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0. 0. 0. 0. 0. 1. 0. 0. 0. 0. 0. 1. 1. 1. Section 3. APPENDIX for the 03vertical lecture These are additional notes which the student interested in the concept of vertical data can use to further study the area. These notes are intended for enrichment. R 11 0 0 0 0 1 0 1

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  1. 0 0 0 0 0 1 0 0 0 0 0 1 1 1 Section 3 APPENDIX for the 03vertical lecture These are additional notes which the student interested in the concept of vertical data can use to further study the area. These notes are intended for enrichment. R11 0 0 0 0 1 0 1 1 Top-down construction of basic P-trees is best for understanding, bottom-up is much faster (once across). Bottom-up construction of 1-Dim, P11, is done using in-order tree traversal, collapsing of pure siblings as we go: P11 R11 R12 R13 R21 R22 R23 R31 R32 R33 R41 R42 R43 0 1 0 1 1 1 1 1 0 0 0 1 0 1 1 1 1 1 1 1 0 0 0 0 0 1 0 1 1 0 1 0 1 0 0 1 0 1 0 1 1 1 1 0 1 1 1 1 1 0 1 0 1 0 0 0 1 1 0 0 0 1 0 0 1 0 0 0 1 1 0 1 1 1 1 0 0 0 0 0 1 1 0 0 1 1 1 0 0 0 0 0 1 1 0 0 0 Siblings are pure0 so collapse!

  2. A Education Database Example Student Courses Enrollments C# CNAME ST TERM S# SNAME GEN C# S# GR In this example database (which is used throughout these notes), there are two entities, Students (a student has a number, S#, a name, SNAME, and gender, GEN Courses (course has a number, C#, name, CNAME, State where the course is offered, ST, TERM and ONE relationship, Enrollments (a student, S#, enrolls in a class, C#, and gets a grade in that class, GR). The horizontal Education (Relational) Database consists of 3 files, each of which consists of a number of instances of identically structured horizontal records: Enrollments Student Courses S#|C#|GR 0 |1 |B 0 |0 |A 3 |1 |A 3 |3 |B 1 |3 |B 1 |0 |D 2 |2 |D 2 |3 |A 4 |4 |B 5 |5 |B S#|SNAME|GEN 0 |CLAY | M 1 |THAD | M 2 |QING | F 3 |AMAL | M 4 |BARB | F 5 |JOAN | F C#|CNAME|ST|TERM 0 |BI |ND| F 1 |DB |ND| S 2 |DM |NJ| S 3 |DS |ND| F 4 |SE |NJ| S 5 |AI |ND| F We have already talked about the process of structuring data in a horizontal database (e.g., develop an Entity-Relationship diagram or ER diagram, etc. - in this case: We will discuss this a little more on the next slide. Note that much more information on the Entity-Relationship model is presented in the lectures on Relational databases.

  3. One way to begin to vertically structure this data is: 1. Code some attributes in binary For numeric fields, we have used standard binary encoding (red indicates the highorder bit, green the middle bit and blue the loworder bit) to the right of each field value encoded). . For gender, F=1 and M=0. For term, Fall=0, Spring=1. For grade, A=11, B=10, C=01, D=00 (which could be called GPAencoding?). We have abreviated STUDENT to S, COURSE to C and ENROLLMENT to E. S:S#___|SNAME|GEN 0000|CLAY |M0 1001|THAD |M0 2010|QING |F1 3 011|BARB |F1 4100|AMAL |M0 5101|JOAN |F1 C:C#___|CNAME|ST|TERM 0 000|BI |ND|F 0 1 001|DB |ND|S 1 2 010|DM |NJ|S 1 3 011|DS |ND|F 0 4 100|SE |NJ|S 1 5 101|AI |ND|F 0 E:S#___|C#___|GR . 0 000|1 001|B 10 0 000|0 000|A 11 3 011|1 001|A 11 3 011|3 011|D 00 1 001|3 011|D 00 1 001|0 000|B 10 2 010|2 010|B 10 2 010|3 011|A 11 4 100|4 100|B 10 5 101|5 101|B 10 The above encoding seem natural. But how did we decide which attributes are to be encoded and which are not? As a term paper topic, that would be one of the main issues to research Note, we have decided not to encode names (our rough reasoning (not researched) is that there would be little advantage and it would be difficult (e.g. if name is a CHAR(25) datatype, then in binary that's 25*8 = 200 bits!). Note that we have decided not to encode State. That may be a mistake! Especially in this case, since it would be so easy (only 2 States ever? so 1 bit), but more generally there could be 50 and that would mean at least 6 bits. 2. Another binary encoding scheme (which can be used for numeric and non-numeric fields) is value map or bitmap encoding. The concept is simple. For each possible value, a, in the domain of the attribute, A, we encode 1=true and 0=false for the predicate A=a. The resulting single bit column becomes a map where a 1 means that row has A-value = a and a 0 means that row or tuple has A-value which is not a. There is a wealth of existing research on bit encoding. There is also quite a bit of research on vertical databases. There is even the first commercial vertical database announced called Vertica (check it out by Googling that name). Vertica was created by the same fellow, Mike Stonebraker, who created one of the first Relational Databases, Ingres.

  4. Method-1 for vertically structuring the Educational Database S:S#___|SNAME|GEN 0 000|CLAY |M 0 1 001|THAD |M 0 2 010|QING |F 1 3 011|BARB |F 1 4 100|AMAL |M 0 5 101|JOAN |F 1 C:C#___|CNAME|ST|TERM 0 000|BI |ND|F 0 1 001|DB |ND|S 1 2 010|DM |NJ|S 1 3 011|DS |ND|F 0 4 100|SE |NJ|S 1 5 101|AI |ND|F 0 E:S#___|C#___|GR . 0 000|1 001|B 10 0 000|0 000|A 11 3 011|1 001|A 11 3 011|3 011|D 00 1 001|3 011|D 00 1 001|0 000|B 10 2 010|2 010|B 10 2 010|3 011|A 11 4 100|4 100|B 10 5 101|5 101|B 10 0 0 0 0 1 1 0 0 1 1 0 0 0 1 0 1 0 1 CLAY THAD QING BARB AMAL JOAN 0 0 1 1 0 1 0 0 0 0 1 1 0 0 1 1 0 0 0 1 0 1 0 1 BI DB DM DS SE AI ND ND NJ ND NJ ND 0 1 1 0 1 0 0 0 0 0 0 0 0 0 1 1 0 0 1 1 0 0 1 1 0 0 0 0 1 1 1 1 0 0 0 1 0 0 0 0 0 0 0 0 1 1 0 0 0 1 1 0 1 1 0 0 1 0 1 1 1 0 0 1 0 1 1 1 1 0 0 1 1 1 1 1 0 1 1 0 0 0 0 1 0 0 The M1 VDBMS would then be stored as: Rather than have to label these (0-Dimensional uncompressed) Ptrees, we will remember the color code, namely: purple border= S (i.e., coming from the Student table); brown border= C (i.e., from the Course table); light blue border= E (Enrollment table); red bits means high-order bit (4’s position bit); green bits means middle bit (2’s position bit); blue bit means low-order (units) bit (on the right).

  5. SELECTS.n, E.g FROMS, E WHERES.s=E.s&E.g=D For the selection,ENROLL.gr = D ( or E.g=D) we we create a ptree mask: EM = E'.g1 AND E'.g2(because we want both bits to be zero for D and E' means "complemented" Ptree). S: S#___ | SNAME | GEN E: S#___ | C#___ | GR 0 0 0 0 1 1 0 0 1 1 0 0 0 1 0 1 0 1 CLAY THAD QING BARB AMAL JOAN 0 0 1 1 0 1 0 0 0 0 0 0 0 0 1 1 0 0 1 1 0 0 1 1 0 0 0 0 1 1 1 1 0 0 0 1 0 0 0 0 0 0 0 0 1 1 0 0 0 1 1 0 1 1 0 0 1 0 1 1 1 0 0 1 0 1 1 1 1 0 0 1 1 1 1 1 0 0 0 1 1 0 0 0 0 0 0 1 1 0 0 0 0 1 0 0 1 0 0 1 1 1 1 0 1 1 EM = AND of E'.g1 and E'.g2 0 0 0 1 1 0 0 0 0 0

  6. SELECTS.n, E.g FROMS, E WHERES.s=E.s&E.g=D For the join, S.s = E.s, we sequence through the masked E tuples and for each, we create a mask for the matching S tuples, concatenate them and output the concatenation. 0 0 0 1 1 0 0 0 0 0 S: S#___ | SNAME | GEN E: S#___ | C#___ | GR EM 0 0 0 1 1 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 0 0 0 0 1 1 0 0 0 0 1 1 0 0 0 1 0 1 0 1 0 1 0 1 0 1 CLAY THAD QING BARB AMAL JOAN 0 0 1 1 0 1 0 0 0 0 0 0 0 0 1 1 0 0 1 1 0 0 1 1 0 0 0 0 1 1 1 1 0 0 0 1 0 0 0 0 0 0 0 0 1 1 0 0 0 1 1 0 1 1 0 0 1 0 1 1 1 0 0 1 0 1 1 1 1 0 0 1 1 1 1 1 0 1 1 0 0 0 0 1 0 0 For S#=(0 11), mask S-tuples with P'S#,2^PS#,1^PS#,0 Concatenate and output (BARB, D) 0 0 0 1 0 0

  7. SELECTS.n, E.g FROMS, E WHERES.s=E.s&E.g=D For the join, S.s = E.s, we sequence through the masked E tuples and for each, we create a mask for the matching S tuples, concatenate them and output the concatenation. 0 0 0 1 1 0 0 0 0 0 S: S#___ | SNAME | GEN E: S#___ | C#___ | GR EM 0 0 0 1 1 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 0 1 0 1 0 1 0 1 0 1 0 1 CLAY THAD QING BARB AMAL JOAN 0 0 1 1 0 1 0 0 0 0 0 0 0 0 1 1 0 0 1 1 0 0 1 1 0 0 0 0 1 1 1 1 0 0 0 1 0 0 0 0 0 0 0 0 1 1 0 0 0 1 1 0 1 1 0 0 1 0 1 1 1 0 0 1 0 1 1 1 1 0 0 1 1 1 1 1 0 1 1 0 0 0 0 1 0 0 For S#=(0 0 1), mask S-tuples with P'S#,2^P'S#,1^PS#,0 0 1 0 0 0 0 Concatenate and output (THAD, D)

  8. SELECTS.n, E.g FROMS, E WHERES.s=E.s Can the join, S.s = E.s, be speeded up? Since there is no selection involved this time, pontentially, we would have to visit every E-tuple, mask the matching S-tuples, concatenate and output. We can speed that up by masking all common E-tuples and retaining partial masks as we go. S: S#___ | SNAME | GEN E: S#___ | C#___ | GR 0 0 0 0 1 1 1 1 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 0 1 0 1 0 1 1 0 1 0 1 0 CLAY THAD QING BARB AMAL JOAN 0 0 1 1 0 1 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 0 0 1 1 1 1 0 0 0 1 1 1 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 1 0 1 1 0 0 1 0 1 1 1 0 0 1 0 1 1 1 1 0 0 1 1 1 1 1 0 1 1 0 0 0 0 1 0 0 For S#=(0 0 0), mask E-tuples with P'S#,2^P'S#,1^P'S#,0 1 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 For S#=(0 0 1), mask S-tuples with P'S#,2^P'S#,1^P'S#,0 Concatenate and output (CLAY, B) and (CLAY, A) Continue down to the next E-tuple and do the same....

