1 Data Warehousing: and OLAP —MIS 542— — Chapter 4 — 2014/2015 Fall

2 Chapter 3: Data Warehousing and OLAP Technology for Data Mining What is a data warehouse? A multi-dimensional data model Data warehouse architecture Data warehouse implementation Further development of data cube technology From data warehousing to data mining

3 What is Data Warehouse? Defined in many different ways, but not rigorously. A decision support database that is maintained separately from the organization’s operational database Support information processing by providing a solid platform of consolidated, historical data for analysis. “A data warehouse is a subject-oriented, integrated, time-variant, and nonvolatile collection of data in support of management’s decision-making process.”—W. H. Inmon Data warehousing: The process of constructing and using data warehouses

4 Data Warehouse—Subject-Oriented Organized around major subjects, such as customer, product, sales. Focusing on the modeling and analysis of data for decision makers, not on daily operations or transaction processing. Provide a simple and concise view around particular subject issues by excluding data that are not useful in the decision support process.

5 Data Warehouse—Integrated Constructed by integrating multiple, heterogeneous data sources relational databases, flat files, on-line transaction records Data cleaning and data integration techniques are applied. Ensure consistency in naming conventions, encoding structures, attribute measures, etc. among different data sources E.g., Hotel price: currency, tax, breakfast covered, etc. When data is moved to the warehouse, it is converted.

6 Data Warehouse—Time Variant The time horizon for the data warehouse is significantly longer than that of operational systems. Operational database: current value data. Data warehouse data: provide information from a historical perspective (e.g., past 5-10 years) Every key structure in the data warehouse Contains an element of time, explicitly or implicitly But the key of operational data may or may not contain “time element”.

7 Data Warehouse—Non-Volatile A physically separate store of data transformed from the operational environment. Operational update of data does not occur in the data warehouse environment. Does not require transaction processing, recovery, and concurrency control mechanisms Requires only two operations in data accessing: initial loading of data and access of data.

8 Data Warehouse vs. Heterogeneous DBMS Traditional heterogeneous DB integration: Build wrappers/mediators on top of heterogeneous databases Query driven approach When a query is posed to a client site, a meta-dictionary is used to translate the query into queries appropriate for individual heterogeneous sites involved, and the results are integrated into a global answer set Complex information filtering, compete for resources Data warehouse: update-driven, high performance Information from heterogeneous sources is integrated in advance and stored in warehouses for direct query and analysis

9 Data Warehouse vs. Operational DBMS OLTP (on-line transaction processing) Major task of traditional relational DBMS Day-to-day operations: purchasing, inventory, banking, manufacturing, payroll, registration, accounting, etc. OLAP (on-line analytical processing) Major task of data warehouse system Data analysis and decision making Distinct features (OLTP vs. OLAP): User and system orientation: customer vs. market Data contents: current, detailed vs. historical, consolidated Database design: ER + application vs. star + subject View: current, local vs. evolutionary, integrated Access patterns: update vs. read-only but complex queries

10 OLTP vs. OLAP

11 Why Separate Data Warehouse? High performance for both systems DBMS— tuned for OLTP: access methods, indexing, concurrency control, recovery Warehouse—tuned for OLAP: complex OLAP queries, multidimensional view, consolidation. Different functions and different data: missing data: Decision support requires historical data which operational DBs do not typically maintain data consolidation: DS requires consolidation (aggregation, summarization) of data from heterogeneous sources data quality: different sources typically use inconsistent data representations, codes and formats which have to be reconciled

12 Chapter 2: Data Warehousing and OLAP Technology for Data Mining What is a data warehouse? A multi-dimensional data model Data warehouse architecture Data warehouse implementation Further development of data cube technology From data warehousing to data mining

