For a Tweetbase construct Docment p Trees indexed

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For a Tweetbase, construct Docment p. Trees (indexed by Term and Position)? Term p. Trees index by Doc, Pos. What about phrases? For 2 word phrases, use 4 D cube. 2… nd 0 0 Why might the positions of words be important? e. g. , “buy” and “AAPL” occur close in tweet position, - a stronger positive sentiment. If multilevel Pos p. Trees, a positive sentiment bloom: buy, tweet 1 AAPL, tweet 1 1 0 0 1 1000 0000 010 determine by level 1 & Doc p. Trees (p. Trees are named or index by Term and Position) Position 1 2 3 4 5 6 0 0 0 0 0 2 wd 1 1 0 0 0 1 0 0 0 0 7 1 0 0 0 0 1 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 00 0 0 0 0 1 0 0 0 0 0 0 1 buy again all always. an and apple April 2 3 4 1 st wd 0 0 0 5 Twt 3 Tweet 2 Tweet 1 2 -word phrase start position 6 2 wd. Phrase. Start. Pos p. Tree index (buy, AAPL, Tweet 1) 1 0 0 Pre-compute, save and catalog? or compute as needed by shifting AAPL TP p. Tree right 1 -bit, then AND with the buy TP p. Tree. Multilevel 2 wd. Phrase. Start. Pos p. Trees strides= D, W, W, W-1, D=#docs, W=#wds Term buy AAPL 0 Do cs Doc Pos 1 Sentiment analysis (by doc) : PSB: Positive Sentiment Bit. Map, 1 iff doc has positive AAPL sentiment. Tweet 3 1 PSV: Positive Sentiment Value. Array, measures positive AAPL sentiment level? Tweet 2 1 0 PSV for each term? Might term context change the sentiment? Tweet 1 1 0 With term position information we should be able to evaluate PSV in context! 0 0 PSB, PSV could be derived by hand (humans read tweets and assign a PSB or PSV). 0 0 Do we need to use PS minus NS measures? (NS=Negative Sentiment) 0 0 0 Research literature on Sentiment Analysis (word/doc sentiment assessment software ? 0 0 Strategy: Each day buy the stock with the greatest Positive Sentiment Tweet Bloom? 0 . . . 1 Tweet 1 AAPL all 0 1 0 0 1 buy 0100… AAPL 0000… buy AAPL… 1 0 0 1 buy AAPL… . . . 0 0 0 . . . always 1 0 0 0 . . . s 0 . D oc 0 Tweet 3 Tweet 2 Tweet 1 1 0 0 0 0 1 0 April 0 0 0 0 . . . are 0 0 0 0 . . . 0 0 1 . . . 0 0 0 0 0 1 0 0 0 0 . . . Etc. 0 . . . 1 0 . . . 0 0 . . . 1 0 apple 0 0 0 and 0 0 0 1 0 0 an 0 0 0 0 Document. Term. Position p. Tree. Set, e. g. , Tweet Sentiment Analysis (1 iff Term in Pos in Doc) Positions Of Terms… 1 2 3 4 5 6 7 . . . Doc Tweet 3 Term a Tweet 2 . . . 1 0 0 Tweet 1 0 0 1 0 . [Term]Pos p. Trees OR row gives Term=a Existential. Term p. Tree. Sum gives Term=a Doc. Freq (df) array

Tp. Trees for 1 st 20 Tweets of the AAPL Tweetbase (100 wds). DOC POS 2015 1 AAPL 2 ahead 3 alerts 4 all 5 Apple 6 area 7 big 8 bond 9 breakout 10 call 11 chart 12 check 13 course 14 custom 15 day 16 email 17 happy 18 helping 19 her 20 https: //t. co/5 rss. X 3 gn. PY 21 https: //t. co/DOTIjn 3 z. Th 22 https: //t. co/gbg. TQasq 22 23 https: //t. co/JCch. Dk. M 3 k. Z 24 https: //t. co/j. GJNzj 6 I 5 i 25 https: //t. co/OSDAg. Xz. NFf 26 https: //t. co/vvd. Qf. CSmj 5 27 https: //t. co/XRD 1 m. Bp. Dd. E 28 https: //t. co/Zc. Zh. F 2 AWSE 29 http: //t. co/o. ZKXEI 87 ZK 30 http: //t. co/Sk. Opa. K 2 v. QS 31 http: //t. co/u 8 Vx. UQot. Tw 32 hype 33 info 34 investor 35 iwatch 36 join 37 keep 38 lessons 39 list 40 looking 41 losing 42 market 43 mentor 44 might 45 moments 46 money 47 more 48 multi-day 49 new 50 nice 51 online 52 out 53 pass 54 pay 55 peeps 56 plans 57 play 58 potential 59 program 60 prosperous 61 reasons 62 run 63 runners 64 safe 65 sale 66 send 67 sensitive 68 shares 69 sign 70 Starbucks 71 stock 72 street 73 subs 74 swing 75 take 76 three 77 today 78 trade 79 up 80 value 81 video 82 visit 83 week 84 winners 85 year 86 $AAPL 87 $BAC 88 $DNKN 89 $FB 90 $GMCR 91 $GOOG 92 $IDRA 93 $JAKK 94 $SBUX 95 $TWTR 96 0. 75 97 2. 7 98 3. 5 99 3. 77 100 1 cts 11 1 1 56 8 a b d e 00 0 0 1 00 0 0 0 0 0 00 0 0 0 0 0 00 0 0 0 0 0 00 0 0 0 0 0 00 0 0 1 0 0 0 0 00 0 0 0 0 0 00 0 0 0 0 0 00 0 0 0 0 0 00 0 1 0 00 0 0 00 0 1 0 0 0 00 0 0 0 0 0 00 0 0 0 00 0 0 01 0 0 0 0 0 00 0 0 0 0 0 00 0 0 0 0 0 00 0 0 0 0 0 00 0 0 0 0 0 00 0 0 00 0 0 10 0 0 00 0 0 0 0 0 00 0 0 0 0 0 00 0 0 0 0 0 00 0 0 0 0 0 00 0 0 0 0 0 00 0 0 0 0 0 00 0 0 0 0 0 00 0 0 00 0 0 11 1 1 2 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 222 245 000 000 000 000 010 000 000 000 000 000 000 000 000 000 000 100 000 000 000 000 000 000 001 000 000 000 000 000 000 111 2 7 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 22 9 a 00 00 00 00 00 00 10 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 01 00 00 00 11 22 bc 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 10 00 00 01 00 00 11 2 d 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 Question: Use p. Trees or use string processing primitives on the actual tweets (only 1 1 -bit per p. Tree)? Use multilevel p. 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The first 20 tweets of the AAPL Tweetbase. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. At Check For Keep Custom Happy Take Get Day Online $SBUX Sign Hot Nice Been Sign Send $IDRA $AAPL one out more losing trading new a stock pass Trading Three up list looking a up me could Apple of my info money classes year trading alerts for Course's Reasons for of daily nice for an run plans those custom on in with to course here our The Starbucks our stocks chart run our email to investor sensitive trading our the us. all! with multi-day trading Street Shares trading out on for daily for the call moments mentorship courses market? https: //t. co Have us! runners! alerts Mentor Might alerts today $JAKK sub's trading the $3. 50 ahead give courses visit Sign $FB a https: //t. co here: https: //t. co Stay here Sign for from alerts 75% area of her a iwatch helping the here: https: //t. co$fb $AAPL $twtr us here: https: //t. co$FB $AAPL up for a custom trading $AAPL $TWTR Peeps #investing #trading safe and prosperous new year! $FB $AAPL $TWTR Peeps #stocks $FB $AAPL $TWTR peeps https: //t. co$FB $AAPL $TWTR #stocks $FB $AAPL $TWTR Traders. Hot In 2015 http: //t. co/#IBDNews Big winners this week. https: //t. co up for our daily alerts a swing trade here. $TWTR $2. 70 trade' at $3. 77 AH_ and weekly video lessons here! off trading courses here $FB today. Nice breakout here. $FB a potential bond sale: WSJ 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. hype peeps. $TWTR mentorship AAPL #stocks program #trading with us today. https: //t. co $FB $AAPL #stocks #investing #trading #investing $FB $AAPL $TWTR peeps. $GOOG $DNKN $GMCR $TWTR play. $AAPL Peeps Join http: //t. co/ $BAC us! u 8 Vx. UQot. Tw $FB $TWTR #daytrading via @IBDinvestor$SBUX $AAPL $FB $AAPL $TWTR Peeps here! https: //t. co$FB $AAPL Peeps value Nice pay day $TWTR https: //t. co$FB $AAPL $TWTR Traders. http: //t. co/via @Yahoo. Finance 19. https: //t. co/vvd. Qf. CSmj 5 It might be useful for the group to focus on Sentiment Analysis as a over-riding topic this term. So would all those who are required to give a presentation please choose a recent substantial (meaning from a the proceedings of a major ACM or IEEE conference or a ACM or IEE journal dated 2010 or after)? Send you choice to me for approval and then we can schedule a presentation week. Thanks so much!

