Frequency domain analysis and simulation of multi-channel complex-valued time series
Bibliographic record
Abstract
Complex-valued representation of a two-component real-valued time series yields additional physical insights that are lost otherwise. The spectral representation theorem allows us to study covariance stationary complex-valued random sequences in the frequency domain, and this is known as rotary spectral analysis. It is a widely-used technique for studying elliptical motions in ocean currents, wind etc. An important and useful parameter in rotary spectral analysis of scalar complex-valued time series is the rotary coefficient. It measures the tendency of vectors to rotate in a clockwise or counter-clockwise manner. We derive the theoretical distribution of the rotary coefficient estimator and apply our results to ocean current speed and direction measurements at six depths in the Labrador Sea. Canonical correlation techniques are commonly employed in the analysis of a pair of vector-valued random variables. We introduce a framework to extend classical multivariate analysis techniques such as canonical correlation analysis, partial least squares, and multivariate linear regression, to define coherence – a measure of correlation in the frequency domain. In the statistical analysis of complex-valued time series, we refer to a time series as proper/improper according to whether it is uncorrelated/correlated with its complex conjugate. In earlier work, complex-valued signals were assumed to be proper for the simple reason that it led to a simpler algebra. However, the loss in performance caused by overlooking the potential impropriety of such data is realized to be significant, and therefore, when the data is improper, information contained in the complementary covariance structure must be considered. Since impropriety in the time domain may not necessarily correspond to impropriety at all frequencies, we propose a generalized likelihood ratio test which may be used to test propriety of a discrete time complex-valued process at a given frequency. Finally, the idea of vector circulant embedding is exploited to yield a frequency domain bootstrap methodology. With the help of three example parameters involved in the study of multi-channel complex-valued time series, we illustrate how our method allows us to draw statistical inference such as confidence intervals. Our method can prove useful in cases where no theoretical distributional results are available, or to check the effect of nuisance parameter estimates where theoretical results are available.
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How this classification was reachedexpand
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.001 |
| Bibliometrics | 0.001 | 0.002 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.001 | 0.000 |
| Insufficient payload (model declined to judge) | 0.008 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".