Properties of the sample
MNIST - handwritten digits/Properties of the sample
Sections
All pixels
The training sample has a size of 60,000. We shall take every eighth event from the sample to make a more convenient size of 7,500. First load the training sample $A$ (see History and HistoryRepa),
:l NISTDev
(uu,hrtr) <- nistTrainIO
let digit = VarStr "digit"
let vv = uvars uu
let vvl = sgl digit
let vvk = vv `minus` vvl
card $ hrvars hrtr
785
hrsize hrtr
60000
Now take a subset,
let hr = hrev [i | i <- [0 .. hrsize hrtr - 1], i `mod` 8 == 0] hrtr
hrsize hr
7500
The system is $U$. The sample substrate variables are $V = \mathrm{vars}(A)$, the label variables are $V_{\mathrm{l}} = \{\mathrm{digit}\}$, and the query variables form the remainder, $V_{\mathrm{k}} = V \setminus V_{\mathrm{l}}$.
The query variable valency is 256, $\{|U_w| : w \in V_{\mathrm{k}}\} = \{256\}$,
rp $ llqq [u | w <- qqll vvk, let u = vol uu (sgl w)]
"{256}"
The label variable valency is $|U_{\mathrm{digit}}| = 10$,
vol uu vvl
10
rpln $ aall $ araa uu $ hr `hrred` vvl
"({(digit,0)},743 % 1)"
"({(digit,1)},822 % 1)"
"({(digit,2)},738 % 1)"
"({(digit,3)},745 % 1)"
"({(digit,4)},746 % 1)"
"({(digit,5)},720 % 1)"
"({(digit,6)},707 % 1)"
"({(digit,7)},767 % 1)"
"({(digit,8)},754 % 1)"
"({(digit,9)},758 % 1)"
We can image any event by creating a bitmap. There are several utility functions in NISTDev for creating and manipulating bitmaps from a HistoryRepa. For example, this generates bitmaps of the first 25 events placed in a row:
let file = "NIST.bmp"
bmwrite file $ bmhstack $ map (bmborder 1 . hrbm 28 1 256) $ [hrev [i] hr' | let hr' = hr `hrhrred` vvk, i <- [0..24]]
The entire history can be averaged together, $\hat{A}\%V_{\mathrm{k}}$,
bmwrite file $ bmborder 1 $ hrbm 28 3 256 $ hr `hrhrred` vvk
The pixel variables can be bucketed to reduce the valency from 256 to 2, by partitioning at 128,
(uu,hrtr) <- nistTrainBucketedIO 2
let digit = VarStr "digit"
let vv = uvars uu
let vvl = sgl digit
let vvk = vv `minus` vvl
hrsize hrtr
60000
let hr = hrev [i | i <- [0 .. hrsize hrtr - 1], i `mod` 8 == 0] hrtr
hrsize hr
7500
The query variable valency is 2, $\{|U_w| : w \in V_{\mathrm{k}}\} = \{2\}$,
rp $ llqq [u | w <- qqll vvk, let u = vol uu (sgl w)]
"{2}"
The first 25 events placed in a row now look slightly coarser,
bmwrite file $ bmhstack $ map (bmborder 1 . hrbm 28 1 2) $ [hrev [i] hr' | let hr' = hr `hrhrred` vvk, i <- [0..24]]
Again, the entire history can be averaged together, $\hat{A}\%V_{\mathrm{k}}$,
let hrbmav = hrbm 28 3 2 $ hr `hrhrred` vvk
bmwrite file $ bmborder 1 $ hrbmav
We can show the average image for each digit, $\{\hat{B}\%V_{\mathrm{k}} : S \in V_{\mathrm{l}}^{\mathrm{CS}},~B = A * \{S\}^{\mathrm{U}}\}$,
bmwrite file $ bmhstack $ map (bmborder 1 . hrbm 28 2 2) $ [hr `hrhrsel` hrs `hrhrred` vvk | ss <- qqll (cart uu vvl), let hrs = aahr uu (single ss 1)]
Now consider how highly aligned variables might be grouped together. See Entropy and alignment. We will use the tupler to group together highly aligned variables in the substrate, $V$.
Create a shuffled sample, $A_{\mathrm{r}}$,
let hrr = historyRepasShuffle_u hr 1
hrsize hrr
7500
bmwrite file $ bmborder 1 $ hrbm 28 3 2 $ hrr `hrhrred` vvk
The average shuffle looks exactly the same as the average sample, but individual events are, in fact, randomised,
bmwrite file $ bmhstack $ map (bmborder 1 . hrbm 28 1 2) $ [hrev [i] hr' | let hr' = hrr `hrhrred` vvk, i <- [0..24]]
Now optimise the shuffle content alignment with the tuple set builder, $I_{P,U,\mathrm{B,ns,me}}$, \[ \{(\mathrm{algn}(A\%K)-\mathrm{algn}(A_{\mathrm{r}}\%K),~K) : ((K,\cdot,\cdot),\cdot) \in I_{P,U,\mathrm{B,ns,me}}^{ * }((V,~\emptyset,~A,~A_{\mathrm{r}}))\} \]
let buildtup xmax omax bmax uu vv xx xxrr = reverse $ sort $ map (\((kk,_),_) -> (algn (araa uu (xx `hrred` kk)) - algn (araa uu (xxrr `hrred` kk)), kk)) $ parametersSystemsBuilderTupleNoSumlayerMultiEffectiveRepa_u xmax omax bmax 1 uu vv fudEmpty xx (hrhx xx) xxrr (hrhx xxrr)
rpln $ buildtup (2^2) 10 10 uu vv hr hrr
"(2292.2815160591927,{<13,9>,<14,9>})"
"(2233.012036338092,{<14,9>,<15,9>})"
"(2188.031695004327,{<14,8>,<15,8>})"
"(2170.9278646040475,{<12,9>,<13,9>})"
"(2137.518853393034,{<23,11>,<23,12>})"
"(2134.735654658769,{<23,12>,<23,13>})"
"(2132.4501082092684,{<12,10>,<13,10>})"
"(2130.122516232521,{<15,9>,<16,9>})"
"(2123.706836400408,{<24,12>,<24,13>})"
"(2123.460469219186,{<17,20>,<18,20>})"
let ll = buildtup (2^12) 10 10 uu vv hr hrr
rpln ll
"(23436.877679734192,{<10,10>,<10,11>,<11,9>,<11,10>,<11,11>,<12,9>,<12,10>,<12,11>,<13,9>,<13,10>,<14,9>,<14,10>})"
"(23274.21784395763,{<10,9>,<10,10>,<10,11>,<11,9>,<11,10>,<11,11>,<12,9>,<12,10>,<12,11>,<13,9>,<13,10>,<14,9>})"
"(23232.752272709135,{<10,10>,<10,11>,<11,9>,<11,10>,<11,11>,<12,9>,<12,10>,<12,11>,<13,9>,<13,10>,<13,11>,<14,9>})"
"(23232.169432601415,{<11,9>,<11,10>,<11,11>,<12,9>,<12,10>,<12,11>,<13,9>,<13,10>,<13,11>,<14,9>,<14,10>,<14,11>})"
"(23201.26418245857,{<10,10>,<11,9>,<11,10>,<11,11>,<12,9>,<12,10>,<12,11>,<13,9>,<13,10>,<13,11>,<14,9>,<14,10>})"
"(23184.294304054132,{<11,9>,<11,10>,<11,11>,<12,9>,<12,10>,<12,11>,<13,9>,<13,10>,<13,11>,<14,9>,<14,10>,<15,9>})"
"(23170.794157965698,{<10,11>,<11,9>,<11,10>,<11,11>,<12,9>,<12,10>,<12,11>,<13,9>,<13,10>,<13,11>,<14,9>,<14,10>})"
"(23146.79351844896,{<10,10>,<10,11>,<11,9>,<11,10>,<11,11>,<12,9>,<12,10>,<12,11>,<13,9>,<13,10>,<14,9>,<15,9>})"
"(23086.94962868454,{<10,10>,<10,11>,<10,12>,<11,9>,<11,10>,<11,11>,<12,9>,<12,10>,<12,11>,<13,9>,<13,10>,<14,9>})"
"(23080.549929231824,{<11,9>,<11,10>,<11,11>,<12,9>,<12,10>,<12,11>,<13,9>,<13,10>,<14,9>,<14,10>,<15,9>,<15,10>})"
Let us image the most highly aligned tuple, \[ Q_1~=~\{\mathrm{<10,10>},\dots,~\mathrm{<14,10>}\} \]
let qqhr d uu vv qq = aahr uu (single (llss ([(v, ValInt (d-1)) | v <- qqll qq] ++ [(v, ValInt 0) | v <- qqll (vv `minus` qq)])) 1)
let qq1 = snd $ head ll
bmwrite file $ bmborder 1 $ hrbm 28 3 2 $ qqhr 2 uu vvk qq1
Note that the coordinates <x,y> are defined so that the x coordinate increases from top row 1 to bottom row 28, and the y coordinate runs from left column 1 to right column 28.
Showing it overlaid on the average,
bmwrite file $ bmborder 1 $ bmmax hrbmav 0 0 $ hrbm 28 3 2 $ qqhr 2 uu vvk qq1
Showing it overlaid on the average for each digit,
bmwrite file $ bmhstack $ map (\av -> bmborder 1 (bmmax (hrbm 28 2 2 av) 0 0 (hrbm 28 2 2 (qqhr 2 uu vvk qq1)))) $ [hr `hrhrsel` hrs `hrhrred` vvk | ss <- qqll (cart uu vvl), let hrs = aahr uu (single ss 1)]
We can see that this substrate tuple appears in the top loop where a three would differ from an eight.
