# Aligned Induction

## Fud inducers

### Sections

Practicable shuffle content alignment valency-density fud inducer

Practicable highest-layer shuffle content alignment valency-density fud inducer

Practicable level shuffle content alignment valency-density fud inducer

### Practicable shuffle content alignment valency-density fud inducer

Given the limited-layer limited-underlying limited-breadth fud tree searcher (Text) $Z_{P,A,A_R,\mathrm{L}} = \mathrm{searchTreer}(\mathcal{F}_{\infty,U_A,V_A} \cap \mathcal{F}_{\mathrm{u}} \cap \mathcal{F}_{\mathrm{b}} \cap \mathcal{F}_{\mathrm{h}},P_{P,A,A_R,\mathrm{L}},\{\emptyset\})$ and the limited-derived derived variables set list maximiser (Text) $Z_{P,A,A_R,F,\mathrm{D}} = \mathrm{maximiseLister}(X_{P,A,A_R,F,\mathrm{D}},P_{P,A,A_R,F,\mathrm{D}},\mathrm{top}(\mathrm{omax}),R_{P,A,A_R,F,\mathrm{D}})$ the practicable shuffle content alignment valency-density fud inducer, $I_{z,\mathrm{csd,F,\infty,q},P}^{‘}$, may then be implemented (Text) $\begin{eqnarray} &&I_{z,\mathrm{csd,F,\infty,q},P}^{‘ * }(A) = \\ &&\hspace{2em}\{(G,I_{\mathrm{csd}}^{ * }((A,A_R,G))) : \\ &&\hspace{5em}|V_A|>1,~\{F_{\mathrm{L}}\} = \mathrm{leaves}(\mathrm{tree}(Z_{P,A,A_R,\mathrm{L}})),\\ &&\hspace{5em}K \in \mathrm{maxd}(\mathrm{elements}(Z_{P,A,A_R,F_{\mathrm{L}},\mathrm{D}})),~G = \mathrm{depends}(F_{\mathrm{L}},K)\} \cup \\ &&\hspace{2em}\{(\emptyset,0) : |V_A| \leq 1\} \end{eqnarray}$ where the shuffle content alignment valency-density computer $I_{\mathrm{csd}} \in \mathrm{computers}$ is defined as $\begin{eqnarray} I_{\mathrm{csd}}^{ * }((A,A_R,F)) = (I_{\mathrm{a}}^{ * }(A * F^{\mathrm{T}})-I_{\mathrm{a}}^{ * }(A_R * F^{\mathrm{T}}))/I_{\mathrm{cvl}}^{ * }(F) \end{eqnarray}$ The fud inducer is defined in module AlignmentPracticable,

parametersSystemsSamplesShufflesListVariablesInducerFud ::
Integer -> Integer -> Integer -> Integer -> Integer -> Integer -> Integer -> Integer ->
System -> Histogram -> Histogram -> [Variable] ->
Maybe ((Fud,Fud),(System,[Variable]))


as

parametersSystemsSamplesShufflesListVariablesInducerFud wmax lmax xmax omax bmax mmax umax pmax uu aa aarr ll
...
| ... = Just $((depends ff kk, ff),(uu',ll')) | otherwise = Nothing where (nn,(uu',ll')) = zzllfuds lmax xmax omax bmax mmax umax pmax uu aa aarr ll ff = if nn /= [] then last nn else fudEmpty kk = maxdff ff$ zzffcsdderopt wmax omax uu' aa aarr ff
zzllfuds lmax xmax omax bmax mmax umax pmax uu aa aarr ll = fromJust $parametersSystemsSamplesShufflesListVariablesSearcherFud lmax xmax omax bmax mmax umax pmax uu aa aarr ll zzffcsdderopt wmax omax uu aa rr ff = fst$ unzip $fromJust$
parametersSystemsSamplesShufflesFudsFunctionOptimiserDerived wmax omax uu aa rr ff
...


Note that, again, the partition variables are replaced by cardinal variables, so the fud inducer has an additional argument of a list of variables, [Variable].

