Vector Time Series, Cross-Sectional Normalization, and Spherical Wandering

Introduction

In many modern applications—finance, signal processing, neuroscience, and machine learning—we encounter vector time series: sequences of observations where each time point $t$ is associated with a vector $x_t \in \mathbb{R}^d$.

Unlike scalar time series, these objects encode both magnitude and directional information, and the interplay between the two often carries the signal of interest.

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What is a Vector Time Series?

A vector time series is:

$$\{x_t\}_{t=1}^T, \quad x_t \in \mathbb{R}^d$$

Each observation contains:

• A length (norm) $\|x_t\|$, capturing scale or intensity
• A direction $\frac{x_t}{\|x_t\|}$, capturing cross-sectional structure

These two components often behave very differently over time.

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Cross-Sectional Normalization

A common transformation is cross-sectional normalization, where each vector is rescaled to unit length:

$$\tilde{x}_t = \frac{x_t}{\|x_t\|}$$

This maps the series onto the unit sphere $S^{d-1}$, removing magnitude and preserving only direction.

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When Should You Normalize?

Cross-sectional normalization is appropriate when:

1. Scale is Uninformative or Noisy

If variation in $\|x_t\|$ is dominated by noise or irrelevant factors, normalization helps isolate signal.

2. Only Relative Structure Matters

When the pattern across components is key (e.g., portfolio weights, normalized signals), direction is sufficient.

3. Multiplicative Effects Dominate

If the system has the form:

$$x_t = a_t \cdot y_t$$

normalization removes the scalar factor $a_t$, isolating $y_t$.

4. Geometric Constraints Are Useful

Some models benefit from bounded inputs or angular structure (e.g., cosine similarity, spherical clustering).

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When Not to Normalize

Avoid normalization when:

• Magnitude carries predictive information
• Scale itself is the signal (e.g., volatility, energy)
• Small vectors would be disproportionately amplified

A useful check: test whether $\|x_t\|$ alone has predictive power.

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Geometry After Normalization

After normalization:

$$\tilde{x}_t \in S^{d-1}$$

The system becomes purely directional:

• Euclidean distance → angular distance
• Covariance → alignment
• Dynamics → motion on the sphere

This is a shift from standard vector analysis to spherical geometry.

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The No-Predictability Regime

Suppose there is no temporal predictability:

$$\tilde{x}_t \perp \tilde{x}_{t-1}, \tilde{x}_{t-2}, \dots$$

Then the process behaves like independent draws on the sphere.

Uniform Case

If:

$$\tilde{x}_t \sim \text{Uniform}(S^{d-1})$$

then the sequence exhibits no structure, no persistence, and no preferred direction.

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Spherical Wandering

In the absence of predictability, the normalized series undergoes what can be thought of as:

Spherical wandering

Each vector is simply another direction on the sphere, with no memory of the past.

Key Properties

1. Near Orthogonality

In high dimensions:

$$\mathbb{E}[\tilde{x}_t^\top \tilde{x}_{t-1}] \approx 0$$

Successive vectors are nearly orthogonal.

2. Concentration of Measure

Random vectors on the sphere:

• Spread uniformly
• Appear almost independent
• Rarely align strongly

3. Illusion of Structure

Even random spherical data can:

• Look clustered in projections
• Show transient alignments
• Produce misleading patterns

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Interpretation

Normalization fundamentally changes the modeling problem:

Before Normalization	After Normalization
Magnitude + Direction	Direction only
Euclidean geometry	Spherical geometry
Scale-dependent models	Angular models

If predictability disappears after normalization, then:

• The system has no directional memory
• Any signal lies in magnitudes or higher-order effects
• The observed process is effectively noise on the sphere

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Conclusion

Cross-sectional normalization is more than a preprocessing step—it is a geometric transformation.

When predictability remains, it reveals directional structure.
When it vanishes, what remains is something simpler and more fundamental:

A sequence of vectors wandering across the sphere,
exploring directions without memory—
like a stochastic spherical code unfolding in time.