Behavioral Feature-Space Representations

Behavioral Feature-Space Representations are structured representations in which each eligible behavioral-evidence instance is encoded by an explicitly ordered collection of coordinates with declared identities, value semantics, units or scales, missingness rules, and comparability conditions. Importantly, a feature-space representation is not merely a numeric row vector or array. It is not synonymous with a Descriptor Set, nor is it defined by its downstream use in a model. Moreover, the presence of numeric coordinates does not automatically endow the space with Euclidean geometry or any other implicit metric. Coordinates within a feature space can originate from diverse sources such as descriptors, direct measurements, transformed quantities, encoded categories, or other declared evidence mappings, each retaining its own provenance and semantic distinctions.

Meaning and Boundaries of Feature-Space Representation

A Behavioral Feature-Space Representation is a representation definition that maps comparable behavioral-evidence instances into an explicitly defined coordinate space. This space is characterized by declared coordinate identities, ordering, domains, scaling conventions, validity rules, and meaningful operations. One Representation Instance is typically one point or structured coordinate vector in that space, while a collection of comparable instances can be arranged as a feature matrix. However, the feature matrix itself is not the scientific definition of the space; it is an arrangement of instances within the defined schema.

To clarify terminology:

• A Descriptor is a standalone property or measurement with inherent scientific semantics.
• A Descriptor Set is a controlled collection of descriptors, often curated for a particular study or domain.
• A Feature is a quantity adopted for an analytical purpose; it may be a descriptor or a derived quantity.
• A Feature-Space Coordinate is one identified axis of a feature-space representation, with explicit semantics.
• A Feature Vector realizes one instance under the full coordinate schema of the feature space.
• A Feature Matrix arranges multiple comparable feature vectors (instances) into a two-dimensional array.
• A Feature-Space Representation is the entire formally defined coordinate space with its schema and semantics.

Membership in one of these objects does not erase the source identity or semantics of its components.

Every coordinate within a feature space has explicit semantics. It must have:

• A stable identity or clear derivation
• An admissible value domain
• Units or a dimensionless interpretation
• Sign or direction semantics where relevant
• Provenance of support (source and method)
• Defined missing/invalid states
• Any normalization or fitted-state dependencies

A column label alone is insufficient to establish coordinate identity when multiple implementations, supports, parameterizations, coordinate frames, or reference sets can produce numerically similar but scientifically non-equivalent values.

Coordinate Schema and Instance Identity

The feature-space schema is the ordered coordinate contract shared by comparable instances. It explicitly declares:

• The number of coordinates
• Coordinate names or unique IDs
• The ordering of coordinates
• Source semantics for each coordinate
• Data type and value domain
• Units and scaling state
• Valid, invalid, and missing semantics
• Any grouping or block structure among coordinates

Reordering coordinates without updating the schema changes the interpretation of the numeric vector even if the actual values remain the same.

Instance identity corresponds to a single declared analytical instance such as a support, event, episode, participant-condition unit, trial, or other scientifically defined object. One feature vector corresponds to exactly one such instance. Mixing coordinates computed from incompatible supports, different entities, future-inclusive evidence, or unmatched observational units into one vector requires an explicit mapping and justification; mere equal row length does not establish row comparability.

A generic feature-vector representation can be rendered as:

Here:

• i is the identified representation-instance index
• p is the declared number of coordinates in the feature-space schema
• z_ik is the value of coordinate k for instance i under that coordinate's declared semantics and validity rule
• Bold z_i is the ordered feature vector

This vector notation does not imply that all coordinates share units, metric meaning, statistical distribution, or equal importance.

Coordinates can be grouped into blocks or hierarchical schema structures according to scientific organizing variables such as modality, descriptor family, anatomical region, frequency band, entity, or temporal support. Preserving block identity is important because treating a structured block as an undifferentiated flat vector can obscure common provenance, duplicated information, scale differences, and block-specific missingness.

