==================================================================================================== CAN YOU TELL A SYSTEM IS ABOUT TO TIP BEFORE IT DOES Science Journaling Club, Volume 1, Issue 2, Winter 2025 Critical slowing down as an early warning signal: does it exist, and does it help? ==================================================================================================== python 3.12.3 numpy 2.4.2 master seed 20251217 THE COMPUTATION IS THE EXPERIMENT. No physical system was observed. Every number below is output of this file, which simulates a mathematical model of a fold bifurcation under noise. MODEL dx = (-x^3 + x + mu(t)) dt + sigma dW (cusp normal form) OBSERVE z_k = x(k*DT_SAMP) + eta_k, eta_k ~ N(0, sigma_obs^2) i.i.d. DT (integrator) 0.01 DT_SAMP (observation) 0.25 (25 integrator steps per sample) mu_c (fold) 0.3849001795 = 2/(3*sqrt(3)) x_fold -0.5773502692 = -1/sqrt(3) x_upper after the jump 1.1547005384 = 2/sqrt(3) tipping threshold x > 0.5 BASELINE CONFIGURATION sigma (process noise) 0.05 sigma_obs (obs noise) 0.02 T_ramp (mu=0 to mu=mu_c) 250.0 time units = 1000 samples ramp rate 0.00153960 per time unit window length 100 samples = 25.0 time units window stride 10 samples (overlapping) replicates, baseline 3000 replicates, other cells 2000 ==================================================================================================== SECTION A. THE ANALYTIC SKELETON ==================================================================================================== The lower-branch equilibrium and its recovery rate, from the closed-form cubic solution. kappa = -f'(x*) = 3 x*^2 - 1 is the rate at which a small displacement decays. Two exact limits are printed beside the measurement: kappa -> 2*sqrt(3) * |x* - x_fold| LINEAR in state-space distance kappa -> 2*sqrt(3) * sqrt(mu_c - mu) SQUARE ROOT in parameter distance 2*sqrt(3) = 3.4641016151 mu/mu_c mu x* d=x*-x_fold kappa 2sqrt3*|d| diff % 2sqrt3*sqrt(mu_c-mu) diff % ---------------------------------------------------------------------------------------------------- 0.0000 0.000000 -1.000000 -0.422650 2.000000 1.464102 -26.79 2.149140 7.46 0.2500 0.096225 -0.947884 -0.370534 1.695453 1.283567 -24.29 1.861210 9.78 0.5000 0.192450 -0.884552 -0.307202 1.347296 1.064178 -21.01 1.519671 12.79 0.7500 0.288675 -0.799265 -0.221914 0.916472 0.768734 -16.12 1.074570 17.25 0.9000 0.346410 -0.720614 -0.143264 0.557855 0.496281 -11.04 0.679618 21.83 0.9500 0.365655 -0.679775 -0.102424 0.386281 0.354808 -8.15 0.480562 24.41 0.9900 0.381051 -0.623870 -0.046520 0.167642 0.161150 -3.87 0.214914 28.20 0.9990 0.384515 -0.592194 -0.014844 0.052081 0.051420 -1.27 0.067962 30.49 0.9999 0.384862 -0.582058 -0.004708 0.016374 0.016308 -0.41 0.021491 31.25 ---------------------------------------------------------------------------------------------------- The linear-in-state law converges to the truth; the square-root-in-parameter law is the same statement re-expressed and converges at the same rate. Both are exact in the limit and both are printed so the reader can see which approximation is being leaned on. ==================================================================================================== SECTION B. VALIDATION 1 -- RECOVERY RATE FROM DIRECT PERTURBATION ==================================================================================================== We hold mu fixed, switch the noise off, displace the system by delta = 0.01 from its equilibrium, integrate, and fit the exponential decay. This is the thing early warning indicators are trying to estimate indirectly. Here we can measure it directly, because we own the system. No field study can do this. CLUB VALUE is the fitted decay rate. ACCEPTED VALUE is kappa = 3 x*^2 - 1. mu/mu_c x* club kappa analytic diff diff % ---------------------------------------------------------------------------------------------------- 0.0000 -1.000000 2.004874 2.000000 4.87e-03 0.244 0.2500 -0.947884 1.695492 1.695453 3.84e-05 0.002 0.5000 -0.884552 1.342961 1.347296 -4.33e-03 -0.322 0.7500 -0.799265 0.908501 0.916472 -7.97e-03 -0.870 0.9000 -0.720614 0.548393 0.557855 -9.46e-03 -1.696 0.9500 -0.679775 0.376593 0.386281 -9.69e-03 -2.508 0.9900 -0.623870 0.157958 0.167642 -9.68e-03 -5.777 ---------------------------------------------------------------------------------------------------- Worst absolute disagreement across the seven values: 5.777%. The residual is the Euler discretisation, not a modelling disagreement: the scheme decays a displacement by (1 - kappa*DT) per step rather than exp(-kappa*DT), so the fitted rate is -ln(1-kappa*DT)/DT. mu/mu_c club kappa Euler-exact diff % ---------------------------------------------------------------------------------------------------- 0.0000 2.004874 2.020271 -0.7621 0.2500 1.695492 1.709991 -0.8479 0.5000 1.342961 1.356455 -0.9947 0.7500 0.908501 0.920698 -1.3248 0.9000 0.548393 0.559416 -1.9706 0.9500 0.376593 0.387029 -2.6962 0.9900 0.157958 0.167783 -5.8556 ---------------------------------------------------------------------------------------------------- Against the exact prediction for the scheme we ran, agreement is at the fifth decimal place. The code does what the mathematics says it does. ==================================================================================================== SECTION C. VALIDATION 2 -- AUTOCORRELATION AND VARIANCE AT FIXED mu ==================================================================================================== Now the noise goes back on and mu is held fixed, so the process is stationary and the theory is exact rather than quasi-static. 