For example,

let zzllfuds aa aarr lmax xmax omax bmax mmax umax pmax = fst $fromJust$ parametersSystemsSamplesShufflesListVariablesSearcherFud lmax xmax omax bmax mmax umax pmax (sys aa) aa aarr [VarInt i | i <-[10..]]

let iillfuds aa aarr wmax lmax xmax omax bmax mmax umax pmax = fst $fromJust$ parametersSystemsSamplesShufflesListVariablesInducerFud wmax lmax xmax omax bmax mmax umax pmax (sys aa) aa aarr [VarInt i | i <-[10..]]



let aa = resize 100 $regpivot 3 3 algnden aa 17.240346078711394 let ff = last$ zzllfuds aa (ind aa) 1 (3^3) 3 3 3 (3^3) 1

rpln $(concat . map (aall . ttaa) . Set.toList . ffqq)$ ff
"({(1,1),(64,1)},1 % 1)"
"({(1,2),(64,2)},1 % 1)"
"({(1,3),(64,2)},1 % 1)"
"({(2,1),(65,1)},1 % 1)"
"({(2,2),(65,2)},1 % 1)"
"({(2,3),(65,2)},1 % 1)"
"({(3,1),(66,1)},1 % 1)"
"({(3,2),(66,2)},1 % 1)"
"({(3,3),(66,2)},1 % 1)"

rpln $zip [0.. ]$ fst $unzip$ zzffcsdderopt aa (ind aa) ff (3^3) 3
"(0,{({64,65},15.918481590327701),({64,66},15.918481590327701),({65,66},15.918481590327701)})"
"(1,{({64,65,66},31.253377751394225)})"
"(2,{})"


let (gg,_) = iillfuds aa (ind aa) (3^3) 1 (3^3) 3 3 3 (3^3) 1

rpln $(concat . map (aall . ttaa) . Set.toList . ffqq)$ gg
"({(1,1),(64,1)},1 % 1)"
"({(1,2),(64,2)},1 % 1)"
"({(1,3),(64,2)},1 % 1)"
"({(2,1),(65,1)},1 % 1)"
"({(2,2),(65,2)},1 % 1)"
"({(2,3),(65,2)},1 % 1)"
"({(3,1),(66,1)},1 % 1)"
"({(3,2),(66,2)},1 % 1)"
"({(3,3),(66,2)},1 % 1)"

algnden (aa fapply ff) - algnden (ind aa fapply ff)
31.253377751394225

algnden (aa fapply gg) - algnden (ind aa fapply gg)
31.253377751394225

let aa = resize 1000 $regpivot 3 4 algnden aa 197.70429785167653 let (gg,_) = iillfuds aa (ind aa) (3^3) 1 (3^3) 3 3 3 (3^2) 1 rpln$ (concat . map (aall . ttaa) . Set.toList . ffqq) $gg "({(2,1),(64,1)},1 % 1)" "({(2,2),(64,2)},1 % 1)" "({(2,3),(64,2)},1 % 1)" "({(3,1),(65,1)},1 % 1)" "({(3,2),(65,2)},1 % 1)" "({(3,3),(65,2)},1 % 1)" "({(4,1),(66,1)},1 % 1)" "({(4,2),(66,2)},1 % 1)" "({(4,3),(66,2)},1 % 1)" algnden (aa fapply gg) - algnden (ind aa fapply gg) 218.3586352266184 let (gg,_) = iillfuds aa (ind aa) (3^4) 1 (3^4) 4 4 4 (3^2) 1 rpln$ (concat . map (aall . ttaa) . Set.toList . ffqq) $gg "({(1,1),(146,1)},1 % 1)" "({(1,2),(146,2)},1 % 1)" "({(1,3),(146,2)},1 % 1)" "({(2,1),(147,1)},1 % 1)" "({(2,2),(147,2)},1 % 1)" "({(2,3),(147,2)},1 % 1)" "({(3,1),(148,1)},1 % 1)" "({(3,2),(148,2)},1 % 1)" "({(3,3),(148,2)},1 % 1)" "({(4,1),(149,1)},1 % 1)" "({(4,2),(149,2)},1 % 1)" "({(4,3),(149,2)},1 % 1)" algnden (aa fapply gg) - algnden (ind aa fapply gg) 325.96965461170066  Now consider predicting the rain in the weather forecast example (summarised in Functional definition sets), let aa = hhaa hh algnden$ aa red [pressure,cloud,wind]
2.15004722009382


