%?%?End-member?modelling?script?
%?%?by?E.?Dietze?A.?Borchers?&?K.?Hartmann?（2009/2010）
%?
%?%?Preconditions?of?input?data:?
%?%?grain?size?classes?as?variables?in?columns?withheader?row?in?1st?row?
%?%?samples?in?rows?with?header?column?as?1st?column
%?%?optional?water?depth?（or?e.g.?depth?in?core）?in?2nd?column
%?
%?%?Locations?of?necessary?parameters
%?%?1.?Constant?sum?value:?c?line?34
%?%?2.?Percentile?weight?for?scaling?and?transformation:?l?line?43
%?%?3.?Loop?（9A）?to?find?optimal?EMmodel:?start?line?90?end?line?202
%?%?Alternative:?number?of?end-members?for?manual?choice?（9B）:?q?line?94.?
%?%?If?9A?is?active?put?loop?lines?90?and?202?in?comment（%）（also?lines?204
%?%?to?231）?and?remove?comment?signs?from?line?234?on?
%?%?4.?Visualisation?steps?（no.?20?and?21?to?22）?only?reasonable?for?loop
%?%?routine?（9A）?and?manually?chosen?end-member?number?（9B）?respectively%%?
clear?all
data?=?load（‘WLG?Surface?sample?GS?001.txt‘）;?%?data?import?including?labels
X?=?data（2:end?3:end）;
[mn]?=?size（X）;?%?size?of?the?matrix
%?X:?（mxn）?matrix?with?original?data
%?m?rows:?observations?（e.g.?sample?ID/?sites/?depths）
%?n?columns:?variables?（e.g.?phi?classes?of?grain?sizes）
?
%?for?labelling
phi?=?data（13:end）‘;?%?variables
dcs?=?data（2:end1）;?%?observations
depth?=?data（2:end2）;?%?additional?information?from?sampling
?
%?Scaling?of?data?to?constant?sum
c?=?100;
%?c:?scalar?value?of?constant?sum?????????
?
for?j?=?1:m
????X（j:）?=?X（j:）?/?sum（X（j:）‘）?*?c;?
end
%?X:?matrix?with?data?scaled?to?constant?sum
?
%%?2.?Weight?transformation
l?=?10;
%?l:?percentile?weight?after?Manson?&?Imbrie?（1964）?Klovan?&?Imbrie?（1971）?
?for?i?=?1:m
????for?j?=?1:n
????????h?=?prctile（X（:j）?l）;
????????g?=?prctile（X（:j）?100-l）;
????????W（ij）?=?（X（ij）?-?h）?/?（g?-?h）;
????end
end
%?W:?（mxn）?matrix?with?weight?transformed?data
?
%%?3.?Definition?of?the?similarity?matrix
A?=?W‘?*?W;?
%?A:?（mxm）?matrix?Gamma?i.e.?minor?product?matrix?after?Weltje?（1997）?
?
%%?4.?Extraction?of?the?eigenspace?（Reyment?&?J?reskog?1993）
[VD]?=?eig（A）;?
%?V:?（nxn）?matrix?with?eigenvectors?
%?D:?（nxn）?matrix?with?eigenvalues?in?the?diagonal?
?
%%?5.?Eigenvalues?in?column?vector
L?=?diag（D）;?
%?L:?vector?Lambda?of?size?n?which?contains?all?eigenvalues?from?D?
?
%%?6.?Calculation?of?eigenvalue?proportions
Ln?=?L?/?sum（L）;?
%?Ln:?vector?Lambda?normalised?which?gives?the?fraction?of?variance?
%?of?the?principal?components
?
%%?7.?Proportion?of?variance?for?decision?on?minimum?number?of?eigenvectors
Lv?=?cumsum（flipud（Ln））;?
%?Lv:?vector?Lambda?cumulative?with?cumulative?fractions?of?the?
%?eigenvector?s?variance?concerning?the?absolute?variance?of?W
?
%%?8.?A?plot?for?visualisation?of?Lv
%?serves?for?evaluating?the?proportions?and?how?many?eigenvectors?are?
%?necessary?for?explaining?a?certain?level?of?the?data’s?variance
?
figure（1）
tickstep?=?1:1:n;
plot（tickstep?Lv）?xlabel（‘Number?of?eigenvectors‘）