cs.LGOct 25, 2026

Exponential Wake-Up Overshoot in Explicit Gaussian Point-Process Filters

Authors: Hanwu Qin, Dianhai Wang

Organizations: Institute of Intelligent Transportation Systems, Zhejiang University, Hangzhou 310058, China

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Abstract

When events resume after a long silence, explicit Gaussian point-process filters amplify the first update exponentially in the silence duration. We give the exact distribution of this update and a computable index that predicts when it overshoots. For the explicit (single-scoring-step) filter and its implicit posterior-mode counterpart, we derive second-order silence expansions of the state variance with opposite-sign logarithmic corrections. The crossing index determines when the post-wake prediction exceeds the resumed rate, separating the bin width at which the amplification exponent reaches one from the larger width at which the true rate is overshot. At fixed bin width, the implicit filter's expected loss grows logarithmically with silence duration, whereas a continuous-time counterexample rules out a bound uniform in bin width and silence duration. Experimental evaluation confirms these predictions in all 73 crossing cells and on exchange trade data, while variance caps reduce explicit overshoot but mostly hurt the implicit filter.

Figures & tables

Fig. 1: Silence. (A) Vn−qn/3V_{n}-qn/3 for the exact recursions (dots) and the second-order law of Proposition 1 (lines), with bjb_{j} taken from independent trajectories of length 10610^{6} ; qc=0.2q_{c}=0.2 /s, Δ=0.05\Delta=0.05 s, initial rate 2020 /s, V0=0.05V_{0}=0.05 . (B) Exact CT Bayes predictive mean against 1/(2t)1/(2t) and the Gaussian median readout 6/(qct2)6/(q_{c}t^{2}) , for the same qcq_{c} and initial state.
Fig. 2: Binning, from initial rate 20/s and V0=0.05V_{0}=0.05 with qc=0.2q_{c}=0.2 /s and L=60L=60 s. (A) Post-wake rate under a mean-count input for Λ=1000\Lambda=1000 /s: the unit-exponent width ( 0.250.25 ms) and crossing width ( ≈2.88\approx 2.88 ms) differ by an order of magnitude. (B) For 50 fixed event times in the 50 ms after the silence, the discrete explicit filter converges to the event-aligned CT filter as Δ↓0\Delta\downarrow 0 (CT peak 40494049 /s).
 HypothesisConfirmedEvidence
H1I << E, crossing cells73/73p≤2.0×10−8p\leq 2.0\times 10^{-8}
H2both caps help ∗2/24I cap fails
H3CTRL, non-inferior † (2%)10/10p≤9.2×10−12p\leq 9.2\times 10^{-12}
R1real, 1 s: I >> Eyes18:2, p=6.0×10−4p=6.0\times 10^{-4}
R2real, 250 ms: I >> Eyes18:2, p=6.0×10−4p=6.0\times 10^{-4}
R3real: I cap non-inferior (0.5%)nop=0.91p=0.91
TABLE I: Confirmatory Results (Holm-Adjusted)
Fig. 3: Median excess loss per cell (64 streams) against the crossing index κ\kappa frozen before the data; the loss axis is doubly logarithmic. The shaded band is the transition region. The cluster of explicit losses near 102710^{27} at κ<0\kappa<0 comes from L=1200L=1200 s, qc=0.2q_{c}=0.2 /s, Δ≤50\Delta\leq 50 ms, where m≪1m\ll 1 and the mean-count index under-predicts the crossing (Corollary 1 ).
Appendix figures & tables12 assets

Supplementary material from the paper’s appendix.

