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LinearRegression
# name: <cell-element>
# type: sq_string
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# length: 1257
-- 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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adsmax
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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.
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# length: 7
battery
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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
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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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# length: 16
initial values
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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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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
http://en.wikipedia.org/wiki/Nonlinear_conjugate_gradient.
See also: bfgsmin.
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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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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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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 File: leasqr (X, Y, PIN, F)
-- Function File: leasqr (X, Y, PIN, F, STOL)
-- Function File: leasqr (X, Y, PIN, F, STOL, NITER)
-- Function File: leasqr (X, Y, PIN, F, STOL, NITER, WT)
-- Function File: leasqr (X, Y, PIN, F, STOL, NITER, WT, DP)
-- Function File: leasqr (X, Y, PIN, F, STOL, NITER, WT, DP, DFDP)
-- Function File: leasqr (X, Y, PIN, F, STOL, NITER, WT, DP, DFDP,
OPTIONS)
-- Function File: [F, P, CVG, ITER, CORP, COVP, COVR, STDRESID, Z, R2]
= leasqr (...)
Levenberg-Marquardt nonlinear regression.
Input arguments:
X
Vector or matrix of independent variables.
Y
Vector or matrix of observed values.
PIN
Vector of initial parameters to be adjusted by leasqr.
F
Name of function or function handle. The function must be of
the form 'y = f(x, p)', with y, x, p of the form Y, X, PIN.
STOL
Scalar tolerance on fractional improvement in scalar sum of
squares, i.e., 'sum ((WT .* (Y-F))^2)'. Set to 0.0001 if
empty or not given;
NITER
Maximum number of iterations. Set to 20 if empty or not
given.
WT
Statistical weights (same dimensions as Y). These should be
set to be proportional to 'sqrt (Y) ^-1', i.e., 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. Set to 'ones (size (Y))'
if empty or not given.
DP
Fractional increment of P for numerical partial derivatives.
Set to '0.001 * ones (size (PIN))' if empty or not given.
* 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 won't change
initial guess: pin(j)
DFDP
Name of partial derivative function in quotes or function
handle. If not given or empty, set to 'dfdp', a slow but
general partial derivatives function. The function must 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
explicitly specified to leasqr (see dfdp.m).
OPTIONS
Structure with multiple options. The following fields are
recognized:
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 'fract_prec', i.e., 'all (abs
(chg) < abs (options.fract_prec .* current_parm_est))' on
two consecutive iterations. Defaults to 'zeros (size
(PIN))'.
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
'max_fract_change' between sucessive iterations, i.e.,
'abs (chg(i)) = abs (min([chg(i),
options.max_fract_change(i) * current param estimate]))'.
Defaults to 'Inf * ones (size (PIN))'.
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.
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.
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.
cpiv
Function for complementary pivot algorithm for inequality
constraints. Defaults to cpiv_bard. No different
function is supplied.
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.
Output:
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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Levenberg-Marquardt nonlinear regression.
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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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lsqcurvefit
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-- Function File: lsqcurvefit (FUN, X0, XDATA, YDATA)
-- Function File: lsqcurvefit (FUN, X0, XDATA, YDATA, LB, UB)
-- Function File: lsqcurvefit (FUN, X0, XDATA, YDATA, LB, UB, OPTIONS)
-- Function File: [X, RESNORM, RESIDUAL, EXITFLAG, OUTPUT, LAMBDA,
JACOBIAN] = lsqcurvefit (...)
Solve nonlinear least-squares (nonlinear data-fitting) problems
min [EuclidianNorm (f(x, xdata) - ydata)] .^ 2
x
The first four input arguments must be provided with non-empty
initial guess X0. For a given input XDATA, YDATA is the observed
output. YDATA must be the same size as the vector (or matrix)
returned by FUN. The optional bounds LB and UB should be the same
size as X0. OPTIONS can be set with 'optimset'. Follwing Matlab
compatible options are recognized:
'Algorithm' String specifying backend algorithm. Currently
available "lm_svd_feasible" only.
'TolFun' Minimum fractional improvement in objective function in an
iteration (termination criterium). Default: 1e-6.
'TypicalX' Typical values of x. Default: 1.
'MaxIter' Maximum number of iterations allowed. Default: 400.
'Jacobian' If set to "on", the objective function must return a
second output containing a user-specified Jacobian. The Jacobian
is computed using finite differences otherwise. Default: "off"
'FinDiffType' "centered" or "forward" (Default) type finite
differences estimation.
