in fields under the respective parameter names can be given. In
this case, rows containing only zeros need not be given.
The vector-based settings 'lbound', 'ubound', 'fixed', 'diffp',
'diff_onesided', 'fract_prec', and 'max_fract_change' can be
replaced by the setting 'param_config'. It is a structure that can
contain fields named in 'param_order'. For each such field, there
may be subfields with the same names as the above vector-based
settings, but containing a scalar value for the respective
parameter. If 'param_config' is specified, none of the above
vector/matrix-based settings may be used.
Additionally, named parameters are allowed to be non-scalar real
arrays. In this case, their dimensions are given by the setting
'param_dims', a cell-array of dimension vectors, each containing at
least two dimensions; if not given, dimensions are taken from the
initial parameters, if these are given in a structure. Any
vector-based settings or not structure-based linear constraints
then must correspond to an order of parameters with all parameters
reshaped to vectors and concatenated in the user-given order of
parameter names. Structure-based settings or structure-based
initial parameters must contain arrays with dimensions reshapable
to those of the respective parameters.
Description of backends
"lm_feasible"
A Levenberg/Marquardt-like optimizer, attempting to honour
constraints throughout the course of optimization. This means that
the initial parameters must not violate constraints (to find an
initial feasible set of parameters, e.g. Octaves 'sqp' can be
used, by specifying an objective function which is constant or
which returns the quadratic distance to the initial values). If
the constraints need only be honoured in the result of the
optimization, Octaves 'sqp' may be preferable. The Hessian is
either supplied by the user or is approximated by the BFGS
algorithm.
Returned value CVG will be '2' or '3' for success and '0' or '-4'
for failure (see above for meaning).
Backend-specific defaults are: 'MaxIter': 20, 'fract_prec': 'zeros
(size (parameters))', 'max_fract_change': 'Inf' for all parameters.
Interpretation of 'Display': if set to '"iter"', currently only
information on applying 'max_fract_change' is printed.
"siman"
A simulated annealing (stochastic) optimizer, changing all
parameters at once in a single step, so being suitable for
non-bound constraints.
No gradient or hessian of the objective function is used. The
settings 'MaxIter', 'fract_prec', 'TolFun', and 'max_fract_change'
are not honoured.
Accepts the additional settings 'T_init' (initial temperature,
default 0.01), 'T_min' (final temperature, default 1.0e-5), 'mu_T'
(factor of temperature decrease, default 1.005), 'iters_fixed_T'
(iterations within one temperature step, default 10),
'max_rand_step' (column vector or structure-based configuration of
maximum random steps for each parameter, default 0.005 * PIN),
'stoch_regain_constr' (if 'true', regain constraints after a random
step, otherwise take new random value until constraints are met,
default false), 'trace_steps' (set field 'trace' of OUTP with a
matrix with a row for each step, first column iteration number,
second column repeat number within iteration, third column value of
objective function, rest columns parameter values, default false),
and 'siman_log' (set field 'log' of OUTP with a matrix with a row
for each iteration, first column temperature, second column value
of objective function, rest columns numbers of tries with decrease,
no decrease but accepted, and no decrease and rejected.
Steps with increase 'diff' of objective function are accepted if
'rand (1) < exp (- diff / T)', where 'T' is the temperature of the
current iteration.
If regaining of constraints failed, optimization will be aborted
and returned value of CVG will be '0'. Otherwise, CVG will be '1'.
Interpretation of 'Display': if set to '"iter"', an informational
line is printed after each iteration.
# name: <cell-element>
# type: sq_string
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Frontend for constrained nonlinear minimization of a scalar objective
function.
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 15
nonlin_residmin
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 13420
-- Function File: [P, RESID, CVG, OUTP] = nonlin_residmin (F, PIN)
-- Function File: [P, RESID, CVG, OUTP] = nonlin_residmin (F, PIN,
SETTINGS)
Frontend for nonlinear minimization of residuals returned by a
model function.
The functions supplied by the user have a minimal interface; any
additionally needed constants (e.g. observed values) can be
supplied by wrapping the user functions into anonymous functions.
The following description applies to usage with vector-based
parameter handling. Differences in usage for structure-based
parameter handling will be explained in a separate section below.
F: function returning the array of residuals. It gets a column
vector of real parameters as argument. In gradient determination,
this function may be called with an informational second argument,
whose content depends on the function for gradient determination.
PIN: real column vector of initial parameters.