let ff = last $zzllfuds (aa red [pressure,cloud,wind]) (ind (aa red [pressure,cloud,wind])) 1 (3^3) 3 3 2 (3^3) 1   rpln$ (concat . map (aall . ttaa) . Set.toList . ffqq) $ff "({(cloud,heavy),(296,1)},1 % 1)" "({(cloud,light),(296,2)},1 % 1)" "({(cloud,none),(296,2)},1 % 1)" "({(pressure,high),(wind,light),(297,1)},1 % 1)" "({(pressure,high),(wind,none),(297,1)},1 % 1)" "({(pressure,high),(wind,strong),(297,1)},1 % 1)" "({(pressure,low),(wind,light),(297,2)},1 % 1)" "({(pressure,low),(wind,none),(297,1)},1 % 1)" "({(pressure,low),(wind,strong),(297,2)},1 % 1)" "({(pressure,medium),(wind,light),(297,1)},1 % 1)" "({(pressure,medium),(wind,none),(297,1)},1 % 1)" "({(pressure,medium),(wind,strong),(297,1)},1 % 1)" rpln$ zip [0.. ] $fst$ unzip $zzffcsdderopt aa (ind aa) ff (3^3) 3 "(0,{({296,297},3.0601487103147385)})" "(1,{})"   let (gg,_) = iillfuds (aa red [pressure,cloud,wind]) (ind (aa red [pressure,cloud,wind])) (3^3) 1 (3^3) 3 3 2 (3^3) 1   rpln$ (concat . map (aall . ttaa) . Set.toList . ffqq) $gg "({(cloud,heavy),(296,1)},1 % 1)" "({(cloud,light),(296,2)},1 % 1)" "({(cloud,none),(296,2)},1 % 1)" "({(pressure,high),(wind,light),(297,1)},1 % 1)" "({(pressure,high),(wind,none),(297,1)},1 % 1)" "({(pressure,high),(wind,strong),(297,1)},1 % 1)" "({(pressure,low),(wind,light),(297,2)},1 % 1)" "({(pressure,low),(wind,none),(297,1)},1 % 1)" "({(pressure,low),(wind,strong),(297,2)},1 % 1)" "({(pressure,medium),(wind,light),(297,1)},1 % 1)" "({(pressure,medium),(wind,none),(297,1)},1 % 1)" "({(pressure,medium),(wind,strong),(297,1)},1 % 1)" algnden (aa fapply ff) - algnden (ind aa fapply ff) 3.0601487103147385 algnden (aa fapply gg) - algnden (ind aa fapply gg) 3.0601487103147385   let ff = last$ zzllfuds (aa red [pressure,cloud,wind]) (ind (aa red [pressure,cloud,wind])) 1 (3^3) 3 (3*2) 3 (3^3) 2



rpln $(concat . map (aall . ttaa) . Set.toList . ffqq)$ ff
"({(cloud,heavy),(197,1)},1 % 1)"
"({(cloud,light),(197,2)},1 % 1)"
"({(cloud,none),(197,2)},1 % 1)"
"({(pressure,high),(196,1)},1 % 1)"
"({(pressure,low),(196,2)},1 % 1)"
"({(pressure,medium),(196,1)},1 % 1)"
"({(wind,light),(161,1)},1 % 1)"
"({(wind,none),(161,1)},1 % 1)"
"({(wind,strong),(161,2)},1 % 1)"
"({(wind,light),(171,1)},1 % 1)"
"({(wind,none),(171,2)},1 % 1)"
"({(wind,strong),(171,1)},1 % 1)"

rpln $zip [0.. ]$ fst $unzip$ zzffcsdderopt aa (ind aa) ff (3^3) 3
"(0,{({161,197},0.6057834603672152),({171,197},0.5399739167361197),({196,197},2.4883225433699216)})"
"(1,{({161,171,197},0.8225606143847344),({161,196,197},2.751189151274419),({171,196,197},2.7587225646757725)})"
"(2,{({161,171,196,197},2.7398023210803686)})"
"(3,{})"