AAPL 2 level p. Trees stride=10 doc pos 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 92 94 95 96 97 98 99 11 1 1 56 8 a b d e 00 0 0 1 00 0 0 0 0 0 00 0 0 0 0 0 00 0 0 0 0 0 00 0 0 0 0 0 00 0 0 1 0 0 0 0 00 0 0 0 0 0 00 0 0 0 0 0 00 0 0 0 0 0 00 0 1 0 00 0 0 00 0 1 0 0 0 00 0 0 0 0 0 00 0 0 0 00 0 0 01 0 0 0 0 0 00 0 0 0 0 0 00 0 0 0 0 0 00 0 0 0 0 0 00 0 0 0 0 0 00 0 0 00 0 0 10 0 0 00 0 0 0 0 0 00 0 0 0 0 0 00 0 0 0 0 0 00 0 0 0 0 0 00 0 0 0 0 0 00 0 0 0 0 0 00 0 0 0 0 0 00 0 0 00 0 0 doc , pos 1, 5: 0 0 0 1 0 0000000100 0000000100 0000000100 1, 6: 0 0 1 0 0 0000000100 0000010000000100 0000000100 1, 8: 0 1 0 0 0 0000000100 0000010000000100 0000000100 1, a: 0 0 0 1 0 0 0000000100 0000010000000100 0000000100 1, b: 0 1 0 0 0 0000000100 000010 000000010000 0000000100 0000000100 1, d: 0 0 0 1 0 0 0000000100 000010 00000001000000000100 0000000100 1, e: 1 0 0 0 0 01000010 00000001000000000100 0000000100 2, 1: 0 1 0 0 0 00000001000000000100 0000010000000100 0000000100 2, 2: 0 0 1 0 0 0000000100 0000010000000 0000000100 2, 4: 0 1 0 0 0 0000000100000 000000010000 0000000100 0000000100 2, 5: 0 0 0 0 1 0 0000000100 0000010000000100 0000000100 2, 7: 0 1 0 0 0 00000001000000000100 0000010000000100 0000000100 2, 9: 0 0 1 0 0 0 0000000100 0000100000000100 0000000100 2, a: 0 0 0 0 1 01000010 00000001000000000100 0000000100 000001 2, b: 0 0 0 0 1 01000010 00000001000000000100 00000001000 2, c: 0 0 0 0 01000010 00000001000000000100 00000001000 2, c: 0 0 0 1 0 01000010 00000001000000000100 0000010000000100 0000001000 0 1 2 3 4 5 6 7 8 0 0 1 00000001002 3 4 0 5 6 00000001007 8 9 10 0 11 12 000000010013 14 15 0 16 17 000000010018 19 20 21 0 22 23 000000010024 25 26 0 27 28 29 000000010030 31 32 33 34 35 36 37 0 38 39 000000010040 41 42 43 0 44 45 0000000100 46 47 48 0 49 50 51 0000000100 52 53 54 0 55 56 000000010057 58 59 0 60 61 000000010062 63 64 65 0 66 67 000000010068 69 70 71 0 72 73 000000010074 75 76 0 77 78 000000010079 80 81 82 1 83 84 000001000085 86 87 88 0 89 90 000010000091 92 92 94 95 96 97 98 99 9 2 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 222 245 000 000 000 000 010 000 000 000 000 000 000 000 000 000 000 100 000 000 000 000 000 000 001 000 000 000 000 000 000 2 7 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 22 9 a 00 00 00 00 00 00 10 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 01 00 00 00 22 bc 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 10 00 00 01 00 00 2 d 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 2015 AAPL ahead alerts all Apple area big bond breakout call chart check course custom day email happy helping her https: //t. co/5 rss. X 3 gn. PY https: //t. co/DOTIjn 3 z. Th https: //t. co/gbg. TQasq 22 https: //t. co/JCch. Dk. M 3 k. Z https: //t. co/j. GJNzj 6 I 5 i https: //t. co/OSDAg. Xz. NFf https: //t. co/vvd. Qf. CSmj 5 https: //t. co/XRD 1 m. Bp. Dd. E https: //t. co/Zc. Zh. F 2 AWSE http: //t. co/o. ZKXEI 87 ZK http: //t. co/Sk. Opa. K 2 v. QS http: //t. co/u 8 Vx. UQot. Tw hype info investor iwatch join keep lessons list looking losing market mentor might moments money more multi-day new nice online out pass pay peeps plans play potential program prosperous reasons runners safe sale send sensitive shares sign Starbucks stock street subs swing take three today trade up value video visit week winners year $AAPL $BAC $DNKN $FB $GMCR $GOOG $IDRA $JAKK $SBUX $TWTR 0. 75 2. 7 3. 5 3. 77