  9. Start here 1 0 0 0 1 1 1 1 1 0 1 0 0 0 0 1 1 0 1 0 0 1 1 1 1 1 1 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 0 0 1 1 1 1 1 0 0 0 1 1 1 1 1 1 0 0 1 1 1 1 1 1 1 0 1 1 1 1 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 0 0 0 0 0 1 1 1 0 0 0 0 Bottom-up construction of the 2-Dimensional P-tree is done using Peano (in order) traversal of a fanout=4, log4(64)= 4 level tree, collapsing pure siblings, as we go: From here on we will take 4 bit positions at a time, for efficiency.

  10. Astronomy Application:(National Virtual Observatory data) What Ptree dimension and ordering should be used for astronomical data?, where all bodies are assumed to lie on the surface of a celestial sphere (shares its origin and equatorial plane with earth but has no specified radius) Hierarchical Triangle Mesh Tree (HTM-tree, seems to be an accepted standard) Peano Triangle Mesh Tree (PTM-tree) is a [better?] alternative - at least for data mining? (Note: RA=Recession Angle (=longitudinal angle); dec=declination (=latitudinal angle) PTM is similar to HTM used in the Sloan Digital Sky Survey project (which is a project to create a National Virtual Observatory of all [?] telescope data integrated into one repository). In both: • The Celestial Sphere is divided into triangles with great circle segment sides. • But PTM differs from HTM in the way in which these triangles are ordered at each level.

  11. 1,2 1,2 1,3,3 1,1,2 1,0 1,3,0 1,1,1 1,0 1,1,0 1,1 1,3 1,3,2 1,1 1.1.3 1,3,1 1,3 The difference between HTM and PTM-trees is in the ordering. 1 1 Ordering of PTM-tree Ordering of HTM Why use a different ordering?

  12. dec RA PTM Triangulation of the Celestial Sphere The following ordering produces a sphere-surface filling curve with good continuity characteristics, The picture at right shows the earth (blue ball at the center) and the celestial sphere out around it. Traverse southern hemisphere in the revere direction (just the identical pattern pushed down instead of pulled up, arriving at the Southern neighbor of the start point. Next, traverse the southern hemisphere in the revere direction (just the identical pattern pushed down instead of pulled up, arriving at the Southern neighbor of the start point. left Equilateral triangle (90o sector) bounded by longitudinal and equatorial line segments right right left turn Traverse the next level of triangulation, alternating again with left-turn, right-turn, left-turn, right-turn..

  13. PTM-triangulation - Next Level LRLR RLRL LRLR RLRL LRLR RLRL LRLR RLRL LRLR RLRL LRLR RLRL LRLR RLRL LRLR RLRL

  14. 90o 0o -90o 0o 360o Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z South Plane  Plane Peano Celestial Coordinates Unlike PTM-trees which initially partition the sphere into the 8 faces of an octahedron, in the PCCtree scheme: The Sphere is tranformed to a cylinder, then into a rectangle, then standard Peano ordering is used on the Celestial Coordinates. • Celestial Coordinates Recession Angle (RA) runs from 0 to 360o dand Declination Angle (dec) runs from -90o to 90o. Sphere  Cylinder

  15. SubCell-Location Myta Ribo Nucl Ribo 17, 78 12, 60 Mi, 40 1, 48 10, 75 0 0 7, 40 0 14, 65 0 0 16, 76 0 9, 45 1, 43 Function apop meio mito apop StopCodonDensity .1 .1 .1 .9 PolyA-Tail 1 1 0 0 Organism Species Vert Genome Size (million bp) Gene Dimension Table g0 g1 g2 g3 o0 human Homo sapiens 1 3000 Organism Dimension Table o1 fly Drosophila melanogaster 0 185 o2 1 1 1 1 1 0 0 1 0 1 0 0 1 0 1 1 o3 0 1 0 1 1 0 0 1 0 1 0 1 1 0 0 0 yeast Saccharomyces cerevisiae 0 12.1 e0 1 0 1 1 0 1 1 1 1 1 0 1 1 0 1 0 e0 mouse Mus musculus 1 3000 e1 e1 e2 e2 e3 LAB PI UNV STR CTY STZ ED AD S H M N e3 Experiment Dimension Table (MIAME) 3 2 a c h 1 2 2 b s h 0 2 4 a c a 1 2 4 a s a 1 PUBLIC (Ptree Unfied BioLogical InformtiCs Data Cube and Dimension Tables) Gene-OrganismDimension Table (chromosome,length) Gene-Experiment-Organism Cube (1 iff that gene from that organism expresses at a threshold level in that experiment.) many-to-many-to-many relationship

  16. Association of Computing Machinery KDD-Cup-02http://www.biostat.wisc.edu/~craven/kddcup/winners.html BIOINFORMATICS Task: Yeast Gene Regulation Prediction • There are now experimental methods that allow biologists to measure some aspect of cellular "activity" for thousands of genes or proteins at a time. A key problem that often arises in such experiments is in interpreting or annotating these thousands of measurements. This KDD Cup task focused on using data mining methods to capture the regularities of genes that are characterized by similar activity in a given high-throughput experiment. To facilitate objective evaluation, this task did not involve experiment interpretation or annotation directly, but instead it involved devising models that, when trained to classify the measurements of some instances (i.e. genes), can accurately predict the response of held aside test instances. • The training and test data came from recent experiments with a set ofS. cerevisiae (yeast) strains in which each strain is characterized by a single gene being knocked out. Each instance in the data set represents a single gene, and the target value for an instance is a discretized measurement of how active some (hidden) system in the cell is when this gene is knocked out. The goal of the task is to learn a model that can accurately predict these discretized values. Such a model would be helpful in understanding how various genes are related to the hidden system. • The best overall score (Kowalczyk) was 1.3217 (summed AROC for   the two partitions).  The best score for the "narrow" partition   was 0.6837 (Denecke et al), and the best score for the "broad"   partition was 0.6781 (Amal Perera, Bill Jockheck, Willy Valdivia Granda, Anne Denton, Pratap Kotala and William Perrizo, North Dakota State UniversityKDD Cup Pagehttp://www.acm.org/sigkdd/explorations/ (Note: It is my policy to always put my name last on publications involving students.)

  17. Association of Computing Machinery KDD-Cup-06http://www.cs.unm.edu/kdd_cup_2006http://www.cs.ndsu.nodak.edu/~datasurg/kddcup06/kdd6News.html MEDICAL INFORMATICS Task: Computer Aided Detection of Pulmonary Embolism Description of CAD systems: Over the last decade, Computer-Aided Detection (CAD) systems have moved from the sole realm of academic publications, to robust commercial systems that are used by physicians in their clinical practice to help detect early cancer from medical images. For example, CAD systems have been employed to automatically detect (potentially cancerous) breast masses and calcifications in X-ray images, detect lung nodules in lung CT (computed tomography) images, and detect polyps in colon CT images to name a few CAD applications. CAD applications lead to very interesting data mining problems. Typical CAD training data sets are large and extremely unbalanced between positive and negative classes. Often, fewer than 1% of the examples are true positives. When searching for descriptive features that can characterize the target medical structures, researchers often deploy a large set of experimental features, which consequently introduces irrelevant and redundant features. Labeling is often noisy as labels are created by expert physicians, in many cases without corresponding ground truth from biopsies or other independent confirmations. In order to achieve clinical acceptance, CAD systems have to meet extremely high performance thresholds to provide value to physicians in their day-to-day practice. Finally, in order to be sold commercially (at least in the United States), most CAD systems have to undergo a clinical trial (in almost exactly the same way as a new drug would). Typically, the CAD system must demonstrate a statistically significant improvement in clinical performance, when used, for example, by community physicians (without any special knowledge of machine learning) on as yet unseen cases – i.e., the sensitivity of physicians with CAD must be (significantly) above their performance without CAD, and without a corresponding marked increase in false positives (which may lead to unnecessary biopsies or expensive tests). In summary, very challenging machine learning and data mining tasks have arisen from CAD systems

  18. Association of Computing Machinery KDD-Cup-06 http://www.cs.unm.edu/kdd_cup_2006http://www.cs.ndsu.nodak.edu/~datasurg/kddcup06/kdd6News.html Challenge of Pulmonary Emboli Detection: Pulmonary embolism (PE) is a condition that occurs when an artery in the lung becomes blocked. In most cases, the blockage is caused by one or more blood clots that travel to the lungs from another part of your body. While PE is not always fatal, it is nevertheless the third most common cause of death in the US, with at least 650,000 cases occurring annually.1 The clinical challenge, particularly in an Emergency Room scenario, is to correctly diagnose patients that have a PE, and then send them on to therapy. This, however, is not easy, as the primary symptom of PE is dysapnea (shortness of breath), which has a variety of causes, some of which are relatively benign, making it hard to separate out the critically ill patients suffering from PE. The two crucial clinical challenges for a physician, therefore, are to diagnose whether a patient is suffering from PE and to identify the location of the PE. Computed Tomography Angiography (CTA) has emerged as an accurate diagnostic tool for PE. However, each CTA study consists of hundreds of images, each representing one slice of the lung. Manual reading of these slices is laborious, time consuming and complicated by various PE look-alikes (false positives) including respiratory motion artifacts, flowrelated artifacts, streak artifacts, partial volume artifacts, stair step artifacts, lymph nodes, and vascular bifurcation, among many others. Additionally, when PE is diagnosed, medications are given to prevent further clots, but these medications can sometimes lead to subsequent hemorrhage and bleeding since the patient must stay on them for a number of weeks after the diagnosis. Thus, the physician must review each CAD output carefully for correctness in order to prevent overdiagnosis. Because of this, the CAD system must provide only a small number of false positives per patient scan. CAD system Goal: To automatically identify PE’s. In an almost universal paradigm for CAD algorithms, this problem is addressed by a 3 stage system: 1. Identification of candidate regions of interest (ROI) from a medical image, 2. Computation of descriptive features for each candidate, and 3. Classification of each candidate (in this case, whether it is a PE or not) based on its features. NPV Task: One of the most useful applications for CAD would be a system with very high (100%?) Negative Predictive Value. In other words, if the CAD system had zero positive candidates for a given patient, we would like to be very confident that the patient was indeed free from PE’s. In a very real sense, this would be the “Holy Grail” of a PE CAD system. The best NPV score was by Amal Perera, William Perrizo, North Dakota State University (twice as high as the next best score!) http://www.acm.org/sigs/sigkdd/explorations/issue.php?volume=8&issue=2&year=2006&month=12

