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 pdf fS(xsoft) as follow
x
S : xsoft I, P[Xsoft<xsoft] = soft du f S (u) = FS(xsoft)
Usually the pdf of soft data points are independent of one another so that fS(xsoft) = fS(x1)*…*fS(xns) .
1
In general the pdf of each individual soft datum fS(x1),…,fS(xns) can have various shapes
fS(x1)
fS(x2)
x1
fS(x3)
x2
fS(x4)
x3
x4
2
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 Xsoft = (X1 X2) ? The answer is as follow
1 ( x1  m1 ) 2
exp(
)=
2
2
1
2  1
1
fS(x1) =
( x1  m) 2
exp(
)
6
6
1
0.5 x2 if 4  x2  6
, and
0
otherwise

fS(x2) = 
fS(xsoft) = fS(x1) fS(x2)
3
Coding soft data in BMEGUI
BMEGUI supports the following three data types.
 Hard data
 Soft data with uniform distribution
 Soft data with Gaussian distribution
When using the default settings, BMEGUI assumes that the data file only contains hard data, and in that case it
uses only the fields described so far (i.e. the X field, the Y field, the T field, the optional ID field, and the Data
field containing the hard data values. However, when using a combination of hard and soft data, then BMEGUI
requires that the Data field be replaced by the following three fields: The Data type field, the Value1 field, and the
Value2 field. The Data type field is used to specify the type of data. The Value1 and Value2 fields are used to
describe the data, as follow:
 Hard data
o Data Type: 0
o Value1 Field: The true value (e.g. a measurement without error)
o Value2 Field: Same as Value 1
 Soft uniform data
o Data Type: 1
o Value1 Field: Lower bound of the interval for the true value
o Value2 Field: Upper bound of the interval for the true value
4
 Soft Gaussian data
o Data Type: 2
o Value1 Field: Mean (also called expectation) of the true value.
o Value2 Field: Standard deviation of the true value around its mean
Example (CSV Format) of hard and soft data
X,Y,T,Type,Val1,Val2
-74.35,40.55,0,0,0.4012,0.4012
-74.35,40.55,1,0,0.5528,0.5528
-74.35,40.55,2,1,0.7637,0.9637
Data type: 1 (Soft uniform data)
Lower Bound: 1.0592
Upper Bound: 1.2592
-74.35,40.55,3,1,1.0592,1.2592
-74.35,40.55,4,0,0.9344,0.9344
-74.35,40.55,5,0,0.98,0.98
-74.35,40.55,6,0,0.96489,0.96489
-74.35,40.55,7,0,0.8023,0.8023
-74.35,40.55,8,2,0.7396,0.1
Data type: 2 (Soft Gaussian data)
Mean: 0.7396
Standard Deviation: 0.1
-74.35,40.55,9,2,0.6551,0.1
-74.35,40.55,10,0,0.562,0.562
5
In order to enter soft data in BMEGUI version 2.0, the user needs to specify in the first BMEGUI dialog box which
columns of the data file correspond to the data type field, the value1 field, and the value2 field. The procedure is:
1) Check the “Use Datatype” check box, then drop down boxes for “Data Type”, “Value1 Field”, and “Value2
Field” will appear
2) Select the appropriate data columns for “Data Type”, “Value1 Field”, and “Value2 Field”
3) Click “Next” to move to the second dialog box.
6
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.
7
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);
8
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);
9
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);
10
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);
11
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
7
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
12
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);
13
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);
14
% 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
15
% 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)
16
17