MDL Multiresolution Linear Regression Framework

In this work, we provide the framework to analyze a multiresolution partition (e.g. country, provinces, subdistrict) where each individual data point belongs to only one partition in each layer (e.g. \(i\) belongs to subdistrict \(A\), province \(P\), and country \(Q\)).

We assume that a partition in a higher layer subsumes lower-layer partitions (e.g. a nation is at the 1st layer subsumes all provinces at the 2nd layer).

Given \(N\) individuals that have a pair of real values \((x,y)\) that generated from independent variable \(X\) and dependent variable \(Y\). Each individual \(i\) belongs to one partition per layer.

Our goal is to find which partition at which highest level that all individuals in the this partition share the same linear model \(Y=f(X)\) where \(f\) is a linear function.

Explanation: FindMaxHomoOptimalPartitions(DataT,gamma)

library(MRReg)
## Loading required package: caret
## Loading required package: lattice
## Loading required package: ggplot2
# Generate simulation data type 4 by having 100 individuals per homogeneous partition.
DataT<-SimpleSimulation(100,type=4)

gamma <- 0.05 # Gamma parameter

out<-FindMaxHomoOptimalPartitions(DataT,gamma)
## Warning in summary.lm(submodels[[inx2]]): essentially perfect fit: summary may
## be unreliable