  19. Association of Computing Machinery KDD-Cup-06 Professor William Perrizo and his PhD student Amal Shehan Perera of the department of computer science at North Dakota State University (NDSU) won the KDD-Cup 2006 Knowledge Discovery and Data Mining competition which was held in conjunction with the ACM SIGKDD International Conference on Knowledge Discovery and Data Mining. The ACM KDD-Cup is the most rigorous annual competition in the field of data mining and machine learning. The competition is open to all academic institutes, industries as well as individuals from around the world. Since its inception in 1997, the KDD-Cup competition has presented practical and challenging data mining problems. Considerable number of researchers and practitioners participate in this annual contest. KDD-Cup datasets have become benchmarks for data mining research over the years. KDD-Cup 2006 was conducted between May and August 2006 by the Association for Computing Machinery(ACM) Special Interest Group on Knowledge Discovery and Data Mining (SIGKDD). This year’s contest was for a Computer-Aided Detection (CAD) system that could identify pulmonary embolisms, or blood clots, in the lung through examinations of the features from Computed Tomography (CT) images. A typical CT study consists of hundreds of images, each representing one slice of the lung. Manual reading of these slices is laborious, time consuming and complicated. It is also very important to be accurate in the prediction. NDSU team won the Negative Predictive Value (NPV) task of the competition, which was characterized by the organizers as the "Holy Grail" of Computer Aided Detection (CAD) of pulmonary embolisms. Siemens Medical Solutions provided dataset for the contest. Over 200 teams from around the world registered for the competition and 65 entries were submitted. This year's tasks were particularly challenging due to multiple instance learning, nonlinear cost functions, skewed class distributions, noisy class labels, and sparse data space. The NDSU team used a combined nearest neighbor and boundary classification with genetic algorithm parameter optimization. Dr. William Perrizo is a senior Professor in Computer Science at the North Dakota State University. He leads the Data Systems Users and Research Group (DataSURG) involved in innovative research on scalable data mining research using vertical data structures in the Computer Science Department at NDSU. DataSURG has been supported by NSF, NASA, DARPA, and GSA. Amal Shehan Perera is a lecturer at the Department of Computer Science and Engineering at the University of Moratuwa, Sri Lanka on study leave where he completed his PhD at NDSU.

  20. A Knowledge Discovery and Datamining Sphere (KDD_Sphere) Instrument? Knowledge discover from the massive datasets produced by modern scientific sensors and other instrumentation is a key enabling technology for the advancement of science in this information-based century. New tools are desperately needed that can extract the useful information and knowledge from these massive raw datasets so that human analysts and scientists can benefit from those data. Too often, the raw data production technology (sensor technology, materials production technology, nano-scale production technology, etc.) precedes and overwhelms the technologies needed to make the raw data useful to science. That is certainly the case in today’s scientific world. To address the problem described in the previous paragraph, it would be useful to build an instrument to extract knowledge and information from massive raw datasets – one which can do so quickly and accurately. The instrument, which might be called the Knowledge Discover and Dataming Sphere or KDD-Sphere, would incorporate (for the first time) a strong computational datamining technology (NDSU’s patented P-tree Vertical Datamining technology) and a strong visual datamining technology (full 3-D visual immersion technologies of the type typified by the stereo-optic CAVE and Immersadesk 3-D immersion instruments developed by the NCSA at The University of Illinois, Urbana-Champagne). CAVE technology was developed in the 1990s so that scientists could be fully immersed in a virtual environment (a cubical room, called a CAVE, in which 4, 5 or 6 of the surfaces are stereo-optic computer images, producing a 3-D virtual immersable environment..). Early on there were high scientific expectations for CAVE like technologies. However, judging from the dates of publications involving CAVE technology, one has to conclude that the scientific benefit has been limited. The reason may be that full immersion technology focuses on producing a virtual environment that replicates the real world. Such environments have been very successful in stimulating interest in science on the part of, say, the K-20 population, however, the ability of these technologies to help in the massive raw data analysis problem described above has been very limited (limited almost exclusively to ad hoc visual data mining). In fact, if one reads the literature surrounding these technologies, one finds that it was never really the intent of the creators of these instruments to do KDD. The focus was on producing a 3-D virtual world, not on knowledge discovery and data mining of massive datasets.

  21. The Knowledge Discovery and Datamining Sphere (KDD_Sphere) A KDD-Sphere would be focused primarily on knowledge discovery and data mining of massive datasets and secondarily on production of a 3-D immersable virtual environment. To accomplish this, a KDD-Sphere would be designed for speed and accuracy of data mining (in particular prediction and classification of a response phenomena from massive raw input datasets, based on the relationship between those inputs and responses.). The KDD-Sphere would combine (for the first time in modern science) both a strong Computational Knowledge Discovery and Data-mining engine (CKDD engine) and a strong Visual Knowledge Discovery and Datamining engine (VKDD engine) and would integrate the two engines into a sophisticated tool for solving many of the most pressing problems of modern 21st century science. Any problem in which a decision has to be made on the most likely response associated with a newly produced input “sample” will be solvable with this instrument. Examples of such problems include: Materials Science – a new material sample, Nano Science – a carbon tube composite or another nano material, Chemistry – a new compound, protein or polymer, Biology – a new gene, protein or other “sequence” Agriculture – a new cultivar or plant variety Energy science – a newly sensed energy source (deposit). Sensor science – e.g., a new nano-sensor dataset Atmospheric science – a new (e.g., the current) atmospheric pattern BioMedical science – a new sample to be diagnosed (Computer Aided Diagnosis) Sociological science – a newly observed sociological phenomenon Anthropological science – discovery of new potential dig sites Remotely Sensed Imagery (RSI) science – a new relationship between RSI values and high potential site identification of a desired (or undesired) phenomenon at that site.

  22. Knowledge Discovery and Datamining Sphere (KDD_Sphere) The initial prototype KDD-Sphere instrument could be a two-foot diameter version in which a scientist would sit in a rotateable and elavatable chair which would elevate her/him up into a spherical two-foot diameter computer display (with an entry hole at the bottom). A high performance computer would drive the display of vertically structured massive raw datasets onto that immersion screen. Rather than focusing almost exclusively on producing a virtual reproduction of a portion of our real world (the focus of CAVE technology) the KDD-Sphere would focus on displaying (quickly and accurately) the raw dataset – so that fast and accurate visual pattern data-mining of that raw dataset can done – which would then be used to direct the subsequent (and immediate) computational knowledge discovery and data-mining of that data. This interaction between visual data-mining (pattern recognition) and computational data-mining (computer-based prediction and classification) could go on in a number of integrated iterations and combinations, limited only by the imagination. The potential of the KDD-Sphere, those focused on scientific use in the areas listed above (and others), would clearly extend to K-20 and graduate education and to stimulating the general public’s interest in and excitement for science. The instrument could be driven by high performance computers at the site of the instrument (if it is a fixed location instrument, such as the prototype described above) or via the Internet2 from any remote location. By using the NDSU patented Ptree Vertical Data technology, the scientist or other user would experience almost no delay other than latency delay (due to the speed of light limitation). Also, several KDD-Sphere instruments could be integrated concurrently for team research and group interaction. The following is a very brief introduction to the KDD-Sphere instrument technology.

  23. KDDsphere The two-foot diameter fixed location “beauty shop” version; or the room-sized (ten foot diameter) fixed location “walk in” version, or a two-foot portable model, would be one “computer screen” displaying [multiple] vertical bit slices of the raw data. Many mathematical and algorithmic questions remain to be studied and determined for this instrument, the main of which may be “How to traverse the KDD-Sphere surface optimally with the vertical Ptree bit slices?”. Already, this question has been preliminarily studied and an important preliminary conclusion has been reached – namely that the “Peano Triangular Mesh Tree or PTM” spherical-space-filling-curve approach may be best (where “best” means that it “best preserves distances or correlations”) or at least it better preserves distance than the standard approach (called the Hierarchical Triangular Mesh or HTM). The PTM traversal of the KDD_Sphere (with an entry hole in the bottom for the scientist head). The PTM ordering produces a sphere filling curve with good continuity characteristics (two “near” points on the surface are more often “near” on the line (of the linear ordering) than in the standard (HTM) way of linearizing the sphere). This is important for effective data mining. Declination (-90 to 90o) and Recession Angle (0 to 360) are on next slide. Points on the sphere re uniquely identified by a (dec, RA) pair. PTM (and the standard HTM) are ways of linearizing these pairs. Uniquely identifying sphere points with numbers on a line). The following slides show a PTM traversal of the northern hemisphere, then the southern hemisphere. An alternative to traversing the southern hemisphere as described, is to the do the northern hemisphere only but expand the dec angle from the 0 to 90 degrees shown here to -75 to 90. The south-pole 15 latitude degree hole would be where the user’s head would enter in the beauty-shop model and would be a flat floor portion of the room-sizedmodel where the scientist would stand.

