Thinking in Waves Episode 60 / Part XIX — Back to Physics, with the Tools
Where did the subtracted part go?
Where a trend ends and a signal begins ──
the boundary is set by the record, not the system
★ Removing a degree-\(q\) trend is a high-pass filter of order \(\alpha=q+1\) ── \(\text{dB/oct}=6\alpha\) again. Episode 1’s yardstick turns up here too.
★★ From it follows the condition for finite leakage, \(\beta\le2q+3\), which explains all five rows of Episode 59 §03.
★★★ And the “variance” of a \(1/f^\beta\) process is not a number ── it runs as \(N^{\beta-1}\) for \(\beta>1\), and as \(\log N\) at exactly \(\beta=1\). The same structure as Episode 29’s renormalisation.
Episode 59 concluded: subtract a degree 1–2 trend and the leakage goes away. ★ So where did the subtracted part go ── and where exactly is the boundary between trend and signal?
01★★★ Detrending is a high-pass filter
Feed a sinusoid at frequency \(f\) [bins] into a record of length 1024 and measure the residual energy.
| \(f\) [bins] | \(q{=}0\) (mean) | \(q{=}1\) (line) | \(q{=}2\) | \(q{=}3\) |
|---|---|---|---|---|
| 0.01 | −34.83 | −76.65 | −122.14 | −170.19 |
| 0.1 | −14.89 | −36.70 | −62.18 | −90.23 |
| 1 | 0.00 | −1.57 | −6.31 | −13.72 |
| 3 | 0.00 | −0.15 | −0.17 | −0.43 |
(The table is gain² in dB; 0 dB means it passes unchanged.)
| \(q\) | low-frequency slope dB/oct | predicted \(6(q{+}1)\) | \(\alpha\) | \(-3\)dB point [bins] |
|---|---|---|---|---|
| 0 | 6.014 | 6.021 | 1 | 0.4436 |
| 1 | 12.036 | 12.041 | 2 | 0.8527 |
| 2 | 18.057 | 18.062 | 3 | 1.2391 |
| 3 | 24.078 | 24.082 | 4 | 1.6152 |
The core of §01
degree-\(q\) detrending \(=\) a high-pass filter of order \(\alpha=q+1\)
★★★ \(\text{dB/oct}=6(q+1)\) ── Episode 1’s yardstick appears here too: the third time, after Episode 57’s edges and Episode 58’s windows.
★★ The \(-3\)dB points are 0.44 / 0.85 / 1.24 / 1.62 bins ── about \(0.4(q+1)\). All of them sit at “roughly one cycle per record.”
★★★ So the boundary between trend and signal is set by the record’s length, not by the system. Make the record ten times longer and what was a trend becomes a signal ── exactly the shape of Episode 5’s “a period longer than the age of the universe is indistinguishable from a constant.”
02★★★ The condition for finite leakage ── \(\beta\le2q+3\)
Hann window (\(\alpha=3\)), leakage into bin \(k=64\). By what factor does it grow when the lower limit drops from 1/16 to 1/64? (Near 1 means converged.)
| \(\beta\) | \(q{=}0\) | \(q{=}1\) | \(q{=}2\) |
|---|---|---|---|
| 3 | 1.018 | 1.000 | 1.000 |
| 4 | 1.136 | 1.002 | 1.000 |
| 5 | 1.797 | 1.018 | 1.000 |
| 6 | 4.805 | 1.134 | 1.002 |
| 7 | 17.145 | 1.790 | 1.017 |
★ Exactly at \(\beta=2q+3\) (\(\beta{=}5,q{=}1\) and \(\beta{=}7,q{=}2\)) the divergence is logarithmic: dropping the lower limit by 4× raises the leakage only 1.02× ── in practice the same as converged.
★★ This explains Episode 59 §03.
| \(\beta\) | \(q\) | \(2q{+}3\) | leakage ratio | Episode 59 measured | ceiling \(2\alpha{=}6\) |
|---|---|---|---|---|---|
| 5 | 0 | 3 | 1.80 diverges | 5.512 (overshoot) | — |
| 5 | 1 | 5 | 1.02 converges | 4.992 (fixed) | — |
| 6 | 1 | 5 | 1.13 diverges | 6.035 (not yet) | — |
| 6 | 2 | 7 | 1.00 converges | 5.994 (fixed) | — |
| 7 | 2 | 7 | 1.02 converges | 6.039 (ceiling) | clamped |
The core of §02 ── two separate conditions
★★★ Only the converging rows are fixed. \(\beta=6,q=1\) has a ratio of 1.13 and the measurement stayed at 6.035.
★★ \(\beta=7,q=2\) has its leakage under control and is still wrong ── because it is hitting the window ceiling \(2\alpha=6\). That is how you tell two causes apart.