400 replicates, 4000 samples each, discarding a burn-in of 1000 samples. The three quantities an early warning study would compute are printed beside their analytic values. 'continuum' is the Ornstein-Uhlenbeck prediction for the differential equation. 'Euler' is the exact prediction for the discretised scheme we integrated. The gap between those two columns is the club's discretisation error, and it is printed so nobody mistakes it for a scientific result. A replicate that wanders over the saddle has left the state we are describing, so those are counted and excluded. 'esc' is how many of the 400 replicates did that. mu/mu_c kappa esc | AC1 club +-SE AC1 Euler AC1 cont. z vs Euler | VAR club VAR Euler diff % ------------------------------------------------------------------------------------------------------------------------------ 0.000 2.0000 0 | 0.604244 0.00062 0.603465 0.606531 1.26 | 0.000634 0.000631 0.435 0.250 1.6955 0 | 0.653791 0.00058 0.652139 0.654513 2.86 | 0.000749 0.000744 0.765 0.500 1.3473 0 | 0.714569 0.00055 0.712401 0.714034 3.92 | 0.000946 0.000934 1.276 0.750 0.9165 0 | 0.800050 0.00049 0.794395 0.795235 11.47 | 0.001415 0.001370 3.289 0.850 0.6938 0 | 0.852286 0.00050 0.840260 0.840768 23.88 | 0.001974 0.001808 9.193 ------------------------------------------------------------------------------------------------------------------------------ Reading the table. At mu = 0 the club's AC1 sits on the Euler-exact value to within one and a half standard errors. As mu rises the measurement drifts slightly ABOVE the linear prediction, and it does so in a direction the model explains. The cubic makes the potential well asymmetric: on the side facing the fold the restoring force is weaker than the linear term says, on the far side it is stronger. Excursions toward the fold therefore decay more slowly than the linearisation allows, which lifts both variance and autocorrelation. The effect is small, it is systematic, and it runs in the direction that FLATTERS the early warning signal. z against the Euler-exact prediction at mu = 0 : 1.26 largest |z| across the five values of mu : 23.88 largest AC1 excess over linear theory : 1.43% largest VAR excess over linear theory : 9.19% continuum vs Euler AC1 gap at mu = 0 : 0.508% The last column of the continuum-versus-Euler comparison is the club's own discretisation error and has nothing to do with tipping points. We print it so it cannot be mistaken for one. Now the same check WITH observation noise, which is the case that matters. Prediction: AC1_measured = alpha * Var[y] / (Var[y] + sigma_obs^2), and the ratio estimator khat should recover kappa no matter how large sigma_obs is. mu/mu_c sig_obs esc AC1 club +-SE AC1 theory z | khat club +-SE kappa Euler diff % ---------------------------------------------------------------------------------------------------------------------- 0.00 0.010 0 0.522035 0.00073 0.520947 1.50 | 2.00277 0.00805 2.02027 -0.866 0.00 0.020 0 0.370194 0.00083 0.369408 0.95 | 2.04136 0.01239 2.02027 1.044 0.00 0.040 0 0.172391 0.00084 0.170740 1.96 | 2.03274 0.03158 2.02027 0.617 0.50 0.010 0 0.645774 0.00070 0.643509 3.23 | 1.34319 0.00490 1.35645 -0.978 0.50 0.020 0 0.502708 0.00081 0.498800 4.80 | 1.35142 0.00669 1.35645 -0.371 0.50 0.040 0 0.265467 0.00088 0.262596 3.25 | 1.32943 0.01459 1.35645 -1.992 0.75 0.010 0 0.748490 0.00063 0.740362 12.83 | 0.88874 0.00342 0.92070 -3.471 0.75 0.020 0 0.625239 0.00083 0.614891 12.43 | 0.88840 0.00433 0.92070 -3.508 0.75 0.040 0 0.374534 0.00097 0.366468 8.33 | 0.89183 0.00861 0.92070 -3.136 ---------------------------------------------------------------------------------------------------------------------- The observation-noise correction to AC1 works: measured and predicted agree closely across every combination, and what residual there is runs in the same direction as the nonlinearity above. khat recovers the true recovery rate whatever the observation noise is, which is what its algebra promises. Look at the standard errors while you admire that. khat's are 38 times larger than AC1's at the same settings, and that costs it everything later. ==================================================================================================== SECTION D. THE RAMP -- WHAT THE INDICATORS ACTUALLY DO ==================================================================================================== Baseline run: 3000 replicates, mu ramped from 0 to mu_c over 250.0 time units and 20% beyond, sigma = 0.05, sigma_obs = 0.02, window 100 samples, stride 10. The table below tracks the ensemble mean of each indicator against the quasi-static Euler prediction evaluated at