Space/Time mapping with Soft data
Soft Data
X(p)=X(s,t) is a S/TRF.
Soft data for X(p) is available at the soft data points psoft = (p1,…, pns).
The vector of random variables xsoft = (x1,…,xns) represents the S/TRF at the soft data point, i.e.
xsoft=(X(p1),…, X(pns))
Soft data of probabilistic type are expressed using the soft pdf fS(soft) as follow
S : soft I, P[xsoft<soft] =
χsoft
 du f S (u) = FS(soft)
Usually the soft pdf is independent between soft data points so that fS(soft) = fS(1)*…*fS(ns) .
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EXAMPLE:
At point p1, the soft pdf is Gaussian with mean m1=2 and variance 12=3, and at point p2, the soft pdf
is uniform from a2=4 to b2=6. What is the soft pdf for soft = (1 2) ? The answer is as follow
fS(1) =
1 (   m1 ) 2
exp( 
)=
2
2  1
 12
1
1
(   m) 2
exp( 
)
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6
0.5  2 if 4   2  6
fS(2) = 
, and
0
otherwise

fS(soft) = fS(1) fS(2)
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Coding soft data in BMElib
In BMElib, the soft pdf fS(soft) = fS(1)*…*fS(ns) is coded in a discretized form using 4 variables: softpdftype, nl, limi, and
probdens. The reader can type: help probasyntax for a detailed explanation of how these variables work. The first variable
(softpdftype) is an integer taking values 1, 2, 3 and 4. It specifies the type of soft pdf as follows: 1 for histogram, 2 for linear, 3 for
histogram on a regular grid, and 4 for linear on a regular grid. Along each of the ns dimension the univariate pdf fS(i) is defined using
intervals of values for i. The interval limits are specified using the matrix limi, and the value of fS(i) in these intervals is specified
by the matrix probdens.
nl: (nsx1) vector of the number of interval limits. nl(i) is the number of interval limits used to define the soft pdf fS(i) at point

pi

limi: (nsxl) matrix of interval limits, where l is equal to either max(nl) or 3, depending on the softpdftype.
If softpdftype =1 or 2 then limi is a ns by max(nl) matrix, and limi(i,1: nl(i)) are the interval limits for soft data i.
If softpdftype =3 or 4 then limi is a ns by 3 matrix. The interval limits are on a regular grid, and limi(i,1:3) are the lower limit,
increment, and upper limit of the interval limits for soft data i