Assembly from Behavioral Evidence

Feature-space coordinates can be assembled from heterogeneous but scientifically compatible sources such as:

• Scalar descriptors (e.g., mean speed)
• Components of vector-valued descriptors (e.g., 3D acceleration axes)
• Direct measurements (e.g., event durations)
• Event counts or rates (e.g., number of vocalizations)
• Relational descriptors (e.g., cross-signal correlations)
• Selected transform coefficients with stable semantics (e.g., power in a frequency band)
• Explicitly encoded categorical evidence (e.g., one-hot encoding of context)

It is critical to preserve the distinction between the source object's original scientific identity and its role as one or more representation coordinates in the feature space.

Concatenation, aggregation, and recomputation are distinct operations:

• Concatenation preserves separate coordinate identities by placing coordinates side-by-side without changing them.
• Aggregation combines multiple values into a new summary coordinate only when an explicit aggregation mapping is defined.
• Recomputation involves recalculating descriptors on pooled or combined evidence.

Simply placing values side by side does not create a new descriptor and should not be described as evidence fusion solely because multiple sources are present.

Coordinates can legitimately summarize different aspects of one declared analytical instance, but their supports, entities, modalities, or temporal anchors must be semantically compatible with the vector's intended identity. For example, one coordinate derived from an entire episode and another from a brief event-centered window can coexist only if the mixed support design is explicit rather than silently treated as homogeneous evidence.

Categorical and ordinal coordinates can be used as feature-space coordinates when their semantics are explicit. Binary indicators, one-hot-like encodings, ordinal codes, counts, and other numerical encodings qualify as coordinates if their meaning and domain are declared. However, arbitrary category IDs do not imply ordered or metric distances. Expanding one categorical variable into several indicator coordinates changes dimensionality and weighting in distance or regularized analyses and therefore changes feature-space geometry.

Scaling, Centering, and Coordinate Conditioning

Coordinate scale is a fundamental part of feature-space geometry. Coordinates measured in different units or with widely different numerical ranges can dominate distance, variance, regularization, projection, or optimization operations if treated naively. Scaling can rebalance these operations but also removes or changes absolute magnitude semantics, so scaling must follow the intended geometry rather than being applied automatically.

Coordinate-wise standardization can be expressed as:

Where:

• z_ik is the original value of coordinate k for instance i
• μ_k is the declared reference-set center for coordinate k
• s_k is a positive declared reference-set scale for coordinate k
• z′_ik is the standardized coordinate

The parameters μ_k and s_k are part of the fitted representation state when estimated from data. Zero or near-zero s_k requires explicit handling rules. Standardization changes coordinate units and distance geometry; it does not universally improve every representation.

Different coordinate conditioning operations include:

• Centering: Subtracting a location statistic (e.g., mean) to align coordinate centers.
• Standardization: Centering plus scaling by standard deviation.
• Min–Max Scaling: Linearly rescaling coordinate values to a fixed range.
• Robust Scaling: Using robust statistics (e.g., median and IQR) for centering and scaling.
• Unit-Norm Scaling (Per-Vector): Normalizing each feature vector to unit length, changing relative scaling across coordinates.
• Domain-Reference Normalization: Scaling based on domain-specific reference values or standards.
• Block Weighting: Applying weights to coordinate blocks to rebalance their contributions.

Per-coordinate scaling changes coordinate axes consistently across instances, whereas per-vector normalization rescales the entire vector for each instance, removing absolute level information but preserving direction.

Fitted-state leakage and comparability must be carefully managed. If scaling, centering, clipping ranges, reference distributions, quantile mappings, or other coordinate-conditioning statistics are estimated from data, the fitted reference set and its version become part of the representation identity. Using future, held-out, or comparison data to fit these statistics can alter the representation and invalidate intended independent comparisons. Independently fitted scalers can place nominally identical coordinates in nonidentical spaces, undermining comparability.

