The Intentional Analytics Model (IAM) has been devised to couple OLAP and analytics by (i) letting users express their analysis intentions on multidimensional data cubes and (ii) returning enhanced cubes, i.e., multidimensional data annotated with knowledge insights in the form of models (e.g., correlations). Five intention operators were proposed to this end; of these, describe and assess have been investigated in previous papers. In this work we enrich the IAM picture by focusing on the explain operator, whose goal is to provide an answer to the user asking "why does a measure show these values?". Specifically, we propose a syntax for the operator and discuss how enhanced cubes are built by (i) finding the polynomials that best approximate the relationship between a measure and the other cube measures, and (ii) highlighting the most interesting one. Finally, we test the operator implementation in terms of efficiency.

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The Whys and
Wherefores of Cubes
Matteo Francia1, Stefano Rizzi1, Patrick Marcel2
1DISI, University of Bologna, Italy 2LIFAT, University of Tours, France
DOLAP 2023: 25th International Workshop on Design, Optimization, Languages and Analytical Processing of Big Data

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Intentional Analytics Model
Context: Intentional Analytics Model (IAM) [1]
- Facilitate OLAP analysis of multidimensional cubes
- Escape from query answers as plain tables
Express high-level intentions, not queries
- Describe, Assess, Explain, etc.
Get cubes enhanced with insights
- Apply (mining/ML) models to data
- Return interesting insights
Explain: finding interesting relationships in cube facts
- Data exploration: automatically extracts meaningful relationships from facts
- Validating user’s belief: check if known relationships hold
- In agriculture, the quantity of potassium is correlated with the quality of Kiwifruits.
Do facts confirm this belief?
Matteo Francia – University of Bologna 2
[1] Panos Vassiliadis, Patrick Marcel, Stefano Rizzi: Beyond roll-up's and drill-down's:
An intentional analytics model to reinvent OLAP. Inf. Syst. 85: 68-91 (2019)

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Classical OLAP
Case study:
- Given the cube of Sales
- Explain monthly revenue against cost and quantity
If we had to do this in plain OLAP
- Query the cube, get a plain table
- Manually identify interesting patterns
But…
- What if we have thousands of cells?
- What if we have many measures?
- Can we have an effective representation?
Matteo Francia – University of Bologna 3
select month, sum(quantity), sum(cost), sum(revenue)
from sales_ft join date_dt on (…)
group by month
product
type
category
customer
gender
store
city
country
date month year
quantity
revenue
cost
SALES
month cost quantity revenue
125 10 12 125
132 20 14 150
12 30 10 60
15 40 5 15
50 50 9 50

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Intentional OLAP: Explain
`Explain` intention:
with cube explain m [ for P ] by l1,…,ln [ against m1, ..., mr ]
“Explained” measure: m
Selection predicate: P (consider all facts if omitted)
Group-by set: l1,…,ln (at least one level)
Measures: m1, ..., mr (compute against all measures if omitted)
Semantics translates into an execution plan
i. Execute query for given cube,
measures, predicate, group-by set
ii. Apply models explaining relationships
through components
iii. Rank components by interestingness
iv. Return effective visualization
Matteo Francia – University of Bologna 4
with sales explain
revenue by month
Analytic dashboard
R² = 0.9901
revenue
quantity
month cost quantity revenue
125 10 12 125
132 20 14 150
12 30 10 60
15 40 5 15
50 50 9 50

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Model
Models are “types” of relationships hiding in the cube facts
- Are made of components, each being a specific relationship…
- … computed on levels/members/measures
To give a proof-of-concept, we restrict to consider
- A single model: polynomial regression
- Each component is a polynomial relationship
between a pair of measures (univariate regression)
- The dependent variable revenue is modeled as an
dth degree polynomial in the independent variable
(e.g., quantity)
Matteo Francia – University of Bologna 5
R² = 0.9901
revenue
quantity R² = 0.6524
revenue
cost
Model: Polynomial regression
A component
(revenue, quantity)
Another component
(revenue, cost)
with sales explain
revenue by month

