Equivalence — The Ability of Data to Preserve Identity Under Multiple Representations

1. Definition — What EQUIVALENCE means in Power BI

In Power BI, Equivalence is the structural property that ensures that different representations, structures, levels of granularity, transformations or calculations describe the same entity, fact, or reality without loss of meaning.

Data possesses equivalence when:

• It can take multiple forms while remaining the same truth.
• Different calculation paths produce the same semantic result.
• Different tabular models represent the same business logic.
• Divergent keys or identifiers can be mapped to the same identity.
• Format conversions do not alter the essential meaning of the data.

Equivalence answers the question:
Does the identity of the data remain intact even when its form changes?

2. Nature — Why EQUIVALENCE exists as a structural property

Equivalence arises from three fundamental principles:

A) Transformation without loss of identity

Change of form ≠ change of meaning. This includes transformations:

• in format (text, date, number)
• in granularity
• in tabular structure
• in model design (star → snowflake)

B) Alternative paths toward the same truth

In tabular models:

• different DAX formulas
• different joins
• different relational routes

can reach the same result.

C) Controlled semantic multiplicity

A real-world entity may have:

• multiple codes
• multiple keys
• multiple references
• multiple valid forms

But all must converge toward a single center of truth.

This makes Equivalence an ontological property: the truth of data exists beyond its particular representation.

3. Function — How EQUIVALENCE operates in the Power BI pipeline

A) In Power Query (Equivalent transformations)

Different transformations may produce equivalent datasets if they preserve:

• identity
• integrity
• semantics
• mapping capability

Examples:

• "01/01/2025", "2025-01-01", "1 Jan 2025" → same date
• splitting and recombining columns without changing meaning
• normalizing or denormalizing tables while preserving entity identity

B) In the Tabular Model (Equivalent structures)

• functionally identical relationships
• keys that represent the same entity
• aggregations that produce identical results

C) In DAX (Equivalent calculations)

Examples:

SUM(Sales[Amount])

and

SUMX(Sales, Sales[Amount])

Or:

[Total Sales]

and

CALCULATE(SUM(Sales[Amount]))

D) In Visualizations

• different aggregations
• different hierarchical levels
• different filters

If they share semantic equivalence → they convey the same truth.

4. Consequences — What happens when Equivalence fails

A) Double counting or loss of identity

Example: two different keys representing the same entity → duplication.

B) Divergent calculation paths

DAX returns different values depending on the method used.

C) Incompatible models

Tables that should represent the same thing do not match in:

• cardinality
• granularity
• keys
• semantics

D) Inconsistency across reports

Different reports show different values for the same metric → equivalence collapse.

5. Interactions — Properties related to EQUIVALENCE

• Consistency
• Quality
• Validity
• Integrability
• Abstraction
• Semantic Congruence

6. Evaluation Methods — How to measure EQUIVALENCE in Power BI

A) Identity Convergence Test

Evaluate whether multiple keys converge into the same logical entity.

B) Result Equivalence Test (RET)

Compare two calculation paths:

• via SUM
• via SUMX
• via CALCULATE
• via different relationships

The result must be identical.

C) Structural Equivalence Check

Verify whether two tabular models produce:

• the same totals
• the same relationships
• the same semantics

Even when the structure differs.

D) Format Equivalence

Different formats must represent the same value:

• "$1,000"
• "1000.00"
• "1K"

E) Domain Mapping Validation

Ensure that different codes map to the same entity.

7. Applicable Models — Frameworks to ensure EQUIVALENCE

A) Equivalence Mapping Matrix (EMM)

• alternative keys
• equivalent domains
• equivalent formats
• equivalent calculation paths

B) Semantic Equivalence Framework (SEF)

• semantic intent
• representation
• granularity
• entity relationships

C) Multi-Key Identity Resolver

Unifies heterogeneous keys → one consistent entity.

D) Functional Equivalence Engine

Validates that different formulas produce the same output.

E) Transformative Equivalence Model

Defines which PQ transformations preserve:

• entity
• identity
• semantics

8. Samples — Implementation in Power Query and DAX

A) POWER QUERY — Equivalent date normalization

DateNormalized = 
Table.TransformColumns(Source, {{"Date", each Date.From(_), type date}})

B) POWER QUERY — Key equivalence mapping

EquivalenceMap =
Table.NestedJoin(
    KeysA, {"ClientID"},
    KeysB, {"AltClient"},
    "Mapped", JoinKind.LeftOuter
)

C) DAX — Functional equivalence of calculations

TotalSales_A = SUM(Sales[Amount])

TotalSales_B = SUMX(Sales, Sales[Amount])

D) DAX — Equivalence validation

Equivalence_OK =
IF(
    [TotalSales_A] = [TotalSales_B],
    TRUE(),
    FALSE()
)

E) DAX — Unified identity mapping

CustomerUnified =
SELECTCOLUMNS(
    Customers,
    "UnifiedID",
        COALESCE(Customers[ID], Customers[LegacyID], Customers[ExternalID])
)