AAPL 2 level p. Trees stride=10 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 92 94 95 96 97 98 99 3333 2367 abcd 00000000 00000000 00000000 0000 00100000000 00000000 000000001000 00000000 00000000 0000 0100000000 00000000 00000000 00000000 100000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000 0001000000000010 00000000100 00000000 00000001 00000000 3, 2: 0 0 1 0 0 0000000100 0000000100 0000000100 3, 3: 0 0 0 1 0 0 0000000100 001000000010000000100 0000000100 3, 6: 0 1 0 0 0 00000001000000000100 0000010000000100 0000000100 3, 7: 0 0 0 0 1 0000000100 0000010000000100 000000010000000 3, a: 0 0 1 0 0 0 0000000100 0000100000010000000100 0000000100 3. b: 0 0 0 0 1 0000000100 000010 00000001000000000100 0000000100 000001 3, c: 0 0 0 0 1 01000010 00000001000000000100 00000001000 3, d: 0 0 0 0 01000010 00000001000000000100 0000000100 4, 1: 0 0 0 1 0 0 00000001000000000100 0000000100 0000000100 4, 2: 0 0 1 0 0 0000000100 0000010000001000 0000000100 4, 3: 0 0 1 0 0 0000000100000 000000010000 0000001000 0000000100 4, 6: 0 0 1 0 0 0000000100 0000010000000 0000000100 4, 7: 0 0 0 1 0 0 00000001000000000100 0000010000000100 000001 0000000100 4, 8: 0 0 0 0 1 0 0000000100 0000100000000100 0000000100 000001 0000000100 4, a: 0 1 0 0 0 010000100000001000000000100 0000000100 000001 4, b: 0 0 0 0 1 0 01000010 00000001000000000100 0000000100 000010 0000001000 4, d: 0 0 0 1 0 01000010 00000001000000000100 000001 00000001000 4, g: 0 0 0 0 1 0 01000010 00000001000000000100 0000010000000100 0000001000 4, h: 0 0 1 0 0 0 01000010 0000010000000 000000010000 00000001000 4, i: 0 0 0 0 1 01000010 00000001000000000100 0000010000000100 000001 4, j: 0 0 0 0 1 01000010 00000001000000000100 0000010000000100 0000001000 0 1 2 3 4 5 6 7 8 4 1 0 0 0000000100 1 0 2 0 0 3 0 4 0 0000000100 5 0 6 0 7 0 0 8 0 9 0 0000000100 110 0 120 130 0000000100 140 150 160 0 170 180 0000000100 190 200 0 210 220 0000000100 230 240 1 250 260 0 0000010000 270 28 290 300 310 320 330 340 350 360 371 380 0 390 400 0000000100 410 420 0 430 440 0000000100 450 460 470 0 480 490 0000000100 510 0 520 530 0000000100 540 550 0 560 570 580 0000000100 590 600 0 610 620 0000000100 630 640 0 650 660 0000000100 670 680 690 0 700 710 0000000100 720 730 0 740 750 0000010000 760 770 780 0 790 800 0000100000 810 820 0 830 840 0000100000 850 860 870 0 880 890 0000100000 910 0 920 0000100000 940 950 960 970 980 990 9 0 444444 23678 abdghij 000000000000 000000000000 000000000000 0000000000001000000000000000000000000000000000000100 000000000000 000000000000 000000000000 000000000000 1000000 001000000000 000000 01000000000000 000000000000 000000000000 00000001000000000000000000000000000000 000100000000000000000000000000001000 00000010000000 000000000000 0000000000001 00000000000010 000000000000 000000000000 000000000000 2015 AAPL ahead alerts all Apple area big bond breakout call chart check course custom day email happy helping her https: //t. co/5 rss. X 3 gn https: //t. co/DOTIjn 3 z https: //t. co/gbg. TQasq https: //t. co/JCch. Dk. M 3 https: //t. co/j. GJNzj 6 I https: //t. co/OSDAg. Xz. N https: //t. co/vvd. Qf. CSm https: //t. co/XRD 1 m. Bp. D https: //t. co/Zc. Zh. F 2 AW http: //t. co/o. ZKXEI 87 Z http: //t. co/Sk. Opa. K 2 v. Q http: //t. co/u 8 Vx. UQot. T hype info investor iwatch join keep lessons list looking losing market mentor might moments money more multi-day new nice online out pass pay peeps plans play potential program prosperous reasons runners safe sale send sensitive shares sign Starbucks stock street subs swing take three today trade up value video visit week winners year $AAPL $BAC $DNKN $FB $GMCR $GOOG $IDRA $JAKK $SBUX $TWTR 0. 75 2. 7 3. 5 3. 77

