36e8ceaInteractive explorer for compressibility of quantized filtered Gaussian time seriesJeremy Magland 1# timeseries-compressibility
3Interactive exploration of how compressible quantized time series are.
5The generating model is: i.i.d. Gaussian noise (std σ, measured in quantization
6steps) → FIR filter → optional additive uniform dither on [-½, ½) → round to
7integers. The app shows the filter (convolution kernel and frequency response,
eb36f3fMake the signal view stationary by default with a play toggleJeremy Magland 8with cutoffs in Hz against a chosen sample rate), a window of the generated
9integer signal (stationary by default, with a play toggle to let it stream
e411dffMake LPC order and compression block size controlsJeremy Magland 10endlessly), and the measured compression of a block of the generated integers
11under nine methods — zlib, zstd, and an rANS entropy coder, each raw,
36e8ceaInteractive explorer for compressibility of quantized filtered Gaussian time seriesJeremy Magland 12delta-coded, and LPC-residual-coded — as bits per sample and as ratio against
e411dffMake LPC order and compression block size controlsJeremy Magland 13raw int16 storage. The predictor order and the block size are controls, so the
14measurement can be pushed from 10k to a million samples and LPC from order 1
15to 128. Each prefilter group also carries a hollow bar: the order-0 entropy of
16the stream being coded, the limit a per-sample entropy coder cannot beat, which
17ANS misses by 1–2% (its symbol table plus its own arithmetic loss).
36e8ceaInteractive explorer for compressibility of quantized filtered Gaussian time seriesJeremy Magland 18
ef6c138Charge rounding at full variance: R = min(spectral rate, one-sample ceiling)Jeremy Magland 19Alongside the measurements it plots a theoretical bits/sample R — the smaller
20of a spectral estimate and a rigorous one-sample ceiling:
36e8ceaInteractive explorer for compressibility of quantized filtered Gaussian time seriesJeremy Magland 21
22```
ef6c138Charge rounding at full variance: R = min(spectral rate, one-sample ceiling)Jeremy Magland 23R = min(Rspec, Rsamp)
24Rspec = ∫₀¹ ½ log₂( 2πe (S(f) + ν) ) df S(f) = σ²|H(f)|², ν = 1/12 (1/6 dithered)
25Rsamp = H( round(N(0, v) [+ U(-½,½) with dither]) ) v = σ² Σ h²
36e8ceaInteractive explorer for compressibility of quantized filtered Gaussian time seriesJeremy Magland 26```
ef6c138Charge rounding at full variance: R = min(spectral rate, one-sample ceiling)Jeremy Magland 28Rspec is the Zamir–Feder rate of the dithered quantizer counted per Fourier
29mode: the signal modes are independent Gaussians of variance S(f), and the
30i.i.d. roundoff(+dither) noise is Gaussianized per mode by the CLT, so it
31enters at its full variance ν — not at the entropy power 1/(2πe) an aligned
32scalar quantizer would charge (the lattice lives in the sample basis; a dead
33band inside a live process costs ≈0.25 bits/mode, not zero). At high SNR it
34reduces to the Kolmogorov rate ½log₂(2πe σ²) + ∫log₂|H| df. Where the whole
35process sits below the quantization step, Rspec bottoms out while the true
36rate collapses; subadditivity H(z) ≤ Σ H(zₙ) makes Rsamp — the exact marginal
37entropy of one stored sample — a true upper bound with the right collapse,
38and the min selects it exactly there. Monte-Carlo puts R within ~0.01–0.02
39bits/sample for v ≳ 0.25 (worst ~+0.03 at the branch crossover). LPC + ANS
40should approach R; probing where the approximation holds is the point.
36e8ceaInteractive explorer for compressibility of quantized filtered Gaussian time seriesJeremy Magland 41
42## Run it
44```sh
45npm install
46npm run dev
47```
49## Layout
51```
5bab85aRatio-first chart, quantization-floor theory formula, line-segment view, fixed latent dataJeremy Magland 52src/model/ the latent source (fixed seeded randomness indexed by sample
53 position, convolved zero-phase with the kernel on demand),
54 FIR presets, and the theoretical-rate formula
36e8ceaInteractive explorer for compressibility of quantized filtered Gaussian time seriesJeremy Magland 55src/compress/ lossless codecs run in the browser: zlib (fflate), zstd (wasm),
56 ans.ts (a bit-identical port of simple_ans), and FLAC-style
57 integer LPC; borrowed from entropy-quantized-linear-transform
58src/worker/ the codecs run off the main thread on a debounced parameter set
5bab85aRatio-first chart, quantization-floor theory formula, line-segment view, fixed latent dataJeremy Magland 59src/components/ controls, filter plots, signal canvas, compression chart
95929e6Monte-Carlo ground-truth script for R, with the command printed in the UIJeremy Magland 60scripts/ true_rate.py — Monte-Carlo ground truth for R (Gibbs over the
61 latent Gaussian given the observed integers, Rao-Blackwellized
62 next-sample pmf); the app prints the exact command to run.
63 Requires numpy only.
36e8ceaInteractive explorer for compressibility of quantized filtered Gaussian time seriesJeremy Magland 64```
66Every reported size round-trips through the decoder and includes whatever the
5bab85aRatio-first chart, quantization-floor theory formula, line-segment view, fixed latent dataJeremy Magland 67decoder needs (ANS symbol table, LPC coefficients). The signal view and the
68compression block read the same fixed latent noise sequence — parameter changes
69transform the same underlying data rather than resampling it, and the first
70window shown is the start of the block that gets compressed.