  24. Traversal method 1: For the KDD-Sphere? left right right left turn dec RA

  25. Traversal method 2 (3rd level traversal?) (Which preserves nearness best?) (Other “best preservation of nearness” ideas?) dec RA

  26. R11 0 0 0 0 1 0 1 1 pTrees construction revisited node_name: (Level, offset left-to-right) E.g., lower left corner node (0,0). Array of nodes at level=L is [L, *] pTree name: Sn-1_..._S1_S0_gteX%OnesPi,j is a n-level pTree, predicate gteX%Ones. S=Stride=# of leaf bits strided by (represented by) the node. The fanout of an inode = its stride / its children's stride. Subscripts i,j specify attribute,bitslice. We may drop the "%Ones": 0 0 0 1 1 0 Sn-1_..._S1_S0_gteXPi,j 0 1 1 0 1 0 1 1 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 • 1-child is pure1 and 0-child is just below 50% (so parent_node=1) • 1-child is just above 50% and the 0-child has almost no 1-bits (so that the parent node=0). (example on next slide). R11 R12 R13 R21 R22 R23 R31 R32 R33 R41 R42 R43 1 1 0 0 0 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 0 0 0 1 0 1 1 1 1 1 1 1 0 0 0 0 0 1 0 1 1 0 1 0 1 0 0 1 0 1 0 1 1 1 1 0 1 1 1 1 1 0 1 0 1 0 0 0 1 1 0 0 0 1 0 0 1 0 0 0 1 1 0 1 1 1 1 0 0 0 0 0 1 1 0 0 1 1 1 0 0 0 0 0 1 1 0 0 Previously, the construction of pure1 pTrees was shown. Here we show that several pTrees (different predicates) can be constructed concurrently during one pass. We also introduce a pTree namine and a node naming convention for pTrees. 8_4_2_1_gte50%Ones_pTree11 0 R11 R12 R13 R21 R22 R23 R31 R32 R33 R41 R42 R43 0 1 0 1 1 1 1 1 0 0 0 1 0 1 1 1 1 1 1 1 0 0 0 0 0 1 0 1 1 0 1 0 1 0 0 1 0 1 0 1 1 1 1 0 1 1 1 1 1 0 1 0 1 0 0 0 1 1 0 0 0 1 0 0 1 0 0 0 1 1 0 1 1 1 1 0 0 0 0 0 1 1 0 0 1 1 1 0 0 0 0 0 1 1 0 0 binary_pure1 _pTree11 = 8_4_2_1_gte100%Ones_pTree11 0 We must record the 1-count of the stride of each inode (e.g., If a child=1 and the other_child=0, it may be the The 1-child is pure1 and the 0-child is just below 50% (so parent_node=1) or the The 1-child is just above 50% and the 0-child has almost no 1-bits (so parent node=0). R11 1 0 0 0 1 0 1 1 8_4_2_1_gte50%Ones_pTree11 1 0 or 1? Need to know left branch OneCount=1, and right branch Onecount=3. So this stride=8 subtree OneCount=4 ( 50%). 0 or 1? OneCount of left branch=1, of right branch=0. So stride=4 subtree OneCount=1 (< 50%). OneCount of right branch=0 (pure0), but OneCount of left branch=?. Finally, recording the OneCounts as we build the tree upwards is a near-zero-extra-cost step.

  27. A few more examples: Given a table and a predicate, p, a raw pTree is a truth map of the predicate applied to each row. Bit slices of numeric attributes and bit map of categorical attributes are typical examples of raw pTrees. Multi-level pTree w fanout f: Divide the raw pTree into f equal segments (final segment short? - the remainder segment). Represent each segment by a 0 or 1 bit placed as f children of the root, depending on the truth of p applied to the segment. Recursively, divide each of those child nodes into f equal segments (with, possibly, a smaller final remainder segment). Represent each segment by a 0 or 1 bit placed as f children of that child depending on the truth of p applied to the segment The canonical example row-set-predicate is pure1. For example, a table (1 column) with 11 values and f=3. Note: with respect to a remainder node pure0 for f=3 means three 0s. Value 3-Bit representation 7 1 1 1 7 1 1 1 2 0 1 0 0 0 0 0 0 0 0 0 7 1 1 1 0 0 0 0 6 1 1 0 5 1 0 1 1 0 0 1 4 1 0 0 For the left-most (high order) raw-bit-slice: pure1 0 11_3_100%: 0 0 0 0 . 1 1 0 0 0 1 0 1 1 0 1 pure0 0 11_3_0%: 0 0 0 0 . 0 0 1 1 1 0 1 0 0 1 0 pure1 0 11_2_100%: 1 0 0 0 0 1 0 1 01 10 1 pure0 0 11_2_0%: 0 1 0 0 0 1 . 1 0 10 01 0 Theorem: (let Tnode= Terminal node and Inode= Interior node). Each Tnode of the Complement of a pure1 pTree is the complement of that Tnode of the pure0 pTree. Each Inode of the Complement of a pure1 pTree is the same as that Inode of the pure0 pTree.

  28. R:r cap |0 00|30 11| |1 01|20 10| |2 10|30 11| |3 11|10 01| C:c ncred |0 00|B|1 01| |1 01|D|3 11| |2 10|M|3 11| |3 11|S|2 10| S:s ngn |0 000|A|M| |1 001|T|M| |2 010|S|F| |3 011|B|F| |4 100|C|M| |5 101|J|F| O :o c r |0 000|0 00|0 01| |1 001|0 00|1 01| |2 010|1 01|0 00| |3 011|1 01|1 01| |4 100|2 10|0 00| |5 101|2 10|2 10| |6 110|2 10|3 11| |7 111|3 11|2 10| E:s o grade |0 000|1 001|2 10| |0 000|0 000|3 11| |3 011|1 001|3 11| |3 011|3 011|0 00| |1 001|3 011|0 00| |1 001|0 000|2 10| |2 010|2 010|2 10| |2 010|7 111|3 11| |4 100|4 100|2 10| |5 101|5 101|2 10| Columns that are not converted to pTrees: C.n B D M S S.n A T S B C J S.gn M M F F M F Oo,2 0 0 0 0 1 1 1 1 Cc,1 0 0 1 1 Cc,0 0 1 0 1 Cr,1 0 1 1 1 Cr,0 1 1 1 0 Ss,2 0 0 1 1 0 0 Ss,1 0 0 0 0 1 1 Ss,0 0 1 0 1 0 1 Rr,1 0 0 1 1 Rr,0 0 1 0 1 Rc,1 1 1 1 0 Rc,0 1 0 1 1 Columns that are converted to pTrees: Oo,1 0 0 1 1 0 0 1 1 Oo,0 0 1 0 1 0 1 0 1 Oo,1 0 0 0 0 1 1 1 1 Oc,0 0 0 1 1 0 0 0 1 Or,1 0 0 0 0 0 1 1 1 Or,0 1 1 0 1 0 0 1 0 Es,2 0 0 0 0 0 0 0 0 1 1 Es,1 0 0 1 1 0 0 1 1 0 0 Es,0 0 0 1 1 1 1 0 0 0 1 Eo,2 0 0 0 0 0 0 0 1 1 1 Eo,1 0 0 0 1 1 0 1 1 0 0 Eo,0 1 0 1 1 1 0 0 1 0 1 Eg,1 1 1 1 0 0 1 1 1 1 1 Eg,0 0 1 1 0 0 0 0 1 0 0 Cn,B 1 0 0 0 Cn,D 0 1 0 0 Cn,M 0 0 1 0 Cn,S 0 0 0 1 Sgn 0 0 1 1 0 1 This is the "Educational" Database consisting of a Student entity table (S), a Course entity table (C), an Enrollment relationship table (E), an Offerings relationship table (O), and Room entity table (R). Some data columns are converted to pTrees (C.name, S.name, S,gender). If S.gender needed to be converted to pTrees, probably the best way to do that would be to first code the gender values, female=1 and male=0, to create pTrees for those coded values (here there is only 1 bit slice so it is the pTree). For C.name=C.n, probably the best way is to bitmap the values (separate t/f pTree for each value).

  29. The IRIS dataset is available from the University of California Machine Learning Repository. It is used ubiquitously for the performance analysis of database and data mining papers. It is a 5-column, 150 row table, IRIS(Type,Sepal_Length,Sepal_Width,Pedal_Length,Pedal_Width). The first 50 rows are Setosa iris flower samples (Setosa is the type), the middle 50 are Virginica and the last 50 are Versicolor. The last 4 numeric columns give the length and width values for the sepals (the part covering the flower blossom before it blooms) and the pedals during full bloom. The following show the 2's bit slice pTree of the Pedal Width (PW) column. The pTree has 5 levels, numbered 0 (the bottom or leaf level), 1, 2, 3 and 4 (the top or root level). The predicate is "greater or equal to 60% 1-bits", thus the pTree is named: s150_s50_s25_s5_gt60%_PPW,1 s150_s50_s25_s5_gt60%_PPW,1 (level_4 each bit strides 150 leaf bits) 0 (level_3 each bit strides 50 leaf bits) 100 (level_2 each bit strides 25 leaf bits) 11 10 01 (level_1 each bit strides 5leaf bits) 11111 10111 10110 11000 10010 11111 11111 01110 11001 00111 10101 10111 10010 11011 11110 10011 11100 00100 11100 10011 00100 11011 11001 01000 01 010 00010 01011 00110 01000 11111 10010 11100 10111 10110 01110 11011 (level_0) This is the first example of a multi-level pTree which is not a binary tree. Practically speaking, for very deep tables (e.g., with billions of rows, not 150 rows), the number of levels and the strides should be tuned to fit the machine architecture. E.g., for the 64-bit computers of today that can AND and OR massive bit strings almost instantly, it might be better (provide faster processing) to have jsut 2 levels, level_0 (we always have level_0 since it is the raw, uncompressed bitslice) and level_1 with a stride of 64=26. If the file was 4 billion rows deep (~ =232), level_1 would consist of ~ 226 = 64 million bits. If testing determined this to be still inefficient, a level_2 could be added with stride 212 = 6464 = 4094. Then level_2 would consist of 220 = 1 million bits. Million bit strings AND and OR rapidly in modern CPUs (or GPUs). If testing determined that this is still inefficient, a level_3 could be added with stride 218 = 643. Level_3 would consist of 214 = 16,000 bits. And so forth.