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FAUST Classification using Class Hulls: Use the inside of a Circumscribed Training Class Hull to approximate each class, i. e. , use a series of (d, a) pairs, each defining a half-space {z | doz(<|>)a}. Then the C-hull is XC &{(d, a)}Pxod(<|>)a&PC The question remaining: How to determine the series of (d. a) pairs? 1. Choose the next d to be perpendicular to all previous (The simplest way is to use as the d series; e 1, e 2, . . . en) 2. User the diagonals, e's, mean-to-median, mean-to-furthest, . . . 3. Start with {ei}. Add a finer and finer grid of unit vectors until diameter of CIRCX is close to the diameter of C Model-based 1 -classifier for Hi. Value, Durable C (e. g. , C=10 yrs normal activity. Looking for anomalous activity). It may be worth the additional training time to continue to better the model in II by trimming the circumscriptor corners further Let CIRC 1 be 1 st Circumscriptor: k, define Lk={x | xk=min. Xk}, Hk={x | xk=max. Xk}. Classify x in C iff x is in CIRC 1 (min. Xk xk max. Xk). (Eliminate outliers 1 st? Replace min. Xk by lowest count change and max. Xk by highest or? ) Does C fill CIRC 1 corners? In high dimensions, corners can be huge. a. Cap each corner with a fitted round cap (r 2, r 4, . . . )? Barrel cap? ? ? b. For EVERY diagonal, cap to it: e. g. , D 12=e 1+e 2 Yo. D 12=Y 1+Y 2), Yo. D 123=Y 1+Y 2+Y 3) etc. Enclosing classes with linear boundaries to sums of dimension unit vectors and their negatives, may be good for multi-classification too c. Use a C-circumscribing barrel wrt each (d, a) (limits the radial reach time, with a round cap on corners. ) d. Use a C-circumscribing sphere. Note that the ultimate Circumscriber is the convex hull. The algorithms for computing convex hull exist but they are complex, even with the VPHD tools that were created over 500 years. Can we do it with our HPVD tools (created over 10 yrs? ) Circumscribe linear pieces using diagonals, {z| dozk>minz 1+mnz 2&dozk<mxz 1+mxz 2} and {z| dozk>minz 1+mxz 2&dozk<mxz 1+mnz 2}. e 3 D=e 1 -e 3 D=e 1+e 2+e 3 3 D example D=e 1+e 3 Convex hull circumsciber, CHX. e 2 e 1 Circumscribe linear pieces {z| dozk>min(zk)& dozk<max(zk)}, k=1, 2 (i. e. , d=e 1, e 2) c c c c cc c

A p. Tree Pillar k-means clustering method (The k is not specified - it reveals itself. ) m 1 m 4 Choose m 1 as a pt that maximizes Distance(X, avg. X) Choose m 2 as a pt that maximizes Distance(X, m 1) Choose m 3 as a pt that maximizes Distance(X, mh) h=1. . 2 Choose m 4 as a pt that maximizes Distance(X, mh) h=1. . 3 m 3 Do until minimumh=1. . k. Distance(X, mh) < Threshold (or Do until mk < Threshold) This gives k. Apply pk-means. (Note we already have all Dis(X, mh)s for the first round. m 2 Note: D=m 1 2 line. Treat PCCs like parentheses - ( corresponds to a PCI and ) corresponds to a PC. m Each matched pair should indicate a cluster somewhere in that slice. Where? One could take the Vo. M as the best-guess centroid? Then proceed by restricting to that slice. Or 1 st apply R and do PCC parenthesizing on R values to identify radial slice where the cluster occurs. Vo. M of that combo slice (linear and radial) as the centroid. Apply S to confirm. Note: A possible clustering method for identifying density clusters (as opposed to round or convex clusters) (Treating PCCs like parentheses) PCI PCD d-line PCI PCD

Why I like the FAUST Linear-Spherical-Radial Serial-Parallel Classifier very much (FAUST LSR SP) The parallel part lets us build a pair of Linear, Spherical and Radial hull segments for every p. Tree computation (the more the merrier) The serial part allows us the possibility of building a hull better than the convex hull! E. g. , in a linear step, if we not only use min and max but also PCIs and PCDs, potentially we could do the following on class=@: d @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ min. L = pci 1 L pcd 1 L pci 2 L max. L = pcd 2 L On each PCC interval (not yet well defined n general, but in this example they are [pci 1 L, pcd 1 L] (pcd 1 L, pci 2 L) [pci 2 L, pcd 2 L] We build hull segments on each interval and OR them. Whereas the convex hull is in orange (lots of false positives)

m 1 Finding the Pillars of X (So, e. g. , the k can be chosen intelligently in k-means) m 4 : Let m 1 be a point in X that maximizes the SPTS, dis 2(X, a)=(X-a)o(X-a) where a Avg. X If m 1 is an outlier (Check using Sm or better using D 2 NN? ), repeat until m 1 is a non-outlier. 1 A point, m 1, found in this manner is called a non-outlier pillar of X wrt a, or nop(X, a) ) Let m 2 nop(X, m 1) In general, if non-outlier pillars m 1. . mi-1 have been chosen, choose mi from 2 nop(X, {m 1, . . . , mi-1}) (i. e. , mi maximizes k=1. . i-1 dis (X, mk) and is a non-outlier). D 2 NN Av. X 1 m 3 m 2 (Instead of using Smi or to eliminate outliers each round, one might get better pillars by constructing Lmi-1 mi: X eliminating outliers that show up on L, then picking the pillar R, to be the mean (or vector of medians) of the slice L-1[(3 PCC 1+PCC 2)/4 , PCC 2) ? ) A PCC Pillar pkmeans clusterer: Assign each (object, class) a Class. Weight Reals (all CW init at 0) As we are identifying pillar m j's, compute Lm = Xo(mj-mj-1) and Classes numbered as they are revealed. j 1. For the next larger PCI in L d(C), left-to-right. 1. 1 a If followed by PCD, Ck Avg(Ld-1[PCI, PCD]) (or Vo. M). If Ck is center of a sphere-gap (or barrel gap), declare Class k and mask off. 1. 1 b If followed by another PCI, declare next Class k=the sphere-gapped set around C k=Avg( Ld-1[ (3 PCI 1+PCI 2)/4, PCI 2) ). Mask it off. 2. For the next smaller PCD in L d from the left side. 2. 1 a If preceded by a PCI, declare next Class k= subset of Ld-1[PCI, PCD] sphere-gapped around Ck=Avg. Mask off. 2. 1 b If preceded by another PCD declare next Class k=subset of same, sphere-gapped around C k=Avg(Ld-1( [PCD 2, (PCD 1+PCD 2)/4] ). Mask off @ @ @ @ A potential advantage of the classifier: @ @ @ @ FAUST Linear-Spherical-Radial (LSR) @ @ The parallel part lets us build a pair of L, S, R hull segments @ for every p. Tree computation (the more the merrier) @ Serial part allows possibility of better hull than Convex. Hull @ @ E. g. , in a linear step, if we not only use min and max but also PCIs @ and PCDs, potentially we could do the following on class=@: @ On each PCC interval (ill-defined but here [pci 1 L, pcd 1 L] (pcd 1 L, pci 2 L) [pci 2 L, pcd 2 L] @ Build hull segments on each interval and OR them? @ @ @ @ Whereas the convex hull in orange (lots of false positives) @ @ @ @ @ @ @ @ d min. L = pci 1 L pcd 1 L pci 2 L max. L = pcd L