13 From Tables and Spreadsheets to Data Cubes A data warehouse is based on a multidimensional data model which views data in the form of a data cube A data cube, such as sales, allows data to be modeled and viewed in multiple dimensions Dimensions A sale data warehouse with respect to dimension time, item, branch, location Dimension tables, such as item (item_name, brand, type), or time(day, week, month, quarter, year) Facts: numerical measures dollars_sold: sales amount in dollars units_sold: number of units sold Fact table contains measures (such as dollars_sold) and keys to each of the related dimension tables

14 A 2-D data cube: A table or spreadsheet for sales from AllElectronics AllElectronics sales data for items sold per quarter in the city of Istanbul Time dimension organized in quarters item dimension by types of items sold The fact or measure is dollar_sold

15 A three dimensional cube Dimensions: time, item, location for the cities

16 A Data Cube Representation of the same data Sales volume as a function of product, month, and region items locations Quarters Dimensions: Product, Location, Time Hierarchical summarization paths ankara izmir istabbul Q1 comp phon

17 A 4-D Cube as a series of 3-D cubes Dimensions: item,time,location,supplier Quarters items locations From Supplier A From Supplier B From Supplier C

18 n-Dimensional Cube Any n-D data as a series of (n-1)-D “cubes” In data warehousing literature, c A data cube is referred to as a cuboid The lattice of cuboids forms a data cube. The cuboid holding the lowest level of summarization is called a base cuboid. the 4-D cuboid is the base cuboid for the given four dimensions The top most 0-D cuboid, which holds the highest-level of summarization, is called the apex cuboid. Here this is the total sales, or dollars_sold summarized over all four dimensions typically denoted by all

19 Cube: A Lattice of Cuboids all timeitemlocationsupplier time,itemtime,location time,supplier item,location item,supplier location,supplier time,item,location time,item,supplier time,location,supplier item,location,supplier time, item, location, supplier 0-D(apex) cuboid 1-D cuboids 2-D cuboids 3-D cuboids 4-D(base) cuboid

20 Conceptual Modeling of Data Warehouses Modeling data warehouses: dimensions & measures Star schema: A fact table in the middle connected to a set of dimension tables Snowflake schema: A refinement of star schema where some dimensional hierarchy is normalized into a set of smaller dimension tables, forming a shape similar to snowflake Fact constellations: Multiple fact tables share dimension tables, viewed as a collection of stars, therefore called galaxy schema or fact constellation

21 Example of Star Schema time_key day day_of_the_week month quarter year time location_key street city province_or_street country location Sales Fact Table time_key item_key branch_key location_key units_sold dollars_sold avg_sales Measures item_key item_name brand type supplier_type item branch_key branch_name branch_type branch

22

23 Example of Snowflake Schema time_key day day_of_the_week month quarter year time location_key street city_key location Sales Fact Table time_key item_key branch_key location_key units_sold dollars_sold avg_sales Measures item_key item_name brand type supplier_key item branch_key branch_name branch_type branch supplier_key supplier_type supplier city_key city province_or_street country city

24 A unnormalized City Table City province and Country is repeated For every steet in İstanbul A Normalized City Table Unnecessary repitations of province end country are eliminated Memory gain but complex queries

25 Example of Fact Constellation time_key day day_of_the_week month quarter year time location_key street city province_or_street country location Sales Fact Table time_key item_key branch_key location_key units_sold dollars_sold avg_sales Measures item_key item_name brand type supplier_type item branch_key branch_name branch_type branch Shipping Fact Table time_key item_key shipper_key from_location to_location dollars_cost units_shipped shipper_key shipper_name location_key shipper_type shipper

26 Measures: Three Categories A multidimensional point in the data cube space dimension-value pairs: (time=“Q1”,location=“Istanbul”,item=“computer”) A data cube measure is a numerical function that can be evaluated at each point in the data cube space computed for a given point by aggregating the data corresponding to the respective dimension-value pairs defining the given point

27 Measures: Three Categories distributive: Suppose the data D is partitioned into n sets D i i =1,..n the computation of the function f on each partition derives one aggregate value A i =f(D i ) i = 1,..,n, f(D)=f(A 1,A 2,..,A n ) if the result derived by applying the function to n aggregate values is the same as that derived by applying the function on all the data without partitioning. The function can be computed in a distributed manner E.g., count(), sum(), min(), max().