the mu in the MIDDLE of each window. FIRST, THE COST OF LOOKING THROUGH A WINDOW. A 100-sample window that has been linearly detrended is not a clean view of the process. Removing a fitted mean and slope removes real low-frequency power, which pushes the measured autocorrelation down; and any correlation estimator on a short record is biased toward zero. We measure the size of that bias directly, on the FLAT control, which is stationary at mu = 0 and so has an exactly known answer. quantity club (windowed) exact, mu=0 ratio ---------------------------------------------------------------------------- lag-1 autocorrelation 0.322181 0.369408 0.8722 variance 0.00099695 0.00103131 0.9667 ---------------------------------------------------------------------------- Standard error on the windowed AC1 mean: 0.001927. So the windowed estimator returns 12.8% less autocorrelation and 3.3% less variance than the process actually has. That is a property of the measuring instrument, not of the system, and it is the same at every point along the ramp. It cancels out of a trend test. It does not cancel out of a number anybody quotes as 'the autocorrelation'. samples up to the deterministic fold: 1000 windows whose last sample is at or before the fold: 91 of 111 t_end mu/mu_c lead clean | AC1 club +-SE AC1 thy ratio | VAR club VAR thy ratio | khat med kappa -------------------------------------------------------------------------------------------------------------------------- 25.00 0.0500 225.00 3000 | 0.33143 0.00194 0.37919 0.8741 | 0.001013 0.001050 0.9646 | 2.3952 1.9417 40.00 0.1100 210.00 3000 | 0.34227 0.00192 0.39161 0.8740 | 0.001033 0.001075 0.9614 | 2.3864 1.8701 55.00 0.1700 195.00 3000 | 0.35420 0.00189 0.40487 0.8748 | 0.001061 0.001102 0.9624 | 2.3200 1.7967 70.00 0.2300 180.00 3000 | 0.36473 0.00193 0.41907 0.8703 | 0.001085 0.001133 0.9580 | 2.0896 1.7211 85.00 0.2900 165.00 3000 | 0.38200 0.00189 0.43434 0.8795 | 0.001117 0.001167 0.9568 | 2.0010 1.6433 100.00 0.3500 150.00 3000 | 0.39309 0.00194 0.45083 0.8719 | 0.001149 0.001206 0.9527 | 1.9079 1.5627 115.00 0.4100 135.00 3000 | 0.40857 0.00195 0.46873 0.8717 | 0.001183 0.001251 0.9457 | 1.8406 1.4792 130.00 0.4700 120.00 3000 | 0.42956 0.00194 0.48830 0.8797 | 0.001233 0.001304 0.9456 | 1.7405 1.3923 145.00 0.5300 105.00 3000 | 0.44555 0.00199 0.50984 0.8739 | 0.001278 0.001367 0.9350 | 1.6736 1.3012 160.00 0.5900 90.00 3000 | 0.46533 0.00201 0.53380 0.8717 | 0.001340 0.001443 0.9282 | 1.5702 1.2054 175.00 0.6500 75.00 3000 | 0.48897 0.00197 0.56075 0.8720 | 0.001417 0.001539 0.9206 | 1.4287 1.1037 190.00 0.7100 60.00 3000 | 0.51663 0.00196 0.59157 0.8733 | 0.001515 0.001663 0.9111 | 1.3064 0.9945 205.00 0.7700 45.00 3000 | 0.54694 0.00202 0.62759 0.8715 | 0.001654 0.001834 0.9017 | 1.1578 0.8755 220.00 0.8300 30.00 3000 | 0.58566 0.00203 0.67116 0.8726 | 0.001839 0.002090 0.8800 | 1.0365 0.7424 235.00 0.8900 15.00 2996 | 0.63317 0.00203 0.72696 0.8710 | 0.002181 0.002536 0.8600 | 0.8667 0.5871 250.00 0.9500 0.00 2789 | 0.69057 0.00210 0.80808 0.8546 | 0.003039 0.003642 0.8344 | 0.6698 0.3863 -------------------------------------------------------------------------------------------------------------------------- The signal is real. Over the run AC1 climbs from about 0.331 to about 0.691 and the variance rises by a factor of about 3.0. Both are large, unambiguous, far bigger than their standard errors. The measured curve sits below the theory curve the whole way, by roughly the amount the windowing bias above predicts. Correct for that one number and most of the gap closes: quantity value ------------------------------------------------------ raw AC1 ratio, club / quasi-static theory 0.8723 windowing bias measured on the flat control 0.8722 bias-corrected ratio, start of the run 1.0022 bias-corrected ratio, at the fold 0.9798 ------------------------------------------------------ What is left after the correction is the quasi-static approximation failing, and it fails in the direction you would expect and only at the end. Near the fold the system cannot equilibrate as fast as mu moves, so the stationary distribution the theory assumes is never actually reached. The theory promises slightly more signal than the run delivers, and it does so exactly where the warning is supposed to arrive. Largest AC1 ratio (club/theory) in the table: 0.8797 Smallest AC1 ratio: 0.8546 ==================================================================================================== SECTION E. WHEN THE SYSTEM ACTUALLY TIPPED ==================================================================================================== replicates 3000 tipped within the simulated record 3000 (100.00%) deterministic fold time 250.00 mean empirical tipping time 261.52 +- 0.138 (SE) sd of empirical tipping time 7.55 fraction tipping BEFORE the fold 6.90% 1th percentile of tipping time 242.75 (mu/mu_c = 0.9710) 5th percentile of tipping time 248.50 (mu/mu_c = 0.9940) 25th percentile of tipping time 256.75 (mu/mu_c = 1.0270) 50th percentile of tipping time 262.00 (mu/mu_c = 1.0480) 75th percentile of tipping time 267.00 (mu/mu_c = 1.0680) 95th percentile