$IRIS "metro map" for the Iris data set. [4] Small fraction of virg mixed$

IRIS "metro map" for the Iris data set. [4] Small fraction of virg mixed w versicolor (looks like 3 of them? )

Improved Oblique FAUST Cuts are made at count changes, not just at gaps. Count changes reveal the entry or exit of a cluster by the perpendicular hyper-plane. This improves Oblique FAUST's ability to cluster big data (compared to cutting only at gaps. ). We tried Improved Oblique FAUST on the Spaeth dataset successfully (produces a full dendogram of sub-clusterings by recursively taking the dot product with the vector from the Mean to the VOM (Vector-Of-Medians) and by cutting at each 25% count change in the interval count distribution produced by the UDR procedure with interval widths of 23. We claim that an appropriate count change will reveal cluster boundaries almost always. i. e. , almost always a precipitous count decrease will occur as the cut hyper-plane enters a cluster and a precipitous count increase will occur as the cut hyper-plane exits a cluster. We also claim that Improved Oblique FAUST will scale up for big data, because entering and leaving clusters "smoothly" (without noticeable count change) is no more likely for big data than for small. (since it's a measure=0 phenomenon). For the count changes to reveal themselves, it may be necessary in some data settings to look for a change pattern over a distribution window because entering a round cluster may not produce a large abrupt change in counts but may produce a noticeable change pattern over a window of counts. It may be sufficient for this purpose to just use a naive windowing in which we stop the UDR count distribution generation process at intervals of width=2 k for some small value of k and look for consecutive count changes in that rough count distribution. This approach appears to be effective and is fast. We built the distribution down to intervals of width 23=8 for the Spaeth dataset, which has diameter=114. So, for Spaeth we stopped UDR at interval widths equal to 7% of the overall diameter (8/114=. 07). Outliers, especially exterior outliers, can produce a bad diameter estimate. To get a good cluster diameter estimate, we should identify and mask off exterior outliers first (before applying the Pythagorean diameter estimation formula). Cluster outliers can be identified as singleton sub-clusters that are sufficiently gapped away from the rest of the cluster. Note that pure outlier or anomaly detection procedure need not use the Improved Oblique FAUST method since outliers are always surrounded by gaps and they do not produce big count changes. Points furthest from [or just far from] the VOM are high probability candidates for exterior outliers. These can be identified and then checked for outliers by creating SPTS, (Yo. VOM)2 and use just the high end of the UDR to mask those candidates. Of course points that project at the extremes of any dot product projection set are outlier candidates too.

APPLYING FAUST TO SPAETH Density Count/ r 2 labeled dendogram on Spaeth with D=Avg. Median DET=. 3 1 y 1 y 2 y 7 2 y 3 y 5 y 8 3 y 4 y 6 y 9 4 ya 5 6 7 8 yf 9 yb a yc b yd ye c d e D-line f 0 1 2 3 4 5 6 7 8 9 a b c d e f 1234 5 67 8 9 a bc def 1 3 1 0 2 0 6 2 Y(. 15) {y 1, y 2, y 3, y 4, y 5}(. 37) {y 6, yf}(. 08) {y 1, y 2, y 3, y 4}(. 63) {y 5}( {y 6}( {y 1, y 2, y 3, y 4}(. 63) {y 5}( ) {y 6}( ) {y 1, y 2, y 3}(2. 54) {yf}( ) MA cut at 7 and 11 {y 7, y 8, y 9, ya, yb. yc. yd. ye}(. 07) {y 7, y 8, y 9, ya}(. 39) {y 7, y 8, y 9}(1. 27) {y 4}( D=AM DET=1 {y 7, y 8, y 9}(1. 27) {y 4}( ) {ya}( ) {yb, yc, yd, ye}(1. 01) D=AM DET=. 5 labeled dendogram for FAUST on Spaeth w D=furthest. Avg, DET=. 3 1 y 1 y 2 y 7 2 y 3 y 5 y 8 3 y 4 y 6 y 9 4 ya 5 6 7 8 yf 9 yb a yc b yd ye 0 1 2 3 4 5 6 7 8 9 a b c d e f D Count/ 2 labeled dendogram for FAUST on Spaeth w r D=cylces thru diagonals nnxx, nxxn, nnxx, nxxn. . . , DET=. 3 Y(. 15) {y 1, y 2, y 3, y 4, y 5}(. 37) Y(. 15) {y 6, y 7, y 8, y 9, ya, yb. yc. yd. ye, yf}(. 09) {y 6, y 7, y 8, y 9, ya}(. 17) y 1, 2, 3, 4, 5(. 37 {y 6, yf}(. 08) {y 6}( ) {yb, yc, yd, ye, yf}(. 25) {y 7, y 8, y 9, ya, yb. yc. yd. ye}(. 07) {yf}( ) {y 7, 8, 9, a}(. 39) {yf}( {yb, yc, yd, ye}(1. 01) {yf}( ) {yb, yc, yd, ye}(1. 01) {y 7, y 8, y 9, ya}(. 39) {y 6}( )

Oblique FAUST Classifiers C = class, X = unclassified samples, r = a chosen minimum gap threshold. Fast, Analytic, Unsupervised and Supervised, Technology FAUST One-Class Spherical (OCS) Construct a sphere around x. Then x is of the C class iff the sphere hits C. That is, classify x as class C iff there exists c C such that (c-x)o(c-x) r 2 FAUST One-Class Linear (OCL) Construct a tight enclosure (= a hull, H) around C. x is class C iff x H. For a series of vectors, D, let lo. D mn. Co. D (or the 1 st PCI); hi. D mx. Co. D (or the last PCD). Classify x C iff lo. D Dox hi. D D. E. g. , let the D-series be the diagonals e 1, e 2, . . . en, e 1+e 2, e 1 -e 2, e 1+e 3, e 1 -e 3, . . . , e 1 -e 2 -. . . -en? (add more Ds until diam. H-diam. C < ε? FAUST Multi-Class Linear (MCL) Construct a tight hull, Hk, enclosing Ck k. Then x isa Ck iff k, x Hk. Classify x isa Ck iff lok, D Dox hik, D D. FAUST MCL allows for a "none of the classes" when x H , k. k In FAUST MCL the D 12=e 1 -e 2 line 2 12 o. D Co mn m c c c c cc c o. D o D 12 B D 1 Ao mn x. A m mx mx. C 1 2 12 mn. C 1 e 3 can be constructed in parallel: Convex hull Our hull, H c c c c cc c e 2 e 1 b b b b bb b 2 2 1 D 2 o. D 1 Ao Co n. A x. C m mx m e 1 line a a a a aa a Hks D=e 1 -e 3 D=e 1+e 2+e 3 3 D example of HULL 1 mn. A 1 mn. B 1 mx. A 1 mx. B 1 D=e 1+e 3 D 12 D 1 Bo B mx m 1 -classification Reference : http: //homepage. tudelft. nl/n 9 d 04/thesis. pdf#appendix*. 7

