Community structure in social and biological networks

https://www.ncbi.nlm.nih.gov/pmc/articles/PMC122977/pdf/pq1202007821.pdf

Introduction to Tensors

https://www.grc.nasa.gov/www/k-12/Numbers/Math/documents/Tensors_TM2002211716.pdf

interchange of the expected value and summation

The expectation of a sum of nonnegative random variables indexed by the integers is equal to the sum of the expectations, whether the resulting number is finite or infinite. This follows from Fubini's theorem for nonnegative functions (often called Tonelli's theorem) applied to the product measure which is the product of P by the counting measure on the integers.

Square of summation

Create an empty list and flatten the list

x<-c(2.3,0.3,5.2,3.1,1.1,0.9,2.0,0.7,1.4,0.3)
y<-c(0.8,2.8,4.0,2.4,1.2,0.0,6.2,1.5,28.8,0.7)

L<-vector("list",10)

for(i in 1:10){

L[[i]]<-y[i]-x #key is [[]] operator here
}

d<-unlist(L)
median(d)  # This is Hodges-Lehmann estimator

Hodges-Lehmann Estimation of Location Shift

data e1041;
input group $ x;
datalines;
1 2.3
1 0.3
1 5.2
1 3.1
1 1.1
1 0.9
1 2.0
1 0.7
1 1.4
1 0.3
2 0.8
2 2.8
2 4.0
2 2.4
2 1.2
2 0.0
2 6.2
2 1.5
2 28.8
2 0.7
;
run;
 proc npar1way hl alpha=.05 data=e1041;
      class group;
      var x;
      exact hl;
      ods select WilcoxonScores HodgesLehmann;
  run;