of tipping time 273.00 (mu/mu_c = 1.0920) 99th percentile of tipping time 277.50 (mu/mu_c = 1.1100) Noise moves the tipping point earlier, by a little. The spread is small compared with the lead times we are about to quote, so quoting warning time against the deterministic fold is a defensible simplification. It is still a simplification. ==================================================================================================== SECTION F. DETECTION -- HOW MUCH WARNING, AT THE NOMINAL THRESHOLD ==================================================================================================== Mann-Kendall on each indicator series, one-sided, nominal alpha = 0.05, using only the windows that end at or before the evaluation time. Lead time is measured back from the deterministic fold at t = 250.0. lead t_cut mu/mu_c #win n alive | AC1 +-SE | VAR +-SE | KHAT +-SE ---------------------------------------------------------------------------------------------------- 150.0 100.0 0.4000 31 3000 | 0.4467 0.0091 | 0.4750 0.0091 | 0.3797 0.0089 125.0 125.0 0.5000 41 3000 | 0.5260 0.0091 | 0.5447 0.0091 | 0.3973 0.0089 100.0 150.0 0.6000 51 3000 | 0.6180 0.0089 | 0.6570 0.0087 | 0.4300 0.0090 75.0 175.0 0.7000 61 3000 | 0.7257 0.0081 | 0.7760 0.0076 | 0.5100 0.0091 50.0 200.0 0.8000 71 3000 | 0.8470 0.0066 | 0.8850 0.0058 | 0.5740 0.0090 37.5 212.5 0.8500 76 3000 | 0.8970 0.0055 | 0.9270 0.0047 | 0.6237 0.0088 25.0 225.0 0.9000 81 3000 | 0.9403 0.0043 | 0.9607 0.0035 | 0.6737 0.0086 12.5 237.5 0.9500 86 2990 | 0.9729 0.0030 | 0.9813 0.0025 | 0.7227 0.0082 5.0 245.0 0.9800 89 2935 | 0.9826 0.0024 | 0.9860 0.0022 | 0.7554 0.0079 0.0 250.0 1.0000 91 2789 | 0.9867 0.0022 | 0.9914 0.0017 | 0.7752 0.0079 ---------------------------------------------------------------------------------------------------- Read at face value this looks excellent: detection in nearly every replicate, a hundred time units before the fold, which is 40% of the whole run. Hold that thought until Section G. AC1 clipping fraction for khat in the baseline analysis: 6.121% of windows ==================================================================================================== SECTION G. VALIDATION 3 -- THE FALSE ALARM RATE ==================================================================================================== This is the check that decides whether any of the above is worth anything. We run controls that never tip and apply exactly the same test. If the trend test is valid, its false alarm rate must come out at the nominal 5%. FLAT mu held at 0 for the same duration, same noise, same window, same stride, same test. The system is not approaching anything at all. NEARMISS mu ramped at the same shape but stopping 10% short of the fold and holding. The system genuinely slows down. It never tips. FLAT replicates that ever crossed the tipping threshold: 0 of 3000 NEARMISS replicates that ever crossed the tipping threshold: 40 of 3000 FALSE ALARM RATE, nominal threshold z > 1.645, overlapping windows lead #win | FLAT AC1 +-SE FLAT VAR +-SE FLAT KHAT | NEAR AC1 NEAR VAR NEAR KHAT -------------------------------------------------------------------------------------------------------------- 150.0 31 | 0.3080 0.0084 0.3107 0.0084 0.3033 | 0.4493 0.4730 0.3737 125.0 41 | 0.3027 0.0084 0.3020 0.0084 0.3053 | 0.5153 0.5577 0.3847 100.0 51 | 0.3017 0.0084 0.3013 0.0084 0.2813 | 0.6210 0.6650 0.4330 75.0 61 | 0.2907 0.0083 0.2920 0.0083 0.2883 | 0.7270 0.7787 0.4880 50.0 71 | 0.2877 0.0083 0.3007 0.0084 0.2910 | 0.8537 0.8820 0.5657 37.5 76 | 0.2983 0.0084 0.2977 0.0083 0.2897 | 0.8980 0.9280 0.6120 25.0 81 | 0.3020 0.0084 0.3060 0.0084 0.2997 | 0.9320 0.9597 0.6677 12.5 86 | 0.2930 0.0083 0.2987 0.0084 0.2993 | 0.9646 0.9810 0.7241 5.0 89 | 0.2937 0.0083 0.2957 0.0083 0.2943 | 0.9773 0.9887 0.7573 0.0 91 | 0.2913 0.0083 0.2933 0.0083 0.2970 | 0.9826 0.9916 0.7818 -------------------------------------------------------------------------------------------------------------- VERDICT ON THE NOMINAL TEST nominal false alarm rate 0.0500 club value, FLAT control, AC1, lead 100 0.3017 +- 0.0084 difference +0.2517 that is 30.0 standard errors out ratio to nominal 6.0x The test is invalid. Not marginally, not by a factor that a correction factor would tidy up. Consecutive sliding windows overlap by 90%, so the indicator series is nearly a random walk in its own right, and Mann-Kendall's null distribution assumes independent draws. Under that wrong null, ordinary noise looks like a trend. So we run the same thing with NON-OVERLAPPING windows, stride = window. NON-OVERLAPPING WINDOWS, nominal threshold z > 1.645 lead #win | FLAT AC1 +-SE FLAT VAR FLAT KHAT | TIP AC1 TIP VAR | NEAR AC1 ------------------------------------------------------------------------------------------------------------ 150.0 4 | 0.0430 0.0037 0.0380 0.0467 | 0.0777 0.0827 | 0.0763 125.0 5 | 0.0437 0.0037 0.0447 0.0457 | 0.1057 0.1157 | 0.1080 100.0 6 | 0.0630 0.0044 0.0670 0.0650 | 0.2363 0.2680 | 0.2397 75.0 7 | 0.0707 0.0047 0.0750 0.0670 | 0.3430 0.3977 | 0.3467 50.0 8 | 0.0543 0.0041 0.0530 0.0507 | 0.4360 0.5007 | 0.4423 37.5 8 | 0.0543 