SETTINGS: structure whose fields stand for optional settings
referred to below. The fields can be set by 'optimset()' with
Octave versions 3.3.55 or greater; with older Octave versions, the
fields must be set directly as structure-fields in the correct
case.
The returned values are the column vector of final parameters P,
the final array of residuals RESID, an integer CVG indicating if
and how optimization succeeded or failed, and a structure OUTP with
additional information, curently with only one field: NITER, the
number of iterations. CVG is greater than zero for success and
less than or equal to zero for failure; its possible values depend
on the used backend and currently can be '0' (maximum number of
iterations exceeded), '2' (parameter change less than specified
precision in two consecutive iterations), or '3' (improvement in
objective function - e.g. sum of squares - less than specified).
SETTINGS:
'Algorithm': String specifying the backend. Default:
'"lm_svd_feasible"'. The latter is currently the only backend
distributed with this package. It is described in a separate
section below.
'dfdp': Function computing the jacobian of the residuals with
respect to the parameters, assuming residuals are reshaped to a
vector. Default: finite differences. Will be called with the
column vector of parameters and an informational structure as
arguments. The structure has the fields 'f': value of residuals
for current parameters, reshaped to a column vector, 'fixed':
logical vector indicating which parameters are not optimized, so
these partial derivatives need not be computed and can be set to
zero, 'diffp', 'diff_onesided', 'lbound', 'ubound': identical to
the user settings of this name, 'plabels': 1-dimensional cell-array
of column-cell-arrays, each column with labels for all parameters,
the first column contains the numerical indices of the parameters.
The default jacobian function will call the model function with the
second argument set with fields 'f': as the 'f' passed to the
jacobian function, 'plabels': cell-array of 1x1 cell-arrays with
the entries of the column-cell-arrays of 'plabels' as passed to the
jacobian function corresponding to current parameter, 'side': '0'
for one-sided interval, '1' or '2', respectively, for the sides of
a two-sided interval, and 'parallel': logical scalar indicating
parallel computation of partial derivatives.
'diffp': column vector of fractional intervals (doubled for central
intervals) supposed to be used by jacobian functions performing
finite differencing. Default: '.001 * ones (size (parameters))'.
The default jacobian function will use these as absolute intervals
for parameters with value zero.
'diff_onesided': logical column vector indicating that one-sided
intervals should be used by jacobian functions performing finite
differencing. Default: 'false (size (parameters))'.
'complex_step_derivative_f', 'complex_step_derivative_inequc',
'complex_step_derivative_equc': logical scalars, default: false.
Estimate Jacobian of model function, general inequality
constraints, and general equality constraints, respectively, with
complex step derivative approximation. Use only if you know that
your model function, function of general inequality constraints, or
function of general equality constraints, respectively, is suitable
for this. No user function for the respective Jacobian must be
specified.
'cstep': scalar step size for complex step derivative
approximation. Default: 1e-20.
'parallel_local': logical scalar, default: false. Estimate
Jacobians of model function and of constraints in parallel
processes. Works for default finite difference Jacobian function
and for complex step derivatives. Due to overhead, a speed
advantage can only be expected if model function or constraint
functions are time consuming enough.
'fixed': logical column vector indicating which parameters should
not be optimized, but kept to their inital value. Fixing is done
independently of the backend, but the backend may choose to fix
additional parameters under certain conditions.
'lbound', 'ubound': column vectors of lower and upper bounds for
parameters. Default: '-Inf' and '+Inf', respectively. The bounds
are non-strict, i.e. parameters are allowed to be exactly equal to
a bound. The default jacobian function will respect bounds (but no
further inequality constraints) in finite differencing.
'inequc': Further inequality constraints. Cell-array containing up
to four entries, two entries for linear inequality constraints
and/or one or two entries for general inequality constraints.