Geometry, Distance, and Neighborhood Semantics

A numeric feature-space representation does not automatically justify Euclidean geometry or any particular distance metric. A distance or similarity measure is scientifically meaningful only when coordinate differences, scales, domains, missingness, and joint structure support that operation.

Different relations answer different questions and can rank neighbors differently:

• Euclidean distance measures straight-line distance assuming metric semantics.
• Manhattan (L1) distance sums absolute coordinate differences.
• Cosine similarity measures angles between vectors, emphasizing direction over magnitude.
• Correlation-like metrics consider standardized relationships.
• Covariance-aware distances adjust for coordinate dependencies.
• Mixed-type or domain-defined metrics handle heterogeneous coordinate types.

Weighted Euclidean distance can be rendered as:

Where:

• Bold z_i and z_j are two comparable feature vectors under one schema
• p is the number of coordinates
• k is the coordinate index
• z_ik and z_jk are comparable coordinate values
• w_k ≥ 0 is the declared weight for coordinate k
• d_w is the weighted Euclidean distance

Equal weights do not imply equal scientific importance when coordinate scales differ. This metric is inappropriate for nominal categories or constrained geometries unless an encoding makes squared coordinate differences meaningful.

Covariance-aware geometry accounts for strong correlations or coordinate blocks. Ordinary Euclidean distance can count shared variation repeatedly. Covariance-aware distances or whitening-like mappings alter geometry to account for dependence but introduce fitted state and assumptions about covariance estimation, rank, support, and stationarity. Such covariance-adjusted spaces represent different representation geometries, not neutral corrections.

Neighborhood and prototype semantics depend on the feature-space geometry. Nearest neighbors, centroids, averages, interpolation, convex combinations, and cluster structures are meaningful only if the geometry supports them. A coordinate-wise mean can be mathematically valid yet not correspond to any physically realizable behavior, valid categorical configuration, or admissible geometric state.

Redundancy, Dependence, and Dimensionality

Different relationships among coordinates include:

• Exact duplication: Identical coordinate values across all instances.
• Algebraic dependence: Coordinates related by deterministic functions.
• Monotonic relation: One coordinate is a monotonic function of another.
• Empirical correlation: Statistical association in data.
• Common-source dependence: Coordinates derived from the same underlying evidence.
• Conceptual redundancy: Overlapping scientific meaning.

Redundant coordinates can overweight one property in distance or model fitting. However, empirical correlation alone does not prove semantic duplication, and low correlation does not prove complementarity.

Dimensionality is the declared number of coordinates versus intrinsic or effective structure. Adding coordinates increases ambient dimensionality even when new coordinates are redundant, nearly constant, deterministic functions of existing coordinates, or dominated by noise. Therefore, “more dimensions” does not mean “more independent behavioral information.”

High-dimensional geometry requires caution. Distances can become less discriminative, neighborhoods sparser, covariance estimation unstable, and sample support inadequate as dimensionality grows. These are data- and geometry-dependent phenomena rather than universal rules.

Dimensionality reduction and feature selection are conceptual boundaries. Selecting coordinates changes which explicitly defined components remain; projecting coordinates constructs a new fitted coordinate system; learned compression can create a latent representation. None of these operations is identical to the original feature-space representation, and original coordinate semantics should not be attributed automatically to projected or latent dimensions.

Missingness, Sparse Coordinates, and Mixed Types

Coordinate states must be carefully distinguished:

• Observed Zero: Measured zero value with scientific meaning.
• True Absence: Structural absence of the property measured.
• Below-Detection Value: Value below measurement resolution or sensitivity.
• Undefined Descriptor: Descriptor not defined for the instance.
• Unavailable Evidence: Evidence not collected or missing.
• Structural Inapplicability: Coordinate not meaningful for instance type.
• Censored Value: Value only partially known due to limits.
• Missing Coordinate: Data point missing or invalid.