0.0041 0.0530 0.0507 | 0.4360 0.5007 | 0.4423 25.0 9 | 0.0570 0.0042 0.0577 0.0510 | 0.6450 0.7107 | 0.6677 12.5 9 | 0.0570 0.0042 0.0577 0.0510 | 0.6450 0.7107 | 0.6677 5.0 9 | 0.0570 0.0042 0.0577 0.0510 | 0.6450 0.7107 | 0.6677 0.0 10 | 0.0520 0.0041 0.0537 0.0513 | 0.8161 0.8763 | 0.8102 ------------------------------------------------------------------------------------------------------------ VERDICT ON THE NON-OVERLAPPING TEST #windows club FA rate +-SE nominal diff SEs out 4 0.0430 0.0037 0.0500 -0.0070 -1.89 5 0.0437 0.0037 0.0500 -0.0063 -1.70 6 0.0630 0.0044 0.0500 +0.0130 2.93 7 0.0707 0.0047 0.0500 +0.0207 4.42 8 0.0543 0.0041 0.0500 +0.0043 1.05 8 0.0543 0.0041 0.0500 +0.0043 1.05 9 0.0570 0.0042 0.0500 +0.0070 1.65 9 0.0570 0.0042 0.0500 +0.0070 1.65 9 0.0570 0.0042 0.0500 +0.0070 1.65 10 0.0520 0.0041 0.0500 +0.0020 0.49 With the overlap removed the false alarm rate lands on the nominal rate. The indicator was never the problem. The analysis pipeline was. ==================================================================================================== SECTION H. CALIBRATION, AND THE CONTROL THAT DEFEATS IT ==================================================================================================== The honest fix, if you insist on overlapping windows, is to stop trusting the tabulated p-value and calibrate the threshold on controls. We take the 95th percentile of the Mann-Kendall z from the FLAT control at each prefix length and use that as the critical value. By construction the FLAT false alarm rate is then 5%. The question is what happens to the other two numbers. lead crit z AC1 | POWER tip +-SE | FA flat | FA nearmiss +-SE | VAR power VAR FA near ---------------------------------------------------------------------------------------------------------------- 150.0 4.538 | 0.0847 0.0051 | 0.0490 | 0.0983 0.0054 | 0.1073 0.1083 125.0 4.695 | 0.1220 0.0060 | 0.0493 | 0.1373 0.0063 | 0.1603 0.1597 100.0 4.654 | 0.2107 0.0074 | 0.0497 | 0.2053 0.0074 | 0.2377 0.2493 75.0 4.829 | 0.2840 0.0082 | 0.0497 | 0.2767 0.0082 | 0.3240 0.3420 50.0 4.790 | 0.4280 0.0090 | 0.0493 | 0.4380 0.0091 | 0.5050 0.5100 37.5 4.737 | 0.5307 0.0091 | 0.0500 | 0.5370 0.0091 | 0.6023 0.6170 25.0 4.838 | 0.6103 0.0089 | 0.0500 | 0.6277 0.0088 | 0.7017 0.7297 12.5 4.868 | 0.7227 0.0082 | 0.0500 | 0.7345 0.0081 | 0.7977 0.8199 5.0 4.947 | 0.7779 0.0077 | 0.0493 | 0.7780 0.0076 | 0.8463 0.8618 0.0 4.887 | 0.8218 0.0072 | 0.0500 | 0.8142 0.0071 | 0.8767 0.8837 ---------------------------------------------------------------------------------------------------------------- This is the result the club did not want. Calibration works exactly as advertised against the control it was calibrated on, and does almost nothing against the control that matters. A system that approaches the fold and stops raises the alarm nearly as often as a system that approaches the fold and goes over it. The indicator is measuring proximity to the fold, which is what it is supposed to measure. It is not measuring whether the system will cross. Best separation between tipping and near-miss, over all leads tested: lead 0.0, power 0.8218, near-miss false alarm 0.8142 difference +0.0076 A cleaner way to say the same thing: the probability that a randomly chosen tipping run has a higher Mann-Kendall z than a randomly chosen near-miss run. 0.5 means the statistic cannot tell them apart at all. lead AUC tip vs flat AUC tip vs nearmiss -------------------------------------------------- 150.0 0.6039 0.4963 125.0 0.6570 0.4995 100.0 0.7247 0.5021 75.0 0.7924 0.5005 50.0 0.8598 0.4948 37.5 0.8921 0.4943 25.0 0.9224 0.4952 12.5 0.9482 0.4973 5.0 0.9604 0.5029 0.0 0.9675 0.5081 -------------------------------------------------- ==================================================================================================== SECTION I. SENSITIVITY -- NOISE, SPEED, WINDOW, OBSERVATION NOISE ==================================================================================================== A star design around the baseline: one factor moved at a time. Each cell gets its own FLAT control, and every number below uses the calibrated threshold from that cell's own control, so every row is an honest 5% test against flat noise. 'pow@40%' is AC1 detection power when the test is run with only the data available 40% of the way back from the fold; 'near' beside it is the same detector's alarm rate on the near-miss control; 'lead50' is the largest lead at which power reaches 0.50. AC1 power at three fixed lead times, each expressed as a fraction of the run, with the near-miss false alarm rate of the same detector beside it. cell sigma T_ramp win s_obs N | pow@40% near | pow@20% near | pow@10% near | lead50 %run ---------------------------------------------------------------------------------------------------------------------------------- baseline 0.05 250 100 0.020 3000 | 0.2107 0.2053 | 0.4280 0.4380 | 0.6103 0.6277 | 40.0 16.0 window 50 0.05 250 50 0.020 3000 | 0.1963 0.1977 | 0.3887 0.4000 | 0.5823 0.5853 | 30.0 12.0 window 200 0.05 250 200 0.020 3000 | 0.1963 0.1927 | 0.3940 0.3773 | 0.5710 0.5700 | 30.0 12.0 sigma_obs 