'FinDiffRelStep' Step size factor. The default is sqrt(eps) for
forward finite differences, and eps^(1/3) for central finite
differences
'OutputFcn' One or more user-defined functions, either as a
function handle or as a cell array of function handles that an
optimization function calls at each iteration. The function
definition has the following form:
'stop = outfun(x, optimValues, state)'
'x' is the point computed at the current iteration. 'optimValues'
is a structure containing data from the current iteration in the
following fields: "iteration"- number of current iteration.
"residual"- residuals. 'state' is the state of the algorithm:
"init" at start, "iter" after each iteration and "done" at the end.
'Display' String indicating the degree of verbosity. Default:
"off". Currently only supported values are "off" (no messages) and
"iter" (some messages after each iteration).
Returned values:
X
Coefficients to best fit the nonlinear function fun(x,xdata)
to the observed values ydata.
RESNORM
Scalar value of objective as squared EuclidianNorm(f(x)).
RESIDUAL
Value of solution residuals f(x).
EXITFLAG
Status of solution:
'0'
Maximum number of iterations reached.
'2'
Change in x was less than the specified tolerance.
'3'
Change in the residual was less than the specified
tolerance.
'-1'
Output function terminated the algorithm.
OUTPUT
Structure with additional information, currently the only
field is 'iterations', the number of used iterations.
LAMBDA
Structure containing Lagrange multipliers at the solution X
sepatared by constraint type (LB and UB).
JACOBIAN
m-by-n matrix, where JACOBIAN(i,j) is the partial derivative
of FUN(I) with respect to X(J) If 'Jacobian' is set to "on" in
OPTIONS then FUN must return a second argument providing a
user-sepcified Jacobian. Otherwise, lsqnonlin approximates
the Jacobian using finite differences.
This function is a compatibility wrapper. It calls the more
general 'nonlin_curvefit' function internally.
See also: lsqnonlin, nonlin_residmin, nonlin_curvefit.
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Solve nonlinear least-squares (nonlinear data-fitting) problems
min [Euclid
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lsqlin
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-- Function File: lsqlin (C, D, A, B)
-- Function File: lsqlin (C, D, A, B, AEQ, BEQ, LB, UB)
-- Function File: lsqlin (C, D, A, B, AEQ, BEQ, LB, UB, X0)
-- Function File: lsqlin (C, D, A, B, AEQ, BEQ, LB, UB, X0, OPTIONS)
-- Function File: [X, RESNORM, RESIDUAL, EXITFLAG, OUTPUT, LAMBDA] =
lsqlin (...)
Solve the linear least squares program
min 0.5 sumsq(C*x - d)
x
subject to
A*X <= B,
AEQ*X = BEQ,
LB <= X <= UB.
The initial guess X0 and the constraint arguments (A and B, AEQ and
BEQ, LB and UB) can be set to the empty matrix ('[]') if not given.
If the initial guess X0 is feasible the algorithm is faster.
OPTIONS can be set with 'optimset', currently the only option is
'MaxIter', the maximum number of iterations (default: 200).
Returned values:
X
Position of minimum.
RESNORM
Scalar value of objective as sumsq(C*x - d).
RESIDUAL
Vector of solution residuals C*x - d.
EXITFLAG
Status of solution:
'0'
Maximum number of iterations reached.
'-2'
The problem is infeasible.
'1'
Global solution found.
OUTPUT
Structure with additional information, currently the only
field is 'iterations', the number of used iterations.
LAMBDA
Structure containing Lagrange multipliers corresponding to the
constraints.
This function calls the more general function 'quadprog'
internally.
See also: quadprog.
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Solve the linear least squares program
min 0.5 sumsq(C*x - d)
x
sub
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lsqnonlin
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-- Function File: lsqnonlin (FUN, X0)
-- Function File: lsqnonlin (FUN, X0, LB, UB)
-- Function File: lsqnonlin (FUN, X0, LB, UB, OPTIONS)
-- Function File: [X, RESNORM, RESIDUAL, EXITFLAG, OUTPUT, LAMBDA,
JACOBIAN] = lsqnonlin (...)
Solve nonlinear least-squares (nonlinear data-fitting) problems
min [EuclidianNorm(f(x))] .^ 2
x
The initial guess X0 must be provided while the bounds LB and UB)
can be set to the empty matrix ('[]') if not given.
OPTIONS can be set with 'optimset'. Follwing Matlab compatible
options are recognized:
'Algorithm' String specifying backend algorithm. Currently
available "lm_svd_feasible" only.
