187 lines
7.3 KiB
Matlab
187 lines
7.3 KiB
Matlab
% function [msyn,a,b,sa,sb,cc,rms,N]=LinReg_O(m1,m2,opt,n) - Octave Compatible Version
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% -----------------------------------------------------------------------------------------------
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% The program calculates the linear regression parameters (a and b)
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% and their uncertainties by applying different regression techniques.
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% -----
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% The program calls the following functions, appended in the scipt:
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% function [a,b,sa,sb,rms,N]=GOR(m1,m2,n) - General Orthogonal Regression
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% function [a b sa sb rms]=OLS(m1,m2) - Ordinary Least Squares
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% function [a b sa sb rms]=ILS(m1,m2) - Inverted Least Squares
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% -----------------------------------------------------------------------------------------------
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% COMPLIED BY:
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% Leptokaropoulos Konstantinos, June 2014, (kleptoka@igf.edu.pl) within IS-EPOS Project,
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% ------------------------------------------------------------------------------------------------
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% DESCRIPTION
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% The program is used to establish relationships between different magnitude scales.
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% These relationships can be then used for converting diverse magnitude scales into a
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% common one, such that a homogeneous magnitude is applied for further analyses.
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% The suggested method is GOR, because the ordinary least-squares assumes that
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% there are no uncertainties in the values of the independent variable, a condition that
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% is never truth when dealing with earthquake magnitudes. This may introduce
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% systematic errors in magnitude conversion, apparent catalog incompleteness, and
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% significant bias in the estimates of the b-value. According to GOR technique, the
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% projection of the independent variable is done along a weighted orthogonal distance
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% from the linear fitting curve. Although the error varriance ratio of the two variables
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% is introduced in calculations, Castellaro et al. (2006) showed that even if the applied
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% values differ from the real ones, GOR method still performs better than the OLS.
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% Despite the GOR superiority, OLS and ILS are also provided as supplementary
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% options in the present program, for comparison.
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% ------------------------------------------------------------------------------------------------
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% INPUT PARAMETERS:
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% m1 - variable 1 (e.g. magnitude 1) [VECTOR]
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% m2 - variable 2 (e.g. magnitude 2) [VECTOR]
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% opt - regression method selection (1,2 or 3): [SCALAR] (Default=1)
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% 1 - General Orthogonal Regression - GOR function
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% 2 - Ordinary Least Squares - OLS function
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% 3 - Inverted Least Squares - ILS function
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% n - error variance ratio (var(m2)/var(m1)) [SCALAR] (FOR GOR only, Default=1)
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% ------------------------------------------------------------------------------------------------
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% OUTPUT:
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% msyn - Converted Magnitude ([VECTOR]
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% a - intercept [SCALAR]
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% b - slope [SCALAR]
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% sa - 95% confidence interval for intercept [SCALAR]
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% sb - 95% confidence interval for slope [SCALAR]
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% cc - Pearson correlation coefficient between vectors m1 and m2 [SCALAR-optional]
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% rms - root mean square error [SCALAR-optional]
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% N - number of data pairs m1,m2 [SCALAR-optional]
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% ---------------------------------------------------------------------------------------------
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% REFERENCES
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% **** (for General Orthogonal Regression) ****
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% Fuller,W. A. (1987). Measurement Error Models,Wiley, New York, 440 pp.
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% Castellaro, S., F. Mulargia, and Y. Y. Kagan (2006). Regression problems for magnitudes,
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% Geophys. J. Int. 165, 913–930.
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% Lolli, B., and P. Gasperini (2012). A comparison among general orthogonal regression methods
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% applied to earthquake magnitude conversions, Geophys. J. Int. 190, 1135–1151.
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% Wason, H. R., R. Das, and M. L. Sharma (2012).Magnitude conversion problem using general
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% orthogonal regression, Geophys. J. Int. 190, 1091–1096.
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% ----------------------------------------------------------------------------------------------
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% LICENSE
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% This file is a part of the IS-EPOS e-PLATFORM.
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%
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% This is free software: you can redistribute it and/or modify it under
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% the terms of the GNU General Public License as published by the Free
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% Software Foundation, either version 3 of the License, or
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% (at your option) any later version.
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%
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% This program is distributed in the hope that it will be useful,
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% but WITHOUT ANY WARRANTY; without even the implied warranty of
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% MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
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% GNU General Public License for more details.