Appendix

tt (s)exact mean1/(2t)1/(2t)1/(2t)+BX/t21/(2t)+B_{X}/t^{2}Gaussian median 6/(qct2)6/(q_{c}t^{2})
200.0803060.0250.0855410.075
600.0148580.0083330.0150600.008333
1000.00737750.0050.00742160.003
5000.00109650.0010.00109690.00012
10005.2417×10−45.2417\times 10^{-4}5×10−45\times 10^{-4}5.2422×10−45.2422\times 10^{-4}3×10−53\times 10^{-5}
100005.0242×10−55.0242\times 10^{-5}5×10−55\times 10^{-5}5.0242×10−55.0242\times 10^{-5}3×10−73\times 10^{-7}
Appendix
TABLE II: Exact CT Bayes mean, λ0=20\lambda_{0}=20 /s, V0=0.05V_{0}=0.05 , qc=0.2q_{c}=0.2 /s ( BX=24.216B_{X}=24.216 s).
LL (s)post-event rate (/s)time to λc\lambda_{c} (s)divergence integral (nats)
2404.9414×1034.9414\times 10^{3}0.2440552.65
3606.5671×1066.5671\times 10^{6}0.02539256.78
6002.1060×10132.1060\times 10^{13}2.0407×10−42.0407\times 10^{-4}1.4366×1041.4366\times 10^{4}
10002.8947×10242.8947\times 10^{24}4.6871×10−84.6871\times 10^{-8}6.2557×1076.2557\times 10^{7}
Appendix
TABLE III: CT first event immediately after a silence of length LL ( qc=0.2q_{c}=0.2 /s, initial rate 20/s, Wi=0.05W_{i}=0.05 ). The divergence integral is exact for the descent to λc\lambda_{c} at Λ=1000\Lambda=1000 /s. The first two rows do not satisfy λc≥6Λ\lambda_{c}\geq 6\Lambda , so ( 33 ) is not used there.
Δ\Delta (ms)qcq_{c}d1d_{1}d2d_{2}pHolmp_{\rm Holm}
10.02−23.59-23.59 (0.29)−23.59-23.59 (0.29)4.0×10−374.0\times 10^{-37}
10.2−40.18-40.18 (0.21)−39.98-39.98 (0.21)2.1×10−482.1\times 10^{-48}
100.02−23.75-23.75 (0.30)−23.77-23.77 (0.30)5.4×10−375.4\times 10^{-37}
100.2−42.01-42.01 (0.23)−40.05-40.05 (0.22)4.7×10−484.7\times 10^{-48}
500.02−24.66-24.66 (0.35)−24.78-24.78 (0.34)3.4×10−353.4\times 10^{-35}
500.2−48.67-48.67 (0.54)−40.05-40.05 (0.26)1.4×10−381.4\times 10^{-38}
Appendix
TABLE IV: H3 per CTRL cell (32 streams): mean (SE) of d1=loss(I)−1.02 loss(E)d_{1}=\mathrm{loss(I)}-1.02\,\mathrm{loss(E)} and d2=loss(Icap)−1.02 loss(I)d_{2}=\mathrm{loss(I_{cap})}-1.02\,\mathrm{loss(I)} , in nats, and the Holm-adjusted intersection–union pp .
RegionMethodQ1MedianQ3Mean>1000>1000 nats
CE1.3×10111.3\times 10^{11}5.9×10405.9\times 10^{40}3.1×102373.1\times 10^{237}103701810^{37018}97.2%
CI48813562467266256.4%
CE cap7614.5×1074.5\times 10^{7}5.2×10395.2\times 10^{39}1045910^{459}74.4%
CI cap51412442464249956.2%
NE9830710896.1×10606.1\times 10^{60}28.1%
NI5714536932012.2%
Appendix
TABLE V: Excess loss by theory region, pooled over cells and streams (C: 4672, N: 5888, T: 2240 records per method).
RegionMethodfirst >Λ>\Lambdafirst >10Λ>10\Lambdafirst updatewindow maxrecovery
CE92.96%91.67%8.9×10398.9\times 10^{39}2.2×10412.2\times 10^{41}3282/4672, 9 s
CI  1232024672/4672, 0.25 s
CE cap  1.3×1081.3\times 10^{8}1.5×1081.5\times 10^{8}4224/4672, 3.5 s
CI cap  861864672/4672, 0.5 s
NE1.89%1.22%0.0531345888/5888, 1.75 s
NI  0.053365888/5888, 1.21 s
Appendix
TABLE VI: Wake-up behavior by region: share of streams whose first post-wake prediction of E exceeds Λ\Lambda and 10Λ10\Lambda ; median first update and median window maximum of the predicted rate (/s); recovery (recovered/total and median delay among recovered streams).
L1→L2L_{1}\to L_{2} (s)qc=0.02q_{c}=0.02qc=0.2q_{c}=0.22mlog⁡(L2/L1)2m\log(L_{2}/L_{1})
5→205\to 20261727192773
20→6020\to 60211721292197
60→30060\to 300314231423219
300→1200300\to 1200272827222773
60→120060\to 1200587058645991
Appendix
TABLE VII: Increase of the median implicit loss (nats) between successive silence lengths, Λ=1000\Lambda=1000 /s, Δ=1\Delta=1 s.
