Tracking Rolling SVD Factors Through Rotation and Rebalancing
Summary
The document examines whether factors estimated by singular value decomposition remain stable when the asset-return matrix is re-estimated in overlapping rolling windows. It compares factor-return vectors over their shared dates using cosine distance. A distance near zero suggests alignment, a distance near two can indicate that the vectors point in opposite directions because SVD may flip signs, and a value near one suggests substantial disagreement. The response cautions that factors from separate decompositions are generally not identical: they can differ through rotations, scaling, and ongoing evolution.
The answer recommends interpreting continuity through similar characteristic portfolios and their economic meaning, rather than requiring factor-return vectors to match exactly. It identifies periodic re-estimation or rebalancing as a general way commercial and academic factor models accommodate changes in assets and loadings. The discussion is conceptual; it does not provide a specific matching algorithm, stability threshold, or empirical test, and cosine similarity alone cannot resolve rotations or establish that a factor retains its interpretation.
Key ideas
- SVD factors estimated independently in rolling windows need not remain identical.
- Cosine distance can flag aligned, opposed, or markedly different factor-return vectors.
- Sign flips can make equivalent directions appear opposed in a direct comparison.
- Factors may evolve through rotations, scaling, and changes in the underlying asset universe.
- Periodic re-estimation is a common way to accommodate evolving factor loadings and portfolios.
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# Statistical factors: stability over time
# Statistical factors: stability over time
I have an asset returns matrix of shape (n, t) with n assets and t days, n > t. I perform SVD decomposition of the matrix into c factors, getting a (c, t) shaped matrix of factor returns.
This process repeats with a sliding window of let's say 63 days, shifting one day at a time. Now the question is, are the factors from the previous window the same as from the current one?
To test this, I compute cosine distance between overlapping parts (62 days) of the subsequent factor return vectors. If the factors are the same, their returns will be the same. Here is the plot of those distances for the first factor.
In most cases, the distance is zero or close to zero, which is what we want.
Sometimes the distance is two, which means that the vectors are opposed. As far as I know, this comes from SVD sign flipping and is a minor nuissance.
The problem is with distances close to one, with vectors being nearly orthogonal. This means that the factor changed between windows. It's an entirely different factor now.
The question is how to ensure that the factors stay the same or close from one window to the next. Is there a popular, robust and simple way of dealing with the problem?
## Answer by krkeane (score 1, accepted)
https://quant.stackexchange.com/a/80635
> are the factors from the previous window the same as from the current one?
In general, no. They are probably better thought of as rotations and scalings (and minor other evolution) of each other. Each matrix you present to SVD is decomposed in isolation. Assuming you use the same universe of securities, some of the characteristic portfolios (weightings across rows) are likely to be similar, and the interpretation of the similar portfolios is likely to be similar. Cosine similarity is a reasonable metric for this.
> how to ensure that the factors stay the same or close from one window to the next.
You probably want the factor semantics to maintain continuity, but companies have birth and death processes. Commercial indices (and factor models) all have mechanisms ("rebalancing") to accommodate evolution of the component assets over time.
While not constraining the factors to be "the same or close", a paper of mine with my PhD advisor tracks the evolution of characteristic portfolios: Keane and Corso, "Maintaining prior distributions across evolving eigenspaces: An application to portfolio construction."
> Is there a popular, robust and simple way of dealing with the problem?
Periodic rebalancing is the most generic task that addresses this "problem". Factor loadings and factor portfolios of well know commercial and academic products are typically re-estimated at some specified frequency.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.