Now let’s image the first 20 states ordered by size descending, $\mathrm{top}(20)(A\%Q_1)$,
let pp = take 20 $ reverse $ sort $ map (\(a,b) -> (b,a)) $ aall $ araa uu $ hr `hrred` qq1
map numerator $ fst $ unzip pp
[2656,317,175,146,123,114,111,94,92,90,89,85,83,73,72,72,68,65,63,60]
bmwrite file $ bmhstack $ map (bmborder 1 . hrbm 28 1 2) $ [hr' `hrhrsel` hrs | let hr' = hr `hrhrred` vvk, (_,ss) <- pp, let hrs = aahr uu (single ss 1)]
The first state is the largest slice. In this case all of the pixels of the tuple are background, so it includes all images where this region is blank. The remaining states all have at least one foreground pixel in this region, for which there are images for most digits except one.
Now optimise again having removed the top tuple from the substrate, and \[ \{(\mathrm{algn}(A\%K)-\mathrm{algn}(A_{\mathrm{r}}\%K),~K) : ((K,\cdot,\cdot),\cdot) \in I_{P,U,\mathrm{B,ns,me}}^{ * }((V \setminus Q_1,~\emptyset,~A,~A_{\mathrm{r}}))\} \]
let ll = buildtup (2^12) 10 10 uu (vv `minus` qq1) hr hrr
rpln ll
"(22975.09671273167,{<21,10>,<21,11>,<21,12>,<21,13>,<22,10>,<22,11>,<22,12>,<22,13>,<23,10>,<23,11>,<23,12>,<23,13>})"
"(22955.22994752979,{<21,10>,<21,11>,<22,10>,<22,11>,<22,12>,<23,10>,<23,11>,<23,12>,<23,13>,<24,11>,<24,12>,<24,13>})"
"(22918.537058146852,{<21,10>,<21,11>,<22,9>,<22,10>,<22,11>,<22,12>,<23,10>,<23,11>,<23,12>,<23,13>,<24,12>,<24,13>})"
"(22845.04074482481,{<22,10>,<22,11>,<22,12>,<23,10>,<23,11>,<23,12>,<23,13>,<23,14>,<24,11>,<24,12>,<24,13>,<24,14>})"
"(22838.5273227061,{<21,10>,<21,11>,<22,10>,<22,11>,<22,12>,<22,13>,<23,10>,<23,11>,<23,12>,<23,13>,<24,12>,<24,13>})"
"(22812.956894269897,{<21,10>,<21,11>,<21,12>,<22,9>,<22,10>,<22,11>,<22,12>,<22,13>,<23,10>,<23,11>,<23,12>,<23,13>})"
"(22805.291225676185,{<22,9>,<22,10>,<22,11>,<22,12>,<22,13>,<23,10>,<23,11>,<23,12>,<23,13>,<24,11>,<24,12>,<24,13>})"
"(22802.779936122715,{<21,10>,<21,11>,<21,12>,<22,10>,<22,11>,<22,12>,<23,10>,<23,11>,<23,12>,<23,13>,<24,12>,<24,13>})"
"(22786.280375971393,{<21,10>,<22,9>,<22,10>,<22,11>,<22,12>,<23,10>,<23,11>,<23,12>,<23,13>,<24,11>,<24,12>,<24,13>})"
"(22773.7718687649,{<22,10>,<22,11>,<22,12>,<22,13>,<23,10>,<23,11>,<23,12>,<23,13>,<24,10>,<24,11>,<24,12>,<24,13>})"
let qq2 = snd $ head ll
bmwrite file $ bmborder 1 $ bmmax hrbmav 0 0 $ hrbm 28 3 2 $ qqhr 2 uu vvk qq2
This second substrate tuple is nearly as highly aligned as the first. Again it is located where a three would differ from an eight, but now appears in the bottom loop.
Again, let’s image the first 20 states ordered by size descending,
let pp = take 20 $ reverse $ sort $ map (\(a,b) -> (b,a)) $ aall $ araa uu $ hr `hrred` qq2
map numerator $ fst $ unzip pp
[2518,230,220,166,142,129,105,105,100,97,93,85,82,79,79,76,75,73,70,68]
bmwrite file $ bmhstack $ map (bmborder 1 . hrbm 28 1 2) $ [hr' `hrhrsel` hrs | let hr' = hr `hrhrred` vvk, (_,ss) <- pp, let hrs = aahr uu (single ss 1)]
The first state is the largest slice. In this case also, all of the pixels of the tuple are background, so it includes all images where this region is blank. It resembles a non-oblique nine.
Continuing on,
let ll = buildtup (2^12) 10 10 uu (vv `minus` qq1 `minus` qq2) hr hrr
rpln ll
"(23011.1811072602,{<17,20>,<18,19>,<18,20>,<18,21>,<19,19>,<19,20>,<19,21>,<20,19>,<20,20>,<20,21>,<21,19>,<21,20>})"
"(22879.335116168455,{<17,20>,<17,21>,<18,20>,<18,21>,<19,19>,<19,20>,<19,21>,<20,19>,<20,20>,<20,21>,<21,19>,<21,20>})"
"(22780.385544374076,{<17,20>,<18,20>,<18,21>,<19,19>,<19,20>,<19,21>,<20,19>,<20,20>,<20,21>,<21,19>,<21,20>,<22,19>})"
"(22780.36943821011,{<17,19>,<18,19>,<18,20>,<18,21>,<19,19>,<19,20>,<19,21>,<20,19>,<20,20>,<20,21>,<21,19>,<21,20>})"
"(22745.02629997328,{<17,20>,<18,19>,<18,20>,<19,19>,<19,20>,<19,21>,<20,19>,<20,20>,<20,21>,<21,19>,<21,20>,<22,19>})"
"(22739.154007140343,{<17,19>,<17,20>,<18,19>,<18,20>,<19,19>,<19,20>,<19,21>,<20,19>,<20,20>,<20,21>,<21,19>,<21,20>})"
"(22725.709724299384,{<17,20>,<17,21>,<18,19>,<18,20>,<18,21>,<19,19>,<19,20>,<19,21>,<20,19>,<20,20>,<21,19>,<21,20>})"
"(22651.541904709375,{<17,20>,<18,20>,<18,21>,<19,19>,<19,20>,<19,21>,<20,19>,<20,20>,<20,21>,<21,19>,<21,20>,<21,21>})"
"(22630.679931470084,{<17,19>,<17,20>,<18,19>,<18,20>,<18,21>,<19,19>,<19,20>,<19,21>,<20,19>,<20,20>,<21,19>,<21,20>})"
"(22629.81021650106,{<17,20>,<18,19>,<18,20>,<18,21>,<19,19>,<19,20>,<19,21>,<20,19>,<20,20>,<21,19>,<21,20>,<22,19>})"
let qq3 = snd $ head ll
bmwrite file $ bmborder 1 $ bmmax hrbmav 0 0 $ hrbm 28 3 2 $ qqhr 2 uu vvk qq3
Again, the third substrate tuple is as highly aligned as the first two. It is located where a five would differ from a two in the bottom loop.
Imaging the first 20 states ordered by size descending,
let pp = take 20 $ reverse $ sort $ map (\(a,b) -> (b,a)) $ aall $ araa uu $ hr `hrred` qq3
map numerator $ fst $ unzip pp
[2693,249,244,160,136,120,93,93,89,80,79,78,76,75,71,70,69,65,64,59]
bmwrite file $ bmhstack $ map (bmborder 1 . hrbm 28 1 2) $ [hr' `hrhrsel` hrs | let hr' = hr `hrhrred` vvk, (_,ss) <- pp, let hrs = aahr uu (single ss 1)]
let ll = buildtup (2^12) 10 10 uu (vv `minus` qq1 `minus` qq2 `minus` qq3) hr hrr
rpln ll
"(16178.865805790956,{digit,<10,18>,<10,19>,<11,18>,<11,19>,<12,18>,<12,19>,<13,18>,<13,19>})"
"(16102.671528899547,{digit,<10,19>,<11,18>,<11,19>,<12,18>,<12,19>,<13,18>,<13,19>,<14,18>})"
"(16071.750779434426,{digit,<10,19>,<11,19>,<12,18>,<12,19>,<13,18>,<13,19>,<14,18>,<14,19>})"
"(16048.253316818988,{digit,<10,19>,<11,18>,<11,19>,<12,18>,<12,19>,<13,18>,<13,19>,<14,19>})"
"(16040.078810291008,{digit,<10,19>,<11,18>,<11,19>,<11,20>,<12,18>,<12,19>,<12,20>,<13,19>})"
"(16030.85158783264,{digit,<10,19>,<11,18>,<11,19>,<11,20>,<12,18>,<12,19>,<13,18>,<13,19>})"
"(15980.493550725612,{digit,<10,19>,<10,20>,<11,18>,<11,19>,<11,20>,<12,18>,<12,19>,<13,19>})"
"(15969.701495503503,{digit,<10,19>,<11,19>,<11,20>,<12,18>,<12,19>,<12,20>,<13,18>,<13,19>})"
"(15953.852976397857,{digit,<10,19>,<11,18>,<11,19>,<12,18>,<12,19>,<12,20>,<13,18>,<13,19>})"
"(15924.575899591051,{digit,<10,19>,<10,20>,<11,19>,<11,20>,<12,18>,<12,19>,<13,18>,<13,19>})"
Now, in the fourth tuple we see that digit has appeared.
let qq4 = snd $ head ll
bmwrite file $ bmborder 1 $ bmmax hrbmav 0 0 $ hrbm 28 3 2 $ qqhr 2 uu vvk (qq4 `minus` vvl)
This fourth substrate tuple is not as highly aligned as the first three. It is located within the top loop at the bottom.