28 Example: Sum() Data set: D:{1,3,6,8,9} Sum(D) = 27 Partition the set into D1 end D2 as D1:{1,3,6), D2:{8.9} Sum(D1) = 10, Sum(D2) = 17 Sum(sum(D1),sum(D2)) = sum(10,17) =27 =sum(D)

29 Measures: Three Categories algebraic: if it can be computed by an algebraic function with M arguments (where M is a bounded integer), each of which is obtained by applying a distributive aggregate function. E.g., avg(), min_N(), standard_deviation(). E.g.,avg() = sum()/count() both sum() and count() are distributive agg. Functions Show that min_N(), standard_deviation().

30 Measures: Three Categories holistic: if there is no constant bound on the storage size needed to describe a subaggregate. There is no an algebric function with M arguments(M being bounded) that characterizes the computation E.g., median(), mode(), rank().

31 Example The relational database scheme for AE: time(time_key,day,day_of_week,month,quarter,year) item(item_key,item_name,brand,type,supplier_type) branch(branch_key,branch_name,branch_type) location(location_key,street,city,province_or_state,cou ntry) sales(time_key,item_key,branch_key,location_key,num ber_of_units_sold,price)

32 Example cont. Select s.time_key,s.item_key,s.branch_key,s.location_key, sum(s.number_of_units_sold*s.price), sum(s.number_of_units_sold) from time t, item i,branch b,location l,sales s, where s.time_key=t.time_key and s.item_key=i.item_key and s.branch_key=b.branch_key and s.location_key=l.location_key group by s.time_key, s.item_key, s.branch_key,s.location_key

33 Example Cont. The cube created is the base cuboid of the sales_star datacube it contains all of the dimensions granularity of each is at the join key level by changing the group by clauses E.g., group by t.month: sum up the measures of each group by month removing group by s.branch_key: generate a higher-level cuboid recoving all group bys: total sum of dollars sold and total count of units_sold zero-dimensional cuboid is apex cuboid

34 Time by month Select t.year,t.month,s.item_key,s.branch_key,s.location_key, sum(s.number_of_units_sold*s.price),sum(s.number_of _units_sold) from time t, item i,branch b,location l,sales s, where s.time_key=t.time_key and s.item_key=i.item_key and s.branch_key=b.branch_key and s.location_key=l.location_key group by t.year, t.month, s.item_key, s.branch_key,s.location_key

35 A three dimensional cuboid Select s.time_key,s.item_key, s.location_key, sum(s.number_of_units_sold*s.price), sum(s.number_of_units_sold) from time t, item i,branch b,location l,sales s, where s.time_key=t.time_key and s.item_key=i.item_key and s.branch_key=b.branch_key and s.location_key=l.location_key group by s.time_key, s.item_key,s.location_key

36 Concept hierarchies Defines a sequence of mappings from a set of low-level concepts to high-level more general concepts E.g., dimension location is described by number,street,city,province_or_state,zipcode and country are related by a total order, forming a concept hierarchy street<city<province_or_state<country The attributes of a dimension may be organized in a partial order, forming a lattice day,week,month,quarter, year day<[month<quarter,week]<year

37 A Concept Hierarchy: Dimension (location) all EuropeNorth_America MexicoCanadaSpainGermany Vancouver M. WindL. Chan... all region office country TorontoFrankfurtcity

38 Partially ordered co The attributes of a dimension may be organized in a partial order, forming a lattice day,week,month,quarter, year day<[month<quarter,week]<year predefined in the data mining system time fiscal year starting on April 1 academic year starting on September 1

39 Multidimensional Data Sales volume as a function of product, month, and region Product Region Month Dimensions: Product, Location, Time Hierarchical summarization paths Industry Region Year Category Country Quarter Product City Month Week Office Day A partially ordered hierarchy