probdens: (nsxp) matrix of pdf values, where p is equal to either max(nl)-1 or max(nl), depending on the softpdftype.
If softpdftype ==1 or 3 then probdens is a ns by max(nl)-1 matrix. The pdf value is constant in each interval, and probdens (i,
nl(i)-1) are the value of the pdf in each interval.
If softpdftype ==2 or 4 then probdens is a ns by max(nl) matrix. The pdf value varies linearly between interval limits, and
probdens (i, nl(i)) are the value of the pdf at each interval limit.
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EXAMPLE with softpdftype =1 (soft pdf of histogram type)
>> softpdftype=1;
>> nl=4;
>> limi=[0 2 3 4];
>> probdens=[1 3 2]/7;
>> h=probaplot(softpdftype,nl,limi,probdens);
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EXAMPLE with softpdftype =2 (soft pdf of linear type)
>> softpdftype=2;
>> nl=4;
>> limi=[0 2 3 4];
>> probdens=[0 4 1 0]/7;
>> h=probaplot(softpdftype,nl,limi,probdens);
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EXAMPLE with softpdftype =3 (soft pdf of histogram type on regular grid)
>> softpdftype=3;
>> nl=5;
>> limi=[0 1 4];
>> probdens=[1 2 3 2]/8;
>> h=probaplot(softpdftype,nl,limi,probdens);
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EXAMPLE with softpdftype =4 (soft pdf of linear type on regular grid)
>> softpdftype=4;
>> nl=5;
>> limi=[0 1 4];
>> probdens=[0 3 1 2 0]/6;
>> h=probaplot(softpdftype,nl,limi,probdens);
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Writing and reading soft data from/to files
The writeProba.m and readProba.m functions allow the user to read and write soft probabilistic
data from/to a file.
Syntax
>>writeProba(cs,isST,softpdftype,nl,limi,probdens,filetitle,datafile);
>>[cs,isST,softpdftype,nl,limi,probdens,filetitle]=readProba(datafile);
EXAMPLE: the following file named ‘somesoftdata.txt’ contains the soft data at two points:
BME Probabilistic data
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s1
s2
code for the variable (equal to 1)
Type of soft pdf (equal to 1, corresponding to histogram)
number of limit values, nl
limits of intervals (nl values)
probability density (nl-1 values)
1 0.9
1 1 4
0.1 0.3 0.7 1.1
1.0 1.5 0.5
0.1 0.2
1 1 2
0.1 0.3
5.0
Plotting soft data
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The probaplot.m function, allowing to plot soft data, has the following syntax
>> h=probaplot(softpdftype,nl,limi,probdens,S,idx);
EXAMPLE:
>> [cs,isST,softpdftype,nl,limi,probdens,filetitle]=readProba('somesoftdata.txt');
>> subplot(2,1,1); h=probaplot(softpdftype,nl,limi,probdens,'-',1);
>> subplot(2,1,2); h=probaplot(softpdftype,nl,limi,probdens, '-',2);
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Generating soft data
The probaGaussian.m and probaUniform.m generate soft data of with Gaussian and uniform distributions,
respectively. For example the following code generate a two soft data points, the first is Gaussian with mean 2 and
variance 3, the second data point is Gaussian with mean 1 and variance 4.
>> [softpdftype,nl,limi,probdens]=probaGaussian([2;10],[3;4]);
>> subplot(2,1,1); h=probaplot(softpdftype,nl,limi,probdens,'-',1);
>> subplot(2,1,2); h=probaplot(softpdftype,nl,limi,probdens, '-',2);
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% The SRF X(s) is a function of space only in a 2D spatial domain
% This SRF has a mean trend equal to zero, and a covariance
% C(r)=c0*exp(-3r/ar) with c0=1, ar=5
% Additionally we have hard data at two hard data points.
% At s=(.1,4) X(s)=1.2 and at s=(5,2) X(s)=1.7
% And we have soft data (1 , .9) and (2,3).
% We want to estimate the posterior pdf and it's moments at (1,1)
% specify the general knowledge
order=NaN;
% The mean trend is equal to zero
covmodel='exponentialC'; % covariance is exponential, C(r)=c0*exp(-3r/ar)
covparam=[1 5];
% parameters for the covariance model, c0=1, ar=5
% specify the specificatory knowledge
ch=[.1 4;5 2];
% Hard data has two data points, at (.1,4) and (5,2)
zh=[1.2;1.7];
% Value of hard data at (0,4) is 1.2, and at (5,2) it is 1.7
cs=[1 .9;2 3]
% Soft data has two data points, at (1 , .9) and (2,3)
softpdftype=2;
% Soft pdf type=2 correspoinding to linear
nl=[4;3];
% Number of limits for each soft data points
limi=[0 2 3 6;1 2 4 NaN]; % Limits for each soft data points
probdens=[[0 2 10 0]/23;[0 2 0 NaN]/3]; % soft pdf value for each limit value
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% specify calculation parameters
nhmax=10;
% max number of hard data in estimation neighborhood
nsmax=10;
% max number of soft data in estimation neighborhood
dmax=[100];
% dmax=max spatial search radius for estimation neighborhood
options=BMEoptions;
% Use default options
% specify the coordinate of estimation point
ck=[1 1];
% The estimation point is (1,1)
% calculate BME posterior pdf using BMEprobaPdf
[z,pdf,info]=BMEprobaPdf([],ck,ch,cs,zh,softpdftype,nl,limi,probdens,covmodel,covparam,nhmax,nsmax,dmax,or
der,options);
% calculate moments of BME posterior pdf using BMEprobaMoments
[moments,info]=BMEprobaMoments(ck,ch,cs,zh,softpdftype,nl,limi,probdens,covmodel,covparam,nhmax,nsmax,
dmax,order,options);
expecvalk=moments(:,1)
vark=moments(:,2)
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