★★★ There are two conditions, and they are separate:
① leakage from waves longer than the record → \(\beta\le2q+3\) (bought with detrending)
② the window’s sidelobe ceiling → \(\beta<2\alpha\) (bought with the window)
★ Episode 58’s “raise the window” and Episode 59’s “subtract a trend” were different conditions ── neither one alone is enough.
03What is the subtracted trend a measurement of?
Segment length 1024. Compare the fitted degree-1 polynomial against the true low-frequency part (the series with only components below the record’s 1-bin resolution).
| \(\beta\) | correlation | trend variance / low-part variance | low part’s share of total variance |
|---|---|---|---|
| 1 | 0.6175 | 0.7906 | 0.1121 |
| 2 | 0.7078 | 0.7744 | 0.5700 |
| 3 | 0.8801 | 0.8194 | 0.9067 |
| 4 | 0.9564 | 0.9350 | 0.9684 |
| 5 | 0.9987 | 0.9968 | 1.0007 |
The core of §03 ── what you subtract is not gone
★ The larger \(\beta\), the higher the correlation ── because the low end dominates (0.999 at \(\beta=5\)).
★★ The subtracted trend is not waste; it is a measurement of “waves longer than the record.” At \(\beta=5\) it reproduces the true low-frequency part almost exactly.
★★★ But a very coarse measurement, with two degrees of freedom ── one straight line’s worth of information. Episode 51’s “a periodogram point has two degrees of freedom” shows up here too.
★ What Episode 59 thought it was throwing away was still in hand ── whether to use it is another matter.
04★★ The “variance” of \(1/f^\beta\) is not a number
| \(\beta\) | \(N{=}256\) | \(N{=}1024\) | \(N{=}4096\) | slope (dec/dec) | predicted |
|---|---|---|---|---|---|
| 0.5 | 3.25e−7 | 3.40e−7 | 3.40e−7 | 0.016 | 0 (converges) |
| 1 | 5.10e−9 | 6.33e−9 | 7.97e−9 | 0.161 | 0.146 (\(\log N\)) |
| 2 | 8.99e−12 | 3.32e−11 | 2.16e−10 | 1.147 | 1.0 |
| 3 | 2.24e−13 | 3.01e−12 | 5.43e−11 | 1.980 | 2.0 |
The core of §04 ── the structure of Episode 29’s renormalisation
★★★ For \(\beta>1\) the variance keeps growing as the record lengthens (\(N^{\beta-1}\)). “What is this signal’s variance?” has no answer ── not until you state the record length.
★ For \(\beta<1\) the high end dominates and it converges (measured 0.016).
★★★ And at exactly \(\beta=1\) it grows as \(\log N\) ── neither a power nor a constant. Over \(N{=}256\to4096\) the prediction \(\ln(\ln4096/\ln256)/\ln16=0.146\) against a measured 0.161. \(1/f\) sits exactly on the boundary here too.
★★ This is the structure of Episode 29’s renormalisation: a physical quantity depends on where you cut, and it runs as you move the cut ── with the logarithmically running point playing the role of the fixed point.
05How much variance does detrending remove?
\(N=1024\). Residual variance, with the original variance set to 1.
| \(\beta\) | \(q{=}0\) | \(q{=}1\) | \(q{=}2\) | \(q{=}3\) | \(q{=}5\) |
|---|---|---|---|---|---|
| 0.5 | 1.0000 | 0.9861 | 0.9768 | 0.9674 | 0.9532 |
| 1 | 1.0000 | 0.8999 | 0.8495 | 0.8064 | 0.7502 |
| 2 | 1.0000 | 0.5517 | 0.3958 | 0.3059 | 0.2130 |
| 3 | 1.0000 | 0.2145 | 0.1032 | 0.0545 | 0.0255 |
| 4 | 1.0000 | 0.0920 | 0.0184 | 0.0077 | 0.0029 |
★★★ At \(\beta=4\), one straight line takes the variance below a tenth. That shows how thoroughly “the variance” was dominated by the low end.
★★ At \(\beta=0.5\) almost nothing changes (0.986) ── subtracting does little harm. “When in doubt, detrend” is sound advice in exactly this sense.
Exercises
- §01 says the record sets the boundary. Does the system itself distinguish trend from signal?
Show answer
A pure \(1/f^\beta\) process does not. It is scale invariant ── if every frequency looks the same, no line can be drawn from the system’s side saying “below here is trend.”
★ A distinction exists only when the system has a characteristic time: Episode 56’s \(\tau_{\max}\), or Episode 44’s band edge.
★★ So “subtract the trend” is not a claim about the system but a declaration about the measurement ── “I have decided not to measure waves longer than this record.”
★★★ In Episode 55’s classification, detrending is not an assumption but a statement of range. What must be written honestly is therefore not a count of assumptions but \(q\) and the record length. - §04 says the variance “runs.” Is it really the same as Episode 29’s renormalisation?