qcq_{c} (1/s)Λ\Lambda (1/s)LL (s)κ\kappa at Δ\Delta = 1, 10, 50, 250, 1000 ms
0.0215−- 0.555 T−- 0.554 T−- 0.551 T−- 0.537 T−- 0.484 T
0.02120−- 1.57 T−- 1.57 T−- 1.56 T−- 1.52 T−- 1.36 T
0.02160−- 3.01 N−- 3.01 N−- 2.99 N−- 2.88 N−- 2.51 N
0.021300−- 5.82 N−- 5.8 N−- 5.71 N−- 5.29 N−- 3.71 N
0.0211200−- 8.5 N−- 8.43 N−- 8.1 N−- 6.48 N−- 0.402 T
0.02105−- 1.34 T−- 1.33 T−- 1.31 T−- 1.2 T−- 0.824 T
Appendix
TABLE VIII: Frozen crossing index κ\kappa of all 200 SW cells (C: crossing, κ≥log⁡10\kappa\geq\log 10 ; N: no crossing, κ≤−log⁡10\kappa\leq-\log 10 ; T: transition).
CellLLΛ\Lambdaqcq_{c}Δ\Delta (ms)W/L/TpHolmp_{\rm Holm}median E / I
SW251000.225064/0/04.0×10−184.0{\times}10^{-18}1896 / 508
SW251000.2100064/0/04.0×10−184.0{\times}10^{-18}2.4×1082.4{\times}10^{8} / 652
SW3510000.0225064/0/04.0×10−184.0{\times}10^{-18}1.5×1041.5{\times}10^{4} / 1913
SW3510000.02100064/0/04.0×10−184.0{\times}10^{-18}4.7×1094.7{\times}10^{9} / 5133
SW3510000.25064/0/04.0×10−184.0{\times}10^{-18}1.3×1061.3{\times}10^{6} / 1678
SW3510000.225064/0/04.0×10−184.0{\times}10^{-18}1.1×10311.1{\times}10^{31} / 2441
Appendix
TABLE IX: H1 per crossing cell: wins/losses/ties of I against E over 64 paired streams, Holm-adjusted pp , and cell median excess loss (nats).
CellLLΛ\Lambdaqcq_{c}Δ\Delta (ms)I cap<{}_{\rm cap}< IE cap<{}_{\rm cap}< EpHolmp_{\rm Holm}
SW1230010.2155/9/064/0/04.1×10−84.1{\times}10^{-8}
SW1230010.2102/62/064/0/01
SW13300100.210/64/064/0/01
SW13300100.2100/64/064/0/01
SW143001000.210/64/064/0/01
SW143001000.2100/64/064/0/01
Appendix
TABLE X: H2 per cell: wins/losses/ties of each component over 64 paired streams; IUT pp is the larger component pp , Holm-adjusted across the 24 cells.
qcq_{c}Λ\LambdaL=5L=520603001200
0.021−- 0.4978−- 1.239−- 1.1031.5774.489
0.0210−- 7.24−- 14.07−- 16.65−- 25.68−- 52.67
0.02100−- 165.3−- 338.6−- 536.1−- 832.3−- 1089
0.021000−- 4142−- 6661−- 8733−- 1.186e+04−- 1.459e+04
0.21−- 0.7225−- 1.971−- 2.706−- 4.369−- 4.901
0.2107.106−- 11.83−- 24.39−- 45.71−- 83.69
Appendix
TABLE XI: Median paired difference loss(X) −- loss(I) per cell at Δ=1\Delta=1 s (nats; negative favours exact Bayes). CTRL: -2044 ( qc=0.02q_{c}=0.02 ) and -1450 ( qc=0.2q_{c}=0.2 ).
Δ\Delta (ms)EIE capI capOU I
1010−3.010^{-3.0}10−2.510^{-2.5}10−3.010^{-3.0} , 0.0625 ∗102.010^{2.0} , 0.5 ∗10−1.010^{-1.0} , 10−110^{-1}
5010−3.510^{-3.5}10−2.510^{-2.5}10−3.510^{-3.5} , 0.0625 ∗101.010^{1.0} , 0.2510−2.010^{-2.0} , 10−210^{-2}
25010−4.010^{-4.0}10−3.010^{-3.0}10−3.510^{-3.5} , 0.0625 ∗100.010^{0.0} , 0.12510−2.510^{-2.5} , 10−210^{-2}
100010−4.010^{-4.0}10−3.010^{-3.0}10−4.010^{-4.0} , 0.12510−1.510^{-1.5} , 0.0625 ∗10−3.010^{-3.0} , 10−310^{-3}
Appendix
TABLE XII: Parameters frozen on the training days: qcq_{c} (1/s); V max for capped filters; κOU\kappa_{\rm OU} (1/s) for OU I . ∗ Selection on the edge of the revised grid. All filters start at θ0=μ=−1.6726\theta_{0}=\mu=-1.6726 , V0=1V_{0}=1 .
Δ\Delta (ms)MethodTotal log scorePost-gap 5 sLarge-rate bins
10E−- 89517782642611
10I−- 89209222675490
10E cap−- 89483922642970
10I cap−- 88769172780770
10OU I−- 89792232605190
50E−- 660246420572012
Appendix
TABLE XIII: Real data, 20 test contract-days: total log score (higher is better), negative log score in the 5 s after gaps of at least 20 s, and bins whose predicted rate exceeds 100 times the day maximum of the fixed-second count.
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