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<html><head><title>[design] 5 Matrices and efficiency measures for block designs</title></head>
<body text="#000000" bgcolor="#ffffff"> [<a href = "chapters.htm">Up</a>] [<a href ="CHAP004.htm">Previous</a>] [<a href ="CHAP006.htm">Nex <h1>5 Matrices and efficiency measures for block designs</h1><p> <P> <H3>Sections</H3> <oL> <li> <A HREF="CHAP005.htm#SECT001">Matrices associated with a block design</a> <li> <A HREF="CHAP005.htm#SECT002">The function BlockDesignEfficiency</a> <li> <A HREF="CHAP005.htm#SECT003">Computing an interval for a certain real zero of a rational po </ol><p> <p> In this chapter we describe functions to calculate certain matrices associated with a block design, and the function <code>BlockDesignEfficiency</code> which determines certain statistical efficiency measures of a 1-design. <p> We also document the utility function <code>DESIGN_IntervalForLeastRealZero</code>, which is used in the calculation of E-efficiency measures, but has much wider application. <p> <p> <h2><a name="SECT001">5.1 Matrices associated with a block design</a></h2> <p><p> <a name = "SSEC001.1"></a> <li><code>PointBlockIncidenceMatrix( </code><var>D</var><code> )</code> <p> This function returns the point-block incidence matrix <var>N</var> of the block design <var>D</var>. This matrix has rows indexed by the points of <var>D</var> and columns by the blocks of <var>D</var>, with the <var>(i,j)</var>-entry of <var>N</var> being the number of times point <var>i</var> occurs in <code></code><var>D</var><code>.blocks[</code><var>j</var><c <p> The returned matrix <var>N</var> is immutable. <p> <pre> 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 ] ] </pre> <p> <a name = "SSEC001.2"></a> <li><code>ConcurrenceMatrix( </code><var>D</var><code> )</code> <p> This function returns the concurrence matrix <var>L</var> of the block design <var>D</var>. This matrix is equal to <var>NN<sup>T</sup></var>, where <var>N</var> is the point-block incidence matrix of <var>D</var> (see <a href="CHAP005.htm#SSEC001.1">PointBlockIncidenceMat <var>N<sup>T</sup></var> is the transpose of <var>N</var>. If <var>D</var> is a binary block design then the <var>(i,j)</var>-entry of its concurrence matrix is the number of blocks containing points <var>i</var> and <var>j</var>. <p> The returned matrix <var>L</var> is immutable. <p> <pre> 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 ] ] </pre> <p> <a name = "SSEC001.3"></a> <li><code>InformationMatrix( </code><var>D</var><code> )</code> <p> This function returns the information matrix <var>C</var> of the block design <var>D</var>. <p> This matrix is defined as follows. Suppose <var>D</var> has <var>v</var> points and <var>b</var> blocks, let <var>R</var> be the <var>vtimesv</var> diagonal matrix whose <var>(i,i)</var>-entry is the replication number of the point <var>i</var>, let <var>N</var> be the point-block incidence matrix of <var>D</var> (see <a href="CHAP005.htm#SSEC001.1">PointBlockIncidenceMat be the <var>btimesb</var> diagonal matrix whose <var>(j,j)</var>-entry is the length of <code></code><var>D</var><code>.blocks[</code><var>j</var><code>]</code>. Then the <strong>information mat <var>C:=R-NK<sup>-1</sup>N<sup>T</sup></var>. If <var>D</var> is a 1-<var>(v,k,r)</var> design then this expr for <var>C</var> simplifies to <var>rI-k<sup>-1</sup>L</var>, where <var>I</var> is the <var>vtimesv</var> ide matrix and <var>L</var> is the concurrence matrix of <var>D</var> (see <a href="CHAP005.htm#SSEC001. <p> The returned matrix <var>C</var> is immutable. <p> <pre> 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 ] ] </pre> <p> <p> <h2><a name="SECT002">5.2 The function BlockDesignEfficiency</a></h2> <p><p> <a name = "SSEC002.1"></a> <li><code>BlockDesignEfficiency( </code><var>D</var><code> )</code> <li><code>BlockDesignEfficiency( </code><var>D</var><code>, </code><var>eps</var><code> )</code> <li><code>BlockDesignEfficiency( </code><var>D</var><code>, </code><var>eps</var><code>, </code><var>includ <p> Let <var>D</var> be a 1-<var>(v,k,r)</var> design with <var>v>1</var>, let <var>eps</var> be a positive rational number (default: <var>10<sup>-6</sup></var>), and let <var>includeMV</var> be a boolean (default: <code>false</code>). Then this function returns a record <var>eff</var> containing information on statistical efficiency measures of <var>D</var>. These measures are defined below. See <a href="biblio.htm#Extrep"><cite>Extrep</cite></a>, <a href="biblio.htm# for further details. All returned results are computed using exact algebraic computation. <p> The component <code></code><var>eff</var><code>.A</code> contains the A-efficiency measure for <var>D</ <code></code><var>eff</var><code>.Dpowered</code> contains the D-efficiency measure of <var>D</var> rai power <var>v-1</var>, and <code></code><var>eff</var><code>.Einterval</code> is a list <var>[a,b]</var> of no rational numbers