  30. 86 69 17 17 17 11 11 13 17 53333 42443 31331 43121 32152 34334 11111 01110 11001 00111 10101 10111 10010 11011 11110 10011 11100 00100 11100 10011 00100 11011 11001 01000 01 010 00010 01011 00110 01000 11111 10010 11100 10111 10110 01110 11011 86 41 31 14 11 11 10 09 08 06 07 10 10 04 4331 3233 2341 4113 1223 2211 1222 1423 1423 22 1111 1011 1011 0010 0111 1010 1101 1110 0101 1011 1111 0100 1111 1000 0100 1110 0100 1100 1001 1011 1100 1010 0001 0100 0010 0101 1001 10 01 0001 1111 1001 0111 0010 1111 0110 0111 0110 11 From the previous discussion, it seem practical to have the same stride increase (AKA fanout) throughout the tree. Otherwise it is very difficult to identify inodes (e.g., what does node 2.2 mean?). Also, it may be more practical to record the one count (1count) of the stride at each inode (instead of the truth value of the predicate applied to the stride). The truth value of most predicates can be determined from these 1counts. The array of 1counts for all level_n inodes is more complex data structure than the vector of truth-bits for those inodes. Since storage capacity is no longer an issue, it might make sense to create and store all uncompressed bitslices (level_0s), and for all n>0 all: level_n 1count arrays, pure1 level_n bitvectors, pure0 level_n bitvectors, gte50% level_n bitvectors. They can all be created and stored concurrently during a "one-time, one-pass" creation step. Each can be used when deemed optimal. In the fanout=5 and fanout=4 examples below, note that 150 (the leaf size) is not a power of 5 or 4, so the right side at each level may not be full. fanout=5, 1-count inode, pTree, 5_25_125_150_1count_IRISPW,1: fanout=4:

  31. 89 41 31 17 11 11 10 09 08 06 07 10 10 07 4331 3233 2341 4113 1223 2211 1222 1423 1423 223 1111 1011 1011 0010 0111 1010 1101 1110 0101 1011 1111 0100 1111 1000 0100 1110 0100 1100 1001 1011 1100 1010 0001 0100 0010 0101 1001 10 01 0001 1111 1001 0111 0010 1111 0110 0111 0110 1100 1011 94 41 31 22 11 11 10 09 08 06 07 10 10 11 01 4331 3233 2341 4113 1223 2211 1222 1423 1423 2234 1 86 41 31 14 11 11 10 09 08 06 07 10 10 04 1111 1011 1011 0010 0111 1010 1101 1110 0101 1011 1111 0100 1111 1000 0100 1110 0100 1100 1001 1011 1100 1010 0001 0100 0010 0101 1001 10 01 0001 1111 1001 0111 0010 1111 0110 0111 0110 1100 1011 1111 01 95 4331 3233 2341 4113 1223 2211 1222 1423 1423 22 41 31 23 11 11 10 09 08 06 07 10 10 11 02 4331 3233 2341 4113 1223 2211 1222 1423 1423 2234 11 1111 1011 1011 0010 0111 1010 1101 1110 0101 1011 1111 0100 1111 1000 0100 1110 0100 1100 1001 1011 1100 1010 0001 0100 0010 0101 1001 10 01 0001 1111 1001 0111 0010 1111 0110 0111 0110 11 1111 1011 1011 0010 0111 1010 1101 1110 0101 1011 1111 0100 1111 1000 0100 1110 0100 1100 1001 1011 1100 1010 0001 0100 0010 0101 1001 10 01 0001 1111 1001 0111 0010 1111 0110 0111 0110 1100 1011 1111 0100 0001 As the table grows, the right side fills out:

  32. 103 41 31 31 11 11 10 09 08 06 07 10 10 11 07 03 4331 3233 2341 4113 1223 2211 1222 1423 1423 2234 1141 3 1111 1011 1011 0010 0111 1010 1101 1110 0101 1011 1111 0100 1111 1000 0100 1110 0100 1100 1001 1011 1100 1010 0001 0100 0010 0101 1001 10 01 0001 1111 1001 0111 0010 1111 0110 0111 0110 1100 1011 1111 0100 0001 1111 0001 1110 108 41 31 36 11 11 10 09 08 06 07 10 10 11 07 08 119 4331 3233 2341 4113 1223 2211 1222 1423 1423 2234 1141 3230 44 41 31 36 11 11 11 10 09 08 06 07 10 10 11 07 08 10 01 1111 1011 1011 0010 0111 1010 1101 1110 0101 1011 1111 0100 1111 1000 0100 1110 0100 1100 1001 1011 1100 1010 0001 0100 0010 0101 1001 10 01 0001 1111 1001 0111 0010 1111 0110 0111 0110 1100 1011 1111 0100 0001 1111 0001 1110 0011 1101 0000 1111 1111 4331 3233 2341 4113 1223 2211 1222 1423 1423 2234 1141 3230 4420 1 95 41 31 23 11 11 10 09 08 06 07 10 10 11 02 1111 1011 1011 0010 0111 1010 1101 1110 0101 1011 1111 0100 1111 1000 0100 1110 0100 1100 1001 1011 1100 1010 0001 0100 0010 0101 1001 10 01 0001 1111 1001 0111 0010 1111 0110 0111 0110 1100 1011 1111 0100 0001 1111 0001 1110 0011 1101 0000 1111 1111 0101 0000 1000 4331 3233 2341 4113 1223 2211 1222 1423 1423 2234 11 1111 1011 1011 0010 0111 1010 1101 1110 0101 1011 1111 0100 1111 1000 0100 1110 0100 1100 1001 1011 1100 1010 0001 0100 0010 0101 1001 10 01 0001 1111 1001 0111 0010 1111 0110 0111 0110 1100 1011 1111 0100 0001 As the table continues to grow:

  33. As the table continues to grow: 125 41 31 36 17 11 11 10 09 08 06 07 10 10 11 07 08 10 06 01 4331 3233 2341 4113 1223 2211 1222 1423 1423 2234 1141 3230 4420 1041 1 1111 1011 1011 0010 0111 1010 1101 1110 0101 1011 1111 0100 1111 1000 0100 1110 0100 1100 1001 1011 1100 1010 0001 0100 0010 0101 1001 10 01 0001 1111 1001 0111 0010 1111 0110 0111 0110 1100 1011 1111 0100 0001 1111 0001 1110 0011 1101 0000 1111 1111 0101 0000 1000 0000 1111 0100 0001 131 41 31 36 23 119 11 11 10 09 08 06 07 10 10 11 07 08 10 06 05 02 41 31 36 11 4331 3233 2341 4113 1223 2211 1222 1423 1423 2234 1141 3230 4420 1041 1040 2 11 11 10 09 08 06 07 10 10 11 07 08 10 01 132 131 01 4331 3233 2341 4113 1223 2211 1222 1423 1423 2234 1141 3230 4420 1 1111 1011 1011 0010 0111 1010 1101 1110 0101 1011 1111 0100 1111 1000 0100 1110 0100 1100 1001 1011 1100 1010 0001 0100 0010 0101 1001 10 01 0001 1111 1001 0111 0010 1111 0110 0111 0110 1100 1011 1111 0100 0001 1111 0001 1110 0011 1101 0000 1111 1111 0101 0000 1000 0000 1111 0100 0001 0000 1111 0000 0110 41 31 36 23 01 11 11 10 09 08 06 07 10 10 11 07 08 10 06 05 02 01 1111 1011 1011 0010 0111 1010 1101 1110 0101 1011 1111 0100 1111 1000 0100 1110 0100 1100 1001 1011 1100 1010 0001 0100 0010 0101 1001 10 01 0001 1111 1001 0111 0010 1111 0110 0111 0110 1100 1011 1111 0100 0001 1111 0001 1110 0011 1101 0000 1111 1111 0101 0000 1000 4331 3233 2341 4113 1223 2211 1222 1423 1423 2234 1141 3230 4420 1041 1040 2000 1 1111 1011 1011 0010 0111 1010 1101 1110 0101 1011 1111 0100 1111 1000 0100 1110 0100 1100 1001 1011 1100 1010 0001 0100 0010 0101 1001 10 01 0001 1111 1001 0111 0010 1111 0110 0111 0110 1100 1011 1111 0100 0001 1111 0001 1110 0011 1101 0000 1111 1111 0101 0000 1000 0000 1111 0100 0001 0000 1111 0000 0110 0000 0000 0000 0100 The next set of slides shows the "yield" column values for the cells of a field. It is used as a real life example for multi-leveled pTrees.

  34. 1 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 5 0 0 0 0 0 0 0 0 0 0 0 6 1 0 0 0 0 0 0 0 0 0 1 7 2 0 0 0 0 0 0 0 0 1 0 8 3 0 0 0 0 0 0 0 0 1 1 9 4 0 0 0 0 0 0 0 1 0 0 10 4 0 0 0 0 0 0 0 1 0 0 11 8 0 0 0 0 0 0 1 0 0 0 12 11 0 0 0 0 0 0 1 0 1 1 13 13 0 0 0 0 0 0 1 1 0 1 14 14 0 0 0 0 0 0 1 1 1 0 15 15 0 0 0 0 0 0 1 1 1 1 16 15 0 0 0 0 0 0 1 1 1 1 17 19 0 0 0 0 0 1 0 0 1 1 18 20 0 0 0 0 0 1 0 1 0 0 19 21 0 0 0 0 0 1 0 1 0 1 20 23 0 0 0 0 0 1 0 1 1 1 21 25 0 0 0 0 0 1 1 0 0 1 22 25 0 0 0 0 0 1 1 0 0 1 23 27 0 0 0 0 0 1 1 0 1 1 24 27 0 0 0 0 0 1 1 0 1 1 25 27 0 0 0 0 0 1 1 0 1 1 26 27 0 0 0 0 0 1 1 0 1 1 27 28 0 0 0 0 0 1 1 1 0 0 28 30 0 0 0 0 0 1 1 1 1 0 29 31 0 0 0 0 0 1 1 1 1 1 30 32 0 0 0 0 1 0 0 0 0 0 31 32 0 0 0 0 1 0 0 0 0 0 32 33 0 0 0 0 1 0 0 0 0 1 33 35 0 0 0 0 1 0 0 0 1 1 34 35 0 0 0 0 1 0 0 0 1 1 35 37 0 0 0 0 1 0 0 1 0 1 36 37 0 0 0 0 1 0 0 1 0 1 37 37 0 0 0 0 1 0 0 1 0 1 38 37 0 0 0 0 1 0 0 1 0 1 39 37 0 0 0 0 1 0 0 1 0 1 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0 1 0 1 0 1673 234 0 0 1 1 1 0 1 0 1 0 1674 236 0 0 1 1 1 0 1 1 0 0 1675 236 0 0 1 1 1 0 1 1 0 0 1676 237 0 0 1 1 1 0 1 1 0 1 1677 238 0 0 1 1 1 0 1 1 1 0 1678 247 0 0 1 1 1 1 0 1 1 1 1679 254 0 0 1 1 1 1 1 1 1 0 1680 265 0 1 0 0 0 0 1 0 0 1 1681 265 0 1 0 0 0 0 1 0 0 1 1682 268 0 1 0 0 0 0 1 1 0 0 1683 269 0 1 0 0 0 0 1 1 0 1 1684 270 0 1 0 0 0 0 1 1 1 0 1685 295 0 1 0 0 1 0 0 1 1 1 1686 346 0 1 0 1 0 1 1 0 1 0 1687 415 0 1 1 0 0 1 1 1 1 1 1688 426 0 1 1 0 1 0 1 0 1 0 1689 435 0 1 1 0 1 1 0 0 1 1 1690 777 1 1 0 0 0 0 1 0 0 1 1691 780 1 1 0 0 0 0 1 1 0 0 The pTrees for the 1691 pixel crop yield data (in value asc order). [Line_numbers, values, pTrees]