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%
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% You should have received a copy of the GNU General Public License
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% along with this program. If not, see <http://www.gnu.org/licenses/>.
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% ------------------------------------------------------------------------------------------------
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function [msyn,a,b,sa,sb,cc,rms,N]=LinReg_O(m1,m2,opt,n)
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%%%%% Remove NaN values - K03JUN2015 %%%%%
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mm1=m1(isnan(m1)==0 & isnan(m2)==0);
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mm2=m2(isnan(m1)==0 & isnan(m2)==0);
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m1=mm1;m2=mm2;
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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if nargin==2, opt=1; n=1;
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elseif nargin==3 , n=1;end
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% CC
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cc=corr(m1,m2);
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N=length(m1);
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if opt==1
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[a,b,sa,sb,rms]=GOR(m1,m2,n);
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elseif opt==2
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[a,b,sa,sb,rms]=OLS(m1,m2);
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elseif opt==3
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[a,b,sa,sb,rms]=ILS(m1,m2);
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end
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msyn=m1*b+a;
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end
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% ********************************************************
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% ------------------------- GOR --------------------------------
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function [a,b,sa,sb,rms,N]=GOR(m1,m2,n)
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if nargin==2 || isempty(n),n=1;end
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%GOR parameters
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f=1; %alternative f=0
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mx=var(m1,f);
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my=var(m2,f);
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mxy=cov(m1,m2,f);
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%mxy=mxy(2); Enable for Matlab version
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b=(my-n*mx+sqrt((my-n*mx)^2+4*n*mxy^2))/(2*mxy);
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a=mean(m2)-b*mean(m1);
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% BASIC STATISTICS
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msyn=b*m1+a;
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rms=sqrt(sum((m2-msyn).^2)/(length(m2)));
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N=length(m1);
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% Confidence Intervals of a and b-values
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d=(my-n*mx);
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sx=(sqrt(d^2+4*n*mxy^2)-d)/(2*n);
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sy=(my+n*mx-sqrt(d^2+4*n*mxy^2))/(2*n);
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s=((N-1)*(n+b^2)*sy)/(N-2);
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Vb=(mx*s-(b*sy)^2)/((N-1)*sx^2);
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Va=s/N+Vb*mean(m1)^2;
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sb=sqrt(Vb)*tinv(0.975,N-2);
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sa=sqrt(Va)*tinv(0.975,N-2);
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end
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% ----------------------------- OLS ----------------------------
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function [a b sa sb rms]=OLS(m1,m2)
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f=1;N=length(m1);
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mxy=cov(m1,m2,f);
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mx=var(m1,f);
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my=var(m2,f);
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p=polyfit(m1,m2,1);
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vb=my/sum((m1-mean(m1)).^2);
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va=my*sum(m1.^2)/(N*sum((m1-mean(m1)).^2));
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sb=tinv(0.975,N-2)*sqrt(vb);
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sa=tinv(0.975,N-2)*sqrt(va);
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b=p(1);a=p(2);
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msyn=b*m1+a;
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rms=sqrt(sum((m2-msyn).^2)/(length(m2)));
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end
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% ----------------------------- ILS ----------------------------
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function [a b sa sb rms]=ILS(m1,m2)
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f=1;N=length(m1);
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mxy=cov(m1,m2,f);
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mx=var(m1,f);
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my=var(m2,f);
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p=polyfit(m2,m1,1);
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q=[1/p(1) -p(2)/p(1)];
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vb=my/sum((m2-mean(m2)).^2);
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va=my*sum(m2.^2)/(N*sum((m2-mean(m2)).^2));
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sb=tinv(0.975,N-2)*sqrt(vb);
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sa=tinv(0.975,N-2)*sqrt(va);
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sq1=abs(sb*q(1)/p(1));
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sq2=abs(sb*q(2)/p(2));
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b=q(1);a=q(2);sb=sq1;sa=sq2;
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msyn=b*m1+a;
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rms=sqrt(sum((m2-msyn).^2)/(length(m2)));
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end
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% Testing plotting
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%plot(c.Ml,c.Mw,'ko')
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%hold on
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%plot([min(c.Ml) max(c.Ml)],[min(c.Ml)*b+a max(c.Ml)*b+a],'r--')
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