let (gg,_) = iillfuds (aa red [pressure,cloud,wind]) (ind (aa red [pressure,cloud,wind])) (3^3) 1 (3^3) 3 (3*2) 3 (3^3) 2



rpln $(concat . map (aall . ttaa) . Set.toList . ffqq)$ gg
"({(cloud,heavy),(197,1)},1 % 1)"
"({(cloud,light),(197,2)},1 % 1)"
"({(cloud,none),(197,2)},1 % 1)"
"({(pressure,high),(196,1)},1 % 1)"
"({(pressure,low),(196,2)},1 % 1)"
"({(pressure,medium),(196,1)},1 % 1)"
"({(wind,light),(171,1)},1 % 1)"
"({(wind,none),(171,2)},1 % 1)"
"({(wind,strong),(171,1)},1 % 1)"

algnden (aa fapply ff) - algnden (ind aa fapply ff)
2.7398023210803686

algnden (aa fapply gg) - algnden (ind aa fapply gg)
2.7587225646757725


let ff = last $zzllfuds (aa red [pressure,cloud,wind]) (ind (aa red [pressure,cloud,wind])) 1 (3^3) 3 (3*2) 2 (3^3) 2   rpln$ (concat . map (aall . ttaa) . Set.toList . ffqq) $ff "({(cloud,heavy),(607,1)},1 % 1)" "({(cloud,light),(607,2)},1 % 1)" "({(cloud,none),(607,2)},1 % 1)" "({(cloud,heavy),(614,1)},1 % 1)" "({(cloud,light),(614,2)},1 % 1)" "({(cloud,none),(614,3)},1 % 1)" "({(cloud,heavy),(631,1)},1 % 1)" "({(cloud,light),(631,1)},1 % 1)" "({(cloud,none),(631,2)},1 % 1)" "({(pressure,high),(wind,light),(577,1)},1 % 1)" "({(pressure,high),(wind,none),(577,1)},1 % 1)" "({(pressure,high),(wind,strong),(577,1)},1 % 1)" "({(pressure,low),(wind,light),(577,2)},1 % 1)" "({(pressure,low),(wind,none),(577,1)},1 % 1)" "({(pressure,low),(wind,strong),(577,2)},1 % 1)" "({(pressure,medium),(wind,light),(577,1)},1 % 1)" "({(pressure,medium),(wind,none),(577,1)},1 % 1)" "({(pressure,medium),(wind,strong),(577,1)},1 % 1)" "({(pressure,high),(606,1)},1 % 1)" "({(pressure,low),(606,2)},1 % 1)" "({(pressure,medium),(606,1)},1 % 1)" "({(wind,light),(615,1)},1 % 1)" "({(wind,none),(615,2)},1 % 1)" "({(wind,strong),(615,3)},1 % 1)" "({(wind,light),(630,1)},1 % 1)" "({(wind,none),(630,1)},1 % 1)" "({(wind,strong),(630,2)},1 % 1)" rpln$ zip [0.. ] $fst$ unzip $zzffcsdderopt aa (ind aa) ff (3^3) 3 "(0,{({577,607},3.0601487103147385),({577,614},2.3923266698540573),({606,607},2.4883225433699216)})" "(1,{({577,606,607},3.891346416043536),({577,606,614},3.358279087934026),({577,607,630},3.1643965626442903)})" "(2,{({577,606,607,630},3.8511125582823595),({577,606,607,631},3.8213440591293804),({577,606,614,630},3.8286774042639347)})" "(3,{})" rpln$ zip [0.. ] $fst$ unzip $zzffcsdderopt aa (ind aa) ff (3^7) 3 "(0,{({577,607},3.0601487103147385),({577,614},2.3923266698540573),({606,607},2.4883225433699216)})" "(1,{({577,606,607},3.891346416043536),({577,606,614},3.358279087934026),({577,607,630},3.1643965626442903)})" "(2,{({577,606,607,630},3.8511125582823595),({577,606,607,631},3.8213440591293804),({577,606,614,630},3.8286774042639347)})" "(3,{({577,606,607,614,630},3.8388619125562538),({577,606,607,630,631},4.215798138230515),({577,606,614,630,631},3.8107275511105954)})" "(4,{({577,606,607,614,615,630},3.01163366978457),({577,606,607,614,630,631},3.7980303142856844),({577,606,607,615,630,631},3.265837542015048)})" "(5,{({577,606,607,614,615,630,631},3.001766092132741)})" "(6,{})"   let (gg,_) = iillfuds (aa red [pressure,cloud,wind]) (ind (aa red [pressure,cloud,wind])) (3^7) 1 (3^3) 3 (3*2) 2 (3^3) 2   rpln$ (concat . map (aall . ttaa) . Set.toList . ffqq) $gg "({(cloud,heavy),(607,1)},1 % 1)" "({(cloud,light),(607,2)},1 % 1)" "({(cloud,none),(607,2)},1 % 1)" "({(cloud,heavy),(631,1)},1 % 1)" "({(cloud,light),(631,1)},1 % 1)" "({(cloud,none),(631,2)},1 % 1)" "({(pressure,high),(wind,light),(577,1)},1 % 1)" "({(pressure,high),(wind,none),(577,1)},1 % 1)" "({(pressure,high),(wind,strong),(577,1)},1 % 1)" "({(pressure,low),(wind,light),(577,2)},1 % 1)" "({(pressure,low),(wind,none),(577,1)},1 % 1)" "({(pressure,low),(wind,strong),(577,2)},1 % 1)" "({(pressure,medium),(wind,light),(577,1)},1 % 1)" "({(pressure,medium),(wind,none),(577,1)},1 % 1)" "({(pressure,medium),(wind,strong),(577,1)},1 % 1)" "({(pressure,high),(606,1)},1 % 1)" "({(pressure,low),(606,2)},1 % 1)" "({(pressure,medium),(606,1)},1 % 1)" "({(wind,light),(630,1)},1 % 1)" "({(wind,none),(630,1)},1 % 1)" "({(wind,strong),(630,2)},1 % 1)" algnden (aa fapply ff) - algnden (ind aa fapply ff) 3.001766092132741 algnden (aa fapply gg) - algnden (ind aa fapply gg) 4.215798138230515 rpln$ aarr $(norm (aa fapply gg)) divide (norm (vvc fapply gg)) "({(577,1),(606,1),(607,2),(630,1),(631,1)},1.35)" "({(577,1),(606,1),(607,2),(630,1),(631,2)},2.7)" "({(577,1),(606,1),(607,2),(630,2),(631,1)},1.35)" "({(577,1),(606,2),(607,2),(630,1),(631,1)},1.35)" "({(577,2),(606,2),(607,1),(630,1),(631,1)},2.7)" "({(577,2),(606,2),(607,1),(630,2),(631,1)},2.7)" "({(577,2),(606,2),(607,2),(630,1),(631,2)},1.35)" algn (aa fapply gg) - algn (ind aa fapply gg) 8.43159627646103 algnden (aa fapply gg) - algnden (ind aa fapply gg) 4.215798138230515 ent (aa fapply gg) 1.6787526304900946 cent (fftt gg) (aa red [cloud,pressure,wind]) 0.805614209483707 rent (aa fapply gg) (vvc fapply gg) 12.077070563340271 tlalgn (fftt gg) aa [rain] 14.034757670921657 tlent (fftt gg) aa [rain] 10.043858601430028 size$ eff (aa fapply gg) mul (vvc fapply gg)
42 % 1


The weather forecast example continues in Practicable highest-layer shuffle content alignment valency-density fud inducer.