  38. 1691_20_gte100, lev1 9 8 7 6 5 4 3 2 1 0 1691_20_gt50 9 8 7 6 5 4 3 2 1 0 values values 1691_60_20_gt50 9 8 7 6 5 4 3 2 1 0 values 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 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 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 1 0 0 0 0 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 0 0 0 0 0 0 0 1 0 0 0 0 0 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 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 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 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 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 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 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 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 0 0 0 0 0 0 0 0 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 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 0 0 0 0 0 0 0 0 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 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 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 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 0 0 1 0 0 0 0 0 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 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 0 0 0 1 0 0 0 0 0 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 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 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 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 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 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 0 0 0 0 0 0 0 0 0 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 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 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 0 0 0 0 0 0 0 0 1 0 0 0 0 0 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 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 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 1 0 1 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 1 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 1 1 1 1 0 1 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 1 1 1 0 0 0 1 0 0 1 0 0 0 0 0 0 1 0 0 1 1 0 1 0 0 0 1 0 0 1 1 0 0 0 0 0 1 0 1 0 0 1 0 0 0 0 1 0 1 0 1 1 0 0 0 0 1 0 1 1 0 0 0 0 0 0 1 0 1 1 0 1 1 0 0 0 1 0 1 1 1 0 1 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 1 0 0 0 0 1 1 0 0 1 0 1 0 0 0 1 1 0 0 1 1 1 0 0 0 1 1 0 1 0 1 0 0 0 0 1 1 0 1 1 0 0 0 0 0 1 1 0 1 1 1 0 0 0 0 1 1 1 0 0 0 0 0 0 0 1 1 1 0 0 1 0 0 0 0 1 1 1 0 1 0 1 0 0 0 1 1 1 0 1 1 0 0 0 0 1 1 1 1 0 0 1 0 0 0 1 1 1 1 0 1 1 0 0 0 1 1 1 1 0 0 0 0 0 0 1 1 1 1 1 0 1 0 0 0 1 1 1 1 1 1 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 1 1 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 1 0 0 0 1 0 0 0 0 1 1 1 0 0 1 0 0 0 1 0 0 0 0 0 1 0 0 0 1 0 0 1 0 0 1 0 0 0 1 0 1 0 0 0 1 0 0 0 1 1 0 0 0 0 1 0 0 0 1 1 0 1 0 0 1 0 0 0 1 1 1 0 0 0 1 0 0 0 1 1 1 1 0 0 1 0 0 1 0 0 0 0 0 0 1 0 0 1 0 0 1 0 0 0 1 0 0 1 0 0 1 1 0 0 1 0 0 1 0 1 0 0 0 0 1 0 0 1 0 1 0 1 0 0 1 0 0 1 0 1 1 0 0 0 1 0 0 1 0 1 1 1 0 0 1 0 0 1 1 0 0 0 0 0 1 0 0 1 1 0 0 1 0 0 1 0 0 1 1 0 1 0 0 0 1 0 0 1 1 0 1 1 0 0 1 0 0 1 1 1 0 0 0 0 1 0 0 1 1 1 0 1 0 0 1 0 0 1 1 1 1 0 0 0 1 0 1 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 1 0 1 0 0 0 0 1 0 0 1 0 1 0 0 0 1 1 0 0 1 0 1 0 0 1 0 0 0 0 1 0 1 0 0 1 1 1 0 0 1 0 1 0 1 0 0 0 0 0 1 0 1 0 1 0 0 1 0 0 1 0 1 0 1 0 1 0 0 0 1 0 1 0 1 1 0 1 0 0 1 0 1 0 1 1 0 1 0 0 1 0 1 0 1 1 1 1 0 0 1 0 1 1 0 0 0 1 0 0 1 0 1 1 0 0 1 1 0 0 1 0 1 1 0 1 1 1 0 0 1 0 1 1 1 0 0 0 0 0 1 0 1 1 1 0 0 0 0 0 1 0 1 1 1 1 1 1 0 0 1 1 0 0 0 0 0 1 0 0 1 1 0 0 1 0 1 0 0 0 1 1 0 0 1 1 0 1 0 0 1 1 0 1 0 1 1 1 0 0 1 1 1 0 0 1 1 0 0 1 0 0 0 0 1 1 1 1 0 33 42 48 52 56 61 66 71 72 77 76 82 86 88 91 93 96 98 101 103 106 108 110 112 114 117 118 121 123 120 125 126 128 129 130 131 132 134 135 136 137 138 140 141 142 143 144 146 147 148 149 150 151 152 153 154 155 156 157 158 160 160 161 163 164 167 168 169 170 173 173 175 177 179 183 184 184 191 193 202 205 215 230 271 0 0 32 32 48 56 0 64 68 72 72 64 80 84 88 88 92 64 96 96 96 104 108 96 112 112 116 118 112 120 120 124 124 0 128 128 131 132 132 134 128 136 138 136 140 140 128 144 144 146 144 148 148 150 144 152 152 154 152 156 158 128 160 160 160 164 164 160 169 168 168 172 160 176 176 180 176 184 128 192 192 192 208 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 1 0 0 0 1 1 1 0 0 0 1 0 0 1 1 0 0 0 0 0 1 0 1 0 0 1 0 0 0 0 1 0 1 1 0 0 1 0 0 0 1 1 0 0 1 1 1 0 0 0 1 1 0 1 1 1 0 0 0 0 1 1 1 0 0 0 0 0 0 0 1 1 1 1 0 1 1 0 0 0 1 1 1 1 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 1 0 0 0 1 0 0 0 1 0 0 1 0 0 1 0 0 0 1 1 0 0 0 0 1 0 0 0 1 1 1 0 0 0 1 0 0 1 0 0 1 0 0 0 1 0 0 1 0 1 1 1 0 0 1 0 0 1 1 0 0 0 0 0 1 0 0 1 1 1 0 1 0 0 1 0 1 0 0 0 0 0 0 0 1 0 1 0 0 0 0 1 0 0 1 0 1 0 1 0 0 1 0 0 1 0 1 0 1 1 0 1 0 0 1 0 1 1 0 0 1 1 0 0 1 0 1 1 1 0 0 0 0 0 1 1 0 0 1 0 1 1 0 0 1 1 0 0 0 1 1 1 32 48 71 76 82 89 103 110 112 123 124 128 134 137 140 142 146 151 152 157 160 161 169 173 179 184 203 199 Using level_1 only, 1691_20_gt50 gives perfect classification of "very low yield" (defined as < 44 bushel per acre). The last 5 "very low yield pixels fall into 4th s20_gt50 pixel and thus are misclassified (as expected) The significance of this is that 1691_20_gt50 works but that 1691_20_gte100 (pure1) doesn't. It fails on others (5 pixels fall into the "very low yield class that are not supposed to be there). Using level_2 only, 1692_60_20_gt50 also gives perfect classification. This discussion involves material from chapter 10 on data mining (classification) so you might want to revisit it after studying classification in chapter 10.