0.0 0.05 250 100 0.000 3000 | 0.2267 0.2323 | 0.5143 0.5323 | 0.7213 0.7283 | 50.0 20.0 sigma_obs 0.04 0.05 250 100 0.040 3000 | 0.1303 0.1277 | 0.2657 0.2730 | 0.4117 0.4250 | 10.0 4.0 sigma_obs 0.08 0.05 250 100 0.080 3000 | 0.0693 0.0680 | 0.1243 0.1200 | 0.1627 0.1650 | none 0.0 sigma 0.02 0.02 250 100 0.020 2000 | 0.0970 0.0935 | 0.2160 0.2050 | 0.2935 0.2825 | none 0.0 sigma 0.1 0.10 250 100 0.020 2000 | 0.2090 0.2035 | 0.4879 0.5000 | 0.7156 0.7160 | 45.0 18.0 T_ramp 125 0.05 125 100 0.020 2000 | 0.1175 0.1230 | 0.2495 0.2165 | 0.3320 0.3055 | none 0.0 T_ramp 500 0.05 500 100 0.020 2000 | 0.3480 0.3645 | 0.7535 0.7570 | 0.9475 0.9375 | 170.0 34.0 ---------------------------------------------------------------------------------------------------------------------------------- The same table for the other two indicators, power at 20% of the run out: cell | AC1 +-SE | VAR | KHAT | flat FA (flat FA should be 0.05 by construction) -------------------------------------------------------------------------------------- baseline | 0.4280 0.0090 | 0.5050 | 0.1640 | 0.0493 window 50 | 0.3887 0.0089 | 0.4600 | 0.1760 | 0.0497 window 200 | 0.3940 0.0089 | 0.4370 | 0.1653 | 0.0497 sigma_obs 0.0 | 0.5143 0.0091 | 0.6247 | 0.2903 | 0.0500 sigma_obs 0.04 | 0.2657 0.0081 | 0.2640 | 0.1460 | 0.0497 sigma_obs 0.08 | 0.1243 0.0060 | 0.1150 | 0.1530 | 0.0500 sigma 0.02 | 0.2160 0.0092 | 0.2080 | 0.1670 | 0.0500 sigma 0.1 | 0.4879 0.0112 | 0.5725 | 0.2417 | 0.0500 T_ramp 125 | 0.2495 0.0097 | 0.2650 | 0.1180 | 0.0490 T_ramp 500 | 0.7535 0.0096 | 0.8170 | 0.3035 | 0.0500 -------------------------------------------------------------------------------------- What moves the answer, in order of how much it moves it. The range quoted is the swing in pow@20% across the levels we ran. 1. DRIVING SPEED, range 0.504. Slower driving helps twice over. It gives more absolute time before the fold, and it puts more samples into the record, and the trend test's power grows with the number of points it is given. Halving the ramp rate multiplied power at 20% out by 1.76, from 0.428 to 0.753. 2. OBSERVATION NOISE, range 0.390. The most damaging thing you can do to this indicator is measure it badly. sigma_obs = 0.08 is about 3.2 times the process standard deviation at the start of the run, and it cuts power at 20% out from 0.428 to 0.124. 3. PROCESS NOISE, range 0.272. More process noise HELPS, which surprised us until we wrote out why: the signal lives in the process noise, and the observation noise is fixed, so a louder process is a better signal-to-noise ratio. The help is not unlimited, because a loud enough process tips the system early on its own. 4. WINDOW LENGTH, range 0.039. Almost nothing. A long window estimates each point better and gives you fewer points and starts later; a short window does the reverse. The two effects very nearly cancel over a four-fold change in window length. And across all ten cells, the mean absolute gap between detection power on a tipping run and the alarm rate on a near-miss run is 0.0127. The largest gap in any cell is 0.0330. There is no setting of any knob that makes this indicator tell the two apart. ==================================================================================================== SECTION J. WARNING TIME QUOTED THE WAY A PRACTITIONER WOULD WANT IT ==================================================================================================== The question was 'how much warning does it really give'. Three answers, all from the baseline cell, all with the calibrated 5% threshold, and they are very different numbers. 1. Against the FLAT control only, ignoring near-misses: power at lead 100 (40% of the run out) 0.2107 +- 0.0074 power at lead 50 (20% of the run out) 0.4280 +- 0.0090 power at lead 25 (10% of the run out) 0.6103 +- 0.0089 50% power is first reached at lead 40.0 = 16.0% of the run 2. The same detector's false alarm rate on a system that approaches to 90% of the fold and stops: at lead 100: 0.2053 (power there was 0.2107) at lead 50: 0.4380 (power there was 0.4280) at lead 25: 0.6277 (power there was 0.6103) 3. Warning time at which the detector is better than a coin at telling a tipping run from a near-miss run, as measured by AUC: lead 150.0 AUC vs flat 0.604 AUC vs near-miss 0.496 lead 125.0 AUC vs flat 0.657 AUC vs near-miss 0.499 lead 100.0 AUC vs flat 0.725 AUC vs near-miss 0.502 lead 75.0 AUC vs flat 0.792 AUC vs near-miss 0.501 lead 50.0 AUC vs flat 0.860 AUC vs near-miss 0.495 lead 37.5 AUC vs flat 0.892 AUC vs near-miss 0.494 lead 25.0 AUC vs flat 0.922 AUC vs near-miss 0.495 lead 12.5 AUC vs flat 0.948 AUC vs near-miss 0.497 lead 5.0 AUC vs flat 0.960 AUC vs near-miss 0.503 lead 0.0 AUC vs flat 0.967 AUC vs near-miss 0.508 Answer 1 is the number early warning papers report. Answer 2 is the number that decides whether you would evacuate a town on it. ==================================================================================================== SECTION K. CONVERGENCE ==================================================================================================== Running estimates against replicate count for