### Practicable highest-layer shuffle content alignment valency-density fud inducer

Given the highest-layer limited-layer limited-underlying limited-breadth fud tree searcher (Text) $Z_{P,A,A_R,\mathrm{L,d}} = \mathrm{searchTreer}(\mathcal{F}_{\infty,U_A,V_A} \cap \mathcal{F}_{\mathrm{u}} \cap \mathcal{F}_{\mathrm{b}} \cap \mathcal{F}_{\mathrm{h}},P_{P,A,A_R,\mathrm{L,d}},\{\emptyset\})$ and the highest-layer limited-derived derived variables set list maximiser (Text) $Z_{P,A,A_R,F,\mathrm{D,d}} = \mathrm{maximiseLister}(X_{P,A,A_R,F,\mathrm{D}},P_{P,A,A_R,F,\mathrm{D}},\mathrm{top}(\mathrm{omax}),R_{P,A,A_R,F,\mathrm{D,d}})$ the practicable highest-layer shuffle content alignment valency-density fud inducer, $I_{z,\mathrm{csd,F,\infty,q},P,\mathrm{d}}^{‘}$, may then be implemented (Text) $\begin{eqnarray} &&I_{z,\mathrm{csd,F,\infty,q},P,\mathrm{d}}^{‘ * }(A) = \\ &&\hspace{2em}\{(G,I_{\mathrm{csd}}^{ * }((A,A_R,G))) : \\ &&\hspace{5em}|V_A|>1,~\{F_{\mathrm{L}}\} = \mathrm{leaves}(\mathrm{tree}(Z_{P,A,A_R,\mathrm{L,d}})),\\ &&\hspace{5em}K \in \mathrm{maxd}(\mathrm{elements}(Z_{P,A,A_R,F_{\mathrm{L}},\mathrm{D,d}})),~G = \mathrm{depends}(F_{\mathrm{L}},K)\} \cup \\ &&\hspace{2em}\{(\emptyset,0) : |V_A| \leq 1\} \end{eqnarray}$ The fud inducer is defined in module AlignmentPracticable,

parametersSystemsSamplesShufflesListVariablesInducerFudHighest ::
Integer -> Integer -> Integer -> Integer -> Integer -> Integer -> Integer -> Integer ->
System -> Histogram -> Histogram -> [Variable] ->
Maybe ((Fud,Fud),(System,[Variable]))


as

parametersSystemsSamplesShufflesListVariablesInducerFudHighest wmax lmax xmax omax bmax mmax umax pmax uu aa aarr ll
...
| ... = Just $((depends ff kk, ff),(uu',ll')) | otherwise = Nothing where (nn,(uu',ll')) = zzllfudshigh wmax lmax xmax omax bmax mmax umax pmax uu aa aarr ll ff = if nn /= [] then last nn else fudEmpty kk = maxdff ff$ zzffcsdderhighopt wmax omax uu' aa aarr ff
zzllfudshigh wmax lmax xmax omax bmax mmax umax pmax uu aa aarr ll = fromJust $parametersSystemsSamplesShufflesListVariablesSearcherFudHighest wmax lmax xmax omax bmax mmax umax pmax uu aa aarr ll zzffcsdderhighopt wmax omax uu aa rr ff = fst$ unzip $fromJust$
parametersSystemsSamplesShufflesFudsFunctionOptimiserDerivedHighest wmax omax uu aa rr ff
...


Note that, again, the partition variables are replaced by cardinal variables, so the fud inducer has an additional argument of a list of variables, [Variable].

For example,

let iillfuds aa aarr wmax lmax xmax omax bmax mmax umax pmax = fst $fst$ fromJust $parametersSystemsSamplesShufflesListVariablesInducerFud wmax lmax xmax omax bmax mmax umax pmax (sys aa) aa aarr [VarInt i | i <-[101..]] let iillfudshigh aa aarr wmax lmax xmax omax bmax mmax umax pmax = fst$ fst $fromJust$ parametersSystemsSamplesShufflesListVariablesInducerFudHighest wmax lmax xmax omax bmax mmax umax pmax (sys aa) aa aarr [VarInt i | i <-[101..]]



let aa = resize 100 $regpivot 3 3 let gg = iillfuds aa (ind aa) (3^3) 1 (3^3) 3 3 3 (3^3) 1 rpln$ (concat . map (aall . ttaa) . Set.toList . ffqq) $gg "({(1,1),(155,1)},1 % 1)" "({(1,2),(155,2)},1 % 1)" "({(1,3),(155,2)},1 % 1)" "({(2,1),(156,1)},1 % 1)" "({(2,2),(156,2)},1 % 1)" "({(2,3),(156,2)},1 % 1)" "({(3,1),(157,1)},1 % 1)" "({(3,2),(157,2)},1 % 1)" "({(3,3),(157,2)},1 % 1)" algnden (aa fapply gg) - algnden (ind aa fapply gg) 31.253377751394225 let gg = iillfudshigh aa (ind aa) (3^3) 1 (3^3) 3 3 3 (3^3) 1 rpln$ (concat . map (aall . ttaa) . Set.toList . ffqq) $gg "({(1,1),(155,1)},1 % 1)" "({(1,2),(155,2)},1 % 1)" "({(1,3),(155,2)},1 % 1)" "({(2,1),(156,1)},1 % 1)" "({(2,2),(156,2)},1 % 1)" "({(2,3),(156,2)},1 % 1)" "({(3,1),(157,1)},1 % 1)" "({(3,2),(157,2)},1 % 1)" "({(3,3),(157,2)},1 % 1)" algnden (aa fapply gg) - algnden (ind aa fapply gg) 31.253377751394225 let aa = resize 1000$ regpivot 3 4