Imaging the first 20 states ordered by size descending,
let pp = take 20 $ reverse $ sort $ map (\(a,b) -> (b,a)) $ aall $ araa uu $ hr `hrred` qq4
let sat ss v = fromJust $ statesVarsValue ss v
rp [(numerator a, ss `sat` digit) | (a,ss) <- pp]
"[(666,1),(399,5),(382,6),(328,0),(205,3),(199,7),(165,9),(154,4),(138,2),(123,2),(116,4),(81,6),(78,8),(74,0),(66,9),(62,4),(59,3),(52,9),(52,8),(52,6)]"
bmwrite file $ bmhstack $ map (bmborder 1 . hrbm 28 1 2) $ [hr `hrhrsel` hrs `hrhrred` vvk | (_,ss) <- pp, let hrs = aahr uu (single ss 1)]
In this case, the images are each associcated with a digit value.
let ll = buildtup (2^12) 10 10 uu (vv `minus` qq1 `minus` qq2 `minus` qq3 `minus` qq4) hr hrr
rpln ll
"(22275.212143169832,{<12,15>,<12,16>,<13,14>,<13,15>,<13,16>,<14,13>,<14,14>,<14,15>,<14,16>,<15,13>,<15,14>,<15,15>})"
"(22166.753255547927,{<12,15>,<12,16>,<13,13>,<13,14>,<13,15>,<13,16>,<14,13>,<14,14>,<14,15>,<14,16>,<15,14>,<15,15>})"
"(22078.817340598478,{<12,14>,<12,15>,<12,16>,<13,14>,<13,15>,<13,16>,<14,14>,<14,15>,<14,16>,<15,14>,<15,15>,<15,16>})"
"(22077.265195860236,{<12,14>,<12,15>,<12,16>,<13,14>,<13,15>,<13,16>,<14,13>,<14,14>,<14,15>,<14,16>,<15,14>,<15,15>})"
"(22042.66798735359,{<12,15>,<13,13>,<13,14>,<13,15>,<13,16>,<14,13>,<14,14>,<14,15>,<14,16>,<15,13>,<15,14>,<15,15>})"
"(22036.275524003177,{<13,13>,<13,14>,<13,15>,<13,16>,<14,13>,<14,14>,<14,15>,<14,16>,<15,13>,<15,14>,<15,15>,<15,16>})"
"(22029.966792075178,{<11,16>,<12,14>,<12,15>,<12,16>,<13,14>,<13,15>,<13,16>,<14,14>,<14,15>,<14,16>,<15,14>,<15,15>})"
"(22028.413553626342,{<11,15>,<11,16>,<12,15>,<12,16>,<13,14>,<13,15>,<13,16>,<14,14>,<14,15>,<14,16>,<15,14>,<15,15>})"
"(22026.690602542018,{<11,16>,<12,15>,<12,16>,<13,14>,<13,15>,<13,16>,<14,14>,<14,15>,<14,16>,<15,14>,<15,15>,<15,16>})"
"(22008.279075414976,{<12,14>,<12,15>,<12,16>,<13,14>,<13,15>,<13,16>,<14,14>,<14,15>,<14,16>,<15,13>,<15,14>,<15,15>})"
let qq5 = snd $ head ll
bmwrite file $ bmborder 1 $ bmmax hrbmav 0 0 $ hrbm 28 3 2 $ qqhr 2 uu vvk qq5
The fifth substrate tuple, qq5, has slightly lower alignment than the first three, but higher than the fourth, qq4.
let ll = buildtup (2^12) 10 10 uu (vv `minus` qq1 `minus` qq2 `minus` qq3 `minus` qq4 `minus` qq5) hr hrr
rpln ll
"(20878.329981278253,{<10,20>,<11,20>,<11,21>,<12,20>,<12,21>,<13,20>,<13,21>,<14,19>,<14,20>,<15,19>,<15,20>,<16,20>})"
"(20855.713802772072,{<10,20>,<11,20>,<12,20>,<12,21>,<13,20>,<13,21>,<14,19>,<14,20>,<14,21>,<15,19>,<15,20>,<16,20>})"
"(20750.789352785985,{<10,20>,<11,20>,<12,20>,<13,20>,<13,21>,<14,19>,<14,20>,<14,21>,<15,19>,<15,20>,<15,21>,<16,20>})"
"(20663.751070455546,{<10,20>,<11,20>,<12,20>,<12,21>,<13,20>,<13,21>,<14,19>,<14,20>,<15,19>,<15,20>,<16,19>,<16,20>})"
"(20631.894394328505,{<10,20>,<11,20>,<11,21>,<12,20>,<12,21>,<13,20>,<14,19>,<14,20>,<15,19>,<15,20>,<16,19>,<16,20>})"
"(20626.36311045546,{<10,20>,<10,21>,<11,20>,<11,21>,<12,20>,<12,21>,<13,20>,<14,19>,<14,20>,<15,19>,<15,20>,<16,20>})"
"(20607.96562172127,{<10,20>,<11,20>,<12,20>,<13,20>,<13,21>,<14,19>,<14,20>,<14,21>,<15,19>,<15,20>,<16,19>,<16,20>})"
"(20589.080650660817,{<10,20>,<11,20>,<11,21>,<12,20>,<12,21>,<13,20>,<14,19>,<14,20>,<14,21>,<15,19>,<15,20>,<16,20>})"
"(20546.598600924262,{<10,20>,<11,20>,<12,20>,<13,20>,<14,19>,<14,20>,<14,21>,<15,19>,<15,20>,<15,21>,<16,19>,<16,20>})"
"(20535.30581799583,{<10,20>,<11,20>,<12,20>,<12,21>,<13,20>,<13,21>,<14,19>,<14,20>,<15,19>,<15,20>,<15,21>,<16,20>})"
let qq6 = snd $ head ll
bmwrite file $ bmborder 1 $ bmmax hrbmav 0 0 $ hrbm 28 3 2 $ qqhr 2 uu vvk qq6
let ll = buildtup (2^12) 10 10 uu (vv `minus` qq1 `minus` qq2 `minus` qq3 `minus` qq4 `minus` qq5 `minus` qq6) hr hrr
rpln ll
"(21783.431050990985,{<13,8>,<14,8>,<15,8>,<15,9>,<15,10>,<16,8>,<16,9>,<16,10>,<17,8>,<17,9>,<17,10>,<18,9>})"
"(21652.508020148773,{<13,8>,<14,8>,<15,8>,<15,9>,<15,10>,<16,8>,<16,9>,<16,10>,<17,8>,<17,9>,<17,10>,<18,8>})"
"(21620.743967537495,{<13,8>,<14,8>,<15,7>,<15,8>,<15,9>,<15,10>,<16,8>,<16,9>,<16,10>,<17,8>,<17,9>,<17,10>})"
"(21579.283500079197,{<13,8>,<14,8>,<15,8>,<15,9>,<16,8>,<16,9>,<16,10>,<17,8>,<17,9>,<17,10>,<18,8>,<18,9>})"
"(21548.238691220504,{<12,8>,<13,8>,<14,8>,<15,8>,<15,9>,<15,10>,<16,8>,<16,9>,<16,10>,<17,8>,<17,9>,<17,10>})"
"(21518.45426088917,{<13,8>,<14,8>,<15,8>,<15,9>,<16,8>,<16,9>,<16,10>,<17,8>,<17,9>,<17,10>,<18,9>,<18,10>})"
"(21516.40862981805,{<13,8>,<14,8>,<15,8>,<15,9>,<15,10>,<16,8>,<16,9>,<16,10>,<16,11>,<17,8>,<17,9>,<17,10>})"
"(21497.093097859324,{<13,8>,<14,8>,<15,8>,<15,9>,<15,10>,<16,8>,<16,9>,<16,10>,<17,8>,<17,9>,<18,8>,<18,9>})"
"(21480.23188336933,{<13,8>,<14,8>,<15,8>,<15,9>,<15,10>,<16,8>,<16,9>,<16,10>,<17,8>,<17,9>,<17,10>,<18,10>})"
"(21479.019454862653,{<13,8>,<14,8>,<15,8>,<15,9>,<15,10>,<16,7>,<16,8>,<16,9>,<16,10>,<17,8>,<17,9>,<17,10>})"
let qq7 = snd $ head ll
bmwrite file $ bmborder 1 $ bmmax hrbmav 0 0 $ hrbm 28 3 2 $ qqhr 2 uu vvk qq7
The seventh substrate tuple, qq7, resembles the first substrate tuple, qq1. When adding them together we obtain,
bmwrite file $ bmborder 1 $ bmmax hrbmav 0 0 $ hrbm 28 3 2 $ qqhr 2 uu vvk (qq1 `union` qq7)
let ll = buildtup (2^12) 10 10 uu (vv `minus` qq1 `minus` qq2 `minus` qq3 `minus` qq4 `minus` qq5 `minus` qq6 `minus` qq7) hr hrr
rpln ll
"(20845.369537783077,{<15,18>,<16,17>,<16,18>,<16,19>,<17,17>,<17,18>,<17,19>,<18,17>,<18,18>,<19,17>,<19,18>,<20,17>})"
