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# 5 Matrices and efficiency measures for block designs

### Sections

1. Matrices associated with a block design
2. The function BlockDesignEfficiency

In this chapter we describe functions to calculate certain matrices associated with a block design, and the function `BlockDesignEfficiency` which determines certain statistical efficiency measures of a 1-design.

## 5.1 Matrices associated with a block design

- `PointBlockIncidenceMatrix(` D`)`

  This function returns the point-block incidence matrix N of the block design D. This matrix has rows indexed by the points of D and columns by the blocks of D, with the (i,j)-entry of N being the number of times point i occurs in ``D`.blocks[`j`]`.

  The returned matrix N is immutable.

      gap> D:=DualBlockDesign(AGPointFlatBlockDesign(2,3,1));;
      gap> BlockDesignBlocks(D);
      [ [ 1, 2, 3, 4 ], [ 1, 5, 6, 7 ], [ 1, 8, 9, 10 ], [ 2, 5, 8, 11 ],
        [ 2, 7, 9, 12 ], [ 3, 5, 10, 12 ], [ 3, 6, 9, 11 ], [ 4, 6, 8, 12 ],
        [ 4, 7, 10, 11 ] ]
      gap> PointBlockIncidenceMatrix(D);
      [ [ 1, 1, 1, 0, 0, 0, 0, 0, 0 ], [ 1, 0, 0, 1, 1, 0, 0, 0, 0 ],
        [ 1, 0, 0, 0, 0, 1, 1, 0, 0 ], [ 1, 0, 0, 0, 0, 0, 0, 1, 1 ],
        [ 0, 1, 0, 1, 0, 1, 0, 0, 0 ], [ 0, 1, 0, 0, 0, 0, 1, 1, 0 ],
        [ 0, 1, 0, 0, 1, 0, 0, 0, 1 ], [ 0, 0, 1, 1, 0, 0, 0, 1, 0 ],
        [ 0, 0, 1, 0, 1, 0, 1, 0, 0 ], [ 0, 0, 1, 0, 0, 1, 0, 0, 1 ],
        [ 0, 0, 0, 1, 0, 0, 1, 0, 1 ], [ 0, 0, 0, 0, 1, 1, 0, 1, 0 ] ]

- `ConcurrenceMatrix(` D`)`

  This function returns the concurrence matrix L of the block design D. This matrix is equal to NNT, where N is the point-block incidence matrix of D (see PointBlockIncidenceMatrix) and NT is the transpose of N. If D is a binary block design then the (i,j)-entry of its concurrence matrix is the number of blocks containing points i and j.

  The returned matrix L is immutable.

      gap> D:=DualBlockDesign(AGPointFlatBlockDesign(2,3,1));;
      gap> BlockDesignBlocks(D);
      [ [ 1, 2, 3, 4 ], [ 1, 5, 6, 7 ], [ 1, 8, 9, 10 ], [ 2, 5, 8, 11 ],
        [ 2, 7, 9, 12 ], [ 3, 5, 10, 12 ], [ 3, 6, 9, 11 ], [ 4, 6, 8, 12 ],
        [ 4, 7, 10, 11 ] ]
      gap> ConcurrenceMatrix(D);
      [ [ 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0 ],
        [ 1, 3, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1 ],
        [ 1, 1, 3, 1, 1, 1, 0, 0, 1, 1, 1, 1 ],
        [ 1, 1, 1, 3, 0, 1, 1, 1, 0, 1, 1, 1 ],
        [ 1, 1, 1, 0, 3, 1, 1, 1, 0, 1, 1, 1 ],
        [ 1, 0, 1, 1, 1, 3, 1, 1, 1, 0, 1, 1 ],
        [ 1, 1, 0, 1, 1, 1, 3, 0, 1, 1, 1, 1 ],
        [ 1, 1, 0, 1, 1, 1, 0, 3, 1, 1, 1, 1 ],
        [ 1, 1, 1, 0, 0, 1, 1, 1, 3, 1, 1, 1 ],
        [ 1, 0, 1, 1, 1, 0, 1, 1, 1, 3, 1, 1 ],
        [ 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 0 ],
        [ 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 3 ] ]

- `InformationMatrix(` D`)`

  This function returns the information matrix C of the block design D.

