A Guided Path Through the Large Deviations Series

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  This post serves as a short guide to the four-part series on large deviations and their applications to stochastic processes, biology, and weak-noise dynamical systems. Each article can be read independently, but together they form a coherent narrative that moves from foundational principles to modern applications. 1. Sanov’s Theorem and the Geometry of Rare Events The series begins with an intuitive introduction to Sanov’s theorem , highlighting how empirical distributions deviate from their expected behavior and how the Kullback-Leibler divergence emerges as the natural rate functional. This post lays the conceptual groundwork for understanding rare events in high-dimensional systems. Read the post → 2. Sanov’s Theorem in Living Systems The second article explores how Sanov’s theorem applies to biological and neural systems . Empirical measures, population variability, and rare transitions in gene expression or neural activity are framed through ...
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Trend Extraction Across Long Timelines in Noisy Cell‑Culture Experiments

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 A follow‑up to “Cell Count Analysis with cycleTrendR” — focused on neurodegenerative drift

In our previous post, we explored how cycleTrendR can be applied to cell‑count trajectories in longitudinal culture experiments. That example sparked an insightful question on LinkedIn:

How does cycleTrendR perform when the signal is slow, noisy, and stretched across long timelines — as in neurodegenerative disease models?

This post is a direct response.

Neurodegenerative phenotypes often evolve gradually, with subtle changes accumulating over weeks or months. Meanwhile, experimental noise — imaging variability, segmentation artefacts, media‑change cycles — fluctuates rapidly and can dominate the signal.

cycleTrendR was designed precisely for this challenge:

to extract slow biological drift from noisy, irregular, and cyclic time series.


How cycleTrendR isolates slow biological drift from fast experimental noise

Longitudinal cell‑culture experiments — especially those modelling neurodegeneration — evolve slowly.
The biological signal accumulates over weeks or months, while the noise fluctuates daily or even hourly.
This mismatch makes long‑term deterioration extremely difficult to visualise.

cycleTrendR was designed specifically for this scenario: slow drift, high noise, irregular sampling, and recurring cycles.

 Figure 1



Legend of Figure 1. cycleTrendR decomposes noisy cell-culture time series into slow drift, recurring cycles, and irregular noise, revealing long-term biological deterioration hidden beneath experimental fluctuations.

1. Why Long‑Horizon Trends Are Hard to See

1.1 High short‑term noise dominates the signal

Cell‑count trajectories fluctuate due to:

  • imaging variability
  • segmentation errors
  • confluency oscillations
  • media‑change effects
  • microenvironmental drift

These fluctuations often exceed the amplitude of the biological trend.

1.2 The biologically meaningful drift is subtle

Neurodegenerative phenotypes typically show:

  • slow, cumulative deterioration
  • partial recoveries
  • plateau phases
  • non‑monotonic stress responses

The result: the pathology is present but visually buried.

1.3 Real experiments are irregularly sampled

Longitudinal datasets rarely follow a perfect schedule:

  • weekends
  • instrument availability
  • batch‑specific sampling frequencies
  • missing days

Many classical time‑series tools assume regular spacing and fail under these conditions.

 

2. How cycleTrendR Reconstructs Slow Biological Drift

2.1 A non‑parametric smoother that respects slow dynamics

cycleTrendR does not impose a logistic, exponential, or Gompertz model.
Instead, it uses a flexible smoother that adapts to:

  • gradual decline
  • long plateaus
  • intermittent recoveries
  • cumulative stress accumulation

The long‑term trajectory emerges organically from the data.

 

3. Native Handling of Irregular Sampling

3.1 No artificial interpolation

cycleTrendR works directly with irregular timestamps, avoiding:

  • interpolation artefacts
  • distortions in late sparse phases
  • bias from uneven sampling density

3.2 A coherent reconstruction of the full timeline

Whether early stages are dense and late stages sparse, the trend remains stable and interpretable.

