Chapter 4: Dimensions, Hierarchies, Operations, Modeling Prof. Bayer, DWH, Ch.4, SS 2000

Chapter 4.1 Hierarchical Dimensions Def: Hierarchical Dimensions are composite keys with an order on the key attributes. Prefixes are allowed as keys. Ex: dimension Time = ( Year, Month, Day) legal keys are: (Year) or (Year, Month) or (Year, Month, Day) Def: Basic facts are values in cells with full foreign keys Prof. Bayer, DWH, Ch.4, SS 2000

Aggregations, Summaries Def: Aggregations are facts in cells with partial keys. These facts are derived by aggregation functions. In a cube with derived facts the aggregation function must be specified. Ex: Sales on a monthly basis Sales (Year, Month) = S Sales (Year, Month, Days) Aggregation Functions: count, sum, avg, min, max, ... Prof. Bayer, DWH, Ch.4, SS 2000

Note on Aggregations Aggregations may be stored explicitely in the cube, but then they should be secured by integrity constraints Aggregations may be virtual and must be computed on demand when needed i.e., classical tradeoff between storage space, performance, flexibility Prof. Bayer, DWH, Ch.4, SS 2000

Relational Modeling Expand and complete partial key by ALL (Year, Month, ALL) (ALL, Month, ALL) (ALL, ALL, ALL) to obtain simple and complete relational keys via special symbol ALL Question: SQL to compute complete cube with all aggregations from base-cube? Prof. Bayer, DWH, Ch.4, SS 2000

Hierarchy Example Prof. Bayer, DWH, Ch.4, SS 2000

Chapter 4.2: OLAP Operations Def: Roll-up computes higher aggregations from lower aggregations or base facts according to hierarchies Ex: for base facts (Year, Month, Day) there are 3 roll-up functions: Roll-up (Year, Month, ALL) Roll-up (Year, ALL, ALL) Roll-up (ALL, ALL, ALL) which are supported in general (canonical roll-ups) Prof. Bayer, DWH, Ch.4, SS 2000

therefore 23 -1 aggregations or in general 2m -1 aggregations Additional Roll-ups: (ALL, Month, ALL) etc. therefore 23 -1 aggregations or in general 2m -1 aggregations for m hierarchy levels Note: see later chapters for the support of arbitrary aggregations Note: for m dimensions with h1, h2, ...hm hierarchy levels there are different aggregations for a given aggregation function. Prof. Bayer, DWH, Ch.4, SS 2000

Dim1: (4, 5) = cardinality of the dimension levels Dim2: (6, 7, 2) Size of base cube 2-dim example Dim1: (4, 5) = cardinality of the dimension levels Dim2: (6, 7, 2) (4 5) ( 6 7 2) 1680 = Size of base cube 42 20 84 Prof. Bayer, DWH, Ch.4, SS 2000

Size of hierarchically aggregated Cube 4 - 6 7 2 336 5 840 84 168 120 42 24 20 1 Number of cells per aggregation function 1645 Prof. Bayer, DWH, Ch.4, SS 2000

Size of completely aggregated cube 4 5 6 7 2 | 1 2 7 14 24 24 x 6 =144 168 5 x 168 = 840 840 + 168 6 x 168 1008 4 x 1008 = 4032 5 x 1008 = 4032 + 1008 = 5040 : : Prof. Bayer, DWH, Ch.4, SS 2000

Computation with binary Tree 4 5 1 20 4 1 6 1 6 24 120 20 4 1 1 1 7 7 1 7 7 20 168 24 28 4 140 840 120 2 1 2 1 1 1 2 1 2 1 2 1 2 2 1 2 120 140 40 20 336 168 48 24 56 28 8 4 1680 840 240 280 Prof. Bayer, DWH, Ch.4, SS 2000

Size of the Cube Lemma: Given a data cube with m dimensions with h1, ..., hm hierarchy levels resp. Let the hierarchy levels of dimension i have Then the base cube has and the cube with all aggregations has Prof. Bayer, DWH, Ch.4, SS 2000

Size of the Cube (2) The aggregated cube is larger than the base cube by the factor Prof. Bayer, DWH, Ch.4, SS 2000

Size of the hierarchically aggregated Cube For a hierarchy i with hi levels and there are hierarchical aggregation possibilities , i.e. Lemma: A hierarchically completely aggregated data cube has Prof. Bayer, DWH, Ch.4, SS 2000

size of the hierarchically aggregated cube plus base cube Ex: (4 5) (6 7 2) size of the hierarchically aggregated cube plus base cube = (1 + 4 + 20) * (1 + 6 + 42 + 84) = 25 * 133 = 3325 Ex: (4 5) (6 7 2) ( 8 3) size of base cube: 40,320 hierarchically aggregated cube plus base: = (1 + 4 + 20) * (1 + 6 + 42 + 84) * (1 + 8 + 24) = 3325 * 33 = 109,725 Prof. Bayer, DWH, Ch.4, SS 2000

hierarchically aggregated cube plus base: Ex: (4 5) (6 7 2) ( 8 3) (5 9) size of base cube: 1 814,400 hierarchically aggregated cube plus base: = 109,725 * (1 + 5 + 45) = 5 595,975 Prof. Bayer, DWH, Ch.4, SS 2000