"(20832.764015826226,{<15,17>,<15,18>,<16,17>,<16,18>,<16,19>,<17,17>,<17,18>,<17,19>,<18,17>,<18,18>,<19,17>,<19,18>})"
"(20608.417374454184,{<16,17>,<16,18>,<16,19>,<17,17>,<17,18>,<17,19>,<18,17>,<18,18>,<19,17>,<19,18>,<20,17>,<20,18>})"
"(20557.67921254835,{<15,18>,<16,17>,<16,18>,<16,19>,<17,17>,<17,18>,<17,19>,<18,17>,<18,18>,<19,17>,<19,18>,<20,18>})"
"(20550.2407579201,{<15,18>,<16,18>,<16,19>,<17,17>,<17,18>,<17,19>,<18,17>,<18,18>,<19,17>,<19,18>,<20,17>,<20,18>})"
"(20545.02772552971,{<14,18>,<15,18>,<16,17>,<16,18>,<16,19>,<17,17>,<17,18>,<17,19>,<18,17>,<18,18>,<19,17>,<19,18>})"
"(20511.45542399987,{<14,18>,<15,17>,<15,18>,<16,17>,<16,18>,<16,19>,<17,17>,<17,18>,<17,19>,<18,17>,<18,18>,<19,18>})"
"(20486.703095425386,{<15,18>,<16,17>,<16,18>,<16,19>,<17,16>,<17,17>,<17,18>,<17,19>,<18,17>,<18,18>,<19,17>,<19,18>})"
"(20475.98333646508,{<15,18>,<16,17>,<16,18>,<16,19>,<17,17>,<17,18>,<17,19>,<18,16>,<18,17>,<18,18>,<19,17>,<19,18>})"
"(20460.397208891693,{<16,18>,<16,19>,<17,17>,<17,18>,<17,19>,<18,17>,<18,18>,<19,17>,<19,18>,<20,17>,<20,18>,<21,17>})"
let qq8 = snd $ head ll
bmwrite file $ bmborder 1 $ bmmax hrbmav 0 0 $ hrbm 28 3 2 $ qqhr 2 uu vvk qq8
Continuing on,
let ll = buildtup (2^12) 10 10 uu (vv `minus` qq1 `minus` qq2 `minus` qq3 `minus` qq4 `minus` qq5 `minus` qq6 `minus` qq7 `minus` qq8) hr hrr
rpln ll
"(22341.597123332398,{<18,14>,<18,15>,<18,16>,<19,14>,<19,15>,<19,16>,<20,14>,<20,15>,<20,16>,<21,14>,<21,15>,<21,16>})"
"(22155.40724854413,{<18,14>,<18,15>,<18,16>,<19,14>,<19,15>,<19,16>,<20,13>,<20,14>,<20,15>,<20,16>,<21,14>,<21,15>})"
"(22083.02080595367,{<18,15>,<18,16>,<19,14>,<19,15>,<19,16>,<20,13>,<20,14>,<20,15>,<20,16>,<21,14>,<21,15>,<21,16>})"
"(22082.130029471875,{<18,15>,<18,16>,<19,13>,<19,14>,<19,15>,<19,16>,<20,13>,<20,14>,<20,15>,<20,16>,<21,14>,<21,15>})"
"(22062.742184575553,{<17,15>,<18,14>,<18,15>,<18,16>,<19,14>,<19,15>,<19,16>,<20,14>,<20,15>,<20,16>,<21,14>,<21,15>})"
"(22024.075862773665,{<17,15>,<17,16>,<18,14>,<18,15>,<18,16>,<19,14>,<19,15>,<19,16>,<20,14>,<20,15>,<21,14>,<21,15>})"
"(22023.176379640623,{<17,15>,<17,16>,<18,15>,<18,16>,<19,14>,<19,15>,<19,16>,<20,14>,<20,15>,<20,16>,<21,14>,<21,15>})"
"(22019.780203420556,{<17,16>,<18,14>,<18,15>,<18,16>,<19,14>,<19,15>,<19,16>,<20,14>,<20,15>,<20,16>,<21,14>,<21,15>})"
"(22011.614728583183,{<18,14>,<18,15>,<18,16>,<19,13>,<19,14>,<19,15>,<19,16>,<20,14>,<20,15>,<20,16>,<21,14>,<21,15>})"
"(21973.620646332893,{<17,16>,<18,15>,<18,16>,<19,14>,<19,15>,<19,16>,<20,14>,<20,15>,<20,16>,<21,14>,<21,15>,<21,16>})"
let qq9 = snd $ head ll
bmwrite file $ bmborder 1 $ bmmax hrbmav 0 0 $ hrbm 28 3 2 $ qqhr 2 uu vvk qq9
let ll = buildtup (2^12) 10 10 uu (vv `minus` qq1 `minus` qq2 `minus` qq3 `minus` qq4 `minus` qq5 `minus` qq6 `minus` qq7 `minus` qq8 `minus` qq9) hr hrr
rpln ll
"(21156.69974273265,{<6,12>,<6,13>,<6,14>,<6,15>,<7,11>,<7,12>,<7,13>,<7,14>,<7,15>,<8,11>,<8,12>,<8,13>})"
"(20892.15406651666,{<6,12>,<6,13>,<6,14>,<6,15>,<7,11>,<7,12>,<7,13>,<7,14>,<7,15>,<8,10>,<8,11>,<8,12>})"
"(20826.463439115603,{<22,15>,<22,16>,<22,17>,<23,14>,<23,15>,<23,16>,<23,17>,<24,12>,<24,13>,<24,14>,<24,15>,<24,16>})"
"(20816.036859972482,{<6,11>,<6,12>,<6,13>,<6,14>,<6,15>,<7,11>,<7,12>,<7,13>,<7,14>,<7,15>,<8,11>,<8,12>})"
"(20803.281775035102,{<22,16>,<22,17>,<23,14>,<23,15>,<23,16>,<23,17>,<24,12>,<24,13>,<24,14>,<24,15>,<24,16>,<24,17>})"
"(20755.76592698853,{<22,14>,<22,15>,<22,16>,<23,14>,<23,15>,<23,16>,<23,17>,<24,12>,<24,13>,<24,14>,<24,15>,<24,16>})"
"(20701.741710050366,{<6,12>,<6,13>,<6,14>,<6,15>,<6,16>,<7,11>,<7,12>,<7,13>,<7,14>,<7,15>,<8,11>,<8,12>})"
"(20693.38377428647,{<22,15>,<22,16>,<23,14>,<23,15>,<23,16>,<23,17>,<24,12>,<24,13>,<24,14>,<24,15>,<24,16>,<24,17>})"
"(20672.2231620883,{<22,14>,<22,15>,<22,16>,<23,14>,<23,15>,<23,16>,<24,12>,<24,13>,<24,14>,<24,15>,<24,16>,<25,14>})"
"(20669.38990640306,{<22,14>,<22,15>,<22,16>,<23,14>,<23,15>,<23,16>,<24,11>,<24,12>,<24,13>,<24,14>,<24,15>,<24,16>})"
let qq10 = snd $ head ll
bmwrite file $ bmborder 1 $ bmmax hrbmav 0 0 $ hrbm 28 3 2 $ qqhr 2 uu vvk qq10
let ll = buildtup (2^12) 10 10 uu (vv `minus` qq1 `minus` qq2 `minus` qq3 `minus` qq4 `minus` qq5 `minus` qq6 `minus` qq7 `minus` qq8 `minus` qq9 `minus` qq10) hr hrr
rpln ll
"(21413.804689507066,{<16,11>,<16,12>,<16,13>,<16,14>,<16,15>,<17,11>,<17,12>,<17,13>,<17,14>,<18,11>,<18,12>,<18,13>})"
"(21312.156855999605,{<16,12>,<16,13>,<16,14>,<16,15>,<17,11>,<17,12>,<17,13>,<17,14>,<17,15>,<18,11>,<18,12>,<18,13>})"
"(21244.05767484347,{<16,12>,<16,13>,<16,14>,<16,15>,<17,12>,<17,13>,<17,14>,<17,15>,<18,12>,<18,13>,<19,12>,<19,13>})"
"(21218.041717893524,{<16,11>,<16,12>,<16,13>,<16,14>,<16,15>,<17,11>,<17,12>,<17,13>,<17,14>,<17,15>,<18,12>,<18,13>})"
"(21207.321040218907,{<16,12>,<16,13>,<16,14>,<16,15>,<17,11>,<17,12>,<17,13>,<17,14>,<18,11>,<18,12>,<18,13>,<19,12>})"
"(21099.822038557955,{<16,11>,<16,12>,<16,13>,<16,14>,<16,15>,<17,11>,<17,12>,<17,13>,<17,14>,<17,15>,<18,11>,<18,12>})"
"(21072.12769174378,{<16,12>,<16,13>,<16,14>,<16,15>,<17,11>,<17,12>,<17,13>,<17,14>,<18,11>,<18,12>,<18,13>,<19,11>})"
"(21009.484731817083,{<16,11>,<16,12>,<16,13>,<16,14>,<16,15>,<17,11>,<17,12>,<17,13>,<17,14>,<18,12>,<18,13>,<19,12>})"
"(20969.383179330696,{<16,12>,<16,13>,<16,14>,<16,15>,<17,12>,<17,13>,<17,14>,<18,11>,<18,12>,<18,13>,<19,12>,<19,13>})"
