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OLAP : Blitzkreig Introduction
3 characteristics of OLAP cubes:
Large data sets ~ Gb, Tb
Expected Query : Aggregation
 Infrequent updates
Star Schema : Hierarchical
Dimensions
Attributes and Measures
Attributes are columns with values
from a fixed domain (foreign keys).
Measures are numerical columns.
Imprecision and Uncertainity
Imprecision in a tuple refers to an attribute instantiated by a
set of values from the domain instead of a single value.
Uncertainity refers to a measure represented by a pdf over
the domain instead of a single value.
Hierarchical Domains : Star Schema
Location
Madhya Pradesh
Maharashtra
Mumbai
Pune
Bhopal
Indore
Restriction on Imprecision
We restrict the sets of values in an imprecise fact to either:
1. A singleton set consisting of a leaf level member of the
hierarchy, or,
2. The set of all the leaf level members under some non-leaf
level member of the hierarchy.
Cells and Regions
A region is a vector of attribute values from an imprecise
domains of each dimension of the cube.
A cell is a region in which all values are leaf level members.
Let reg(R) represent the set of cells in a region R.
Queries on precise data
A query Q = (R, M, A) refers to a region R, a measure M, and
an aggregate function A. Eg : (<Ambassador, Location>,
Repairs, Sum)
The result of the query in a precise database is obtained by
applying A on the measure M of all cells in R.
For the example above, the result is (P1 + P2)
Queries on imprecise data
Consider the query region <Pune, Model> in the figure. It
overlaps two imprecise facts P4 and P5. Here, we need to
make a decision between 3 strategies :
None : Ignore both P4 and P5 because of their imprecision
Contains : Take P5 because it is contained inside the query
Overlaps : Take P5, and somehow, P4 as well.
Contains option : Consistency
Intuitively, consistency means that the answer to a query should be
consistent with the aggregates from individual partitions of the
query.
Using the Contains option could give rise to inconsistent results.
For example, consider the sum aggregate of the query above and
that of its individual cells. With the Contains option, will the
individual results add up to be the same as the collective?
None option
Essentially, the none option ignores the imprecise facts, even if a
fact is completely inside the region.
Lays waste to the whole notion of having imprecise facts.
Overlaps option : Possible Worlds
Query semantics on Possible worlds
With each possible world, assign a weight wi such that the
sum of all weights is 1. Intuitively, the weight of a particular
world is like probability that it is the correct underlying data.
Given a query Q, we can calculate the result for each vi for
each world. Thus, we can return a pdf over the answer Z as
P[Z = o] = ∑ i : v_i = o wi
A neat short answer could be the expected value of Z
E[Z] =∑i wi * vi
Problem with this is : number of possible worlds is
exponential in number of imprecise facts!
Solution : Extended data model
With each cell c in a region r, we add a probability pr, c, called
the allocation of r to c.
The probability of a possible world becomes the multiple of
allocations of ranges to cells that have been populated in the
world.
This leads to a (reasonable) restriction on the kind of
probability distributions on possible worlds.
Advantages of EDM
No extra infrastructure required for representing imprecision
Efficient algorithms for aggregate queries :
SUM and COUNT : linear time algo.
3
AVERAGE : slightly complicated algorithm running in O(m + n )
for m precise facts and n imprecise facts.
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Allocation Policies
For every region r in the database, we want to assign an
allocation pc, r to each cell c in Reg(r), such that
∑c Reg(r) pc, r = 1
Three ways of doing so:
1. Uniform : Assign each cell c in a region r an equal
probability.
pc, r = 1 / |Reg(r)|
Allocation Policies
For every region r in the database, we want to assign an
allocation pc, r to each cell c in Reg(r), such that
∑c Reg(r) pc, r = 1
However, we can do better. Some cells may be naturally
inclined to have more probability than others. Eg : Mumbai
will clearly have more repairs than Bhopal. We can do this
automatically by giving more probability to cells with higher
number of precise facts.
2. Count based :
where Nc is the number of precise facts in cell c
Allocation Policies
For every region r in the database, we want to assign an
allocation pc, r to each cell c in Reg(r), such that
∑c Reg(r) pc, r = 1
Again, we can arguably get a better result by looking at not
just the count, but rather than the actual value of the
measure in question.
3. Measure based : next slide.
Measure Based Allocation
Assumes the following model :
The given database D with imprecise facts has been
generated by randomly injecting imprecision in a precise
database D'.
D' assigns value o to a cell c according to some unknown
pdf P(o, c).
If we could determine this pdf, the allocation is simply
pc, r = P(c) / ∑ c' in Reg(r) P(c')
Maximum Likelihood Principle
A reasonable estimate for this function P can be that which
maximises the probability of generating the given imprecise
data set D.
Example :
Suppose the pdf depends only on the cells and is
independent of the measure values. Thus, the pdf is a
mapping : C  where C is the set of cells.
This pdf can be found by maximising the likelihood function :
() = r D ∑c Reg(r) (c)
EM Algorithm
The Expectation Maximization algorithm provides a standard
way of maximizing the likelihood, when we have some
unknown variables in the observation set.
Expectation step (compute data): Calculate the expected
value of the unknown variables, given the current estimate of
variables.
Maximization step (compute generator): Calculate the
distribution that maximizes the probability of the current
estimated data set.
EM Algorithm : Example
Initialization Step: Data: [4, 10, ?, ?] Initial mean value: 0
New Data: [4, 10, 0, 0]
Step 1: New Mean: 3.5
New Data:[4, 10, 3.5, 3.5]
Step 4: New Mean: 6.5625
New Data: [4, 10, 6.5625, 6.5625]
Step 2: New Mean: 5.25
New Data: [4, 10, 5.25, 5.25]
Step 5: New Mean: 6.7825
New Data: [4, 10, 6.7825, 6.7825]
Step 3: New Mean: 6.125
New Data: [4, 10, 6.125, 6.125]
Result: New Mean: 6.890625
EM Algorithm : Application
Experiments : Allocation run time
Experiments : Query run time
Experiments : Accuracy
Summary
Model for ambiguity : Imprecision, Uncertainity
Querying on uncertain data :
 None v/s Contains v/s Overlaps option
 Consistency, Faithfulness
Possible Worlds interpretation : size blowup
Extended databases : allocation
Aggregation algorithms on Extended databases
Allocation policies :
 Uniform
 Count
 Measure : EM algorithm
Experiments : Allocation time, query time, accuracy
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References :
OLAP over uncertain and imprecise data (Doug Burdick et al.)
- The VLDB Journal (2007) 16:123–144
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OLAP over uncertain and imprecise data(Doug Burdick et al.) - The VLDB Journal (2005)
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http://en.wikipedia.org/wiki/Expectationmaximization_algorithm
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