'TolFun' Minimum fractional improvement in objective function in an
iteration (termination criterium). Default: 1e-6.
'TypicalX' Typical values of x. Default: 1.
'MaxIter' Maximum number of iterations allowed. Default: 400.
'Jacobian' If set to "on", the objective function must return a
second output containing a user-specified Jacobian. The Jacobian
is computed using finite differences otherwise. Default: "off"
'FinDiffType' "centered" or "forward" (Default) type finite
differences estimation.
'FinDiffRelStep' Step size factor. The default is sqrt(eps) for
forward finite differences, and eps^(1/3) for central finite
differences
'OutputFcn' One or more user-defined functions, either as a
function handle or as a cell array of function handles that an
optimization function calls at each iteration. The function
definition has the following form:
'stop = outfun(x, optimValues, state)'
'x' is the point computed at the current iteration. 'optimValues'
is a structure containing data from the current iteration in the
following fields: "iteration"- number of current iteration.
"residual"- residuals. 'state' is the state of the algorithm:
"init" at start, "iter" after each iteration and "done" at the end.
'Display' String indicating the degree of verbosity. Default:
"off". Currently only supported values are "off" (no messages) and
"iter" (some messages after each iteration).
Returned values:
X
Position of minimum.
RESNORM
Scalar value of objective as squared EuclidianNorm(f(x)).
RESIDUAL
Value of solution residuals f(x).
EXITFLAG
Status of solution:
'0'
Maximum number of iterations reached.
'2'
Change in x was less than the specified tolerance.
'3'
Change in the residual was less than the specified
tolerance.
'-1'
Output function terminated the algorithm.
OUTPUT
Structure with additional information, currently the only
field is 'iterations', the number of used iterations.
LAMBDA
Structure containing Lagrange multipliers at the solution X
sepatared by constraint type (LB and UB).
JACOBIAN
m-by-n matrix, where JACOBIAN(I,J) is the partial derivative
of FUN(I) with respect to X(J) Default: lsqnonlin approximates
the Jacobian using finite differences. If 'Jacobian' is set
to "on" in OPTIONS then FUN must return a second argument
providing a user-sepcified Jacobian .
This function is a compatibility wrapper. It calls the more
general 'nonlin_residmin' function internally.
See also: lsqcurvefit, nonlin_residmin, nonlin_curvefit.
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Solve nonlinear least-squares (nonlinear data-fitting) problems
min [Euclid
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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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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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nlinfit
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-- Function File: nlinfit (X, Y, MODELFUN, BETA0)
-- Function File: nlinfit (X, Y, MODELFUN, BETA0, OPTIONS)
-- Function File: nlinfit (..., NAME, VALUE)
-- Function File: [BETA, R, J, COVB, MSE] = nlinfit (...)
Nonlinear Regression.
min [EuclidianNorm (Y - modelfun (beta, X))] ^ 2
beta
X is a matrix of independents, Y is the observed output and
MODELFUN is the nonlinear regression model function. MODELFUN
should be specified as a function handle, which accepts two inputs:
an array of coefficients and an array of independents - in that
order. The first four input arguments must be provided with
non-empty initial guess of the coefficients BETA0. Y and X must be
the same size as the vector (or matrix) returned by FUN. OPTIONS
is a structure containing estimation algorithm options. It can be
set using 'statset'. Follwing Matlab compatible options are
recognized:
'TolFun' Minimum fractional improvement in objective function in an
iteration (termination criterium). Default: 1e-6.
'MaxIter' Maximum number of iterations allowed. Default: 400.
'DerivStep' Step size factor. The default is eps^(1/3) for finite
differences gradient calculation.
'Display' String indicating the degree of verbosity. Default:
"off". Currently only supported values are "off" (no messages) and
"iter" (some messages after each iteration).
Optional NAME, VALUE pairs can be provided to set additional
options. Currently the only applicable name-value pair is
'Weights', w, where w is the array of real positive weight factors
for the squared residuals.
Returned values:
BETA
Coefficients to best fit the nonlinear function modelfun
(beta, X) to the observed values Y.
R
Value of solution residuals: 'modelfun (beta, X) - Y'. If
observation weights are specified then R is the array of
weighted residuals: 'sqrt (weights) .* modelfun (beta, X) -
Y'.
J
A matrix where 'J(i,j)' is the partial derivative of
'modelfun(i)' with respect to 'beta(j)'. If observation
weights are specified, then J is the weighted model function
Jacobian: 'diag (sqrt (weights)) * J'.