## Warning in summary.lm(submodels[[inx2]]): essentially perfect fit: summary may
## be unreliable

## Warning in summary.lm(submodels[[inx2]]): essentially perfect fit: summary may
## be unreliable

## Warning in summary.lm(submodels[[inx2]]): essentially perfect fit: summary may
## be unreliable

## Warning in summary.lm(submodels[[inx2]]): essentially perfect fit: summary may
## be unreliable

## Warning in summary.lm(submodels[[inx2]]): essentially perfect fit: summary may
## be unreliable

## Warning in summary.lm(submodels[[inx2]]): essentially perfect fit: summary may
## be unreliable

## Warning in summary.lm(submodels[[inx2]]): essentially perfect fit: summary may
## be unreliable

## Warning in summary.lm(submodels[[inx2]]): essentially perfect fit: summary may
## be unreliable

## Warning in summary.lm(submodels[[inx2]]): essentially perfect fit: summary may
## be unreliable

## Warning in summary.lm(submodels[[inx2]]): essentially perfect fit: summary may
## be unreliable

## Warning in summary.lm(submodels[[inx2]]): essentially perfect fit: summary may
## be unreliable

## Warning in summary.lm(submodels[[inx2]]): essentially perfect fit: summary may
## be unreliable

## Warning in summary.lm(submodels[[inx2]]): essentially perfect fit: summary may
## be unreliable

## Warning in summary.lm(submodels[[inx2]]): essentially perfect fit: summary may
## be unreliable

## Warning in summary.lm(submodels[[inx2]]): essentially perfect fit: summary may
## be unreliable

## Warning in summary.lm(submodels[[inx2]]): essentially perfect fit: summary may
## be unreliable

## Warning in summary.lm(submodels[[inx2]]): essentially perfect fit: summary may
## be unreliable

## Warning in summary.lm(submodels[[inx2]]): essentially perfect fit: summary may
## be unreliable

## Warning in summary.lm(submodels[[inx2]]): essentially perfect fit: summary may
## be unreliable

## Warning in summary.lm(submodels[[inx2]]): essentially perfect fit: summary may
## be unreliable

## Warning in summary.lm(submodels[[inx2]]): essentially perfect fit: summary may
## be unreliable

## Warning in summary.lm(submodels[[inx2]]): essentially perfect fit: summary may
## be unreliable

## Warning in summary.lm(submodels[[inx2]]): essentially perfect fit: summary may
## be unreliable

## Warning in summary.lm(submodels[[inx2]]): essentially perfect fit: summary may
## be unreliable

## Warning in summary.lm(submodels[[inx2]]): essentially perfect fit: summary may
## be unreliable

## Warning in summary.lm(submodels[[inx2]]): essentially perfect fit: summary may
## be unreliable

## Warning in summary.lm(submodels[[inx2]]): essentially perfect fit: summary may
## be unreliable

## Warning in summary.lm(submodels[[inx2]]): essentially perfect fit: summary may
## be unreliable

## Warning in summary.lm(submodels[[inx2]]): essentially perfect fit: summary may
## be unreliable

## Warning in summary.lm(submodels[[inx2]]): essentially perfect fit: summary may
## be unreliable

## Warning in summary.lm(submodels[[inx2]]): essentially perfect fit: summary may
## be unreliable

## Warning in summary.lm(submodels[[inx2]]): essentially perfect fit: summary may
## be unreliable

## Warning in summary.lm(submodels[[inx2]]): essentially perfect fit: summary may
## be unreliable

## Warning in summary.lm(submodels[[inx2]]): essentially perfect fit: summary may
## be unreliable

## Warning in summary.lm(submodels[[inx2]]): essentially perfect fit: summary may
## be unreliable

## Warning in summary.lm(submodels[[inx2]]): essentially perfect fit: summary may
## be unreliable

## Warning in summary.lm(submodels[[inx2]]): essentially perfect fit: summary may
## be unreliable

## Warning in summary.lm(submodels[[inx2]]): essentially perfect fit: summary may
## be unreliable

## Warning in summary.lm(submodels[[inx2]]): essentially perfect fit: summary may
## be unreliable

## Warning in summary.lm(submodels[[inx2]]): essentially perfect fit: summary may
## be unreliable

## Warning in summary.lm(submodels[[inx2]]): essentially perfect fit: summary may
## be unreliable

## Warning in summary.lm(submodels[[inx2]]): essentially perfect fit: summary may
## be unreliable

## Warning in summary.lm(submodels[[inx2]]): essentially perfect fit: summary may
## be unreliable

## Warning in summary.lm(submodels[[inx2]]): essentially perfect fit: summary may
## be unreliable

## Warning in summary.lm(submodels[[inx2]]): essentially perfect fit: summary may
## be unreliable

## Warning in summary.lm(submodels[[inx2]]): essentially perfect fit: summary may
## be unreliable

## Warning in summary.lm(submodels[[inx2]]): essentially perfect fit: summary may
## be unreliable

## Warning in summary.lm(submodels[[inx2]]): essentially perfect fit: summary may
## be unreliable

## Warning in summary.lm(submodels[[inx2]]): essentially perfect fit: summary may
## be unreliable

## Warning in summary.lm(submodels[[inx2]]): essentially perfect fit: summary may
## be unreliable

## Warning in summary.lm(submodels[[inx2]]): essentially perfect fit: summary may
## be unreliable

## Warning in summary.lm(submodels[[inx2]]): essentially perfect fit: summary may
## be unreliable

## Warning in summary.lm(submodels[[inx2]]): essentially perfect fit: summary may
## be unreliable

## Warning in summary.lm(submodels[[inx2]]): essentially perfect fit: summary may
## be unreliable

## Warning in summary.lm(submodels[[inx2]]): essentially perfect fit: summary may
## be unreliable

## Warning in summary.lm(submodels[[inx2]]): essentially perfect fit: summary may
## be unreliable

## Warning in summary.lm(submodels[[inx2]]): essentially perfect fit: summary may
## be unreliable

#Plotting optimal homogeneous tree The red nodes are homogeneous partitions. All children of a homogeneous partition node share the same linear model.

plotOptimalClustersTree(out)

plot of chunk unnamed-chunk-2

#Printing optimal homogeneous partitions Selected features: 1 is reserved for an intercept, and d is a selected feature if Y[i] ~ X[i,d-1] in linear model Note that the clustInfoRecRatio values are always NA for last-layer partitions.

PrintOptimalClustersResult(out, selFeature = TRUE)
## [1] "========== List of Optimal Clusters =========="
## [1] "Layer2,ClS-C1:clustInfoRecRatio=0.08,modelInfoRecRatio=0.69, eta(C)cv=1.00"
## [1] "Selected features"
## [1] 2
## [1] "Layer3,ClS-C11:clustInfoRecRatio=0.10,modelInfoRecRatio=0.65, eta(C)cv=1.00"
## [1] "Selected features"
## [1] 2
## [1] "Layer3,ClS-C12:clustInfoRecRatio=0.10,modelInfoRecRatio=0.66, eta(C)cv=1.00"
## [1] "Selected features"
## [1] 3
## [1] "Layer3,ClS-C13:clustInfoRecRatio=0.10,modelInfoRecRatio=0.71, eta(C)cv=1.00"
## [1] "Selected features"
## [1] 4
## [1] "Layer3,ClS-C14:clustInfoRecRatio=0.10,modelInfoRecRatio=0.73, eta(C)cv=1.00"
## [1] "Selected features"
## [1] 5
## [1] "Layer4,ClS-C21:clustInfoRecRatio=NA,modelInfoRecRatio=0.65, eta(C)cv=1.00"
## [1] "Selected features"
## [1] 2
## [1] "Layer4,ClS-C22:clustInfoRecRatio=NA,modelInfoRecRatio=0.65, eta(C)cv=1.00"
## [1] "Selected features"
## [1] 3
## [1] "Layer4,ClS-C23:clustInfoRecRatio=NA,modelInfoRecRatio=0.50, eta(C)cv=1.00"
## [1] "Selected features"
## [1] 4
## [1] "Layer4,ClS-C24:clustInfoRecRatio=NA,modelInfoRecRatio=0.43, eta(C)cv=1.00"
## [1] "Selected features"
## [1] 5
## [1] "Layer4,ClS-C25:clustInfoRecRatio=NA,modelInfoRecRatio=0.67, eta(C)cv=1.00"
## [1] "Selected features"
## [1] 6
## [1] "Layer4,ClS-C26:clustInfoRecRatio=NA,modelInfoRecRatio=0.66, eta(C)cv=1.00"
## [1] "Selected features"
## [1] 7
## [1] "Layer4,ClS-C27:clustInfoRecRatio=NA,modelInfoRecRatio=0.40, eta(C)cv=1.00"
## [1] "Selected features"
## [1] 8
## [1] "Layer4,ClS-C28:clustInfoRecRatio=NA,modelInfoRecRatio=0.67, eta(C)cv=1.00"
## [1] "Selected features"
## [1] 9
## [1] "min eta(C)cv:1.000000"