Either linear or general constraints may be the first entries, but
the two entries for linear constraints must be adjacent and, if two
entries are given for general constraints, they also must be
adjacent. The two entries for linear constraints are a matrix (say
'm') and a vector (say 'v'), specifying linear inequality
constraints of the form 'm.' * parameters + v >= 0'. The first
entry for general constraints must be a differentiable
column-vector valued function (say 'h'), specifying general
inequality constraints of the form 'h (p[, idx]) >= 0'; 'p' is the
column vector of optimized paraters and the optional argument 'idx'
is a logical index. 'h' has to return the values of all
constraints if 'idx' is not given. It may choose to return only
the indexed constraints if 'idx' is given (so computation of the
other constraints can be spared); in this case, the additional
setting 'f_inequc_idx' has to be set to 'true'. In gradient
determination, this function may be called with an informational
third argument, whose content depends on the function for gradient
determination. If a second entry for general inequality
constraints is given, it must be a function computing the jacobian
of the constraints with respect to the parameters. For this
function, the description of 'dfdp' above applies, with 2
exceptions: 1) it is called with 3 arguments since it has an
additional argument 'idx', a logical index, at second position,
indicating which rows of the jacobian must be returned (if the
function chooses to return only indexed rows, the additional
setting 'df_inequc_idx' has to be set to 'true'). 2) the default
jacobian function calls 'h' with 3 arguments, since the argument
'idx' is also supplied. Note that specifying linear constraints as
general constraints will generally waste performance, even if
further, non-linear, general constraints are also specified.
'equc': Equality constraints. Specified the same way as inequality
constraints (see 'inequc').
'cpiv': Function for complementary pivoting, usable in algorithms
for constraints. Default: cpiv_bard. Only the default function
is supplied with the package.
'weights': Array of weights for the residuals. Dimensions must
match.
'TolFun': Minimum fractional improvement in objective function
(e.g. sum of squares) in an iteration (termination criterium).
Default: .0001.
'MaxIter': Maximum number of iterations (termination criterium).
Default: backend-specific.
'fract_prec': Column Vector, minimum fractional change of
parameters in an iteration (termination criterium if violated in
two consecutive iterations). Default: backend-specific.
'max_fract_change': Column Vector, enforced maximum fractional
change in parameters in an iteration. Default: backend-specific.
'Display': String indicating the degree of verbosity. Default:
'"off"'. Possible values are currently '"off"' (no messages) and
'"iter"' (some messages after each iteration). Support of this
setting and its exact interpretation are backend-specific.
'plot_cmd': Function enabling backend to plot results or
intermediate results. Will be called with current computed
residuals. Default: plot nothing.
'debug': Logical scalar, default: 'false'. Will be passed to the
backend, which might print debugging information if true.
Structure-based parameter handling
The setting 'param_order' is a cell-array with names of the
optimized parameters. If not given, and initial parameters are a
structure, all parameters in the structure are optimized. If
initial parameters are a structure, it is an error if 'param_order'
is not given and there are any non-structure-based configuration
items or functions.
The initial parameters PIN can be given as a structure containing
at least all fields named in 'param_order'. In this case the
returned parameters P will also be a structure.
Each user-supplied function can be called with the argument
containing the current parameters being a structure instead of a
column vector. For this, a corresponding setting must be set to
'true': 'f_pstruct' (model function), 'df_pstruct' (jacobian of
model function), 'f_inequc_pstruct' (general inequality
constraints), 'df_inequc_pstruct' (jacobian of general inequality
constraints), 'f_equc_pstruct' (general equality constraints), and
'df_equc_pstruct' (jacobian of general equality constraints). If a
jacobian-function is configured in such a way, it must return the
columns of the jacobian as fields of a structure under the
respective parameter names.
Similarly, for specifying linear constraints, instead of the matrix
(called 'm' above), a structure containing the rows of the matrix
in fields under the respective parameter names can be given. In
this case, rows containing only zeros need not be given.
The vector-based settings 'lbound', 'ubound', 'fixed', 'diffp',
'diff_onesided', 'fract_prec', and 'max_fract_change' can be
replaced by the setting 'param_config'. It is a structure that can
contain fields named in 'param_order'. For each such field, there
may be subfields with the same names as the above vector-based
settings, but containing a scalar value for the respective
parameter. If 'param_config' is specified, none of the above
vector/matrix-based settings may be used.
Additionally, named parameters are allowed to be non-scalar real
arrays. In this case, their dimensions are given by the setting
'param_dims', a cell-array of dimension vectors, each containing at
least two dimensions; if not given, dimensions are taken from the
initial parameters, if these are given in a structure. Any
vector-based settings or not structure-based linear constraints
then must correspond to an order of parameters with all parameters
reshaped to vectors and concatenated in the user-given order of
parameter names. Structure-based settings or structure-based
initial parameters must contain arrays with dimensions reshapable
to those of the respective parameters.
Description of backends (currently only one)
"lm_svd_feasible"
A Levenberg/Marquardt algorithm using singular value decomposition
and featuring constraints which must be met by the initial
parameters and are attempted to be kept met throughout the
optimization.