COVB
Estimated covariance matrix of the fitted coefficients.
MSE
Scalar valued estimate of the variance of error term. If the
model Jacobian is full rank, then MSE = (R' * R)/(N-p), where
N is the number of observations and p is the number of
estimated coefficients.
This function is a compatibility wrapper. It calls the more
general 'nonlin_curvefit' and 'curvefit_stat' functions internally.
See also: nonlin_residmin, nonlin_curvefit, residmin_stat,
curvefit_stat.
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Nonlinear Regression.
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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
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.
Also, if the setting 'user_interaction' is given, additional
information is passed to these functions. Type 'optim_doc ("Common
optimization options")' for this setting.
See also: nonlin_residmin.
# name: <cell-element>
# type: sq_string
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Frontend for nonlinear fitting of values, computed by a model function,
to obser
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 10
nonlin_min
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 2457
-- Function File: [P, OBJF, CVG, OUTP] = nonlin_min (F, PIN)
-- Function File: [P, OBJF, CVG, OUTP] = nonlin_min (F, PIN, SETTINGS)
Frontend for 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 separately.
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()'.
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 possible
fields: 'niter', the number of iterations, 'nobjf', the number of
objective function calls (indirect calls by gradient function not
counted), 'lambda', the lambda of constraints at the result, and
'user_interaction', information on user stops (see settings). The
backend may define additional fields. 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' (success without further
specification of criteria), '2' (parameter change less than
specified precision in two consecutive iterations), '3'
(improvement in objective function less than specified), '-1'
(algorithm aborted by a user function), or '-4' (algorithm got
stuck).
For settings, type 'optim_doc ("nonlin_min")'.
For desription of structure-based parameter handling, type
'optim_doc ("parameter structures")'.
For description of individual backends (currently only one), type
'optim_doc ("scalar optimization")' and choose the backend in the
menu.
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 67
Frontend for 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: 2725
-- 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 separately.
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()'.
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 the fields: 'niter', the
number of iterations and 'user_interaction', information on user
stops (see settings). The backend may define additional fields.
If the backend supports it, OUTP has a field 'lambda' with
determined Lagrange multipliers of any constraints, seperated into
subfields 'lower' and 'upper' for bounds, 'eqlin' and 'ineqlin' for
linear equality and inequality constraints (except bounds),
respectively, and 'eqnonlin' and 'ineqnonlin' for general equality
and inequality constraints, respectively. 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), or '-1' (algorithm aborted by a user function).
For settings, type 'optim_doc ("nonlin_residmin")'.
For desription of structure-based parameter handling, type
'optim_doc ("parameter structures")'.
For description of individual backends (currently only one), type
'optim_doc ("residual optimization")' and choose the backend in the
menu.
See also: nonlin_curvefit.
# name: <cell-element>
# type: sq_string
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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: 152
-- 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: 9
optim_doc
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 398
-- Function File: optim_doc ()
-- Function File: optim_doc (KEYWORD)
Show optim package documentation.
Runs the info viewer Octave is configured with on the documentation
in info format of the installed optim package. Without argument,
the top node of the documentation is displayed. With an argument,
the respective index entry is searched for and its node displayed.
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 33
Show optim package documentation.
# 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
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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.
This function will be removed from future versions of the optim
package since it is not related to optimization.
# name: <cell-element>
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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
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[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
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# length: 25
[y,dy] = polyconf(p,x,s)
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# type: sq_string
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polyfitinf
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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
# name: <cell-element>
# type: sq_string
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function [A,REF,HMAX,H,R,EQUAL] = polyfitinf(M,N,K,X,Y,EPSH,MAXIT,REF0)
B
# name: <cell-element>
# type: sq_string
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powell
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# type: sq_string
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# length: 2682
-- 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)
# name: <cell-element>
# type: sq_string
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Multidimensional minimization (direction-set method).
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 8
quadprog
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 2147
-- Function File: quadprog (H, F)
-- Function File: quadprog (H, F, A, B)
-- Function File: quadprog (H, F, A, B, AEQ, BEQ)
-- Function File: quadprog (H, F, A, B, AEQ, BEQ, LB, UB)
-- Function File: quadprog (H, F, A, B, AEQ, BEQ, LB, UB, X0)
-- Function File: quadprog (H, F, A, B, AEQ, BEQ, LB, UB, X0, OPTIONS)
-- Function File: [X, FVAL, EXITFLAG, OUTPUT, LAMBDA] = quadprog (...)