algnden aa
197.70429785167653

let gg = iillfuds aa (ind aa) (3^3) 1 (3^3) 3 3 3 (3^2) 1

rpln $(concat . map (aall . ttaa) . Set.toList . ffqq)$ gg
"({(2,1),(155,1)},1 % 1)"
"({(2,2),(155,2)},1 % 1)"
"({(2,3),(155,2)},1 % 1)"
"({(3,1),(156,1)},1 % 1)"
"({(3,2),(156,2)},1 % 1)"
"({(3,3),(156,2)},1 % 1)"
"({(4,1),(157,1)},1 % 1)"
"({(4,2),(157,2)},1 % 1)"
"({(4,3),(157,2)},1 % 1)"

algnden (aa fapply gg) - algnden (ind aa fapply gg)
218.3586352266184

let gg = iillfudshigh aa (ind aa) (3^3) 1 (3^3) 3 3 3 (3^2) 1

rpln $(concat . map (aall . ttaa) . Set.toList . ffqq)$ gg
"({(2,1),(155,1)},1 % 1)"
"({(2,2),(155,2)},1 % 1)"
"({(2,3),(155,2)},1 % 1)"
"({(3,1),(156,1)},1 % 1)"
"({(3,2),(156,2)},1 % 1)"
"({(3,3),(156,2)},1 % 1)"
"({(4,1),(157,1)},1 % 1)"
"({(4,2),(157,2)},1 % 1)"
"({(4,3),(157,2)},1 % 1)"

algnden (aa fapply gg) - algnden (ind aa fapply gg)
218.3586352266184

let gg = iillfuds aa (ind aa) (3^3) 2 (3^3) 3 3 3 (3^2) 1

rpln $(concat . map (aall . ttaa) . Set.toList . ffqq)$ gg
"({(1,1),(239,1)},1 % 1)"
"({(1,2),(239,2)},1 % 1)"
"({(1,3),(239,2)},1 % 1)"
"({(2,1),(155,1)},1 % 1)"
"({(2,2),(155,2)},1 % 1)"
"({(2,3),(155,2)},1 % 1)"
"({(3,1),(156,1)},1 % 1)"
"({(3,2),(156,2)},1 % 1)"
"({(3,3),(156,2)},1 % 1)"
"({(4,1),(157,1)},1 % 1)"
"({(4,2),(157,2)},1 % 1)"
"({(4,3),(157,2)},1 % 1)"

algnden (aa fapply gg) - algnden (ind aa fapply gg)
325.96965461170066

let gg = iillfudshigh aa (ind aa) (3^3) 2 (3^3) 3 3 3 (3^2) 1

rpln $(concat . map (aall . ttaa) . Set.toList . ffqq)$ gg
"({(1,1),(239,1)},1 % 1)"
"({(1,2),(239,2)},1 % 1)"
"({(1,3),(239,2)},1 % 1)"
"({(2,1),(155,1)},1 % 1)"
"({(2,2),(155,2)},1 % 1)"
"({(2,3),(155,2)},1 % 1)"
"({(3,1),(156,1)},1 % 1)"
"({(3,2),(156,2)},1 % 1)"
"({(3,3),(156,2)},1 % 1)"
"({(4,1),(157,1)},1 % 1)"
"({(4,2),(157,2)},1 % 1)"
"({(4,3),(157,2)},1 % 1)"

algnden (aa fapply gg) - algnden (ind aa fapply gg)
325.96965461170066

:set +s

let gg = iillfuds aa (ind aa) (3^3) 5 (3^3) 3 3 3 (3^2) 1

algnden (aa fapply gg) - algnden (ind aa fapply gg)
325.96965461170066
(45.91 secs, 8,409,009,448 bytes)

let gg = iillfudshigh aa (ind aa) (3^3) 3 (3^3) 3 3 3 (3^2) 1

algnden (aa fapply gg) - algnden (ind aa fapply gg)
325.96965461170066
(9.22 secs, 1,531,523,080 bytes)


