AI_Bmult_NA: Builds a simple (unweighted) equitable transform.

In celphin/Equitable: Equitable Transform

Builds a simple (unweighted) equitable transform.

Description

It is based on equitable slope (A) and shift (B) matrices and data set I Both A and B can be tored efficiently by only retaining 1 column from each matrix NA values are allowed If a transform exists, this function can construct a complete data set from either: 1. a complete data set 2. A data set of only an average profile in variable t or a profile for some individual or location 3. profile at one rbitrary location is available 4. scattered observations across the different "individuals" or locations x for different values of t Examples of each are given. Only 2*length of x dimension + length of t dimension are needed to reconstruct system If the data is not precisely equitable then more data provides a better estimate. # Errors are also given to show how closely the system matches the equitbale one that is assumed.

Usage

AI_Bmult_NA(A, I, B = NULL, zero = 0, maxA = NULL)

Arguments

A	equitable slope matrix NxN
I	two dimensional data set MxN that can contain many NA values
B	equitable shift matrix MxM
zero	Value to be subtracted from I to generate B default NULL (0)
maxA	Values in the slope matrix larger than this threshold are not used in the transform Default NULL

Value

Equitable transform of data that fills in many of the NA values based on the tranformed values

Examples

#case 1 not shown. It is less interesting unless it is a noisy or corrupted system
#case 2: Use an average profile with 2 vectors from the matrices to construct
#        transformed data. (2N+M data points)
# To illustrate the idea first use an an equitable data set to construct
#            equitable matrix A and shift matrix B
#from which vectors f an u are extracted. These are used to reconstruct the matrices when desired.

d<-eg4(2,2); Td<-transformE(d)  # make a data set and find the transform and associated matrices
f<-Td$E.s[,1]                   # form
u<-Td$E.b[,1]
I<-Td$smat;I[]<-NA; I[,"Row_Ave"]<-Td$smat[,"Row_Ave"]
# the 3 necessary components are made so show what they look like and then construct system
plot(f,type="b",main="f(x) profile used for Equitable matrix A")
plot(u,type="b",main="u(x) profile used with matrix A\nto make  for shift matrix B")
plot(I[, "Row_Ave"],type="b",,main="Average profile used for reconstruction")
imagenan(I,main="Only the average profile is defined")
A<-make_A(f=f)
B<-make_B(xref=1,orig=u,A=A,uflag=TRUE)
TI<-AI_Bmult_NA(A=A,I=I,B=B)
attributes(TI)
imagenan(TI$xme,main="Reconstructed data set from Average profile")
         #data successfully reconstructed

# case 3  profile at x=10 available but it is somewhat noisy
xref<-10; I<-Td$smat;I[]<-NA;
I[,xref]<-Td$smat[,xref]
plot(I[, xref],type="b",main="Profile (no Noise)")
I[,xref]<-I[,xref]+rnorm(nrow(I),mean=0,sd=(1/10*sd(I[,xref],na.rm=TRUE)))
plot(I[, xref],type="b",main="Profile used for reconstruction (with noise)")
imagenan(I,main=paste("data only from profile at location x=",xref))
TI<-AI_Bmult_NA(A=A,I=I,B=B)
attributes(TI)
imagenan(TI$xme,main=paste("Reconstructed data set from noisy profile at location x=",xref))
# case 3  profile at x=10 available but is completely different from original
xref<-10; I<-Td$smat;I[]<-NA;

I[,xref]<-eg5(2,2)[,xref]
plot(I[, xref],type="b",main="Average profile used for reconstruction")
imagenan(I,main=paste("data only from profile at location x=",xref))
TI<-AI_Bmult_NA(A=A,I=I,B=B)
attributes(TI)
imagenan(TI$xme,main=paste("Reconstructed data set from profile at location x=",xref))

#case 5 scattered information around the data set
#using random points over the data set only works
#if the system has the same equitable sense as the matrices
I<-Td$smat
numspaces<-nrow(I)
column_location<-sample(1:ncol(I), numspaces, replace=TRUE)
I_NA<-I; I_NA[]<-NA;g<-rep(NA,nrow(I))
for(t in 1:nrow(I_NA)){I_NA[t,column_location[t]]<-I[t,column_location[t]];
 g[t]<-I[t,column_location[t]]}
plot(g,type="b",main=paste("Data set from random points on x as function of t" ))
imagenan(I_NA,main=paste("Data set from random points on x" ))
A<-make_A(f=f)
B<-make_B(xref=1,orig=u,A=A,uflag=TRUE)
TI<-AI_Bmult_NA(A=A,I=I_NA,B=B)
attributes(TI)
imagenan(TI$xme,
main=paste("Data set reconstructed from random points on x as function of t" ))

#case 5 scattered information around the data set that is noisy
#using random points over the data set assumes the system has
# the same equitable sense as the matrices
I<-Td$smat ;g<-rep(NA,nrow(I))
numspaces<-10*nrow(I) ;g<-rep(NA,numspaces)
column_location<-sample(1:(ncol(I)-1), numspaces, replace=TRUE)
I_NA<-I; I_NA[]<-NA;g<-rep(NA,nrow(I))
Ieg4<-eg4(2,2)
Ieg4<-Ieg4+  rnorm(prod(dim(Ieg4)),mean=0,sd=(1/10*sd(Ieg4,na.rm=TRUE)))
imagenan(Ieg4,main="Noise added")
for(t in 1:numspaces){
  I_NA[t%%nrow(I)+1,column_location[t]]<-Ieg4[t%%nrow(I)+1,column_location[t]]
                           g[t]<-Ieg4[t%%nrow(I)+1,column_location[t]]
                           }
imagenan(I_NA,main=paste("Data set from random points on x" ))
A<-make_A(f=f)
B<-make_B(xref=1,orig=u,A=A,uflag=TRUE)
TI<-AI_Bmult_NA(A=A,I=I_NA,B=B)
attributes(TI)
imagenan(TI$xme,
main=paste("Data set reconstructed from random points",
 "(from eg4 with noise) on x as function of t" ))
imagenan(TI$xsd,
main=paste("Std Dev in reconstruction from random points\n",
" (from eg4 with noise) on x as function of t" ))
#error values show where the system was not equitable

celphin/Equitable documentation built on Jan. 23, 2023, 11:02 p.m.