"(20965.63549083451,{<16,12>,<16,13>,<16,14>,<16,15>,<17,11>,<17,12>,<17,13>,<17,14>,<18,12>,<18,13>,<19,12>,<19,13>})"
let qq11 = snd $ head ll
bmwrite file $ bmborder 1 $ bmmax hrbmav 0 0 $ hrbm 28 3 2 $ qqhr 2 uu vvk qq11
let ll = buildtup (2^12) 10 10 uu (vv `minus` qq1 `minus` qq2 `minus` qq3 `minus` qq4 `minus` qq5 `minus` qq6 `minus` qq7 `minus` qq8 `minus` qq9 `minus` qq10 `minus` qq11) hr hrr
rpln ll
"(20826.463439115603,{<22,15>,<22,16>,<22,17>,<23,14>,<23,15>,<23,16>,<23,17>,<24,12>,<24,13>,<24,14>,<24,15>,<24,16>})"
"(20803.281775035102,{<22,16>,<22,17>,<23,14>,<23,15>,<23,16>,<23,17>,<24,12>,<24,13>,<24,14>,<24,15>,<24,16>,<24,17>})"
"(20755.76592698853,{<22,14>,<22,15>,<22,16>,<23,14>,<23,15>,<23,16>,<23,17>,<24,12>,<24,13>,<24,14>,<24,15>,<24,16>})"
"(20693.38377428647,{<22,15>,<22,16>,<23,14>,<23,15>,<23,16>,<23,17>,<24,12>,<24,13>,<24,14>,<24,15>,<24,16>,<24,17>})"
"(20672.2231620883,{<22,14>,<22,15>,<22,16>,<23,14>,<23,15>,<23,16>,<24,12>,<24,13>,<24,14>,<24,15>,<24,16>,<25,14>})"
"(20669.38990640306,{<22,14>,<22,15>,<22,16>,<23,14>,<23,15>,<23,16>,<24,11>,<24,12>,<24,13>,<24,14>,<24,15>,<24,16>})"
"(20622.80386694959,{<22,18>,<23,14>,<23,15>,<23,16>,<23,17>,<23,18>,<24,12>,<24,13>,<24,14>,<24,15>,<24,16>,<24,17>})"
"(20600.248759519392,{<22,14>,<22,15>,<22,16>,<23,14>,<23,15>,<23,16>,<24,12>,<24,13>,<24,14>,<24,15>,<24,16>,<25,15>})"
"(20599.691512665893,{<22,14>,<22,15>,<22,16>,<22,17>,<23,14>,<23,15>,<23,16>,<24,12>,<24,13>,<24,14>,<24,15>,<24,16>})"
"(20592.98908195017,{<22,14>,<22,15>,<23,14>,<23,15>,<23,16>,<23,17>,<24,12>,<24,13>,<24,14>,<24,15>,<24,16>,<24,17>})"
let qq12 = snd $ head ll
bmwrite file $ bmborder 1 $ bmmax hrbmav 0 0 $ hrbm 28 3 2 $ qqhr 2 uu vvk qq12
Having removed 12 tuples, there is plenty of alignment in the remaining substrate. The union of the 12 tuples is
bmwrite file $ bmborder 1 $ bmmax hrbmav 0 0 $ hrbm 28 3 2 $ qqhr 2 uu vvk (qq1 `union` qq2 `union` qq3 `union` qq4 `union` qq5 `union` qq6 `union` qq7 `union` qq8 `union` qq9 `union` qq10 `union` qq11 `union` qq12 `minus` vvl)
We can continue in this way,
let ll = buildtup (2^12) 10 10 uu (vv `minus` qq1 `minus` qq2 `minus` qq3 `minus` qq4 `minus` qq5 `minus` qq6 `minus` qq7 `minus` qq8 `minus` qq9 `minus` qq10 `minus` qq11 `minus` qq12) hr hrr
let qq13 = snd $ head ll
bmwrite file $ bmborder 1 $ bmmax hrbmav 0 0 $ hrbm 28 3 2 $ qqhr 2 uu vvk qq13
let ll = buildtup (2^12) 10 10 uu (vv `minus` qq1 `minus` qq2 `minus` qq3 `minus` qq4 `minus` qq5 `minus` qq6 `minus` qq7 `minus` qq8 `minus` qq9 `minus` qq10 `minus` qq11 `minus` qq12 `minus` qq13) hr hrr
let qq14 = snd $ head ll
bmwrite file $ bmborder 1 $ bmmax hrbmav 0 0 $ hrbm 28 3 2 $ qqhr 2 uu vvk qq14
let ll = buildtup (2^12) 10 10 uu (vv `minus` qq1 `minus` qq2 `minus` qq3 `minus` qq4 `minus` qq5 `minus` qq6 `minus` qq7 `minus` qq8 `minus` qq9 `minus` qq10 `minus` qq11 `minus` qq12 `minus` qq13 `minus` qq14) hr hrr
let qq15 = snd $ head ll
bmwrite file $ bmborder 1 $ bmmax hrbmav 0 0 $ hrbm 28 3 2 $ qqhr 2 uu vvk qq15
let ll = buildtup (2^12) 10 10 uu (vv `minus` qq1 `minus` qq2 `minus` qq3 `minus` qq4 `minus` qq5 `minus` qq6 `minus` qq7 `minus` qq8 `minus` qq9 `minus` qq10 `minus` qq11 `minus` qq12 `minus` qq13 `minus` qq14 `minus` qq15) hr hrr
let qq16 = snd $ head ll
bmwrite file $ bmborder 1 $ bmmax hrbmav 0 0 $ hrbm 28 3 2 $ qqhr 2 uu vvk qq16
let ll = buildtup (2^12) 10 10 uu (vv `minus` qq1 `minus` qq2 `minus` qq3 `minus` qq4 `minus` qq5 `minus` qq6 `minus` qq7 `minus` qq8 `minus` qq9 `minus` qq10 `minus` qq11 `minus` qq12 `minus` qq14 `minus` qq15 `minus` qq16) hr hrr
let qq17 = snd $ head ll
bmwrite file $ bmborder 1 $ bmmax hrbmav 0 0 $ hrbm 28 3 2 $ qqhr 2 uu vvk qq17
let ll = buildtup (2^12) 10 10 uu (vv `minus` qq1 `minus` qq2 `minus` qq3 `minus` qq4 `minus` qq5 `minus` qq6 `minus` qq7 `minus` qq8 `minus` qq9 `minus` qq10 `minus` qq11 `minus` qq12 `minus` qq14 `minus` qq15 `minus` qq16 `minus` qq17) hr hrr
rpln ll
"(19820.65239757364,{<20,18>,<21,18>,<21,21>,<22,18>,<22,19>,<22,20>,<22,21>,<23,18>,<23,19>,<23,20>,<24,18>,<24,19>})"
"(19539.81867588329,{<20,18>,<21,18>,<21,21>,<21,22>,<22,18>,<22,19>,<22,20>,<22,21>,<23,18>,<23,19>,<23,20>,<24,18>})"
"(19518.87254378562,{<20,18>,<21,17>,<21,18>,<21,21>,<22,18>,<22,19>,<22,20>,<22,21>,<23,18>,<23,19>,<23,20>,<24,18>})"
"(19512.985036130944,{<20,18>,<21,17>,<21,18>,<22,18>,<22,19>,<22,20>,<23,18>,<23,19>,<23,20>,<24,17>,<24,18>,<24,19>})"
"(19471.80974950829,{<20,18>,<21,18>,<21,21>,<22,18>,<22,19>,<22,20>,<22,21>,<23,18>,<23,19>,<23,20>,<24,17>,<24,18>})"
"(19465.866754705872,{<20,18>,<21,18>,<21,21>,<22,18>,<22,19>,<22,20>,<22,21>,<23,18>,<23,19>,<23,20>,<23,21>,<24,18>})"
"(19444.622703231304,{<20,18>,<21,18>,<22,18>,<22,19>,<22,20>,<22,21>,<23,18>,<23,19>,<23,20>,<23,21>,<24,18>,<24,19>})"
"(19440.703144773594,{<20,18>,<21,17>,<21,18>,<22,18>,<22,19>,<22,20>,<22,21>,<23,18>,<23,19>,<23,20>,<24,18>,<24,19>})"
"(19424.908999583487,{<20,18>,<21,18>,<22,18>,<22,19>,<22,20>,<22,21>,<23,18>,<23,19>,<23,20>,<24,17>,<24,18>,<24,19>})"
"(19420.152751294787,{<20,18>,<21,18>,<22,18>,<22,19>,<22,20>,<22,21>,<23,18>,<23,19>,<23,20>,<24,18>,<24,19>,<24,20>})"
let qq18 = snd $ head ll
bmwrite file $ bmborder 1 $ bmmax hrbmav 0 0 $ hrbm 28 3 2 $ qqhr 2 uu vvk qq18
Now the alignments are beginning to decrease and the clusters of pixels are less round.