40 Cuboids Corresponding to the Cube all item time location item,timeitem,locationtime, location item, time, location 0-D(apex) cuboid 1-D cuboids 2-D cuboids 3-D(base) cuboid

41 Set grouping hierarchy Set-grouping hierarchy: discretizing or grouping values for a given dimension or attribute Ex: price There may be more than one concept hierarchy for a given attribute or dimension based on different user viewpoints price by defining ranges for inexpensive, moderately_priced,expensive

42 How defined? provided by manually by system users domain experts knowledge engineers or automatically generated based on statistical analysis of the data distribution

43 Typical OLAP Operations In the multidimensional model data are organized into multiple dimensions each dimension contains multiple levels of abstraction defined by concept hierarchies This organization provides users with the flexibility to view data from different perspectives

44 Example Refer to figure 2.10 in Han’s book data cube for AllElls sales dimensions: location,time,item location -- city time -- quarters item -- item types measure displayed is dollars-sold

45 Cuboids Corresponding to the Cube all item time location item,timeitem,locationtime, location item, time, location 0-D(apex) cuboid 1-D cuboids 2-D cuboids 3-D(base) cuboid

46 Roll-up (drill-up) Climbing up a concept hierarchy for a dimension or by dimension reduction Ex:roll-up operation aggregates data by ascending the location hierarchy from the level of city to the level of country rather than grouping the data by city,the cubes groups the data by country

47 roll up By a drill up opperation examine sales By country rather than city level

48 when performed by dimension reduction one or more dimensions are removed from the cube Ex a sales cube with location and time roll-up may remove the time dimension aggregation of total sales by location rather than by location and by time Two dimensional cuboidOne dim. cuboid

49 Drill-down (roll-down) reverse of roll-up navigates from less detailed data to more detailed data from higher level summary to lower level summary or detailed data, or stepping down a concept hierarchy for a dimension introducing new dimensions Ex: drill-down for time day<month<quarter<year form the level of quarter to the more detailed level of month Adding a new dimension to the data

50 Drill down

51 Slice and dice Slice: a selection on one dimension of the cube resulting in subcube Ex: sale data are selected for dimension time using time =Q1 dice: defines a subcube by performing a selection on two or more dimensions Ex: a dice opp. Based on location=“toronto” or “vencover” and time =Q1 or Q2 and item = “home entertainment” or “computer”

52 slice dice

53 Pivot (rotate) Visualization opp. Rotates the data axes in view to provide an alternative presentation of data Ex:item and location axes in a 2-D slice are rotated or transforming a 3-D cube into a series of 2-D planes

54 Other OLAP operations drill across: involving (across) more than one fact table drill through: through the bottom level of the cube to its back-end relational tables (using SQL) ranking the top N or bottom N items in lists moving averages growth rates interests

55 Parent Child Dimensions Based on two dimension table columns that together define the lieage relationships among the members of the dimension Member key column:identifies each member Parent key column: identifies the parent of each member

56 Poal West James Smith Amy Joens Jill Kelly John Grande Jo Brown Example: A HR dimension

57 A Concept Hierarchy: Dimension (location) all EuropeNorth_America MexicoCanadaSpainGermany Vancouver M. WindL. Chan... all region office country TorontoFrankfurtcity

58 Example: Parent Child Dimension Emp._Name Emp_Number Man_Emp_Num James Smith13 Amy Jones23 Paul West33 Jill Kelly43 John Grande51 Jo Brown61 Emp_Num identifies each member Man_Emp_Nun identifies the parent of each member

59 A Star-Net Query Model Radial lines from a central point each line represents a concept hierarchy for a dimension each abstraction level is called a footprint granularities available for use by OLAP Ex:figure 2.11 four radial lines,for concept hierarchies location,customer,item,time time line has 4 footprints: day,month,quarter,year