  39. s20_gt50 - perfect classification of "high yield" also (which was defined as > 170 bpa). 1401 170 0 0 1 0 1 0 1 0 1 0 1402 170 0 0 1 0 1 0 1 0 1 0 1403 171 0 0 1 0 1 0 1 0 1 1 1404 171 0 0 1 0 1 0 1 0 1 1 1405 171 0 0 1 0 1 0 1 0 1 1 1406 171 0 0 1 0 1 0 1 0 1 1 1407 171 0 0 1 0 1 0 1 0 1 1 1408 172 0 0 1 0 1 0 1 1 0 0 1409 172 0 0 1 0 1 0 1 1 0 0 1410 172 0 0 1 0 1 0 1 1 0 0 1411 172 0 0 1 0 1 0 1 1 0 0 1412 172 0 0 1 0 1 0 1 1 0 0 1413 172 0 0 1 0 1 0 1 1 0 0 1414 173 0 0 1 0 1 0 1 1 0 1 1415 173 0 0 1 0 1 0 1 1 0 1 1416 173 0 0 1 0 1 0 1 1 0 1 1417 173 0 0 1 0 1 0 1 1 0 1 1418 173 0 0 1 0 1 0 1 1 0 1 1419 173 0 0 1 0 1 0 1 1 0 1 1420 173 0 0 1 0 1 0 1 1 0 1 1421 173 0 0 1 0 1 0 1 1 0 1 1422 173 0 0 1 0 1 0 1 1 0 1 1423 173 0 0 1 0 1 0 1 1 0 1 1424 173 0 0 1 0 1 0 1 1 0 1 1425 173 0 0 1 0 1 0 1 1 0 1 1426 173 0 0 1 0 1 0 1 1 0 1 1427 173 0 0 1 0 1 0 1 1 0 1 1428 173 0 0 1 0 1 0 1 1 0 1 1429 173 0 0 1 0 1 0 1 1 0 1 1430 173 0 0 1 0 1 0 1 1 0 1 1431 173 0 0 1 0 1 0 1 1 0 1 1432 174 0 0 1 0 1 0 1 1 1 0 1433 174 0 0 1 0 1 0 1 1 1 0 1434 174 0 0 1 0 1 0 1 1 1 0 1435 174 0 0 1 0 1 0 1 1 1 0 1436 174 0 0 1 0 1 0 1 1 1 0 1437 174 0 0 1 0 1 0 1 1 1 0 1438 174 0 0 1 0 1 0 1 1 1 0 1439 174 0 0 1 0 1 0 1 1 1 0 1440 174 0 0 1 0 1 0 1 1 1 0 1441 174 0 0 1 0 1 0 1 1 1 0 1442 174 0 0 1 0 1 0 1 1 1 0 1443 175 0 0 1 0 1 0 1 1 1 1 1444 175 0 0 1 0 1 0 1 1 1 1 1445 175 0 0 1 0 1 0 1 1 1 1 1446 175 0 0 1 0 1 0 1 1 1 1 1447 175 0 0 1 0 1 0 1 1 1 1 1448 175 0 0 1 0 1 0 1 1 1 1 1449 175 0 0 1 0 1 0 1 1 1 1 1450 175 0 0 1 0 1 0 1 1 1 1 1451 175 0 0 1 0 1 0 1 1 1 1 1452 175 0 0 1 0 1 0 1 1 1 1 1453 175 0 0 1 0 1 0 1 1 1 1 1454 176 0 0 1 0 1 1 0 0 0 0 1455 176 0 0 1 0 1 1 0 0 0 0 1456 176 0 0 1 0 1 1 0 0 0 0 1457 176 0 0 1 0 1 1 0 0 0 0 1458 176 0 0 1 0 1 1 0 0 0 0 1459 176 0 0 1 0 1 1 0 0 0 0 1460 176 0 0 1 0 1 1 0 0 0 0 1461 176 0 0 1 0 1 1 0 0 0 0 1462 176 0 0 1 0 1 1 0 0 0 0 1463 176 0 0 1 0 1 1 0 0 0 0 1464 176 0 0 1 0 1 1 0 0 0 0 1465 177 0 0 1 0 1 1 0 0 0 1 1466 177 0 0 1 0 1 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180 0 0 1 0 1 1 0 1 0 0 1502 180 0 0 1 0 1 1 0 1 0 0 1503 181 0 0 1 0 1 1 0 1 0 1 1504 181 0 0 1 0 1 1 0 1 0 1 1505 181 0 0 1 0 1 1 0 1 0 1 1506 181 0 0 1 0 1 1 0 1 0 1 1507 181 0 0 1 0 1 1 0 1 0 1 1508 182 0 0 1 0 1 1 0 1 1 0 1509 182 0 0 1 0 1 1 0 1 1 0 1510 182 0 0 1 0 1 1 0 1 1 0 1511 182 0 0 1 0 1 1 0 1 1 0 1512 182 0 0 1 0 1 1 0 1 1 0 1513 182 0 0 1 0 1 1 0 1 1 0 1514 182 0 0 1 0 1 1 0 1 1 0 1515 183 0 0 1 0 1 1 0 1 1 1 1516 183 0 0 1 0 1 1 0 1 1 1 1517 183 0 0 1 0 1 1 0 1 1 1 1518 183 0 0 1 0 1 1 0 1 1 1 1519 183 0 0 1 0 1 1 0 1 1 1 1520 183 0 0 1 0 1 1 0 1 1 1 1521 183 0 0 1 0 1 1 0 1 1 1 1522 183 0 0 1 0 1 1 0 1 1 1 1523 183 0 0 1 0 1 1 0 1 1 1 1524 184 0 0 1 0 1 1 1 0 0 0 1525 184 0 0 1 0 1 1 1 0 0 0 1526 184 0 0 1 0 1 1 1 0 0 0 1527 184 0 0 1 0 1 1 1 0 0 0 1528 184 0 0 1 0 1 1 1 0 0 0 1529 184 0 0 1 0 1 1 1 0 0 0 1530 184 0 0 1 0 1 1 1 0 0 0 1531 184 0 0 1 0 1 1 1 0 0 0 1532 184 0 0 1 0 1 1 1 0 0 0 1533 185 0 0 1 0 1 1 1 0 0 1 1534 185 0 0 1 0 1 1 1 0 0 1 1535 185 0 0 1 0 1 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0 1 0 1 0 1674 236 0 0 1 1 1 0 1 1 0 0 1675 236 0 0 1 1 1 0 1 1 0 0 1676 237 0 0 1 1 1 0 1 1 0 1 1677 238 0 0 1 1 1 0 1 1 1 0 1678 247 0 0 1 1 1 1 0 1 1 1 1679 254 0 0 1 1 1 1 1 1 1 0 1680 265 0 1 0 0 0 0 1 0 0 1 1681 265 0 1 0 0 0 0 1 0 0 1 1682 268 0 1 0 0 0 0 1 1 0 0 1683 269 0 1 0 0 0 0 1 1 0 1 1684 270 0 1 0 0 0 0 1 1 1 0 1685 295 0 1 0 0 1 0 0 1 1 1 1686 346 0 1 0 1 0 1 1 0 1 0 1687 415 0 1 1 0 0 1 1 1 1 1 1688 426 0 1 1 0 1 0 1 0 1 0 1689 435 0 1 1 0 1 1 0 0 1 1 1690 777 1 1 0 0 0 0 1 0 0 1 1691 780 1 1 0 0 0 0 1 1 0 0 s20gt50 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 1 0 1 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 1 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 1 1 1 1 0 1 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 1 1 1 0 0 0 1 0 0 1 0 0 0 0 0 0 1 0 0 1 1 0 1 0 0 0 1 0 0 1 1 0 0 0 0 0 1 0 1 0 0 1 0 0 0 0 1 0 1 0 1 1 0 0 0 0 1 0 1 1 0 0 0 0 0 0 1 0 1 1 0 1 1 0 0 0 1 0 1 1 1 0 1 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 1 0 0 0 0 1 1 0 0 1 0 1 0 0 0 1 1 0 0 1 1 1 0 0 0 1 1 0 1 0 1 0 0 0 0 1 1 0 1 1 0 0 0 0 0 1 1 0 1 1 1 0 0 0 0 1 1 1 0 0 0 0 0 0 0 1 1 1 0 0 1 0 0 0 0 1 1 1 0 1 0 1 0 0 0 1 1 1 0 1 1 0 0 0 0 1 1 1 1 0 0 1 0 0 0 1 1 1 1 0 1 1 0 0 0 1 1 1 1 0 0 0 0 0 0 1 1 1 1 1 0 1 0 0 0 1 1 1 1 1 1 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 1 1 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 1 0 0 0 1 0 0 0 0 1 1 1 0 0 1 0 0 0 1 0 0 0 0 0 1 0 0 0 1 0 0 1 0 0 1 0 0 0 1 0 1 0 0 0 1 0 0 0 1 1 0 0 0 0 1 0 0 0 1 1 0 1 0 0 1 0 0 0 1 1 1 0 0 0 1 0 0 0 1 1 1 1 0 0 1 0 0 1 0 0 0 0 0 0 1 0 0 1 0 0 1 0 0 0 1 0 0 1 0 0 1 1 0 0 1 0 0 1 0 1 0 0 0 0 1 0 0 1 0 1 0 1 0 0 1 0 0 1 0 1 1 0 0 0 1 0 0 1 0 1 1 1 0 0 1 0 0 1 1 0 0 0 0 0 1 0 0 1 1 0 0 1 0 0 1 0 0 1 1 0 1 0 0 0 1 0 0 1 1 0 1 1 0 0 1 0 0 1 1 1 0 0 0 0 1 0 0 1 1 1 0 1 0 0 1 0 0 1 1 1 1 0 0 0 1 0 1 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 1 0 1 0 0 0 0 1 0 0 1 0 1 0 0 0 1 1 0 0 1 0 1 0 0 1 0 0 0 0 1 0 1 0 0 1 1 1 0 0 1 0 1 0 1 0 0 0 0 0 1 0 1 0 1 0 0 1 0 0 1 0 1 0 1 0 1 0 0 0 1 0 1 0 1 1 0 1 0 0 1 0 1 0 1 1 0 1 0 0 1 0 1 0 1 1 1 1 0 0 1 0 1 1 0 0 0 1 0 0 1 0 1 1 0 0 1 1 0 0 1 0 1 1 0 1 1 1 0 0 1 0 1 1 1 0 0 0 0 0 1 0 1 1 1 0 0 0 0 0 1 0 1 1 1 1 1 1 0 0 1 1 0 0 0 0 0 1 0 0 1 1 0 0 1 0 1 0 0 0 1 1 0 0 1 1 0 1 0 0 1 1 0 1 0 1 1 1 0 0 1 1 1 0 0 1 1 0 0 1 0 0 0 0 1 1 1 1 0 33 42 48 52 56 61 66 71 72 77 76 82 86 88 91 93 96 98 101 103 106 108 110 112 114 117 118 121 123 120 125 126 128 129 130 131 132 134 135 136 137 138 140 141 142 143 144 146 147 148 149 150 151 152 153 154 155 156 157 158 160 160 161 163 164 167 168 169 170 173 173 175 177 179 183 184 184 191 193 202 205 215 230 271 s3_s20_gt50 32 48 71 76 82 89 103 110 112 123 124 128 134 137 140 142 146 151 152 157 160 161 169 173 179 184 203 199 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 1 0 0 0 1 1 1 0 0 0 1 0 0 1 1 0 0 0 0 0 1 0 1 0 0 1 0 0 0 0 1 0 1 1 0 0 1 0 0 0 1 1 0 0 1 1 1 0 0 0 1 1 0 1 1 1 0 0 0 0 1 1 1 0 0 0 0 0 0 0 1 1 1 1 0 1 1 0 0 0 1 1 1 1 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 1 0 0 0 1 0 0 0 1 0 0 1 0 0 1 0 0 0 1 1 0 0 0 0 1 0 0 0 1 1 1 0 0 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  40. s10_gt75 (level_1) 9 8 7 6 5 4 3 2 1 0 s10_gt75 (level_1 cont.) 9 8 7 6 5 4 3 2 1 0 s20_gt75 (level_1) 9 8 7 6 5 4 3 2 1 0 s10_lt25 (level_1) 9 8 7 6 5 4 3 2 1 0 s10_lt25 (level_1 cont.) 9 8 7 6 5 4 3 2 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 0 0 0 1 0 0 0 0 0 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 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 1 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 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 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 1 0 1 0 0 1 0 0 1 0 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0 0 0 0 0 0 1 1 0 1 0 0 0 0 0 0 1 1 0 1 0 0 0 0 0 0 1 1 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 1 0 0 0 1 0 0 1 1 1 0 0 0 0 0 0 0 1 1 0 1 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 1 1 1 0 1 0 0 0 0 0 1 1 0 1 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 1 0 0 0 0 1 0 1 1 0 0 0 0 1 0 0 0 1 1 0 1 1 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 1 1 0 0 1 0 0 1 1 0 1 0 0 0 0 0 0 1 1 0 1 0 0 0 1 0 0 1 1 0 1 0 1 0 0 0 0 1 1 1 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 1 1 0 1 1 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 1 1 0 1 0 1 0 0 0 0 1 0 0 1 0 0 0 1 0 0 1 1 0 1 1 0 0 0 0 0 1 1 0 1 0 1 0 0 0 0 1 1 1 0 1 0 0 0 0 0 1 1 0 1 1 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 1 1 0 0 0 1 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 1 0 1 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 1 0 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 1 0 0 0 0 0 0 1 1 0 1 0 0 1 0 0 0 1 1 0 1 0 0 0 0 0 0 1 1 0 1 1 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1 1 1 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 1 0 0 1 1 0 1 1 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 1 1 1 0 0 0 0 1 1 0 1 0 0 1 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 1 0 0 0 0 0 0 1 1 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 1 0 0 0 0 1 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 1 0 0 0 0 0 0 1 1 0 1 0 0 1 0 0 0 1 1 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 1 1 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 s20_lt25 (level_1) 9 8 7 6 5 4 3 2 1 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 1 0 0 0 0 0 0 1 1 0 1 0 0 0 0 0 0 1 1 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 1 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 1 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 1 0 0 0 0 1 0 1 1 0 0 0 0 0 0 0 0 1 1 0 1 0 0 0 0 0 0 1 1 0 1 1 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 1 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 1 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 1 1 0 1 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 1 0 0 0 0 0 0 1 1 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 1 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 1 1 0 1 0 0 0 0 0 0 1 1 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 1 0 0 0 0 0 0 1 1 0 1 0 0 0 0 0 0 1 1 0 1 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 1 1 0 1 0 0 0 0 0 0 1 1 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 1 0 0 1 1 1 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 Note that there are many more "trues" here than there were for pure pTrees (i.e., there is more information at the upper levels). IRIS (next): On this and the next few slides, we do a similar thing for the IRIS dataset and apply the simplest FAUST algorithm for clustering the IRIS classes, setosa, versicolor and virginica. We find that rough pTrees reveal massive amounts of data mining information at their upper levels! Do you agree? It would be great to have the mid-season RGB for this yield dataset. Then we could really test whether rough upper levels hold enough data to do good classification. Should we try on Netflix data?