Parameters with identical lower and upper bounds will be fixed.
Returned value CVG will be '0', '2', or '3'.
Backend-specific defaults are: 'MaxIter': 20, 'fract_prec': 'zeros
(size (parameters))', 'max_fract_change': 'Inf' for all parameters.
Interpretation of 'Display': if set to '"iter"', currently
'plot_cmd' is evaluated for each iteration, and some further
diagnostics may be printed.
Specific option: 'lm_svd_feasible_alt_s': if falling back to nearly
gradient descent, do it more like original Levenberg/Marquardt
method, with descent in each gradient component; for testing only.
See also: nonlin_curvefit.
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 78
Frontend for nonlinear minimization of residuals returned by a model
function.
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 3
nrm
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 154
-- Function File: XMIN = nrm( F,X0)
Using X0 as a starting point find a minimum of the scalar function
F. The Newton-Raphson method is used.
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 69
Using X0 as a starting point find a minimum of the scalar function F.
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 14
optim_problems
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 64
Problems for testing optimizers. Documentation is in the code.
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 33
Problems for testing optimizers.
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 9
poly_2_ex
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 353
ex = poly_2_ex (l, f) - Extremum of a 1-var deg-2 polynomial
l : 3 : Values of variable at which polynomial is known.
f : 3 : f(i) = Value of the degree-2 polynomial at l(i).
ex : 1 : Value for which f reaches its extremum
Assuming that f(i) = a*l(i)^2 + b*l(i) + c = P(l(i)) for some a, b, c,
ex is the extremum of the polynome P.
# name: <cell-element>
# type: sq_string
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ex = poly_2_ex (l, f) - Extremum of a 1-var deg-2 polynomial
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 8
polyconf
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 1439
[y,dy] = polyconf(p,x,s)
Produce prediction intervals for the fitted y. The vector p
and structure s are returned from polyfit or wpolyfit. The
x values are where you want to compute the prediction interval.
polyconf(...,['ci'|'pi'])
Produce a confidence interval (range of likely values for the
mean at x) or a prediction interval (range of likely values
seen when measuring at x). The prediction interval tells
you the width of the distribution at x. This should be the same
regardless of the number of measurements you have for the value
at x. The confidence interval tells you how well you know the
mean at x. It should get smaller as you increase the number of
measurements. Error bars in the physical sciences usually show
a 1-alpha confidence value of erfc(1/sqrt(2)), representing
one standandard deviation of uncertainty in the mean.
polyconf(...,1-alpha)
Control the width of the interval. If asking for the prediction
interval 'pi', the default is .05 for the 95% prediction interval.
If asking for the confidence interval 'ci', the default is
erfc(1/sqrt(2)) for a one standard deviation confidence interval.
Example:
[p,s] = polyfit(x,y,1);
xf = linspace(x(1),x(end),150);
[yf,dyf] = polyconf(p,xf,s,'ci');
plot(xf,yf,'g-;fit;',xf,yf+dyf,'g.;;',xf,yf-dyf,'g.;;',x,y,'xr;data;');
plot(x,y-polyval(p,x),';residuals;',xf,dyf,'g-;;',xf,-dyf,'g-;;');
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 25
[y,dy] = polyconf(p,x,s)
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 10
polyfitinf
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 4862
function [A,REF,HMAX,H,R,EQUAL] = polyfitinf(M,N,K,X,Y,EPSH,MAXIT,REF0)
Best polynomial approximation in discrete uniform norm
INPUT VARIABLES:
M : degree of the fitting polynomial
N : number of data points
X(N) : x-coordinates of data points
Y(N) : y-coordinates of data points
K : character of the polynomial:
K = 0 : mixed parity polynomial
K = 1 : odd polynomial ( X(1) must be > 0 )
K = 2 : even polynomial ( X(1) must be >= 0 )
EPSH : tolerance for leveling. A useful value for 24-bit
mantissa is EPSH = 2.0E-7
MAXIT : upper limit for number of exchange steps
REF0(M2): initial alternating set ( N-vector ). This is an
OPTIONAL argument. The length M2 is given by:
M2 = M + 2 , if K = 0
M2 = integer part of (M+3)/2 , if K = 1
M2 = 2 + M/2 (M must be even) , if K = 2
OUTPUT VARIABLES:
A : polynomial coefficients of the best approximation
in order of increasing powers:
p*(x) = A(1) + A(2)*x + A(3)*x^2 + ...