60 A Star-Net Query Model Shipping Method AIR-EXPRESS TRUCK ORDER Customer Orders CONTRACTS Customer Product PRODUCT GROUP PRODUCT LINE PRODUCT ITEM SALES PERSON DISTRICT DIVISION OrganizationPromotion CITY COUNTRY REGION Location DAILYQTRLYANNUALY Time Each circle is called a footprint

61 Chapter 2: Data Warehousing and OLAP Technology for Data Mining What is a data warehouse? A multi-dimensional data model Data warehouse architecture Data warehouse implementation Further development of data cube technology From data warehousing to data mining

62 Design of a Data Warehouse: A Business Analysis Framework Four views regarding the design of a data warehouse Top-down view allows selection of the relevant information necessary for the data warehouse Data source view exposes the information being captured, stored, and managed by operational systems Data warehouse view consists of fact tables and dimension tables Business query view sees the perspectives of data in the warehouse from the view of end-user

63 Data Warehouse Design Process Top-down, bottom-up approaches or a combination of both Top-down: Starts with overall design and planning (mature) Bottom-up: Starts with experiments and prototypes (rapid) From software engineering point of view Waterfall: structured and systematic analysis at each step before proceeding to the next Spiral: rapid generation of increasingly functional systems, short turn around time, quick turn around Typical data warehouse design process Choose a business process to model, e.g., orders, invoices, etc. Choose the grain (atomic level of data) of the business process Choose the dimensions that will apply to each fact table record Choose the measure that will populate each fact table record

64 Multi-Tiered Architecture Data Warehouse Extract Transform Load Refresh OLAP Engine Analysis Query Reports Data mining Monitor & Integrator Metadata Data Sources Front-End Tools Serve Data Marts Operational DBs other sources Data Storage OLAP Server

65 Three Data Warehouse Models Enterprise warehouse collects all of the information about subjects spanning the entire organization Data Mart a subset of corporate-wide data that is of value to a specific groups of users. Its scope is confined to specific, selected groups, such as marketing data mart Independent vs. dependent (directly from warehouse) data mart Virtual warehouse A set of views over operational databases Only some of the possible summary views may be materialized

66 Data Warehouse Development: A Recommended Approach Define a high-level corporate data model Data Mart Distributed Data Marts Multi-Tier Data Warehouse Enterprise Data Warehouse Model refinement

67 Storage of the cube Cuboids are referred as aggregations One factor affecting storage requirements Sparsity: the amount of empty cells in a cube The base cuboid is likely to contain many empty cells it is a spares cube or array the 0 or lower dimensional cuboids are less spares than the higher dimensional ones it is not likely that they contain empty cells Moving along higher levels for the dimension hierarchy the cuboids becomes less spares or more dense

68 PCPrtCDDV Two dimensional sparse cuboid items One dimensional Densed cuboid

69 OLAP Server Architectures Relational OLAP (ROLAP) Use relational or extended-relational DBMS to store and manage warehouse data and OLAP middle ware to support missing pieces query response is generally slower low storage requirement Include optimization of DBMS backend, implementation of aggregation navigation logic, and additional tools and services greater scalability appropriate for large data sets that are infrequently queried historical data from less recent previous years

70 Multidimensional OLAP (MOLAP) Array-based multidimensional storage engine (sparse matrix techniques) fast indexing to pre-computed summarized data a two-level storage representation dense subcubes are stored as array structures spars subcubes are stored by compression techniques appropriate for cubes with frequent use and rapid query response

71 Hybrid OLAP (HOLAP) combines ROLAP and MOLAP benefiting from greater scalability of ROLAP faster computation of MOLAP Large volumes of data base cuboid is stored in a relational database aggregations are stored as arrays appropriate for for cubes that requre rapid query response for summaries based on a large amount of base data