FAUST OCS One Class Spherical classifier on the Spaeth dataset as a "lazy" classifier Let the Class be {yb, yc. yd, ye}. OCS classify x=yf. 1 y 1 y 2 y 7 2 y 3 y 5 y 8 3 y 4 y 6 y 9 4 ya 5 6 7 8 yf 9 yb a yc b yd ye 0 1 2 3 4 5 6 7 8 9 a b c d e f C yb yc yd ye Let r=3. C 1 p. Trees C 1 13 12 11 10 10 1 0 11 1 0 1 1 9 1 0 0 1 11 1 0 1 1 yf 7 0 C 2 p. Trees C 2 23 22 21 20 9 1 0 0 1 10 1 0 11 1 0 1 1 1 1 1 8 1 0 C 1 13 12 11 10 yb 10 1 0 C as a 11 1 0 1 1 PTS is: yc yd 9 1 0 0 1 ye 11 1 0 1 1 and x=yf: x=yf 7 0 1 1 1 #7 3 2 1 0 #8 3 2 1 0 1 a Compute SPTS (C-x)o(C-x): 7 0 1 1 1 8 1 0 0 0 1. b Form cosntant SPTSs 7 0 1 1 1 8 1 0 0 0 4 3 2 1. c Construct (C-x)o(C-x) by SPTS arithmetic: 10 0 1 0 20 1 (C-x)o(C-x)=(C 1 -#7) + (C 2 -#8) 13 0 1 1 25 1 1 0 0 0 C 2 23 9 1 10 1 11 1 8 1 #9 3 9 1 9 1 1 0 0 0 0 1 So yf is not in C since it is spherically gapped away from C by r=3 units. How expensive is the algorithm? For each x (c-x)o(c-x) 1. Compute SPTS, (C-x)o(C-x) 10 2. Compute mask p. Tree (C-x)o(C-x) < r 2 20 3. Count 1 -bits in that mask p. Tree. 13 25 Shortcut for 1. d, e, f by some comparison of 0 hi bitslices of (C-x)o(C-x) with #9? 22 21 20 (I think Yue Cui has a shortcut ? ? ? ) 0 0 1 0 Shortcut for 1. a, b, c, d, e, f: 0 1 1 2 2 0 1 1 (C-x)o(C-x) = Co. C -2 Cox +|x| < r 0 0 0 |x|2 -r 2 + Co. C < 2 Cox 2 0 0 1 0 0 0 1 1 1. f Conclude that yf C 1. e Count the 1 bits = 0 1. d Construct the mask p. Tree (C-x)o(C-x)<#9 0 0 Precompute (1 time) SPTS Co. C and PTS 2 C (2 C is just a re-labeling (shift left) of p. Trees of C). For each new unclassified sample, x, add |x|2 -r 2 to Co. C (adding one constant to one SPTS) compute 2 Cox (n multiplications of one SPTS, 2 Ci, by one constant, xi' then add the n resulting SPTSs. 2 -r 2 +Co. C to 2 Cox giving us a mask p. Tree. Count 1 -bits in this mask p. Tree (shortcuts? , shortcuts? ) compare |x| Co. C 181 221 202 242 Co. C p. Trees Co 7 6 5 4 1 0 1 1 1 0 0 1 1 Co. C+a Co Co. C+a 8 7 285 1 0 325 1 0 306 1 0 346 1 0 p. Trees 6 5 4 0 0 1 1 0 0 0 1 1 1 0 1 3 0 1 1 0 2 1 1 0 0 3 1 0 0 1 1 2 1 1 0 0 2 C 1 20 22 18 22 1 0 0 1 1 0 0 2 C 1 p. Trees 14 13 12 11 2 C 2 1 0 18 1 0 1 1 20 1 0 0 1 22 1 0 1 1 22 2 Cox 284 314 302 330 0 1 1 0 0 2 Cox p. Trees 2 Co 8 7 6 1 0 0 1 5 0 1 1 0 2 C 2 p. Trees 24 23 22 21 1 0 0 1 1 0 1 0 1 1 4 1 1 0 0 3 1 1 2 1 0 1 0 1 1 1 a = |x|2 -r 2 = 104 Co. C+a>2 Cox 0 0 0 0 0 Ct=0 so x not in C 1 classify unclassified sample, x=(a, 9). Let r=3. Co. C+a 724 884 808 968 Co. C+a p. Trees Co 9 8 7 6 5 1 0 1 1 0 0 1 1 1 0 4 1 1 0 0 3 0 0 1 1 2 1 1 0 0 0 2 Cox 362 400 378 418 2 Cox p. Trees 2 Co 8 7 6 1 0 1 1 1 0 5 1 0 1 1 4 0 1 1 0 3 1 0 2 0 0 1 1 0 0 0 Co. C+a>2 Cox 1 1 Ct=4 so x in C 1 1