 

4. Separating Progressive Drift from Recurring Cycles

4.1 Biological and experimental cycles are everywhere

Neural cultures often contain periodic components:

  • media‑change rhythms
  • circadian‑like oscillations
  • weekly handling effects
  • segmentation/OCR bias cycles

These cycles can mimic or mask slow deterioration.

4.2 A two‑layer decomposition

cycleTrendR separates the signal into:

  1. Long‑term drift — the disease‑accumulation signature
  2. Recurring cycles — periodic experimental or biological rhythms
  3. Irregular noise — non‑repeating artefacts and spikes

4.3 How the decomposition works

  • Estimate a robust smooth trend
  • Compute residuals: observed – trend
  • Apply cycle‑aware smoothing to detect recurring patterns
  • Recombine components into a clean decomposition

 

5. Robustness to Spikes, Crashes, and Local Outliers

5.1 Why classical methods fail

LOESS, linear regression, and standard splines are highly sensitive to:

  • transient metabolic crashes
  • sudden aggregation events
  • imaging artefacts
  • partial differentiation waves

These events distort the fitted curve, pulling it toward short‑lived anomalies.

 

5.2 How cycleTrendR’s LOESS differs from standard LOESS

Although cycleTrendR uses a LOESS‑inspired smoother, it diverges from classical LOESS in several important ways. These differences are essential for biological time series, which violate many assumptions of standard LOESS.

Figure 2.



Legend of Figure 2. Standard LOESS bends toward spikes and irregular sampling, while cycleTrendR separates long‑term drift from cycles and remains stable under noise.

5.2.1 Stronger robustness to spikes and artefacts

Standard LOESS uses mild robustness iterations.
cycleTrendR instead applies strong M‑estimation‑style weighting, aggressively down‑weighting:

  • imaging artefacts
  • one‑day metabolic crashes
  • segmentation outliers

A single spike cannot bend the curve.

5.2.2 Time‑aware bandwidth for irregular sampling

Standard LOESS uses a fixed proportion of nearest neighbours, which fails when sampling is uneven.
cycleTrendR uses time‑based adaptive bandwidth, ensuring:

  • dense early regions retain detail
  • sparse late regions are smoothed appropriately
  • gaps do not create instability

5.2.3 Curvature‑penalised smoothing for biological plausibility

Standard LOESS can introduce artificial wiggles.
cycleTrendR discourages biologically implausible oscillations through curvature penalisation, while remaining fully non‑parametric.

5.2.4 Two‑stage smoothing: trend first, cycles second

Standard LOESS produces a single curve.
cycleTrendR performs:

  1. Trend smoothing
  2. Residual smoothing to extract cycles

This separation is essential for long‑horizon neurodegenerative drift.

5.2.5 Stability across biological phases

cycleTrendR maintains continuity across:

  • plateau → decline
  • decline → partial recovery
  • recovery → deterioration

Standard LOESS often overfits early dense phases and underfits late sparse ones.

 

6. Clearer Interpretation Across Conditions

6.1 Comparing lines or treatments

cycleTrendR enables:

  • comparison of deterioration rates
  • identification of early vs. late divergence
  • quantification of progressive burden via slope
  • analysis of periodic fragility

6.2 Cleaner phenotyping

Once cycles and noise are removed, the long‑term drift becomes a clean, interpretable phenotype.

 

7. Why This Matters for Neurodegeneration

Neurodegenerative phenotypes:

  • accumulate slowly
  • are easily masked by experimental cycles
  • are distorted by handling artefacts
  • require long‑horizon interpretation

cycleTrendR isolates the slow, cumulative component — the part that actually reflects pathology.

 

8. Conclusion: Making the Invisible Visible

By combining:

  • non‑parametric smoothing
  • irregular‑sampling support
  • cycle decomposition
  • robust outlier handling
  • a LOESS variant tailored to biological data

cycleTrendR extracts long‑term biological structure from extremely noisy data.

It directly addresses the scenario raised in the above-mentioned comment:
long timelines, high noise, irregular sampling, and hidden slow dynamics.

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