These states can have different scientific meanings even when software stores them identically. A complete feature vector requires validity and availability semantics, not just numeric values.

Missing-coordinate policies include:

• Explicit Masks: Separate indicators for missingness.
• Complete-Case Restriction: Using only fully observed instances.
• Pairwise-Valid Operations: Using available data per coordinate pair.
• Imputation: Filling missing values with estimates.
• Model-Based Completion: Predicting missing values within models.
• Missing-Category Indicators: Adding missingness as a categorical state.
• Partial-Distance Rules: Adjusting distances to ignore missing coordinates.

Imputed values are constructed, not observed. Adding missingness indicators enlarges feature space and preserves informative missingness patterns.

Sparse feature spaces have many structurally zero or absent coordinates, such as large indicator vocabularies or event occurrence vectors. Sparse zero differs from missing value. Centering, dense transformations, and some metrics can destroy computational sparsity or alter absence semantics. Sparse storage is an implementation detail unless zero has declared representational meaning.

Mixed-type spaces contain continuous, bounded, count, binary, ordinal, or categorical-derived coordinates. Ordinary arithmetic and a single default metric may be invalid across mixed domains. Representation design should preserve type semantics or use explicitly justified mappings. Numeric encoding alone does not imply coordinates share the same mathematical scale.

Comparability, Quality, and Provenance

Comparability across participants, sessions, devices, sites, contexts, supports, and representation versions requires compatible:

• Schema (coordinate IDs, ordering)
• Source definitions and versions
• Units and support semantics
• Scaling and centering reference states
• Missingness rules and handling
• Coordinate ordering

A shared column name or identical vector length does not establish comparability if any of these differ.

Feature-space quality and sensitivity are properties of the representation itself, not solely determined by downstream model performance. Examination should include:

• Coordinate validity and support adequacy
• Dynamic range and degeneracy
• Scaling sensitivity and redundancy
• Missingness burden
• Geometry stability under preprocessing choices
• Transportability of schema and fitted state

Strong classification or regression performance can coexist with poor representational validity if nuisance variables, leakage, device identity, or context dominate the space.

Worked Example

Consider a behavioral episode described by the following descriptor coordinates:

• Movement speed (m/s)
• Event rate (events/min)
• Spectral concentration (dimensionless ratio)
• Linguistic count (number of words)
• Cross-signal correlation (correlation coefficient)
• Categorical context indicator (encoded as one-hot vector)

Construct a feature vector for one episode:

Multiple such episodes can be arranged as rows in a feature matrix with identical schema.

Without scaling, the linguistic count coordinate (with value 120) can dominate Euclidean distance calculations compared to spectral concentration (0.45). Correlated coordinates, if present (e.g., event rate and movement speed), can overweight a behavioral property in distance or model fitting. An undefined coordinate (e.g., spectral concentration missing from one episode) differs fundamentally from zero and needs explicit missingness handling. Independently fitted scaling on different subsets can place nominally identical coordinates into non-comparable spaces.

Projecting or selecting coordinates creates different representations or representation versions rather than retroactively changing the original coordinate meanings. This preserves provenance and interpretability.

Feature-Space Representation provenance includes information necessary to reproduce and scientifically interpret both the coordinate space and its instances:

• Representation definition and version
• Analytical-instance semantics
• Coordinate IDs and ordering
• Coordinate source definitions and versions
• Support and entity mapping
• Units and value domains
• Scaling, centering, normalization policy, and fitted reference state
• Block structure and weights
• Distance and similarity conventions
• Categorical encodings
• Missing, invalid, and imputed state definitions
• Sparsity semantics
• Coordinate-selection status if applicable
• Implementation and version
• Comparability conditions
• Uncertainty and quality/sensitivity findings
• Intended analytical operations

A defensible feature space is defined by coordinate semantics and geometry, not by array shape or downstream model compatibility.