REF : selected alternating set of points
HMAX : maximum deviation ( uniform norm of p* - f )
H : pointwise approximation errors
R : total number of iterations
EQUAL : success of failure of algorithm
EQUAL=1 : succesful
EQUAL=0 : convergence not acheived
EQUAL=-1: input error
EQUAL=-2: algorithm failure
Relies on function EXCH, provided below.
Example:
M = 5; N = 10000; K = 0; EPSH = 10^-12; MAXIT = 10;
X = linspace(-1,1,N); % uniformly spaced nodes on [-1,1]
k=1; Y = abs(X).^k; % the function Y to approximate
[A,REF,HMAX,H,R,EQUAL] = polyfitinf(M,N,K,X,Y,EPSH,MAXIT);
p = polyval(A,X); plot(X,Y,X,p) % p is the best approximation
Note: using an even value of M, e.g., M=2, in the example above, makes
the algorithm to fail with EQUAL=-2, because of collocation, which
appears because both the appriximating function and the polynomial are
even functions. The way aroung it is to approximate only the right half
of the function, setting K = 2 : even polynomial. For example:
N = 10000; K = 2; EPSH = 10^-12; MAXIT = 10; X = linspace(0,1,N);
for i = 1:2
k = 2*i-1; Y = abs(X).^k;
for j = 1:4
M = 2^j;
[~,~,HMAX] = polyfitinf(M,N,K,X,Y,EPSH,MAXIT);
approxerror(i,j) = HMAX;
end
end
disp('Table 3.1 from Approximation theory and methods, M.J.D.POWELL, p. 27');
disp(' ');
disp(' n K=1 K=3');
disp(' '); format short g;
disp([(2.^(1:4))' approxerror']);
ALGORITHM:
Computation of the polynomial that best approximates the data (X,Y)
in the discrete uniform norm, i.e. the polynomial with the minimum
value of max{ | p(x_i) - y_i | , x_i in X } . That polynomial, also
known as minimax polynomial, is obtained by the exchange algorithm,
a finite iterative process requiring, at most,
n
( ) iterations ( usually p = M + 2. See also function EXCH ).
p
since this number can be very large , the routine may not converge
within MAXIT iterations . The other possibility of failure occurs
when there is insufficient floating point precision for the input
data chosen.
CREDITS: This routine was developed and modified as
computer assignments in Approximation Theory courses by
Prof. Andrew Knyazev, University of Colorado Denver, USA.
Team Fall 98 (Revision 1.0):
Chanchai Aniwathananon
Crhistopher Mehl
David A. Duran
Saulo P. Oliveira
Team Spring 11 (Revision 1.1): Manuchehr Aminian
The algorithm and the comments are based on a FORTRAN code written
by Joseph C. Simpson. The code is available on Netlib repository:
http://www.netlib.org/toms/501
See also: Communications of the ACM, V14, pp.355-356(1971)
NOTES:
1) A may contain the collocation polynomial
2) If MAXIT is exceeded, REF contains a new reference set
3) M, EPSH and REF can be altered during the execution
4) To keep consistency to the original code , EPSH can be
negative. However, the use of REF0 is *NOT* determined by
EPSH< 0, but only by its inclusion as an input parameter.
Some parts of the code can still take advantage of vectorization.
Revision 1.0 from 1998 is a direct human translation of
the FORTRAN code http://www.netlib.org/toms/501
Revision 1.1 is a clean-up and technical update.