  41. Multi-level pTree theory and examples Regarding the previous table column (crop yields): There are about 1700 pixels in the field.  The pTrees were created using the given raster ordering (not the best choice for spatial datasets of course, but the easiest, since the table came ordered that way. The pTree compression level is much lower that it would be if the ordering had been Peano or ?) It could be converted to Peano Ordering (look at lat-lon or x-y values and [roughly] re-order based on it)? I created separate 3-level pTrees using a segment lengths of 20, 10, 5 respectively. I need to explain how I’m using the term “segment”. There are [more than] two ways to create pTrees. Method-1 is to decide upon a global pTree fanout, f, (e.g., f = 2,4,8,16,32,64,…) and let the segmentation [segment lengths] be determined from that.  Note that the stride of an inode is the union of all leaf segments below it. Method-2 is to decide upon a standard segment size base and let the fanouts vary as the table grows (I am assuming, in our new age of infinite storage, that we use the “historical database” approach – i.e., never update anything in place, but keep all old [timestamped] version forever.  How to handle the timestamp column is future research. e.g., Uncompressed_bit_vector_length=2011, 3 levels, level-0 (leaf), level-1 and level-2 (I'm not including the root which would be a single inode at level-3). (Note that an uncompressed pTree then is simply one level in which the fanout=segment length, and there is the single bit at the root (level_1)) the [starting] level_2-fanout=4; the standard segment length for level_2 (top level) is then floor(2011/4) = 502.  So the level-2 segment lengths are 502,502,502,505. Note: I have chosen to make the last segment longer (length=505), rather than using standard segments length=roof(2011/4)=503 and having a last {remainder segment] of length=502.  The final “remainder” segment length and fanout of each inode has to be recorded separately anyway and, in using floor, as the table grows, we can split off a standard segment as soon as the remainder segment length equals or exceeds twice the standard segment length. Level-1 standard segment length is floor(502/4)=125, and the segment lengths are 125,125,125,127. Level-0 standard segment lengths are always 1. QUESTIONS: Is the roof or floor method better? How do we tune the “standard segment length” choices to best fit a table? By table type? By data area (e.g., spatial, RSI, Netflix ratings, precision ag, comp aided medical decisioning, commodity algorithmic equity or commodity trading, …)?

  42. 20p1 @min(a1..a20) 9 8 7 6 5 4 3 2 1 0 20p0 @if(@max(a1..a20)=0,1,0) 9 8 7 6 5 4 3 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 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 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 0 0 0 0 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 0 0 0 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 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 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 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 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 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 0 0 0 0 0 0 0 0 0 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 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 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 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 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 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 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 0 0 0 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 0 0 0 0 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 0 0 0 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 0 0 0 0 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 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 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 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 0 0 0 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 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 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 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 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 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 1 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0

  43. 10p0 L-1 continued-1 9 8 7 6 5 4 3 2 1 0 10p0 L-1 continued-2 9 8 7 6 5 4 3 2 1 0 10p1 level-1 continued-1 9 8 7 6 5 4 3 2 1 0 10p1 level-1 continued-2 9 8 7 6 5 4 3 2 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 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 0 0 0 0 0 0 0 0 1 1 0 0 0 0 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 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 1 0 0 0 0 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 0 0 0 0 0 0 0 0 1 0 0 0 0 0 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 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 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 0 0 0 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 0 0 0 0 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 0 0 0 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 0 0 0 0 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 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 0 0 0 0 0 0 0 0 0 0 0 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1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 10p1 level-2 9 8 7 6 5 4 3 2 1 0 10p0 Level-2 9 8 7 6 5 4 3 2 1 0 10p1 level-1 @min(a1..a10) 9 8 7 6 5 4 3 2 1 0 10p0 L-1 @if(@max(a1..a10)=0,1,0) 9 8 7 6 5 4 3 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 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 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 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 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 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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  44. 5p1 level-2 9 8 7 6 5 4 3 2 1 0 5p1 level-3 9 8 7 6 5 4 3 2 1 0 5p1 level-1 continued-1 9 8 7 6 5 4 3 2 1 0 5p1 level-1 continued-2 9 8 7 6 5 4 3 2 1 0 5p1 level-1 continued-3 9 8 7 6 5 4 3 2 1 0 5p1 level-1 @min(a1..a10) 9 8 7 6 5 4 3 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 0 0 0 0 0 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 0 0 0 0 0 0 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 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 0 0 0 0 0 0 0 0 1 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 0 0 0 0 1 0 0 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 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 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 0 0 0 0 0 0 0 1 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 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 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 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 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 0 0 0 0 0 0 0 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  46. Two ways of implementing multi-level pTrees. 1. bit vectors at each level 2. Path addresses of "predicate=true" leaves that occur above level=0 and do required level=0 bit vector processing as was done for uncompressed pTrees. E.g., 64-bit pure1 pTrees, p1, p2, to be ANDed (level0 size=8, fanout=8). 0 0000 1100 10010011 00000001 00001010 11000011 0 0100 1000 00000001 10101010 11000000 00010000 10010000 1001 0011 0000 0001 0000 1010 0000 0000 1111 1111 1111 1111 0000 0000 1100 0011 0000 0001 1111 1111 1010 1010 1100 0000 1111 1111 0001 0000 0000 0000 1001 0000 1 4 level 0 mixed: 0=00000001 2=10101010 3=11000000 5=00010000 7=10010000 4 5 level 0 mixed: 0=10010011 1=00000001 2=00001010 7=11000011 Pure1's Level1: 4 is common so 8*1=8. 5-p1 is pure1 so retrieve 5-p2 and count =1 1-p2 is pure1 so retrieve 1-p1 and count =1 Done processing level-1 1,4,5. Process (retrieve, AND and count) remaining common mixed level-1's, 0,2,7: 0=00000001 2=10101010 7=10010000 0=10010011 2=00001010 7=11000011 00000001 =1 00001010 =2 10000000 =1 4+8+1+1 = 14 and =

  47. R2 000 01001100 01010011 11100101 01011010 R2 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0101 0011 1111 1111 0000 0000 1110 0101 1111 1111 1111 1111 0101 1010 0000 0000 R1 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 1100 1100 1111 1111 1111 1111 0101 0011 1111 1111 1001 0101 0000 0000 R1 000 00110100 11001100 01010011 10010101 G2 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 1010 1011 0000 1010 1111 1111 0011 0010 1111 1111 1111 1111 1111 0011 1001 0100 G1 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 1010 1011 0000 1010 1111 1100 0011 0010 0100 1111 1111 1111 1111 0011 1001 0100 B2 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 1111 1111 0000 0000 1001 0110 0000 0000 1111 1111 1111 0101 1010 1001 0010 0100 B1 G1 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 1010 0111 1000 0101 1111 1111 0000 0000 0100 1101 0101 1010 1001 1001 0110 0001 0 2.2 2.4 2.5 2.0: 11001100 2.4: 01010011 2.6: 10010101 G2 G2 0 2.2 2.4 2.5 2.0: 10101011 2.1: 00001010 2.3: 00110010 2.6: 11110011 2.7: 10010100 100 00101100 10101011 00001010 00110010 11110011 10010100 R2 R1 2.2 2.4 2.5 2.0: 01010011 2.3: 11100101 2.6: 01011010 2.2 2.3 2.5 2.1: 11001100 2.4: 01010011 2.6: 10010101 G1 100 00110100 11001100 01010011 10010101 B2 B2 2.0 2.4 2.2: 10010110 2.5: 11110101 2.6: 10101001 2.7: 00100100 B1 2.2 2.0: 10100111 2.1: 10000101 2.4: 01001101 2.5: 01011010 2.6: 10011001 2.7: 01100001 000 10001000 10010110 11110101 10101001 00100100 B1 000 00100000 10100111 10000101 01001101 01011010 10101001 01100001 To AND G2 with B2 for example, 2.4 (since it is a common pure1) 2.0: 10101011 (since 2.0 is pure1 in B2) 2.2: 10010110 (since 2.2 is pure1 in G2) 2.5: 11110101 (since 2.5 is pure1 in G2) 2.6: 11110011 (mixed in G2) 10101001 (mixed in B2) 10100001 2.7: 10010100 (mixed in G2) 00100100 (mixed in B2) 00000100

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