Show answer
The shape is the same; the substance is not.
★ The same: in both, an observable depends on the cutoff and runs as you move it, and the point that runs logarithmically is special (Episode 29’s fixed point, this episode’s \(\beta=1\)).
★★ Different: in field theory you can fix the running quantity (a coupling) to an experimental value and thereby remove the cutoff dependence from observables. There is no such step here ── the variance simply keeps running.
★★★ Honestly stated: this is “the same structure as renormalisation,” not “renormalisation is available.” This series enjoys finding analogies, but as Episode 55 insisted, an analogy is not a claim until you say how far it goes. - §03 calls the trend a two-degree-of-freedom measurement. Can it be used?
Show answer
Barely, from one record ── but usefully from many.
★ One record gives two degrees of freedom ── Episode 51’s formula gives \(\mathrm{sd}=1.2825/\sqrt2\approx0.91\) nats, just under 4 dB of scatter. Poor for a single point.
★★ But \(M\) records improve it as \(\sqrt M\): pooling the trends of 100 independent records would measure the power one decade below the record to about 0.4 dB.
★★★ That is a fourth way to buy range, alongside Episode 54’s three ── ① measure longer ② take more samples ③ change the system, plus ④ collect many short records and turn their “trends” into a statistic. ★ It requires the records to be independent, which in \(1/f\) is itself doubtful. Not verified here.
Summary — the boundary belongs to the record, not the system
Episode 59 said “subtract the trend.” This time we looked at it from the side of what was subtracted.
★★★ degree-\(q\) detrending \(=\) a high-pass filter of order \(\alpha=q+1\).
\(\text{dB/oct}=6(q+1)\) (measured 6.014 / 12.036 / 18.057 / 24.078).
★ Episode 1’s yardstick turns up for the third time ── Episode 57’s edges, Episode 58’s windows, and now detrending. ★★ The \(-3\)dB point is \(0.4(q+1)\) bins, i.e. about one cycle per record ── the boundary is set by the record, and lengthening it tenfold turns a trend into a signal. Exactly Episode 5’s shape.
★★ From this follows the leakage condition \(\beta\le2q+3\) ── which explains all five rows of Episode 59 §03. And \(\beta=7,q=2\) has its leakage under control yet is still wrong, because it hits the window ceiling \(2\alpha=6\): the two conditions are separate. ① leakage: \(\beta\le2q+3\), bought with detrending; ② the ceiling: \(\beta<2\alpha\), bought with the window. Neither alone suffices.
★★★ The subtracted trend had not vanished ── at \(\beta=5\) it correlates 0.999 with the true low-frequency part. It is a measurement of “waves longer than the record.” A very coarse one, with two degrees of freedom (Episode 51).
★★★ And the “variance” of \(1/f^\beta\) is not a number. It runs as \(N^{\beta-1}\) for \(\beta>1\) (measured 1.147 / 1.980 against 1 / 2), converges for \(\beta<1\), and grows as \(\log N\) at exactly \(\beta=1\) (0.146 predicted, 0.161 measured). ★ “What is this signal’s variance?” has no answer without the record length ── the structure of Episode 29’s renormalisation, with the logarithmic point as the fixed point. ★★ But “the same structure” is not “renormalisation is available” (Exercise 2).
★ Next episode closes it ── the five results of Episodes 56–60, folded back into the handbooks of Episodes 49 and 55.
★★ And we count how many times this series’ \(\alpha\) appeared across those five episodes ── how far one line from Episode 1, “differentiate a sound and it goes high,” has travelled.
This document is Episode 60 of the “Thinking in Waves” series. The computations are in kenshou/torendo.js (own FFT, zero dependencies). On the numbers in this episode: the detrending gain, its low-frequency slope and \(-3\)dB points, the leakage-integral convergence ratios, the correlation between trend and low-frequency part, the \(N\) dependence of the variance, and the variance removed by detrending are all computed here. That polynomial detrending acts as a high-pass, and that a \(1/f\) process’s variance depends on record length, are well-known facts and are not discoveries of this episode ── what this episode did was re-measure them on Episode 1’s \(\text{dB/oct}=6\alpha\) yardstick, derive the leakage condition \(\beta\le2q+3\) and check it against Episode 59’s table, and compare the \(\log N\) growth at \(\beta=1\) against the predicted 0.146. §01 uses a record of 1024; §02 uses a Hann window at \(k=64\) comparing lower limits 1/16 and 1/64; §§03–05 cut segments from a synthesis of length 65536 ── all are condition-dependent. The “converges/diverges” verdict in §02 uses a convenience threshold of 1.1, and exactly \(\beta=2q+3\) diverges logarithmically (ratio 1.02). The §04 slopes are fitted from only three points (\(N=256,1024,4096\)). This episode also uses synthetic data only, and has not been applied to real measurements. ── To print, use your browser’s Print → Save as PDF.