  This matrix is defined as follows. Suppose D has v points and b blocks, let R be the vtimesv diagonal matrix whose (i,i)-entry is the replication number of the point i, let N be the point-block incidence matrix of D (see PointBlockIncidenceMatrix), and let K be the btimesb diagonal matrix whose (j,j)-entry is the length of ``D`.blocks[`j`]`. Then the information matrix of D is C:=R-NK-1NT. If D is a 1-(v,k,r) design then this expression for C simplifies to rI-k-1L, where I is the vtimesv identity matrix and L is the concurrence matrix of D (see ConcurrenceMatrix).

  The returned matrix C is immutable.

      gap> D:=DualBlockDesign(AGPointFlatBlockDesign(2,3,1));;
      gap> BlockDesignBlocks(D);
      [ [ 1, 2, 3, 4 ], [ 1, 5, 6, 7 ], [ 1, 8, 9, 10 ], [ 2, 5, 8, 11 ],
        [ 2, 7, 9, 12 ], [ 3, 5, 10, 12 ], [ 3, 6, 9, 11 ], [ 4, 6, 8, 12 ],
        [ 4, 7, 10, 11 ] ]
      gap> InformationMatrix(D);
      [ [ 9/4, -1/4, -1/4, -1/4, -1/4, -1/4, -1/4, -1/4, -1/4, -1/4, 0, 0 ],
        [ -1/4, 9/4, -1/4, -1/4, -1/4, 0, -1/4, -1/4, -1/4, 0, -1/4, -1/4 ],
        [ -1/4, -1/4, 9/4, -1/4, -1/4, -1/4, 0, 0, -1/4, -1/4, -1/4, -1/4 ],
        [ -1/4, -1/4, -1/4, 9/4, 0, -1/4, -1/4, -1/4, 0, -1/4, -1/4, -1/4 ],
        [ -1/4, -1/4, -1/4, 0, 9/4, -1/4, -1/4, -1/4, 0, -1/4, -1/4, -1/4 ],
        [ -1/4, 0, -1/4, -1/4, -1/4, 9/4, -1/4, -1/4, -1/4, 0, -1/4, -1/4 ],
        [ -1/4, -1/4, 0, -1/4, -1/4, -1/4, 9/4, 0, -1/4, -1/4, -1/4, -1/4 ],
        [ -1/4, -1/4, 0, -1/4, -1/4, -1/4, 0, 9/4, -1/4, -1/4, -1/4, -1/4 ],
        [ -1/4, -1/4, -1/4, 0, 0, -1/4, -1/4, -1/4, 9/4, -1/4, -1/4, -1/4 ],
        [ -1/4, 0, -1/4, -1/4, -1/4, 0, -1/4, -1/4, -1/4, 9/4, -1/4, -1/4 ],
        [ 0, -1/4, -1/4, -1/4, -1/4, -1/4, -1/4, -1/4, -1/4, -1/4, 9/4, 0 ],
        [ 0, -1/4, -1/4, -1/4, -1/4, -1/4, -1/4, -1/4, -1/4, -1/4, 0, 9/4 ] ]

  ## 5.2 The function BlockDesignEfficiency

- `BlockDesignEfficiency(` D`)`
- `BlockDesignEfficiency(` D`,` eps`)`
- `BlockDesignEfficiency(` D`,` eps`,` includeMV`)`

  Let D be a 1-(v,k,r) design with v>1, let eps be a positive rational number (default: 10-6), and let includeMV be a boolean (default: `false`). Then this function returns a record eff containing information on statistical efficiency measures of D. These measures are defined below. See Extrep, BaCa and BaRo for further details. All returned results are computed using exact algebraic computation.