Tested on MATLAB Version 7.11.0.584 (R2010b) and
GNU Octave Version 3.2.4
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function [A,REF,HMAX,H,R,EQUAL] = polyfitinf(M,N,K,X,Y,EPSH,MAXIT,REF0)
B
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powell
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-- Function File: [ P , OBJ_VALUE, CONVERGENCE, ITERS, NEVS] = powell
(F, P0, CONTROL)
Multidimensional minimization (direction-set method). Implements a
direction-set (Powell's) method for multidimensional minimization
of a function without calculation of the gradient [1, 2]
Arguments
--------------
* F: name of function to minimize (string or handle), which
should accept one input variable (see example for how to pass
on additional input arguments)
* P0: An initial value of the function argument to minimize
* OPTIONS: an optional structure, which can be generated by
optimset, with some or all of the following fields:
- MaxIter: maximum iterations (positive integer, or -1 or
Inf for unlimited (default))
- TolFun: minimum amount by which function value must
decrease in each iteration to continue (default is 1E-8)
- MaxFunEvals: maximum function evaluations (positive
integer, or -1 or Inf for unlimited (default))
- SearchDirections: an n*n matrix whose columns contain the
initial set of (presumably orthogonal) directions to
minimize along, where n is the number of elements in the
argument to be minimized for; or an n*1 vector of
magnitudes for the initial directions (defaults to the
set of unit direction vectors)
Examples
-------------
y = @(x, s) x(1) ^ 2 + x(2) ^ 2 + s;
o = optimset('MaxIter', 100, 'TolFun', 1E-10);
s = 1;
[x_optim, y_min, conv, iters, nevs] = powell(@(x) y(x, s), [1 0.5], o); %pass y wrapped in an anonymous function so that all other arguments to y, which are held constant, are set
%should return something like x_optim = [4E-14 3E-14], y_min = 1, conv = 1, iters = 2, nevs = 24
Returns:
-------------
* P: the minimizing value of the function argument
* OBJ_VALUE: the value of F() at P
* CONVERGENCE: 1 if normal convergence, 0 if not
* ITERS: number of iterations performed
* NEVS: number of function evaluations
References
---------------
1. Powell MJD (1964), An efficient method for finding the minimum
of a function of several variables without calculating
derivatives, 'Computer Journal', 7 :155-162
2. Press, WH; Teukolsky, SA; Vetterling, WT; Flannery, BP (1992).
'Numerical Recipes in Fortran: The Art of Scientific
Computing' (2nd Ed.). New York: Cambridge University Press
(Section 10.5)
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Multidimensional minimization (direction-set method).
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residmin_stat
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-- Function File: INFO = residmin_stat (F, P, SETTINGS)
Frontend for computation of statistics for a residual-based
minimization.
SETTINGS is a structure whose fields can be set by 'optimset' with
Octave versions 3.3.55 or greater; with older Octave versions, the
fields must be set directly and in the correct case. With SETTINGS
the computation of certain statistics is requested by setting the
fields 'ret_<name_of_statistic>' to 'true'. The respective
statistics will be returned in a structure as fields with name
'<name_of_statistic>'. Depending on the requested statistic and on
the additional information provided in SETTINGS, F and P may be
empty. Otherwise, F is the model function of an optimization (the
interface of F is described e.g. in 'nonlin_residmin', please see
there), and P is a real column vector with parameters resulting
from the same optimization.
Currently, the following statistics (or general information) can be
requested:
'dfdp': Jacobian of model function with respect to parameters.
'covd': Covariance matrix of data (typically guessed by applying a
factor to the covariance matrix of the residuals).
'covp': Covariance matrix of final parameters.
'corp': Correlation matrix of final parameters.
Further SETTINGS
The functionality of the interface is similar to 'nonlin_residmin'.
In particular, structure-based, possibly non-scalar, parameters and
flagging parameters as fixed are possible. The following settings
have the same meaning as in 'nonlin_residmin' (please refer to
there): 'param_order', 'param_dims', 'f_pstruct', 'df_pstruct',
'diffp', 'diff_onesided', 'complex_step_derivative_f', 'cstep',
'fixed', and 'weights'. Similarly, 'param_config' can be used, but
only with fields corresponding to the settings 'fixed', 'diffp',
and 'diff_onesided'.
'dfdp' can be set in the same way as in 'nonlin_residmin', but
alternatively may already contain the computed Jacobian of the
model function at the final parameters in matrix- or
structure-form. Users may pass information on the result of the
optimization in 'residuals' (self-explaining) and 'covd'
(covariance matrix of data). In many cases the type of objective
function of the optimization must be specified in 'objf_type';
currently, there is only a backend for the type "wls" (weighted
least squares).
Backend-specific information
The backend for 'objf_type == "wls"' (currently the only backend)
computes 'covd' (due to user request or as a prerequisite for
'covp' and 'corp') as a diagonal matrix by assuming that the
variances of data points are proportional to the reciprocal of the
squared 'weights' and guessing the factor of proportionality from
the residuals. If 'covp' is not defined (e.g. because the
Jacobian has no full rank), it makes an attempt to still compute
its uniquely defined elements, if any, and to find the additional
defined elements (being '1' or '-1'), if any, in 'corp'.
See also: curvefit_stat.
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Frontend for computation of statistics for a residual-based
minimization.
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rosenbrock
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Rosenbrock function - used to create example obj. fns.
Function value and gradient vector of the rosenbrock function
The minimizer is at the vector (1,1,..,1),
and the minimized value is 0.