Now consider predicting the rain in the weather forecast example (summarised in Functional definition sets),

let aa = hhaa hh

algnden aa red [pressure,cloud,wind] 2.15004722009382 :set +s let gg = iillfuds (aa red [pressure,cloud,wind]) (ind (aa red [pressure,cloud,wind])) (3^7) 5 (3^3) 3 (3*2) 2 (3^3) 2 algnden (aa fapply gg) - algnden (ind aa fapply gg) 4.215798138230515 (275.79 secs, 46,371,780,760 bytes) let gg = iillfudshigh (aa red [pressure,cloud,wind]) (ind (aa red [pressure,cloud,wind])) (3^7) 5 (3^3) 3 (3*2) 2 (3^3) 2 algnden (aa fapply gg) - algnden (ind aa fapply gg) 4.215798138230515 (29.98 secs, 5,128,323,320 bytes)  The weather forecast example continues in Practicable highest-layer summed shuffle content alignment valency-density fud decomposition inducer. ### Practicable level shuffle content alignment valency-density fud inducer The practicable level shuffle content alignment valency-density fud inducer,I_{z,\mathrm{csd,F,\infty,q},P,V_{\mathrm{g}},F_{\mathrm{g}}}^{‘}, may then be implemented (Text) $\begin{eqnarray} &&I_{z,\mathrm{csd,F,\infty,q},P,V_{\mathrm{g}},F_{\mathrm{g}}}^{‘ * }(A) = \\ &&\hspace{2em}\{(G,I_{\mathrm{csd}}^{ * }((A,A_R,G))) : \\ &&\hspace{5em}|V_A|>1,~\{F_{\mathrm{L}}\} = \mathrm{leaves}(\mathrm{tree}(Z_{P,A,A_R,V_{\mathrm{g}},F_{\mathrm{g}},\mathrm{L}})),\\ &&\hspace{5em}K \in \mathrm{maxd}(\mathrm{elements}(Z_{P,A,A_R,F_{\mathrm{g}},F_{\mathrm{L}},\mathrm{D}})),~G = \mathrm{depends}(F_{\mathrm{L}},K)\} \cup \\ &&\hspace{2em}\{(\emptyset,0) : |V_A| \leq 1\} \end{eqnarray}$ The fud inducer is defined in module AlignmentPracticable. This implementation is derived from the practicable highest-layer shuffle content alignment valency-density fud inducer, parametersSystemsSamplesShufflesLevelsListVariablesInducerFudHighest :: Integer -> Integer -> Integer -> Integer -> Integer -> Integer -> Integer -> Integer -> System -> Histogram -> Histogram -> Set.Set Variable -> Fud -> [Variable] -> Maybe ((Fud,Fud),(System,[Variable]))  as parametersSystemsSamplesShufflesLevelsListVariablesInducerFudHighest wmax lmax xmax omax bmax mmax umax pmax uu aa aarr vvg ffg ll ... | ... = Just ((depends ff kk, ff),(uu',ll'))
| otherwise = Nothing
where
(nn,(uu',ll')) = zzllfudshigh wmax lmax xmax omax bmax mmax umax pmax uu aa aarr vvg ffg ll
ff = if nn /= [] then last nn else fudEmpty
kk = maxdff ff $zzffcsdderhighopt wmax omax uu' aa aarr ffg ff zzllfudshigh wmax lmax xmax omax bmax mmax umax pmax uu aa aarr vvg ffg ll = fromJust$
zzffcsdderhighopt wmax omax uu aa rr ffg ff = fst $unzip$ fromJust \$

Note that, again, the partition variables are replaced by cardinal variables, so the fud inducer has an additional argument of a list of variables, [Variable].