Additional comments on aggregations 1. In addition to the size of the complete cube there is a factor of 5 for the various aggregation functions, e.g. sum, avg, min, max, count, ... 2. So far we did not consider general restrictions, e.g. „all Saturdays in March“ or „vacation months July and August“, which cross bounds of hierarchy levels Interactive query formulation results in an unlimited number of aggregations Optimization: restrictions corresponding to hierarchy levels shoud be pushed down, since they lead to query boxes Prof. Bayer, DWH, Ch.4, SS 2000

Roll-up (Year, Month, ALL) Roll-up (Year, ALL, ALL) Note: See later chapters for multidimensional indexes and MHC techniques and optimization of ROLAP-algebra to support hierarchical canonical aggregations like Roll-up (Year, Month, ALL) Roll-up (Year, ALL, ALL) Roll-up (ALL, ALL, ALL) but not Roll-up ( ALL, Month, ALL) Prof. Bayer, DWH, Ch.4, SS 2000

Non-hierarchical aggregation, e.g. March for all years Optimization Problem Non-hierarchical aggregation, e.g. March for all years decompose into union of several restrictions, e.g. S Sales (Year, Month, Day) where Month = March and (Year = 1996 or Year = 1997 or Year = 1998) see later for translation into ROLAP expression and transformations for optimization Prof. Bayer, DWH, Ch.4, SS 2000

Aggregation for month e.g. by covering QB of weeks and postfiltering Multiple Hierarchies e.g. the time hierarchy Aggregation for month e.g. by covering QB of weeks and postfiltering Prof. Bayer, DWH, Ch.4, SS 2000

Navigation Operations Drill Down: first show single result for aggregated value, e.g. sales per day, then show: hourly values for days with very high or very low sales in order to plan working hours for sales people better Other Examples: daily sales during Christmas season vacation bookings for skiing on fasching Prof. Bayer, DWH, Ch.4, SS 2000

Roll-up: Compute Aggregations Prof. Bayer, DWH, Ch.4, SS 2000

Slicing Selection of a smaller data cube or even reduction of a multidimensional datacube to fewer dimensions by a point restriction in some dimension (becomes pivot element) Prof. Bayer, DWH, Ch.4, SS 2000

Dicing (würfeln) rotate result, to show another view, e.g. exchanging rows and columns Slice management precomputing and caching of several slices for later or special use, e.g. for a special sales person Prof. Bayer, DWH, Ch.4, SS 2000

Chapter 4.3 Modeling Purpose: analysis of business processes, characteristic facts (Kennzahlen) for managers to support decisions (DSS) Steps of Decision Process: 1. Which business processes to model and analyze? 2. What are the measures, where do they come from? 3. Which degree of details, e.g. minutes like in SAP? Which precision is required for OLAP? 4. Common properties of measures to determine dimensions? Brand, Time, geogr. Region, Productgroup? Dependencies between levels of hierarchies? Prof. Bayer, DWH, Ch.4, SS 2000

5. Attributes of dimensions, e.g. screen size of TV cc and PS for cars focal length for camera Problem: how common are properties and dimensions? Non common properties cannot be modeled by levels of dimensions, are called features at GfK (up to 50), are numbered with meaning dependent on specific dimension element, e.g. TV: screen size color audio system Car: transmission cc PS #cyl ... Prof. Bayer, DWH, Ch.4, SS 2000

6. Constant or changing attributes of dimensions? E.g. New models of car makers new powersource: electrical, hydrogen, solar attributes are rather stable, but still should be planned ahead! (mergers like Daimler-Crysler) 7. Sparsity: one hypercube or several, i.e. multicube model? Influences storage requirements, query formulation and performance, cannot be hidden easily from user, maybe by views? Prof. Bayer, DWH, Ch.4, SS 2000

8. Caching and management of aggregates? Number of aggregates Maintenance costs Avg. Response time 100% 0% Total costs Time Optimal Number of aggregates Prof. Bayer, DWH, Ch.4, SS 2000

Chapter 4.4 Comparison of OLAP Architectures MOLAP: Multidimensional OLAP ROLAP: Relational OLAP 3. HOLAP: Hybrid OLAP Prof. Bayer, DWH, Ch.4, SS 2000

MOLAP Architecture Prof. Bayer, DWH, Ch.4, SS 2000

MDDBMS in ANSI-X3-Sparc Prof. Bayer, DWH, Ch.4, SS 2000

Logical components of a MDDBMS Prof. Bayer, DWH, Ch.4, SS 2000

ROLAP Architecture Prof. Bayer, DWH, Ch.4, SS 2000

HOLAP Architecture Prof. Bayer, DWH, Ch.4, SS 2000

flexible precomputations, partial aggregates parallelism Reasons for MOLAP performance write access Data Marts functional power Reasons for ROLAP scalability flexible precomputations, partial aggregates parallelism DB-mamagement and ACID Prof. Bayer, DWH, Ch.4, SS 2000