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Rosenbrock function - used to create example obj.
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samin_example
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dimensionality
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dimensionality
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test_min_1
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[x,v,niter] = feval (optim_func, "testfunc","dtestf", xinit);
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[x,v,niter] = feval (optim_func, "testfunc","dtestf", xinit);
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test_min_2
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# elements: 1
# length: 60
[xlev,vlev,nlev] = feval(optim_func, "ff", "dff", xinit) ;
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[xlev,vlev,nlev] = feval(optim_func, "ff", "dff", xinit) ;
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# type: sq_string
# elements: 1
# length: 10
test_min_3
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# elements: 1
# length: 166
[xlev,vlev,nlev] = feval (optim_func, "ff", "dff", xinit, "extra", extra) ;
[xlev,vlev,nlev] = feval \
(optim_func, "ff", "dff", list (xinit, obsmat, obses));
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# type: sq_string
# elements: 1
# length: 80
[xlev,vlev,nlev] = feval (optim_func, "ff", "dff", xinit, "extra", extra) ;
[x
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test_min_4
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Plain run, just to make sure ######################################
Minimum wrt 'x' is y0
[xlev,vlev,nlev] = feval (optim_func, "ff", "dff", {x0,y0,1});
ctl.df = "dff";
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Plain run, just to make sure ######################################
Minimum wr
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# elements: 1
# length: 15
test_minimize_1
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Plain run, just to make sure ######################################
Minimum wrt 'x' is y0
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Plain run, just to make sure ######################################
Minimum wr
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test_nelder_mead_min_1
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Use vanilla nelder_mead_min
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Use vanilla nelder_mead_min
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test_nelder_mead_min_2
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# length: 70
Test using volume #################################################
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Test using volume #################################################
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test_wpolyfit
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x y dy
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x y dy
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# length: 6
vfzero
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-- Function File: vfzero (FUN, X0)
-- Function File: vfzero (FUN, X0, OPTIONS)
-- Function File: [X, FVAL, INFO, OUTPUT] = vfzero (...)
A variant of 'fzero'. Finds a zero of a vector-valued multivariate
function where each output element only depends on the input
element with the same index (so the Jacobian is diagonal).
FUN should be a handle or name of a function returning a column
vector. X0 should be a two-column matrix, each row specifying two
points which bracket a zero of the respective output element of
FUN.
If X0 is a single-column matrix then several nearby and distant
values are probed in an attempt to obtain a valid bracketing. If
this is not successful, the function fails. OPTIONS is a structure
specifying additional options. Currently, 'vfzero' recognizes
these options: '"FunValCheck"', '"OutputFcn"', '"TolX"',
'"MaxIter"', '"MaxFunEvals"'. For a description of these options,
see *note optimset: doc-optimset.
On exit, the function returns X, the approximate zero and FVAL, the
function value thereof. INFO is a column vector of exit flags that
can have these values:
* 1 The algorithm converged to a solution.
* 0 Maximum number of iterations or function evaluations has
been reached.
* -1 The algorithm has been terminated from user output
function.
* -5 The algorithm may have converged to a singular point.
OUTPUT is a structure containing runtime information about the
'fzero' algorithm. Fields in the structure are:
* iterations Number of iterations through loop.
* nfev Number of function evaluations.
* bracketx A two-column matrix with the final bracketing of the
zero along the x-axis.
* brackety A two-column matrix with the final bracketing of the
zero along the y-axis.
See also: optimset, fsolve.
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A variant of 'fzero'.
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wpolyfit
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-- Function File: [P, S] = wpolyfit (X, Y, DY, N)
Return the coefficients of a polynomial P(X) of degree N that
minimizes 'sumsq (p(x(i)) - y(i))', to best fit the data in the
least squares sense. The standard error on the observations Y if
present are given in DY.
The returned value P contains the polynomial coefficients suitable
for use in the function polyval. The structure S returns
information necessary to compute uncertainty in the model.
To compute the predicted values of y with uncertainty use
[y,dy] = polyconf(p,x,s,'ci');
You can see the effects of different confidence intervals and
prediction intervals by calling the wpolyfit internal plot function
with your fit:
feval('wpolyfit:plt',x,y,dy,p,s,0.05,'pi')
Use DY=[] if uncertainty is unknown.