  The component ``eff`.A` contains the A-efficiency measure for D, ``eff`.Dpowered` contains the D-efficiency measure of D raised to the power v-1, and ``eff`.Einterval` is a list [a,b] of non-negative rational numbers such that if x is the E-efficiency measure of D then alexleb, b-ale eps, and if x is rational then a=x=b. Moreover ``eff`.CEFpolynomial` contains the monic polynomial over the rationals whose zeros (counting multiplicities) are the canonical efficiency factors of the design D. If ``includeMV`=true` then additional work is done to compute the MV- (also called E'-) efficiency measure, and then ``eff`.MV` contains the value of this measure. (This component may be set even if ``includeMV`=false`, as a byproduct of other computation.)

  We now define the canonical efficiency factors and the A-, D-, E-, and MV-efficiency measures of a 1-design.

  Let D be a 1-(v,k,r) design with vge2, let C be the information matrix of D (see InformationMatrix), and let F:=r-1C. The eigenvalues of F are all real and lie in the interval [0,1]. At least one of these eigenvalues is zero: an associated eigenvector is the all-1 vector. The remaining eigenvalues delta1ledelta2le cdotsledeltav-1 of F are called the canonical efficiency factors of D. These are all non-zero if and only if D is connected (that is, the point-block incidence graph of D is a connected graph).

  If D is not connected, then the A-, D-, E-, and MV-efficiency measures of D are all defined to be zero. Otherwise, the A-efficiency measure is (v-1)/sumi=1v-11/deltai (the harmonic mean of the canonical efficiency factors), the D-efficiency measure is (prodi=1v-1deltai)1/(v-1) (the geometric mean of the canonical efficiency factors), and the E-efficiency measure is delta1 (the minimum of the canonical efficiency factors).

  If D is connected, and the MV-efficiency measure is required, then it is computed as follows. Let F:=r-1C be as before, and let P:=v-1J, where J is the vtimesv all-1 matrix. Set M:=(F+P)-1-P, making M the ``Moore-Penrose inverse'' of F (see BaCa). Then the MV-efficiency measure of D is the minimum value (over all i,jin{1,...,v}, inot=j) of 2/(Mii+Mjj-Mij-Mji).

      gap> D:=DualBlockDesign(AGPointFlatBlockDesign(2,3,1));;
      gap> BlockDesignBlocks(D);
      [ [ 1, 2, 3, 4 ], [ 1, 5, 6, 7 ], [ 1, 8, 9, 10 ], [ 2, 5, 8, 11 ],
        [ 2, 7, 9, 12 ], [ 3, 5, 10, 12 ], [ 3, 6, 9, 11 ], [ 4, 6, 8, 12 ],
        [ 4, 7, 10, 11 ] ]
      gap> BlockDesignEfficiency(D);
      rec( A := 33/41,
        CEFpolynomial := x_1^11-9*x_1^10+147/4*x_1^9-719/8*x_1^8+18723/128*x_1^7-106\
      47/64*x_1^6+138159/1024*x_1^5-159813/2048*x_1^4+2067201/65536*x_1^3-556227/655\
      36*x_1^2+89667/65536*x_1-6561/65536, Dpowered := 6561/65536,
        Einterval := [ 3/4, 3/4 ] )
      gap> BlockDesignEfficiency(D,10^(-4),true);
      rec( A := 33/41,
        CEFpolynomial := x_1^11-9*x_1^10+147/4*x_1^9-719/8*x_1^8+18723/128*x_1^7-106\
      47/64*x_1^6+138159/1024*x_1^5-159813/2048*x_1^4+2067201/65536*x_1^3-556227/655\
      36*x_1^2+89667/65536*x_1-6561/65536, Dpowered := 6561/65536,
        Einterval := [ 3/4, 3/4 ], MV := 3/4 )

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