FAUST OCL One Class Linear classifier applied to IRIS, SEEDS, WINE, CONCRETE datasets For series of D's = diagonals e 1, e 2, . . . en, e 1+e 2, e 1 -e 2, e 1+e 3, e 1 -e 3, . . . , e 1 -e 2 -. . . -en For IRIS with C=Vers, outliers=Virg, FAUST 1 D: SLcutpts (49, 70); Trim outliers: 20; 34 SWcutpts(22, 32); PLcutpts(33, 49); PW Ctpts(10, 16) 44 vers correct. 7 virg errors 30: 50, 51 18 The 3 persistent virg errors virg 24 63 27 49 18 virg 27 62 28 48 18 virg 28 61 30 49 18 1 D_2 D model classifies 50 vers (no eliminated outliers) and 3 virg in the 1 class 1 D_2 D_3 D_4 D model classifies 50 vers (no eliminated outliers) and 3 virg in the 1 class For SEEDS with C=class 1 and outliers=class 2 The 1 D model classifies 1 D The 1 D_2 D model classifies 1 D_2 D The 1 D_2 D_3 D model classifies 1 D_2 D_3 D The 1 D_2 D_3 D_4 D model classifies 1 D_2 D_3 D_4 D 50 50 class 1 and 15 class 2 incorrectly as and 8 class 2 incorrectly as class 1. For SEEDS with C=class 1 and outliers=class 3 The 1 D model classifies 1 D The 1 D_2 D model classifies 1 D_2 D The 1 D_2 D_3 D model classifies 1 D_2 D_3 D The 1 D_2 D_3 D_4 D model classifies 1 D_2 D_3 D_4 D 50 50 class 1 and and 30 27 27 27 class 3 incorrectly as as class 1. For SEEDS with C=class 2 and outliers=class 3 The 1 D model classifies 1 D The 1 D_2 D model classifies 1 D_2 D The 1 D_2 D_3 D model classifies 1 D_2 D_3 D The 1 D_2 D_3 D_4 D model classifies 1 D_2 D_3 D_4 D 50 50 class 1 and and 0 0 class 3 incorrectly as as class 1. For WINE with C=class 4 and outliers=class 7 (Class 4 was enhanced with 3 class 3's to fill out the 50) The 1 D model classifies 50 class 1 and 48 class 3 incorrectly as class 1. 1 D The 1 D_2 D model classifies 50 class 1 and 43 class 3 incorrectly as class 1. 1 D_2 D The 1 D_2 D_3 D model classifies 50 class 1 and 43 class 3 incorrectly as class 1. 1 D_2 D_3 D The 1 D_2 D_3 D_4 D model classifies 50 class 1 and 42 class 3 incorrectly as class 1. 1 D_2 D_3 D_4 D For CONCRETE, conc. LH with C=class(8 -40) and outliers=class(43 -67) The 1 D model classifies 50 class 1 and 43 class 3 incorrectly as class 1. 1 D The 1 D_2 D model classifies 50 class 1 and 35 class 3 incorrectly as class 1. 1 D_2 D The 1 D_2 D_3 D model classifies 50 class 1 and 30 class 3 incorrectly as class 1. 1 D_2 D_3 D The 1 D_2 D_3 D_4 D model classifies 50 class 1 and 27 class 3 incorrectly as class 1. 1 D_2 D_3 D_4 D For CONCRETE, conc. M (class is the middle range of strengths) The 1 D model classifies 50 class 1 and 47 class 3 incorrectly as class 1. 1 D The 1 D_2 D model classifies 50 class 1 and 37 class 3 incorrectly as class 1. 1 D_2 D The 1 D_2 D_3 D model classifies 50 class 1 and 30 class 3 incorrectly as class 1. 1 D_2 D_3 D The 1 D_2 D_3 D_4 D model classifies 50 class 1 and 26 class 3 incorrectly as class 1. 1 D_2 D_3 D_4 D

FAUST MCL C y 1 y 2 y 3 y 4 y 7 y 8 y 9 yb yc yd ye mn 1 mx 1 mn 2 mx 2 mn 1+2 mx 1+2 mn 1 -2 mx 1 -2 x Ck iff lok, D Dox hik, D D. 1 y 1 y 2 2 y 3 y 5 3 y 4 4 5 6 7 8 yf 9 a b 0 1 2 3 4 5 6 7 8 y 7 y 8 y y 9 ya y 6 yb x yc yd ye 9 a b c d e f Versicolor 1 D min 49 20 max 70 34 n 1 n 2 x 1 x 2 33 51 n 3 x 3 10 18 n 4 x 4 1 D MCL Hversicolor has 7 virginica! Versicolor min 70 max 102 n 12 x 12 Versicolor min 105 max 149 n 123 x 123 Versicolor min 115 max 164 n 1234 x 1234 2 D 82 118 n 13 x 13 3 D 59 84 n 14 x 14 55 80 n 23 x 23 80 116 n 124 x 124 92 134 n 134 x 134 95 135 n 123 -4 x 123 -4 59 84 n 24 x 24 45 69 n 12 -34 x 12 -34 4 D 43 67 n 34 x 34 65 98 n 234 x 234 e 1 13 12 11 10 1 0 0 0 1 3 0 0 1 1 2 0 0 1 0 3 0 0 1 1 15 1 1 14 1 1 1 0 15 1 1 10 1 0 11 1 0 1 1 9 1 0 0 1 11 1 0 1 1 1 3 24 40 n 1 -2 x 1 -2 35 55 n 12 -3 x 12 -3 68 104 n 1 -234 x 1 -234 -1 1 9 23 n 1 -3 x 1 -3 58 88 n 1 -23 x 1 -23 20 41 n 12 -3 -4 x 12 -3 -4 13 5 -1 19 10 It is in class 3 (red) only On the basis of e 1 it is "none-or-the-above" 2 f, 2 On the basis of e 1 it is "none-or-the-above" 15 13 17 2 It is in class 2 (green) only 9 11 -24 -7 n 2 -3 x 2 -3 -21 -2 n 1 -2 -3 x 1 -2 -3 48 74 n 1 -23 -4 x 1 -23 -4 9 1 y 5 3 20 22 38 56 n 1 -4 x 1 -4 Shortcuts for MCL? Precompute all diagonal minimums and maximums; e 1, e 2, e 1+e 2, e 1 -e 2. Then, , in fact, there is no p. Tree processing left to do (just straight forward number comparisons). 7 9, a 16 18 Class 1=C 1={y 1, y 2. y 3, y 4. Class 2=C 2={y 7, y 8. y 9}. Class 3=C 3={yb, yc. yd, ye} On basis of e 1 it is "none-or-the-above" xf ya 12 14 9 11 1 3 2 6 0 2 14 15 e 2 23 22 21 20 1 0 0 0 1 2 0 0 1 0 3 0 0 1 1 9 1 0 0 1 10 1 0 11 1 0 1 1 7 18 n 2 -4 x 2 -4 23 35 n 3 -4 x 3 -4 60 88 n 12 -4 x 12 -4 -6 10 n 1 -2 -34 x 1 -2 -34 35 55 n 1 -24 x 1 -24 -39 -12 n 1 -2 -3 -4 x 1 -2 -3 -4 1 D_2 D MCL Hversicolor has 3 virginica! 1 D_2 D_3 D MCL Hversicolor has 3 virginica! 9 28 n 1 -2 -4 x 1 -2 -4 72 103 n 13 -4 x 13 -4 24 37 n 1 -34 x 1 -34 -7 12 n 1 -3 -4 x 1 -3 -4 45 65 n 23 -4 x 23 -4 -9 6 n 2 -34 x 2 -34 -40 -19 n 2 -3 -4 x 2 -3 -4 1 D_2 D_3 D_4 D MCL Hversicolor has 3 virginica (24, 27, 28) 1 D_2 D_3 D_4 D MCL Hvirginica has 20 versicolor errors!! Look at removing outliers (gapped>=3) from Hullvirginica e 1 Ct gp e 2 Ct gp e 3 Ct gp e 4 Ct gp 12 49 56. . . 77 79 1 1 7 1 1 4 3 1 2 2 18 45. . . 67 69 1 27 1 3 4 1 22 25. . . 36 38 14 24 25 1 3 3 74 82. . . 117 2 1 1 1 D MCL Hvirginica only 16 versicolors! Ct gp 1 2 1 8 2 13 78 94 104. . . 146 Ct gp 1 16 1 10 Hvirginica 15 versic 14 Ct gp 66 75. . . 102 1 9 1 23 Ct gp 48 70 72 75 ''' 96 97 102 105 1 22 1 3 2 1 1 1 5 3 24 Ct gp no outliers Hvirginica 12 versic 34 36 62 Hvirginica 65. . . 89 3 versic 92 Ct gp 1 1 1 26 3 1 1 1 3 One possibility would be to keep track of those that are outliers to their class but are not in any other class hull and put a sphere around them. Then any unclassified sample that doesn't fall in any class hull would be checked to see if it falls in any of the class outlier spheres? ? ?