You can use a chi^2 test to reject the polynomial fit:
p = 1-chi2cdf(s.normr^2,s.df);
p is the probability of seeing a chi^2 value higher than that which
was observed assuming the data are normally distributed around the
fit. If p < 0.01, you can reject the fit at the 1% level.
You can use an F test to determine if a higher order polynomial
improves the fit:
[poly1,S1] = wpolyfit(x,y,dy,n);
[poly2,S2] = wpolyfit(x,y,dy,n+1);
F = (S1.normr^2 - S2.normr^2)/(S1.df-S2.df)/(S2.normr^2/S2.df);
p = 1-f_cdf(F,S1.df-S2.df,S2.df);
p is the probability of observing the improvement in chi^2 obtained
by adding the extra parameter to the fit. If p < 0.01, you can
reject the lower order polynomial at the 1% level.
You can estimate the uncertainty in the polynomial coefficients
themselves using
dp = sqrt(sumsq(inv(s.R'))'/s.df)*s.normr;
but the high degree of covariance amongst them makes this a
questionable operation.
-- Function File: [P, S, MU] = wpolyfit (...)
If an additional output 'mu = [mean(x),std(x)]' is requested then
the X values are centered and normalized prior to computing the
fit. This will give more stable numerical results. To compute a
predicted Y from the returned model use 'y = polyval(p,
(x-mu(1))/mu(2)'
-- Function File: wpolyfit (...)
If no output arguments are requested, then wpolyfit plots the data,
the fitted line and polynomials defining the standard error range.
Example
x = linspace(0,4,20);
dy = (1+rand(size(x)))/2;
y = polyval([2,3,1],x) + dy.*randn(size(x));
wpolyfit(x,y,dy,2);
-- Function File: wpolyfit (..., 'origin')
If 'origin' is specified, then the fitted polynomial will go
through the origin. This is generally ill-advised. Use with
caution.
Hocking, RR (2003). Methods and Applications of Linear Models.
New Jersey: John Wiley and Sons, Inc.
See also: polyfit,polyconf.
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Return the coefficients of a polynomial P(X) of degree N that minimizes
'sumsq (
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wrap_f_dfdp
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[ret1, ret2] = wrap_f_dfdp (f, dfdp, varargin)
f and dftp should be the objective function (or "model function" in
curve fitting) and its jacobian, respectively, of an optimization
problem. ret1: f (varagin{:}), ret2: dfdp (varargin{:}). ret2 is
only computed if more than one output argument is given. This
manner of calling f and dfdp is needed by some optimization
functions.
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[ret1, ret2] = wrap_f_dfdp (f, dfdp, varargin)
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wsolve
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[x,s] = wsolve(A,y,dy)
Solve a potentially over-determined system with uncertainty in
the values.
A x = y +/- dy
Use QR decomposition for increased accuracy. Estimate the
uncertainty for the solution from the scatter in the data.
The returned structure s contains
normr = sqrt( A x - y ), weighted by dy
R such that R'R = A'A
df = n-p, n = rows of A, p = columns of A
See polyconf for details on how to use s to compute dy.
The covariance matrix is inv(R'*R). If you know that the
parameters are independent, then uncertainty is given by
the diagonal of the covariance matrix, or
dx = sqrt(N*sumsq(inv(s.R'))')
where N = normr^2/df, or N = 1 if df = 0.
Example 1: weighted system
A=[1,2,3;2,1,3;1,1,1]; xin=[1;2;3];
dy=[0.2;0.01;0.1]; y=A*xin+randn(size(dy)).*dy;
[x,s] = wsolve(A,y,dy);
dx = sqrt(sumsq(inv(s.R'))');
res = [xin, x, dx]
Example 2: weighted overdetermined system y = x1 + 2*x2 + 3*x3 + e
A = fullfact([3,3,3]); xin=[1;2;3];
y = A*xin; dy = rand(size(y))/50; y+=dy.*randn(size(y));
[x,s] = wsolve(A,y,dy);
dx = s.normr*sqrt(sumsq(inv(s.R'))'/s.df);
res = [xin, x, dx]
Note there is a counter-intuitive result that scaling the
uncertainty in the data does not affect the uncertainty in
the fit. Indeed, if you perform a monte carlo simulation
with x,y datasets selected from a normal distribution centered
on y with width 10*dy instead of dy you will see that the
variance in the parameters indeed increases by a factor of 100.
However, if the error bars really do increase by a factor of 10
you should expect a corresponding increase in the scatter of
the data, which will increase the variance computed by the fit.
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[x,s] = wsolve(A,y,dy)
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