let ll = buildtup (2^12) 10 10 uu (vv `minus` qq1 `minus` qq2 `minus` qq3 `minus` qq4 `minus` qq5 `minus` qq6 `minus` qq7 `minus` qq8 `minus` qq9 `minus` qq10 `minus` qq11 `minus` qq12 `minus` qq14 `minus` qq15 `minus` qq16 `minus` qq17 `minus` qq18) hr hrr
let qq19 = snd $ head ll
bmwrite file $ bmborder 1 $ bmmax hrbmav 0 0 $ hrbm 28 3 2 $ qqhr 2 uu vvk qq19
let ll = buildtup (2^12) 10 10 uu (vv `minus` qq1 `minus` qq2 `minus` qq3 `minus` qq4 `minus` qq5 `minus` qq6 `minus` qq7 `minus` qq8 `minus` qq9 `minus` qq10 `minus` qq11 `minus` qq12 `minus` qq14 `minus` qq15 `minus` qq16 `minus` qq17 `minus` qq18 `minus` qq19) hr hrr
let qq20 = snd $ head ll
bmwrite file $ bmborder 1 $ bmmax hrbmav 0 0 $ hrbm 28 3 2 $ qqhr 2 uu vvk qq20
let ll = buildtup (2^12) 10 10 uu (vv `minus` qq1 `minus` qq2 `minus` qq3 `minus` qq4 `minus` qq5 `minus` qq6 `minus` qq7 `minus` qq8 `minus` qq9 `minus` qq10 `minus` qq11 `minus` qq12 `minus` qq14 `minus` qq15 `minus` qq16 `minus` qq17 `minus` qq18 `minus` qq19 `minus` qq20) hr hrr
let qq21 = snd $ head ll
bmwrite file $ bmborder 1 $ bmmax hrbmav 0 0 $ hrbm 28 3 2 $ qqhr 2 uu vvk qq21
let ll = buildtup (2^12) 10 10 uu (vv `minus` qq1 `minus` qq2 `minus` qq3 `minus` qq4 `minus` qq5 `minus` qq6 `minus` qq7 `minus` qq8 `minus` qq9 `minus` qq10 `minus` qq11 `minus` qq12 `minus` qq14 `minus` qq15 `minus` qq16 `minus` qq17 `minus` qq18 `minus` qq19 `minus` qq20 `minus` qq21) hr hrr
let qq22 = snd $ head ll
bmwrite file $ bmborder 1 $ bmmax hrbmav 0 0 $ hrbm 28 3 2 $ qqhr 2 uu vvk qq22
Now the tuples tend to be located around the periphery.
let ll = buildtup (2^12) 10 10 uu (vv `minus` qq1 `minus` qq2 `minus` qq3 `minus` qq4 `minus` qq5 `minus` qq6 `minus` qq7 `minus` qq8 `minus` qq9 `minus` qq10 `minus` qq11 `minus` qq12 `minus` qq14 `minus` qq15 `minus` qq16 `minus` qq17 `minus` qq18 `minus` qq19 `minus` qq20 `minus` qq21 `minus` qq22) hr hrr
rpln ll
"(14977.746849732315,{<12,17>,<13,17>,<13,23>,<14,17>,<14,18>,<14,23>,<15,16>,<15,17>,<15,23>,<16,16>,<17,15>,<17,16>})"
"(14850.547162153118,{<12,17>,<12,23>,<13,17>,<13,23>,<14,17>,<14,18>,<14,23>,<15,16>,<15,17>,<16,16>,<17,15>,<17,16>})"
"(14724.3481495799,{<12,17>,<13,17>,<14,17>,<14,18>,<14,23>,<15,16>,<15,17>,<15,23>,<15,24>,<16,16>,<17,15>,<17,16>})"
"(14694.637140807843,{<12,17>,<13,17>,<13,22>,<13,23>,<14,17>,<14,18>,<14,23>,<15,16>,<15,17>,<16,16>,<17,15>,<17,16>})"
"(14669.459029120844,{<11,23>,<12,17>,<12,23>,<13,17>,<13,23>,<14,17>,<14,18>,<15,16>,<15,17>,<16,16>,<17,15>,<17,16>})"
"(14625.31652953853,{<12,17>,<13,17>,<14,17>,<14,18>,<14,23>,<14,24>,<15,16>,<15,17>,<15,23>,<16,16>,<17,15>,<17,16>})"
"(14618.409135971351,{<12,17>,<12,23>,<13,17>,<14,17>,<14,18>,<14,23>,<15,16>,<15,17>,<15,23>,<16,16>,<17,15>,<17,16>})"
"(14599.846077212673,{<12,17>,<12,23>,<13,17>,<13,22>,<13,23>,<14,17>,<14,18>,<15,16>,<15,17>,<16,16>,<17,15>,<17,16>})"
"(14591.823506121247,{<12,17>,<13,17>,<13,23>,<14,17>,<14,18>,<14,23>,<14,24>,<15,16>,<15,17>,<16,16>,<17,15>,<17,16>})"
"(14576.681402161907,{<12,17>,<13,17>,<13,22>,<14,17>,<14,18>,<14,23>,<15,16>,<15,17>,<15,23>,<16,16>,<17,15>,<17,16>})"
let qq23 = snd $ head ll
bmwrite file $ bmborder 1 $ bmmax hrbmav 0 0 $ hrbm 28 3 2 $ qqhr 2 uu vvk qq23
The union of the first 23 tuples is
bmwrite file $ bmborder 1 $ bmmax hrbmav 0 0 $ hrbm 28 3 2 $ qqhr 2 uu vvk (qq1 `union` qq2 `union` qq3 `union` qq4 `union` qq5 `union` qq6 `union` qq7 `union` qq8 `union` qq9 `union` qq10 `union` qq11 `union` qq12 `union` qq14 `union` qq15 `union` qq16 `union` qq17 `union` qq18 `union` qq19 `union` qq20 `union` qq21 `union` qq22 `union` qq23 `minus` vvl)
let ll = buildtup (2^12) 10 10 uu (vv `minus` qq1 `minus` qq2 `minus` qq3 `minus` qq4 `minus` qq5 `minus` qq6 `minus` qq7 `minus` qq8 `minus` qq9 `minus` qq10 `minus` qq11 `minus` qq12 `minus` qq14 `minus` qq15 `minus` qq16 `minus` qq17 `minus` qq18 `minus` qq19 `minus` qq20 `minus` qq21 `minus` qq22 `minus` qq23) hr hrr
let qq24 = snd $ head ll
bmwrite file $ bmborder 1 $ bmmax hrbmav 0 0 $ hrbm 28 3 2 $ qqhr 2 uu vvk qq24
let ll = buildtup (2^12) 10 10 uu (vv `minus` qq1 `minus` qq2 `minus` qq3 `minus` qq4 `minus` qq5 `minus` qq6 `minus` qq7 `minus` qq8 `minus` qq9 `minus` qq10 `minus` qq11 `minus` qq12 `minus` qq14 `minus` qq15 `minus` qq16 `minus` qq17 `minus` qq18 `minus` qq19 `minus` qq20 `minus` qq21 `minus` qq22 `minus` qq23 `minus` qq24) hr hrr
let qq25 = snd $ head ll
bmwrite file $ bmborder 1 $ bmmax hrbmav 0 0 $ hrbm 28 3 2 $ qqhr 2 uu vvk qq25
let ll = buildtup (2^12) 10 10 uu (vv `minus` qq1 `minus` qq2 `minus` qq3 `minus` qq4 `minus` qq5 `minus` qq6 `minus` qq7 `minus` qq8 `minus` qq9 `minus` qq10 `minus` qq11 `minus` qq12 `minus` qq14 `minus` qq15 `minus` qq16 `minus` qq17 `minus` qq18 `minus` qq19 `minus` qq20 `minus` qq21 `minus` qq22 `minus` qq23 `minus` qq24 `minus` qq25) hr hrr
let qq26 = snd $ head ll
bmwrite file $ bmborder 1 $ bmmax hrbmav 0 0 $ hrbm 28 3 2 $ qqhr 2 uu vvk qq26
let ll = buildtup (2^12) 10 10 uu (vv `minus` qq1 `minus` qq2 `minus` qq3 `minus` qq4 `minus` qq5 `minus` qq6 `minus` qq7 `minus` qq8 `minus` qq9 `minus` qq10 `minus` qq11 `minus` qq12 `minus` qq14 `minus` qq15 `minus` qq16 `minus` qq17 `minus` qq18 `minus` qq19 `minus` qq20 `minus` qq21 `minus` qq22 `minus` qq23 `minus` qq24 `minus` qq25 `minus` qq26) hr hrr
let qq27 = snd $ head ll
bmwrite file $ bmborder 1 $ bmmax hrbmav 0 0 $ hrbm 28 3 2 $ qqhr 2 uu vvk qq27
let ll = buildtup (2^12) 10 10 uu (vv `minus` qq1 `minus` qq2 `minus` qq3 `minus` qq4 `minus` qq5 `minus` qq6 `minus` qq7 `minus` qq8 `minus` qq9 `minus` qq10 `minus` qq11 `minus` qq12 `minus` qq14 `minus` qq15 `minus` qq16 `minus` qq17 `minus` qq18 `minus` qq19 `minus` qq20 `minus` qq21 `minus` qq22 `minus` qq23 `minus` qq24 `minus` qq25 `minus` qq26 `minus` qq27) hr hrr
let qq28 = snd $ head ll
bmwrite file $ bmborder 1 $ bmmax hrbmav 0 0 $ hrbm 28 3 2 $ qqhr 2 uu vvk qq28