72 Efficient Data Cube Computation Data cube can be viewed as a lattice of cuboids The bottom-most cuboid is the base cuboid The top-most cuboid (apex) contains only one cell Example: A data cube containing item,city,year as dimensions and sales_in_$ as measure typical queries are compute sum of sales, grouping by item and city compute sum of sales, grouping by item compute sum of sales, grouping by city what is the total number of cuboids or group by s for this data cube 2 3 =8

73 Cuboids Corresponding to the Cube all product date country product,dateproduct,countrydate, country product, date, country 0-D(apex) cuboid 1-D cuboids 2-D cuboids 3-D(base) cuboid

74 Efficient Data Cube Computation How many cuboids in an n-dimensional cube with 1 levels? 2 n cuboids including the base cuboid in OLAP compute all or at least some of the cuboids in advance fast response time avoids some redundant computation

75 Number of cuboids if the cube has many dimensions with multiple level hierarchies T is total number of cuboids L i is the number of levels associated with dimension i excluding the top level all as generalizing to all is equivalent to the removal of a dimension

76 Materialization of data cube There are three choices for data cube materialization given a base cuboid: Materialize every (cuboid) (full materialization), huge amounts of memory space none (no materialization), or slow processing of queries some (partial materialization) trade-off between storage space and response time Selection of which cuboids to materialize Based on size, sharing, access frequency, etc. A heuristic approach for cuboid selection materialize the set of cuboids on which other popularly referenced cuboids are based

77 Processing Cubes Complete load of the cube: all dimension and fact table data is read and all specified aggregations cuboids are calculated process a cube when its structure is new or its dimensions or measures have been edited Incrementally updating a cube: new data is added but existing data not changed and cube structure si the same Refreshing: data cleared and reloaded its aggregations recalculated faster then processing:no design of aggregation tables

78 Calculated Members Dimension member or measure whose value is computed at run time using an expression Only the definitions are stored but values exists only in memory upon a query do not increase in cube size Ex: if sales and cost are included in the base fact table a profit measure can be a calculated member profit = sales – cost Average_sales = sales/#_items_sold

79 Virtual cubes Combination of multiple cubes in one logical cube can be based on a single cube to expose only selected subsets of measures and dimensions Require no physical space store only the dimensions information not actual data provide a valuable security functiton limiting the access of some users

80 Member Properties Attribute of a dimension member provides additional information about the member a column in the same dimension table as the associated members used in queries provide users more options when analysing cube data

81 Example: time table A typical time table: (time_id,day,month,quarter,year,business day,leap,day of the week) dimension levels day<month<quarter<year member properties for day: weekend or business day:0 or 1 day of the week:1,2,3,...,7 a member property for year is whether it is leap year or not:0 or 1

82 Virtual Dimensions Logical dimension based on a member property of a level in a physical dimension enables users to analyze cube data based on the member properties of dimension levels add a virtual dimension to a cube only if the dimension that supplies its member property is also included in the cube adding a virtual dimension does not increase cube size not affect cube processing time calculated in memory when needed query processing time is slower

83 Example The business day column was a member property for day level of the time dimension the user may want to investigate sales by type of day (business or weekend) makes business day member property as a virtual dimension of the sale cube

84 Exercise How do you represent the same organizational chart by treditional concept hyerarchies?

85 Data Warehouse Usage Three kinds of data warehouse applications Information processing supports querying, basic statistical analysis, and reporting using crosstabs, tables, charts and graphs Analytical processing multidimensional analysis of data warehouse data supports basic OLAP operations, slice-dice, drilling, pivoting Data mining knowledge discovery from hidden patterns supports associations, constructing analytical models, performing classification and prediction, and presenting the mining results using visualization tools. Differences among the three tasks

86 From On-Line Analytical Processing to On Line Analytical Mining (OLAM) Why online analytical mining? High quality of data in data warehouses DW contains integrated, consistent, cleaned data Available information processing structure surrounding data warehouses ODBC, OLEDB, Web accessing, service facilities, reporting and OLAP tools OLAP-based exploratory data analysis mining with drilling, dicing, pivoting, etc. On-line selection of data mining functions integration and swapping of multiple mining functions, algorithms, and tasks. Architecture of OLAM