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Components
Each component is a polynomial relationship αd
( ) between a pair of measures
- How to choose the “best” polynomial and avoid overfitting?
- E.g., consider revenue = αd
(𝑐𝑜𝑠𝑡)
We need an error function weighting the degree (d): fact αd fact.m −fact.m
2
facts −d −1
- αd
( ) is the polynomial with degree d fitted with OrdinaryLeastSquares method
- The error is computed against a test set containing 30% of the facts
Matteo Francia – University of Bologna 6
Too simple
(high error, low polynomial degree)
Too complex
(lower error, higher degree)

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Computing components
Matteo Francia – University of Bologna 7
Start with d=0 and fit the polynomial

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Iterate:
- Increase the degree…
- … until we find a minimum of the error
To ensure training on “sufficient” facts
- Apply the one-to-ten rule of thumb
d=1
d=2
d=3
Computing components
Matteo Francia – University of Bologna 8

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Computing components
Matteo Francia – University of Bologna 9
Iterate:
- Increase the degree…
- … until we find a minimum of the error
d=2

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Computing components
Matteo Francia – University of Bologna 10
Iterate:
- Increase the degree…
- … until we find a minimum of the error
d=2
This could be a local minimum, but we
prefer to return a simpler model
• y = α2 x = a + bx + cx2
• y’ = α4
x = a + bx + … + ex4

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Interestingness
GOAL: given components, return the most interesting one
Interestingness: how variation in the dependent variable is predictable from the independent variable
- This is encoded by the coefficient of determination R2
- The better the model, the closer the value of R2 to 1
Matteo Francia – University of Bologna 11
R² = 0.9901
revenue
quantity R² = 0.6524
revenue
cost
Model: Polynomial regression
with sales explain
revenue by month
R² = 0.9901
revenue
quantity
month cost quantity revenue
125 10 12 125
132 20 14 150
12 30 10 60
15 40 5 15
50 50 9 50

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Visualization
Matteo Francia – University of Bologna 12
Matteo Francia, Matteo Golfarelli, Stefano Rizzi. Describing and Assessing Cubes Through Intentional Analytics. EDBT 2023 (demo)
Notebook-like interface

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(b) Computing on 106 facts (Synth. dataset)
scales linearly wrt the measures in the cube
Evaluation
(a) Computing the results on ~90K facts
(Foodmart dataset) takes 0.5 seconds
Matteo Francia – University of Bologna 13
Implemented in Python with numpy and sk-learn libraries
- The tests were run on an Intel(R) Core(TM)i7-6700 CPU@3.40GHz CPU with 8GB RAM
https://github.com/big-unibo/explain

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Discussion
Overall, this paper is not about:
- (Polynomial) Regression optimization
- “Yet Another” explainability approach
We propose a modular framework where approaches to aggregate data explanation can be plugged
- Regression: return relationships between a dependent variable and one or more independent variables [4]
- Data lineage: which database tuple(s) caused that output to the query? [1]
- Intervention: an input is a cause to an output if a change affects the output [2, 3]
The added value is in the IAM paradigm and augmented analytics
- Data scientists can express high-level intentions…
- … and the system (automatically) selects the most interesting explanations
- … coupled with data and visualization
14
[1] Alexandra Meliou et al. 2010. The Complexity of Causality and Responsibility for Query Answers and non-Answers. VLDB
[2] Sudeepa Roy et al. 2014. A formal approach to finding explanations for database queries. SIGMOD
[3] Zhengjie Miao et al. 2019. LensXPlain: Visualizing and Explaining Contributing Subsets for Aggregate Query Answers. VLDB
[4] Fotis Savva et al. 2018. Explaining Aggregates for Exploratory Analytics. BigData.
https://xkcd.com/605/

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Conclusion & research directions
We have given a proof-of-concept for explain intentions
- Syntax is flexible enough to suit users who wish to verify a specific hypothesis they made
- Intention processing takes a few seconds even on very large query results
- Performances are in line with the interactivity requirements of OLAP sessions
Future research directions
- Explain relationships between a measure and two or more other measures (e.g., multivariate regression)
- Evaluate the effectiveness of the approach by experimenting it with real users
- Generalize the definition of model to cope with additional model types from the literature
- Experiment other interestingness metrics
- Conciseness: large explanations will probably be not well understandable
- Interpretability: the suitability of an explanation will depend on the target users
- Actionability: explanations should point to actionable suggestions
Matteo Francia – University of Bologna 15

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Questions?
Matteo Francia – University of Bologna 16
Thank you.