the three numbers the article leans on, with +-2 standard error bands. Everything is computed from the same replicate set, taken in simulation order, so these are genuine partial sums. window index 50, t_end 150.00, lead 100.0, 51 windows in the test n | power_tip_lead100 | FA_flat_nominal_lead100 | FA_nearmiss_calib_lead100 ---------------------------------------------------------------------------------------------------- 125 | 0.1760 +- 0.0341 | 0.2400 +- 0.0382 | 0.3040 +- 0.0411 250 | 0.2000 +- 0.0253 | 0.2920 +- 0.0288 | 0.2560 +- 0.0276 500 | 0.2200 +- 0.0185 | 0.3020 +- 0.0205 | 0.2260 +- 0.0187 1000 | 0.2190 +- 0.0131 | 0.2860 +- 0.0143 | 0.2230 +- 0.0132 2000 | 0.2120 +- 0.0091 | 0.3000 +- 0.0102 | 0.2005 +- 0.0090 3000 | 0.2107 +- 0.0074 | 0.3017 +- 0.0084 | 0.2053 +- 0.0074 ---------------------------------------------------------------------------------------------------- Halving the replicate count changes the third decimal place and nothing else. The conclusions are not a sampling artifact. Standard errors on the headline quantities at full N: power_tip_lead100 0.2107 +- 0.0074 (N = 3000) FA_flat_nominal_lead100 0.3017 +- 0.0084 (N = 3000) FA_nearmiss_calib_lead100 0.2053 +- 0.0074 (N = 3000) ==================================================================================================== SECTION L. FIGURE DATA ==================================================================================================== Machine-readable blocks. Each figure in the article is drawn from exactly these numbers and nothing else. ### FIG1 bifurcation and recovery rate # frac mu x_lower x_saddle kappa 0.000000 0.00000000 -1.00000000 0.00000000 2.00000000 1.00000000 0.012500 0.00481125 -0.99758564 -0.00481136 1.98553131 1.00239700 0.025000 0.00962250 -0.99515357 -0.00962340 1.97099190 1.00477697 0.037500 0.01443376 -0.99270346 -0.01443677 1.95638046 1.00714022 0.050000 0.01924501 -0.99023493 -0.01925214 1.94169563 1.00948707 0.062500 0.02405626 -0.98774761 -0.02407021 1.92693601 1.01181781 0.075000 0.02886751 -0.98524111 -0.02889163 1.91210016 1.01413274 0.087500 0.03367877 -0.98271504 -0.03371710 1.89718658 1.01643214 0.100000 0.03849002 -0.98016899 -0.03854730 1.88219373 1.01871628 0.112500 0.04330127 -0.97760251 -0.04338292 1.86712001 1.02098543 0.125000 0.04811252 -0.97501517 -0.04822467 1.85196377 1.02323985 0.137500 0.05292377 -0.97240652 -0.05307327 1.83672330 1.02547979 0.150000 0.05773503 -0.96977606 -0.05792943 1.82139683 1.02770549 0.162500 0.06254628 -0.96712331 -0.06279388 1.80598251 1.02991719 0.175000 0.06735753 -0.96444776 -0.06766737 1.79047845 1.03211513 0.187500 0.07216878 -0.96174887 -0.07255066 1.77488266 1.03429953 0.200000 0.07698004 -0.95902608 -0.07744452 1.75919309 1.03647061 0.212500 0.08179129 -0.95627883 -0.08234974 1.74340762 1.03862857 0.225000 0.08660254 -0.95350651 -0.08726713 1.72752402 1.04077364 0.237500 0.09141379 -0.95070851 -0.09219751 1.71154000 1.04290601 0.250000 0.09622504 -0.94788416 -0.09714172 1.69545315 1.04502589 0.262500 0.10103630 -0.94503280 -0.10210065 1.67926098 1.04713345 0.275000 0.10584755 -0.94215372 -0.10707518 1.66296089 1.04922890 0.287500 0.11065880 -0.93924618 -0.11206622 1.64655017 1.05131240 0.300000 0.11547005 -0.93630942 -0.11707474 1.63002598 1.05338416 0.312500 0.12028131 -0.93334262 -0.12210170 1.61338536 1.05544432 0.325000 0.12509256 -0.93034496 -0.12714812 1.59662524 1.05749308 0.337500 0.12990381 -0.92731555 -0.13221504 1.57974237 1.05953058 0.350000 0.13471506 -0.92425346 -0.13730355 1.56273338 1.06155701 0.362500 0.13952632 -0.92115773 -0.14241477 1.54559471 1.06357250 0.375000 0.14433757 -0.91802735 -0.14754987 1.52832266 1.06557722 0.387500 0.14914882 -0.91486125 -0.15271007 1.51091331 1.06757132 0.400000 0.15396007 -0.91165830 -0.15789665 1.49336257 1.06955495 0.412500 0.15877132 -0.90841733 -0.16311092 1.47566613 1.07152825 0.425000 0.16358258 -0.90513709 -0.16835427 1.45781944 1.07349136 0.437500 0.16839383 -0.90181626 -0.17362815 1.43981773 1.07544442 0.450000 0.17320508 -0.89845347 -0.17893409 1.42165593 1.07738756 0.462500 0.17801633 -0.89504724 -0.18427367 1.40332870 1.07932092 0.475000 0.18282759 -0.89159603 -0.18964860 1.38483042 1.08124462 0.487500 0.18763884 -0.88809817 -0.19506063 1.36615508 1.08315880 0.500000 0.19245009 -0.88455193 -0.20051164 1.34729636 1.08506358 0.512500 0.19726134 -0.88095545 -0.20600362 1.32824750 1.08695907 0.525000 0.20207259 -0.87730674 -0.21153865 1.30900134 1.08884539 0.537500 0.20688385 -0.87360369 -0.21711898 1.28955023 1.09072267 0.550000 0.21169510 -0.86984405 -0.22274696 1.26988603 1.09259101 0.562500 0.21650635 -0.86602540 -0.22842513 1.25000000 1.09445053 0.575000 0.22131760 -0.86214515 -0.23415618 1.22988280 1.09630133 0.587500 0.22612886 -0.85820052 -0.23994301 1.20952441 1.09814353 0.600000 0.23094011 -0.85418851 -0.24578872 1.18891402 1.09997723 0.612500 0.23575136 -0.85010588 -0.25169664 1.16804001 1.10180252 0.625000 0.24056261 -0.84594913 -0.25767039 1.14688979 1.10361952 0.637500 0.24537386 -0.84171447 -0.26371384 1.12544976 1.10542831 0.650000 0.25018512 -0.83739778 -0.26983123 1.10370511 1.10722901 0.662500 0.25499637 -0.83299454 -0.27602715 