87 An OLAM Architecture Data Warehouse Meta Data MDDB OLAM Engine OLAP Engine User GUI API Data Cube API Database API Data cleaning Data integration Layer3 OLAP/OLAM Layer2 MDDB Layer1 Data Repository Layer4 User Interface Filtering&IntegrationFiltering Databases Mining queryMining result

88 Summary Data warehouse A subject-oriented, integrated, time-variant, and nonvolatile collection of data in support of management’s decision-making process A multi-dimensional model of a data warehouse Star schema, snowflake schema, fact constellations A data cube consists of dimensions & measures OLAP operations: drilling, rolling, slicing, dicing and pivoting OLAP servers: ROLAP, MOLAP, HOLAP Efficient computation of data cubes Partial vs. full vs. no materialization Multiway array aggregation Bitmap index and join index implementations Further development of data cube technology Discovery-drive and multi-feature cubes From OLAP to OLAM (on-line analytical mining)

89 References (I) S. Agarwal, R. Agrawal, P. M. Deshpande, A. Gupta, J. F. Naughton, R. Ramakrishnan, and S. Sarawagi. On the computation of multidimensional aggregates. In Proc Int. Conf. Very Large Data Bases, , Bombay, India, Sept D. Agrawal, A. E. Abbadi, A. Singh, and T. Yurek. Efficient view maintenance in data warehouses. In Proc ACM-SIGMOD Int. Conf. Management of Data, , Tucson, Arizona, May R. Agrawal, J. Gehrke, D. Gunopulos, and P. Raghavan. Automatic subspace clustering of high dimensional data for data mining applications. In Proc ACM-SIGMOD Int. Conf. Management of Data, , Seattle, Washington, June R. Agrawal, A. Gupta, and S. Sarawagi. Modeling multidimensional databases. In Proc Int. Conf. Data Engineering, , Birmingham, England, April K. Beyer and R. Ramakrishnan. Bottom-Up Computation of Sparse and Iceberg CUBEs. In Proc ACM-SIGMOD Int. Conf. Management of Data (SIGMOD'99), , Philadelphia, PA, June S. Chaudhuri and U. Dayal. An overview of data warehousing and OLAP technology. ACM SIGMOD Record, 26:65-74, OLAP council. MDAPI specification version 2.0. In J. Gray, S. Chaudhuri, A. Bosworth, A. Layman, D. Reichart, M. Venkatrao, F. Pellow, and H. Pirahesh. Data cube: A relational aggregation operator generalizing group-by, cross-tab and sub- totals. Data Mining and Knowledge Discovery, 1:29-54, 1997.

90 References (II) V. Harinarayan, A. Rajaraman, and J. D. Ullman. Implementing data cubes efficiently. In Proc ACM-SIGMOD Int. Conf. Management of Data, pages , Montreal, Canada, June Microsoft. OLEDB for OLAP programmer's reference version 1.0. In K. Ross and D. Srivastava. Fast computation of sparse datacubes. In Proc Int. Conf. Very Large Data Bases, , Athens, Greece, Aug K. A. Ross, D. Srivastava, and D. Chatziantoniou. Complex aggregation at multiple granularities. In Proc. Int. Conf. of Extending Database Technology (EDBT'98), , Valencia, Spain, March S. Sarawagi, R. Agrawal, and N. Megiddo. Discovery-driven exploration of OLAP data cubes. In Proc. Int. Conf. of Extending Database Technology (EDBT'98), pages , Valencia, Spain, March E. Thomsen. OLAP Solutions: Building Multidimensional Information Systems. John Wiley & Sons, Y. Zhao, P. M. Deshpande, and J. F. Naughton. An array-based algorithm for simultaneous multidimensional aggregates. In Proc ACM-SIGMOD Int. Conf. Management of Data, , Tucson, Arizona, May 1997.