let ll = buildtup (2^12) 10 10 uu (vv `minus` qq1 `minus` qq2 `minus` qq3 `minus` qq4 `minus` qq5 `minus` qq6 `minus` qq7 `minus` qq8 `minus` qq9 `minus` qq10 `minus` qq11 `minus` qq12 `minus` qq14 `minus` qq15 `minus` qq16 `minus` qq17 `minus` qq18 `minus` qq19 `minus` qq20 `minus` qq21 `minus` qq22 `minus` qq23 `minus` qq24 `minus` qq25 `minus` qq26 `minus` qq27 `minus` qq28) hr hrr
let qq29 = snd $ head ll
bmwrite file $ bmborder 1 $ bmmax hrbmav 0 0 $ hrbm 28 3 2 $ qqhr 2 uu vvk qq29
let ll = buildtup (2^12) 10 10 uu (vv `minus` qq1 `minus` qq2 `minus` qq3 `minus` qq4 `minus` qq5 `minus` qq6 `minus` qq7 `minus` qq8 `minus` qq9 `minus` qq10 `minus` qq11 `minus` qq12 `minus` qq14 `minus` qq15 `minus` qq16 `minus` qq17 `minus` qq18 `minus` qq19 `minus` qq20 `minus` qq21 `minus` qq22 `minus` qq23 `minus` qq24 `minus` qq25 `minus` qq26 `minus` qq27 `minus` qq28 `minus` qq29) hr hrr
rpln ll
"(8880.090350884573,{<6,8>,<6,9>,<7,7>,<7,8>,<8,6>,<8,7>,<8,8>,<9,6>,<9,7>,<10,5>,<10,6>,<11,6>})"
"(8869.269672126837,{<6,8>,<6,9>,<7,7>,<7,8>,<8,6>,<8,7>,<8,8>,<9,6>,<9,7>,<10,6>,<11,5>,<11,6>})"
"(8841.548488548702,{<6,8>,<6,9>,<7,7>,<7,8>,<8,6>,<8,7>,<8,8>,<9,5>,<9,6>,<9,7>,<10,6>,<11,6>})"
"(8811.666737187988,{<5,10>,<6,8>,<6,9>,<7,7>,<7,8>,<8,6>,<8,7>,<8,8>,<9,6>,<9,7>,<10,6>,<11,6>})"
"(8792.207908629527,{<6,7>,<6,8>,<6,9>,<7,7>,<7,8>,<8,6>,<8,7>,<8,8>,<9,6>,<9,7>,<10,6>,<11,6>})"
"(8771.51758510567,{<6,8>,<6,9>,<7,6>,<7,7>,<7,8>,<8,6>,<8,7>,<8,8>,<9,6>,<9,7>,<10,6>,<11,6>})"
"(8740.400156745367,{<6,9>,<7,7>,<7,8>,<8,6>,<8,7>,<8,8>,<9,6>,<9,7>,<10,5>,<10,6>,<11,5>,<11,6>})"
"(8733.872056226115,{<5,9>,<6,8>,<6,9>,<7,7>,<7,8>,<8,6>,<8,7>,<8,8>,<9,6>,<9,7>,<10,6>,<11,6>})"
"(8717.620830479595,{<6,8>,<6,9>,<7,7>,<7,8>,<8,6>,<8,7>,<8,8>,<9,6>,<9,7>,<10,6>,<11,6>,<12,5>})"
"(8704.861816530327,{<6,8>,<6,9>,<7,7>,<7,8>,<8,6>,<8,7>,<8,8>,<9,6>,<9,7>,<10,6>,<11,6>,<23,21>})"
let qq30 = snd $ head ll
bmwrite file $ bmborder 1 $ bmmax hrbmav 0 0 $ hrbm 28 3 2 $ qqhr 2 uu vvk qq30
Now, after 30 tuples, the alignments are much decreased and the union is
bmwrite file $ bmborder 1 $ bmmax hrbmav 0 0 $ hrbm 28 3 2 $ qqhr 2 uu vvk (qq1 `union` qq2 `union` qq3 `union` qq4 `union` qq5 `union` qq6 `union` qq7 `union` qq8 `union` qq9 `union` qq10 `union` qq11 `union` qq12 `union` qq14 `union` qq15 `union` qq16 `union` qq17 `union` qq18 `union` qq19 `union` qq20 `union` qq21 `union` qq22 `union` qq23 `union` qq24 `union` qq25 `union` qq26 `union` qq27 `union` qq28 `union` qq29 `union` qq30 `minus` vvl)
Averaged pixels
The pixel variables can be averaged as well as bucketed to reduce the dimension from 28x28 to 9x9. Here 3x3 square regions are bucketed to have 8 values,
(uu,hrtr) <- nistTrainBucketedAveragedIO 8 9 0
let digit = VarStr "digit"
let vv = uvars uu
let vvl = sgl digit
let vvk = vv `minus` vvl
let hr = hrev [i | i <- [0 .. hrsize hrtr - 1], i `mod` 8 == 0] hrtr
hrsize hr
7500
The query variable valency is 8, $\{|U_w| : w \in V_{\mathrm{k}}\} = \{8\}$,
rp $ llqq [u | w <- qqll vvk, let u = vol uu (sgl w)]
"{8}"
The first 25 events placed in a row now look blurrier,
bmwrite file $ bmhstack $ map (bmborder 1 . hrbm 9 3 8) $ [hrev [i] hr' | let hr' = hr `hrhrred` vvk, i <- [0..24]]
Again, the entire history can be averaged together, $\hat{A}\%V_{\mathrm{k}}$,
let hrbmav = hrbm 9 (3*3) 8 $ hr `hrhrred` vvk
bmwrite file $ bmborder 1 $ hrbmav
We can show the average image for each digit, $\{\hat{B}\%V_{\mathrm{k}} : S \in V_{\mathrm{l}}^{\mathrm{CS}},~B = A * \{S\}^{\mathrm{U}}\}$,
bmwrite file $ bmhstack $ map (bmborder 1 . hrbm 9 (2*3) 8) $ [hr `hrhrsel` hrs `hrhrred` vvk | ss <- qqll (cart uu vvl), let hrs = aahr uu (single ss 1)]
Again, consider how highly aligned variables might be grouped together. Create a shuffled sample, $A_{\mathrm{r}}$,
let hrr = historyRepasShuffle_u hr 1
hrsize hrr
7500
Now optimise the shuffle content alignment with the tuple set builder, $I_{P,U,\mathrm{B,ns,me}}$, \[ \{(\mathrm{algn}(A\%K)-\mathrm{algn}(A_{\mathrm{r}}\%K),~K) : ((K,\cdot,\cdot),\cdot) \in I_{P,U,\mathrm{B,ns,me}}^{ * }((V,~\emptyset,~A,~A_{\mathrm{r}}))\} \]
let buildtup xmax omax bmax uu vv xx xxrr = reverse $ sort $ map (\((kk,_),_) -> (algn (araa uu (xx `hrred` kk)) - algn (araa uu (xxrr `hrred` kk)), kk)) $ parametersSystemsBuilderTupleNoSumlayerMultiEffectiveRepa_u xmax omax bmax 1 uu vv fudEmpty xx (hrhx xx) xxrr (hrhx xxrr)
rpln $ buildtup (8^2) 10 10 uu vvk hr hrr
"(1776.1172125281591,{<5,3>,<6,3>})"
"(1545.8573948188132,{<5,8>,<6,8>})"
"(1507.5649470190838,{<4,3>,<5,3>})"
"(1452.0630556805627,{<6,7>,<7,7>})"
"(1421.1282539067906,{<8,3>,<8,4>})"
"(1247.7813479567485,{<7,7>,<8,7>})"
"(1200.1703380912804,{<6,8>,<7,8>})"
"(1188.7902099063358,{<5,7>,<6,7>})"
"(1175.2736813378651,{<4,8>,<5,8>})"
"(1171.2449428311447,{<8,4>,<8,5>})"
let ll = buildtup (8^4) 10 10 uu vvk hr hrr
rpln ll
"(5705.540324840418,{<5,3>,<5,8>,<6,3>,<6,8>})"
"(5583.217843046474,{<4,3>,<5,3>,<5,8>,<6,3>})"
"(5341.973726252629,{<5,3>,<5,8>,<6,3>,<7,3>})"
"(5294.712842043013,{<4,3>,<5,3>,<6,3>,<6,8>})"
"(5280.452073191616,{<4,8>,<5,3>,<5,8>,<6,3>})"
"(5157.719449275057,{<4,3>,<5,3>,<6,3>,<7,3>})"
"(5130.214215739274,{<5,3>,<6,3>,<6,8>,<7,3>})"
"(5094.733149115389,{<5,8>,<6,3>,<6,8>,<7,3>})"
"(5029.909759006136,{<4,3>,<5,3>,<5,7>,<6,3>})"
"(5004.033909756206,{<4,3>,<4,4>,<5,3>,<6,3>})"
let ll = buildtup (8^5) 10 10 uu vvk hr hrr
rpln ll
"(8104.741491017787,{<4,3>,<5,3>,<5,8>,<6,3>,<6,8>})"
"(8009.148902931618,{<5,3>,<5,8>,<6,3>,<6,8>,<7,3>})"
"(7789.983742761848,{<4,8>,<5,3>,<5,8>,<6,3>,<6,8>})"
"(7788.850798522242,{<5,3>,<5,8>,<6,3>,<6,8>,<7,8>})"
"(7703.103655342755,{<4,3>,<4,8>,<5,3>,<5,8>,<6,3>})"
"(7584.881419085439,{<4,8>,<5,3>,<5,8>,<6,3>,<7,3>})"
"(7541.137140903713,{<5,3>,<5,7>,<5,8>,<6,3>,<6,8>})"
"(7470.02043815628,{<4,3>,<5,3>,<5,8>,<6,3>,<7,3>})"
"(7443.429731564436,{<5,8>,<6,3>,<6,8>,<7,3>,<7,8>})"
"(7435.267678433767,{<5,3>,<5,8>,<6,3>,<6,7>,<6,8>})"
Let us image the most highly aligned tuple,
let qq1 = snd $ head ll
bmwrite file $ bmborder 1 $ bmmax hrbmav 0 0 $ hrbm 9 (3*3) 8 $ qqhr 8 uu vvk qq1
The first tuple is split into two clusters on the left and right of the averaged image.