such that if <var>x</var> is the E-efficiency measure of <var>D</var> then <var>alexleb</var>, <var>b-ale</var><var>eps</var>, and if <var>x</var> is rational then <var>a=x=b</va Moreover <code></code><var>eff</var><code>.CEFpolynomial</code> contains the monic polynomial over rationals whose zeros (counting multiplicities) are the canonical efficiency factors of the design <var>D</var>. If <code></code><var>includeMV</var><code>=true</code> th additional work is done to compute the MV- (also called E<var>'-) efficiency measure, and then <code></code><var>eff</var><code>.MV</code> contains the value of this measure. (Thi component may be set even if <code></code><var>includeMV</var><code>=false</code>, as a byproduct of other computation.) <p> We now define the canonical efficiency factors and the A-, D-, E-, and MV-efficiency measures of a 1-design. <p> Let <var>D</var> be a 1-<var>(v,k,r)</var> design with <var>vge2</var>, let <var>C</var> be the information matrix of <var>D</var> (see <a href="CHAP005.htm#SSEC001.3">InformationMatrix</a>), and let <var>F The eigenvalues of <var>F</var> are all real and lie in the interval <var>[0,1]</var>. At least one of these eigenvalues is zero: an associated eigenvector is the all-<var>1</var> vector. The remaining eigenvalues <var>delta<sub>1</sub>ledelta<sub>2</sub>le cdotsledelta<sub>v-1</sub></var> of <var>F</var> are called the <strong>canonical efficiency factors</strong> of <var>D</var>. These are all non-zero if and only if <var>D</var> is connected (that is, the point-block incidence graph of <var>D</var> is a connected graph). <p> If <var>D</var> is not connected, then the A-, D-, E-, and MV-efficiency measures of <var>D</var> are all defined to be zero. Otherwise, the <strong>A-efficiency measure</strong> is <var>(v-1)/sum<sub>i=1</sub><sup>v-1</sup>1/delta<sub>i</sub></var> (the harmonic m of the canonical efficiency factors), the <strong>D-efficiency measure</strong> is <var>(prod<sub>i=1</sub><sup>v-1</sup>delta<sub>i</sub>)<sup>1/(v-1)</sup></var> (the geometric mean o the canonical efficiency factors), and the <strong>E-efficiency measure</strong> is <var>delta<sub>1</sub></var> (the minimum of the canonical efficiency factors). <p> If <var>D</var> is connected, and the MV-efficiency measure is required, then it is computed as follows. Let <var>F:=r<sup>-1</sup>C</var> be as before, and let <var>P:=v<sup>-1</sup>J</var>, where <var>J</var> is the <var>vtimesv</var> all-1 matrix. Set <var>M:=(F+P)<sup>-1</sup>-P</var>, making <var>M</var> the ``Moore-Penrose inverse'' of <var>F</var> (s <a href="biblio.htm#BaCa"><cite>BaCa</cite></a>). Then the <strong>MV-efficiency measure</strong> value (over all <var>i,jin{1,...,v}</var>, <var>inot=j</var>) of <var>2/(M<sub>ii</sub>+M<sub>jj</sub>-M<sub>ij</sub>-M<sub>ji</sub>)</var>. <p> <pre> 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 ) </pre> <p> <p> <h2><a name="SECT003">5.3 Computing an interval for a certain real zero of a rational polynomial</ <p><p> We document a DESIGN package utility function used in the calculation of the <code>Einterval</code> component above, but is more widely applicable. <p> <a name = "SSEC003.1"></a> <li><code>DESIGN_IntervalForLeastRealZero( </code><var>f</var><code>, </code><var>a</var><code>, </code><v <p> Suppose that <var>f</var> is a univariate polynomial over the rationals, <var>a</var>, <var>b</var> are rational numbers with <var><var>a</var>le<var>b</var></var>, and <var>eps</var> is a positive rational number. <p> If <var>f</var> has no real zero in the closed interval <var>[<var>a</var>,<var>b</var>]</var>, then this function returns the empty list. Otherwise, let <var>alpha</var> be the least real zero of <var>f</var> such that <var><var>a</var>lealphale<var>b</var></var>. Then this function returns a list <var>[c,d]</var> of rational numbers, with <var>clealphaled</var> and <var>d-cle<var>eps</var></var>. Moreover, <var>c=d</var> if and only if <var>alpha</var> is rational (in which case <var>alpha=c=d</var>). <p> <pre> gap> x:=Indeterminate(Rationals,1); x_1 gap> f:=(x+3)*(x^2-3); x_1^3+3*x_1^2-3*x_1-9 gap> L:=DESIGN_IntervalForLeastRealZero(f,-5,5,10^(-3)); [ -3, -3 ] gap> L:=DESIGN_IntervalForLeastRealZero(f,-2,5,10^(-3)); [ -14193/8192, -7093/4096 ] gap> List(L,Float); [ -1.73254, -1.73169 ] gap> L:=DESIGN_IntervalForLeastRealZero(f,0,5,10^(-3)); [ 14185/8192, 7095/4096 ] gap> List(L,Float); [ 1.73157, 1.73218 ] gap> L:=DESIGN_IntervalForLeastRealZero(f,0,5,10^(-5)); [ 454045/262144, 908095/524288 ] gap> List(L,Float); [ 1.73204, 1.73205 ] gap> L:=DESIGN_IntervalForLeastRealZero(f,2,5,10^(-5)); [ ] </pre> <p> <p> [<a href = "chapters.htm">Up</a>] [<a href ="CHAP004.htm">Previous</a>] [<a href ="CHAP006.htm">Nex <P> <address>design manual<br>November 2024 </address></body></html> ¤ Dauer der Verarbeitung: 0.34 Sekunden (vorverarbeitet) ¤*© Formatika GbR, Deutschland |
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