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view main/optim/inst/polyconf.m @ 9930:d30cfca46e8a octave-forge
optim: upgrade license to GPLv3+ and mention on DESCRIPTION the other package licenses
author | carandraug |
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date | Fri, 30 Mar 2012 15:14:48 +0000 |
parents | ff480e262776 |
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## Author: Paul Kienzle <pkienzle@gmail.com> ## This program is granted to the public domain. ## [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-;;'); function [y,dy] = polyconf(p,x,varargin) alpha = s = []; typestr = 'pi'; for i=1:length(varargin) v = varargin{i}; if isstruct(v), s = v; elseif ischar(v), typestr = v; elseif isscalar(v), alpha = v; else s = []; end end if (nargout>1 && (isempty(s)||nargin<3)) || nargin < 2 print_usage; end if isempty(s) y = polyval(p,x); else ## For a polynomial fit, x is the set of powers ( x^n ; ... ; 1 ). n=length(p)-1; k=length(x(:)); if columns(s.R) == n, ## fit through origin A = (x(:) * ones (1, n)) .^ (ones (k, 1) * (n:-1:1)); p = p(1:n); else A = (x(:) * ones (1, n+1)) .^ (ones (k, 1) * (n:-1:0)); endif y = dy = x; [y(:),dy(:)] = confidence(A,p,s,alpha,typestr); end end %!test %! # data from Hocking, RR, "Methods and Applications of Linear Models" %! temperature=[40;40;40;45;45;45;50;50;50;55;55;55;60;60;60;65;65;65]; %! strength=[66.3;64.84;64.36;69.70;66.26;72.06;73.23;71.4;68.85;75.78;72.57;76.64;78.87;77.37;75.94;78.82;77.13;77.09]; %! [p,s] = polyfit(temperature,strength,1); %! [y,dy] = polyconf(p,40,s,0.05,'ci'); %! assert([y,dy],[66.15396825396826,1.71702862681486],200*eps); %! [y,dy] = polyconf(p,40,s,0.05,'pi'); %! assert(dy,4.45345484470743,200*eps); ## [y,dy] = confidence(A,p,s) ## ## Produce prediction intervals for the fitted y. The vector p ## and structure s are returned from wsolve. The matrix A is ## the set of observation values at which to evaluate the ## confidence interval. ## ## confidence(...,['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. ## ## confidence(...,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. ## ## Confidence intervals for linear system are given by: ## x' p +/- sqrt( Finv(1-a,1,df) var(x' p) ) ## where for confidence intervals, ## var(x' p) = sigma^2 (x' inv(A'A) x) ## and for prediction intervals, ## var(x' p) = sigma^2 (1 + x' inv(A'A) x) ## ## Rather than A'A we have R from the QR decomposition of A, but ## R'R equals A'A. Note that R is not upper triangular since we ## have already multiplied it by the permutation matrix, but it ## is invertible. Rather than forming the product R'R which is ## ill-conditioned, we can rewrite x' inv(A'A) x as the equivalent ## x' inv(R) inv(R') x = t t', for t = x' inv(R) ## Since x is a vector, t t' is the inner product sumsq(t). ## Note that LAPACK allows us to do this simultaneously for many ## different x using sqrt(sumsq(X/R,2)), with each x on a different row. ## ## Note: sqrt(F(1-a;1,df)) = T(1-a/2;df) ## ## For non-linear systems, use x = dy/dp and ignore the y output. function [y,dy] = confidence(A,p,S,alpha,typestr) if nargin < 4, alpha = []; end if nargin < 5, typestr = 'ci'; end y = A*p(:); switch typestr, case 'ci', pred = 0; default_alpha=erfc(1/sqrt(2)); case 'pi', pred = 1; default_alpha=0.05; otherwise, error("use 'ci' or 'pi' for interval type"); end if isempty(alpha), alpha = default_alpha; end s = tinv(1-alpha/2,S.df)*S.normr/sqrt(S.df); dy = s*sqrt(pred+sumsq(A/S.R,2)); end