Now let’s image the first 20 states ordered by size descending, $\mathrm{top}(20)(A\%Q_1)$,
let pp = take 20 $ reverse $ sort $ map (\(a,b) -> (b,a)) $ aall $ araa uu $ hr `hrred` qq1
map numerator $ fst $ unzip pp
[3165,177,154,125,108,95,94,87,72,69,49,44,42,41,40,35,33,31,27,27]
bmwrite file $ bmhstack $ map (bmborder 1 . hrbm 9 3 8) $ [hr' `hrhrsel` hrs | let hr' = hr `hrhrred` vvk, (_,ss) <- pp, let hrs = aahr uu (single ss 1)]
Now optimise again having removed the top tuple from the substrate,
let ll = buildtup (8^5) 10 10 uu (vvk `minus` qq1) hr hrr
rpln ll
"(6572.6547679746145,{<2,6>,<7,3>,<7,4>,<8,3>,<8,4>})"
"(6488.3042785712005,{<3,3>,<6,7>,<7,7>,<7,8>,<8,7>})"
"(6395.7932020613425,{<5,7>,<6,7>,<7,7>,<7,8>,<8,7>})"
"(6354.816586886205,{<3,7>,<7,3>,<7,4>,<8,3>,<8,4>})"
"(6328.2316907764,{<2,4>,<6,7>,<7,7>,<7,8>,<8,7>})"
"(6313.35657709419,{<3,3>,<3,4>,<7,7>,<7,8>,<8,7>})"
"(6238.528984816963,{<3,4>,<6,7>,<7,7>,<7,8>,<8,7>})"
"(6205.672198558159,{<6,7>,<7,3>,<7,7>,<7,8>,<8,7>})"
"(6172.896596761955,{<6,7>,<7,7>,<7,8>,<8,7>,<8,8>})"
"(6104.923863975762,{<2,6>,<3,7>,<7,3>,<8,3>,<8,4>})"
let qq2 = snd $ head ll
bmwrite file $ bmborder 1 $ bmmax hrbmav 0 0 $ hrbm 9 (3*3) 8 $ qqhr 8 uu vvk qq2
Again, let’s image the first 20 states ordered by size descending,
let pp = take 20 $ reverse $ sort $ map (\(a,b) -> (b,a)) $ aall $ araa uu $ hr `hrred` qq2
map numerator $ fst $ unzip pp
[2034,242,131,127,115,82,82,75,71,55,55,45,41,34,29,23,22,22,20,19]
bmwrite file $ bmhstack $ map (bmborder 1 . hrbm 9 3 8) $ [hr' `hrhrsel` hrs | let hr' = hr `hrhrred` vvk, (_,ss) <- pp, let hrs = aahr uu (single ss 1)]
Continuing on,
let ll = buildtup (8^5) 10 10 uu (vvk `minus` qq1 `minus` qq2) hr hrr
rpln ll
"(6488.3042785712005,{<3,3>,<6,7>,<7,7>,<7,8>,<8,7>})"
"(6395.7932020613425,{<5,7>,<6,7>,<7,7>,<7,8>,<8,7>})"
"(6328.2316907764,{<2,4>,<6,7>,<7,7>,<7,8>,<8,7>})"
"(6313.35657709419,{<3,3>,<3,4>,<7,7>,<7,8>,<8,7>})"
"(6238.528984816963,{<3,4>,<6,7>,<7,7>,<7,8>,<8,7>})"
"(6172.896596761955,{<6,7>,<7,7>,<7,8>,<8,7>,<8,8>})"
"(6060.136427505366,{<2,5>,<6,7>,<7,7>,<7,8>,<8,7>})"
"(6018.961715466963,{<3,3>,<3,4>,<6,7>,<7,7>,<8,7>})"
"(5943.431729848828,{<2,4>,<3,3>,<7,7>,<7,8>,<8,7>})"
"(5905.2733534884355,{<2,4>,<3,4>,<7,7>,<7,8>,<8,7>})"
let qq3 = snd $ head ll
bmwrite file $ bmborder 1 $ bmmax hrbmav 0 0 $ hrbm 9 (3*3) 8 $ qqhr 8 uu vvk qq3
We can continue in this way,
let ll = buildtup (8^5) 10 10 uu (vvk `minus` qq1 `minus` qq2 `minus` qq3) hr hrr
rpln ll
let qq4 = snd $ head ll
bmwrite file $ bmborder 1 $ bmmax hrbmav 0 0 $ hrbm 9 (3*3) 8 $ qqhr 8 uu vvk qq4
let ll = buildtup (8^5) 10 10 uu (vvk `minus` qq1 `minus` qq2 `minus` qq3 `minus` qq4) hr hrr
rpln ll
let qq5 = snd $ head ll
bmwrite file $ bmborder 1 $ bmmax hrbmav 0 0 $ hrbm 9 (3*3) 8 $ qqhr 8 uu vvk qq5
let ll = buildtup (8^5) 10 10 uu (vvk `minus` qq1 `minus` qq2 `minus` qq3 `minus` qq4 `minus` qq5) hr hrr
rpln ll
let qq6 = snd $ head ll
bmwrite file $ bmborder 1 $ bmmax hrbmav 0 0 $ hrbm 9 (3*3) 8 $ qqhr 8 uu vvk qq6
let ll = buildtup (8^5) 10 10 uu (vvk `minus` qq1 `minus` qq2 `minus` qq3 `minus` qq4 `minus` qq5 `minus` qq6) hr hrr
rpln ll
let qq7 = snd $ head ll
bmwrite file $ bmborder 1 $ bmmax hrbmav 0 0 $ hrbm 9 (3*3) 8 $ qqhr 8 uu vvk qq7
let ll = buildtup (8^5) 10 10 uu (vvk `minus` qq1 `minus` qq2 `minus` qq3 `minus` qq4 `minus` qq5 `minus` qq6 `minus` qq7) hr hrr
rpln ll
"(2143.66264664146,{<3,6>,<4,6>,<5,6>,<6,6>,<9,4>})"
"(2118.4376825965483,{<3,6>,<4,6>,<5,6>,<6,6>,<9,3>})"
"(2104.8979422774164,{<3,6>,<4,6>,<5,6>,<6,6>,<9,8>})"
"(2103.539781824804,{<3,2>,<3,6>,<4,6>,<5,6>,<6,6>})"
"(2103.194294433093,{<3,6>,<4,6>,<5,6>,<6,6>,<9,7>})"
"(2100.8651522076125,{<3,6>,<4,1>,<4,6>,<5,6>,<6,6>})"
"(2099.9437463645263,{<3,6>,<4,6>,<5,6>,<6,6>,<9,9>})"
"(2099.89495619852,{<3,6>,<4,6>,<5,6>,<6,6>,<7,1>})"
"(2099.5562997076604,{<3,6>,<4,6>,<5,6>,<6,6>,<9,2>})"
"(2097.84368553738,{<3,6>,<4,6>,<5,1>,<5,6>,<6,6>})"
let qq8 = snd $ head ll
bmwrite file $ bmborder 1 $ bmmax hrbmav 0 0 $ hrbm 9 (3*3) 8 $ qqhr 8 uu vvk qq8
Now the alignment has decreased significantly.
let ll = buildtup (8^5) 10 10 uu (vvk `minus` qq1 `minus` qq2 `minus` qq3 `minus` qq4 `minus` qq5 `minus` qq6 `minus` qq7 `minus` qq8) hr hrr
rpln ll
let qq9 = snd $ head ll
bmwrite file $ bmborder 1 $ bmmax hrbmav 0 0 $ hrbm 9 (3*3) 8 $ qqhr 8 uu vvk qq9
let ll = buildtup (8^5) 10 10 uu (vvk `minus` qq1 `minus` qq2 `minus` qq3 `minus` qq4 `minus` qq5 `minus` qq6 `minus` qq7 `minus` qq8 `minus` qq9) hr hrr
rpln ll
"(836.6038538031753,{<2,7>,<2,8>,<4,5>,<7,5>,<8,2>})"
"(778.1487164161663,{<2,7>,<2,8>,<7,5>,<8,2>,<9,3>})"
"(757.7250531224563,{<2,7>,<2,8>,<2,9>,<7,5>,<8,2>})"
"(750.3849735902622,{<7,5>,<7,9>,<8,8>,<9,7>,<9,8>})"
"(746.8050492020193,{<1,7>,<2,7>,<2,8>,<7,5>,<8,2>})"
"(746.3037210041439,{<2,7>,<2,8>,<7,5>,<8,2>,<9,2>})"
"(741.0944619696747,{<2,7>,<2,8>,<4,5>,<8,2>,<9,3>})"
"(737.8425886507466,{<1,6>,<2,7>,<2,8>,<7,5>,<8,2>})"
"(734.0971770242686,{<1,7>,<2,7>,<2,8>,<4,5>,<7,5>})"
"(729.5423314803847,{<2,7>,<2,8>,<2,9>,<4,5>,<7,5>})"
let qq10 = snd $ head ll
bmwrite file $ bmborder 1 $ bmmax hrbmav 0 0 $ hrbm 9 (3*3) 8 $ qqhr 8 uu vvk qq10
Now, after 10 tuples, the alignments are much decreased and the union is
bmwrite file $ bmborder 1 $ bmmax hrbmav 0 0 $ hrbm 9 (3*3) 8 $ qqhr 8 uu vvk (qq1 `union` qq2 `union` qq3 `union` qq4 `union` qq5 `union` qq6 `union` qq7 `union` qq8 `union` qq9 `union` qq10)