UDR Univariate Distribution Revealer (on Spaeth: ) Y y 2 y 3 y 4 y 5 y 6 y 7 y 8 y 9 ya pb yc yd ye yf f= y 1 1 3 2 3 6 9 15 14 15 13 10 11 9 11 7 y 2 1 1 2 3 4 9 10 11 11 8 yof. M 11 27 23 34 53 80 118 114 125 114 110 121 109 125 83 applied to S, a column of numbers in bistlice format (an Sp. TS), will produce the Distribution. Tree of S DT(S) 5 p 6 p 5 p 4 p 3 p 2 p 1 p 0 0 1 0 1 1 1 0 0 0 1 1 0 0 0 0 1 1 1 0 0 1 1 1 1 1 0 0 1 1 1 1 0 0 1 1 1 0 1 1 0 0 1 1 p 6'p 5'p 4 p 3' 0 1 1 1 0 1 0 1 0 0 1 1 0 1[8, 16) 2/16[16, 32) 1[16, 24) 1[24, 32) p 6 p 5'p 4'p 3' 1 0 0 1 0 1 1 0 0 1 0 1 10/64 [64, 128) p 6 p 5'p 4'p 3 1 0 0 1 1 0 0 0 1 0 1 1 0 p 6 p 5'p 4 p 3' 0 1 0 1 0 0 1 1 0 0 1 0 1 1 2[80, 96) 2[80, 88) 10 depth=h=1 and its 1 count p 6'p 5'p 4'p 3 1 0 1 0 1 0 0 1 0 1 1 0 2/32[64, 96) depth=h=0 p 6' p 5' p 4' p 3' p 2' p 1' p 0' 3 2 2 8 node 2, 3 1 1 1 0 0 [96. 128) 1 1 0 0 0 1 1 1 0 1 1 2 1 1 0 2 2 6 1 0 0 1 0 1 1 0 0 0 1 0 0 1 1 0 1 0 0 0 2 3 3 0 0 0 1 1 0 0 0 1 depth. DT(S) b≡Bit. Width(S) h=depth of a node k=node offset 0 0 1 1 0 0 0 1 0 Node has a ptr to p. Tree{x S | F(x) [k 2 b-h+1, (k+1)2 b-h+1)} 0 0 0 1 0 h, k 0 1 1 0 0 p 6'p 5'p 4'p 3' 1 0 1 0 0 1 0 1 0 5/64 [0, 64) 3/32[0, 32) 1/16[0, 16) 0[0, 8) 0[64, 80) 15 p 6'p 5'p 4 p 3 0 1 1 1 0 1 0 1 0 0 1 0 1 1 0 p 6'p 5 p 4'p 3' 1 0 1 0 1 1 0 0 1 0 1 0 1 p 6'p 5 p 4'p 3 1 0 1 0 1 0 0 0 1 0 1 0 p 6'p 5 p 4 p 3' 0 1 0 1 1 1 0 1 0 1 0 p 6'p 5 p 4 p 3 0 1 1 0 1 0 0 1 0 1 0 1[32, 48) 1[32, 40) 0[40, 48) 1[48, 64) 1[48, 56) 0[56, 64) p 6 p 5'p 4 p 3 0 1 1 0 0 1 0 1 0 1 0 0 1 1 0 1 0 p 6 p 5 p 4'p 3' 0 1 0 1 0 0 1 0 1 0 1 1 0 p 6 p 5 p 4'p 3 0 1 0 0 1 1 0 1 1 0 p 6 p 5 p 4 p 3' 0 0 1 1 0 1 0 1 1 0 p 6 p 5 p 4 p 3 0 1 0 1 0 1 1 0 1 1 0 0[88, 96) ¼[96, 128) 2[96, 112) 0[96, 104) 2[194, 112) 6[112, 128) 3[112, 120) 3[120, 128) 2/32[32, 64) Pre-compute and enter into the To. C, all DT(Y k) plus those for selected Linear Functionals (e. g. , d=main diagonals, Mode. Vector. Suggestion: In our p. Tree-base, every p. Tree (basic, mask, . . . ) should be referenced in To. C( p. Tree, p. Tree. Location. Pointer, p. Tree. One. Count ). and these One. Cts should be repeated everywhere (e. g. , in every DT). The reason is that these One. Cts help us in selecting the pertinent p. Trees to access - and in fact are often all we need to know about the p. Tree to get the answers we are after. ).

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