Solve the quadratic program
min 0.5 x'*H*x + x'*f
x
subject to
A*X <= B,
AEQ*X = BEQ,
LB <= X <= UB.
The initial guess X0 and the constraint arguments (A and B, AEQ and
BEQ, LB and UB) can be set to the empty matrix ('[]') if not given.
If the initial guess X0 is feasible the algorithm is faster.
OPTIONS can be set with 'optimset', currently the only option is
'MaxIter', the maximum number of iterations (default: 200).
Returned values:
X
Position of minimum.
FVAL
Value at the minimum.
EXITFLAG
Status of solution:
'0'
Maximum number of iterations reached.
'-2'
The problem is infeasible.
'-3'
The problem is not convex and unbounded
'1'
Global solution found.
'4'
Local solution found.
OUTPUT
Structure with additional information, currently the only
field is 'iterations', the number of used iterations.
LAMBDA
Structure containing Lagrange multipliers corresponding to the
constraints. For equality constraints, the sign of the
multipliers is chosen to satisfy the equation
0.5 H * x + f + A' * lambda_inequ + Aeq' * lambda_equ = 0 .
If lower and upper bounds are equal, or so close to each other
that they are considered equal by the algorithm, only one of
these bounds is considered active when computing the solution,
and a positive lambda will be placed only at this bound.
This function calls Octave's '__qp__' back-end algorithm
internally.
# name: <cell-element>
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Solve the quadratic program
min 0.5 x'*H*x + x'*f
x
subject to
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# length: 13
residmin_stat
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 1534
-- 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 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.
For further settings, type 'optim_doc ("residmin_stat")'.
For desription of structure-based parameter handling, type
'optim_doc ("parameter structures")'.
For backend information, type 'optim_doc ("residual optimization")'
and choose the backends type in the menu.
See also: curvefit_stat.
# name: <cell-element>
# type: sq_string
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Frontend for computation of statistics for a residual-based
minimization.
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 10
rosenbrock
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 195
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.
# name: <cell-element>
# type: sq_string
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# length: 50
Rosenbrock function - used to create example obj.
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 13
samin_example
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 16
dimensionality
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 16
dimensionality
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 7
statget
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 342
-- Function File: statget (OPTIONS, PARNAME)
-- Function File: statget (OPTIONS, PARNAME, DEFAULT)
Return the specific option PARNAME from the statistics options
structure OPTIONS created by 'statset'.
If PARNAME is not defined then return DEFAULT if supplied,
otherwise return an empty matrix.
See also: statset.
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 80
Return the specific option PARNAME from the statistics options structure
OPTIONS
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 7
statset
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 931
-- Function File: statset ()
-- Function File: OPTIONS = statset ()
-- Function File: OPTIONS = statset (PAR, VAL, ...)
-- Function File: OPTIONS = statset (OLD, PAR, VAL, ...)
-- Function File: OPTIONS = statset (OLD, NEW)
Create options structure for statistics functions.
When called without any input or output arguments, 'statset' prints
a list of all valid statistics parameters.
When called with one output and no inputs, return an options
structure with all valid option parameters initialized to '[]'.
When called with a list of parameter/value pairs, return an options
structure with only the named parameters initialized.
When the first input is an existing options structure OLD, the
values are updated from either the PAR/VAL list or from the options
structure NEW.
Please see individual statistics functions for valid settings.
See also: statget.
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 50
Create options structure for statistics functions.
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 10
test_min_1
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 63
[x,v,niter] = feval (optim_func, "testfunc","dtestf", xinit);
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 63
[x,v,niter] = feval (optim_func, "testfunc","dtestf", xinit);
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 10
test_min_2
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 60
[xlev,vlev,nlev] = feval(optim_func, "ff", "dff", xinit) ;
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 60
[xlev,vlev,nlev] = feval(optim_func, "ff", "dff", xinit) ;
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 10
test_min_3
# name: <cell-element>
# type: sq_string
# 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));
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 80
[xlev,vlev,nlev] = feval (optim_func, "ff", "dff", xinit, "extra", extra) ;
[x
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 10
test_min_4
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 173
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";
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 80
Plain run, just to make sure ######################################
Minimum wr
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 22
test_nelder_mead_min_1
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 29
Use vanilla nelder_mead_min
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 29
Use vanilla nelder_mead_min
# name: <cell-element>
# type: sq_string
# elements: 1
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test_nelder_mead_min_2
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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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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 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.
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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.
See also: 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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