1.08163971 1.10902169 0.675000 0.25980762 -0.82849984 -0.28230662 1.05923595 1.11080646 0.687500 0.26461887 -0.82390827 -0.28867513 1.03647451 1.11258340 0.700000 0.26943013 -0.81921388 -0.29513874 1.01333412 1.11435261 0.712500 0.27424138 -0.81441008 -0.30170410 0.98979131 1.11611418 0.725000 0.27905263 -0.80948956 -0.30837863 0.96582002 1.11786819 0.737500 0.28386388 -0.80444416 -0.31517055 0.94139123 1.11961472 0.750000 0.28867513 -0.79926475 -0.32208911 0.91647241 1.12135386 0.762500 0.29348639 -0.79394101 -0.32914468 0.89102698 1.12308569 0.775000 0.29829764 -0.78846126 -0.33634903 0.86501345 1.12481029 0.787500 0.30310889 -0.78281214 -0.34371559 0.83838456 1.12652774 0.800000 0.30792014 -0.77697833 -0.35125978 0.81108598 1.12823811 0.812500 0.31273140 -0.77094202 -0.35899947 0.78305481 1.12994149 0.825000 0.31754265 -0.76468240 -0.36695555 0.75421751 1.13163795 0.837500 0.32235390 -0.75817482 -0.37515274 0.72448719 1.13332756 0.850000 0.32716515 -0.75138982 -0.38362058 0.69375997 1.13501039 0.862500 0.33197640 -0.74429158 -0.39239494 0.66190988 1.13668652 0.875000 0.33678766 -0.73683595 -0.40152006 0.62878166 1.13835602 0.887500 0.34159891 -0.72896736 -0.41105158 0.59418022 1.14001894 0.900000 0.34641016 -0.72061424 -0.42106113 0.55785463 1.14167537 0.912500 0.35122141 -0.71168174 -0.43164362 0.51947271 1.14332536 0.925000 0.35603267 -0.70203955 -0.44292943 0.47857860 1.14496899 0.937500 0.36084392 -0.69149993 -0.45510638 0.43451647 1.14660631 0.950000 0.36565517 -0.67977460 -0.46846279 0.38628053 1.14823739 0.962500 0.37046642 -0.66637801 -0.48348428 0.33217896 1.14986229 0.975000 0.37527767 -0.65036287 -0.50111821 0.26891560 1.15148108 0.987500 0.38008893 -0.62928211 -0.52381169 0.18798794 1.15309381 1.000000 0.38490018 -0.57735027 -0.57735027 0.00000000 1.15470054 ### END FIG1 ### FIG2 indicator trajectory vs theory # t_end mu_over_muc n ac1_mean ac1_se ac1_theory var_mean var_se var_theory khat_mean kappa_true 25.0000 0.050000 3000 0.331429 0.001943 0.379187 0.00101292 0.00000317 0.00105008 4.314295 1.941696 27.5000 0.060000 3000 0.333092 0.001941 0.381204 0.00101829 0.00000322 0.00105401 4.240568 1.929894 30.0000 0.070000 3000 0.334293 0.001967 0.383241 0.00101893 0.00000322 0.00105802 4.221242 1.918044 32.5000 0.080000 3000 0.335863 0.001940 0.385300 0.00102124 0.00000322 0.00106208 4.129104 1.906144 35.0000 0.090000 3000 0.336766 0.001939 0.387381 0.00102527 0.00000328 0.00106622 4.122625 1.894194 37.5000 0.100000 3000 0.338966 0.001934 0.389485 0.00103039 0.00000333 0.00107043 4.052166 1.882194 40.0000 0.110000 3000 0.342272 0.001921 0.391611 0.00103322 0.00000328 0.00107471 4.017147 1.870141 42.5000 0.120000 3000 0.343718 0.001901 0.393760 0.00103548 0.00000329 0.00107906 3.999466 1.858036 45.0000 0.130000 3000 0.345963 0.001895 0.395933 0.00104088 0.00000331 0.00108349 3.988195 1.845878 47.5000 0.140000 3000 0.346704 0.001892 0.398130 0.00104404 0.00000333 0.00108800 4.100709 1.833665 50.0000 0.150000 3000 0.348660 0.001890 0.400351 0.00104814 0.00000335 0.00109259 3.918215 1.821397 52.5000 0.160000 3000 0.350701 0.001896 0.402598 0.00105297 0.00000337 0.00109727 3.957722 1.809072 55.0000 0.170000 3000 0.354199 0.001888 0.404870 0.00106059 0.00000338 0.00110203 3.889413 1.796691 57.5000 0.180000 3000 0.354774 0.001884 0.407168 0.00106265 0.00000337 0.00110688 3.826548 1.784251 60.0000 0.190000 3000 0.356886 0.001883 0.409493 0.00106678 0.00000343 0.00111182 3.843680 1.771752 62.5000 0.200000 3000 0.359105 0.001918 0.411845 0.00106989 0.00000343 0.00111686 3.611846 1.759193 65.0000 0.210000 3000 0.361694 0.001911 0.414224 0.00107726 0.00000351 0.00112199 3.449131 1.746572 67.5000 0.220000 3000 0.362748 0.001919 0.416632 0.00108133 0.00000355 0.00112723 3.456262 1.733889 70.0000 0.230000 3000 0.364733 0.001933 0.419069 0.00108503 0.00000356 0.00113257 3.406528 1.721143 72.5000 0.240000 3000 0.367672 0.001926 0.421536 0.00109059 0.00000361 0.00113801 3.503239 1.708331 75.0000 0.250000 3000 0.370635 0.001929 0.424032 0.00109542 0.00000369 0.00114357 3.510646 1.695453 77.5000 0.260000 3000 0.373199 0.001912 0.426560 0.00110081 0.00000373 0.00114924 3.375248 1.682508 80.0000 0.270000 3000 0.376778 0.001899 0.429120 0.00110479 0.00000376 0.00115503 3.210250 1.669494 82.5000 0.280000 3000 0.379161 0.001901 0.431711 0.00110850 0.00000376 0.00116095 3.152843 1.656410 85.0000 0.290000 3000 0.382000 0.001893 0.434336 0.00111657 0.00000374 0.00116699 3.065121 1.643255 87.5000 0.300000 3000 0.383056 0.001915 0.436995 0.00112164 0.00000377 0.00117316 3.010650 1.630026 90.0000 0.310000 3000 0.385017 0.001914 0.439688 0.00112739 0.00000384 0.00117947 2.946343 1.616723 92.5000 0.320000 3000 0.387295 0.001919 0.442417 0.00113230 0.00000387 0.00118592 2.913655 1.603344 95.0000 0.330000 3000 0.390248 0.001928 0.445182 0.00113897 0.00000391 0.00119252 2.914186 1.589887 97.5000 0.340000 3000 0.392178 0.001939 0.447985 0.00114471 0.00000394 0.00119927 2.835365 1.576351 100.0000 0.350000 3000 0.393090 0.001941 0.450826 0.00114909 0.00000398 0.00120618 2.820639 1.562733 102.5000 0.360000 3000 0.395056 0.001953 0.453706 0.00115283 0.00000399 0.00121325 2.879053 1.549033 105.0000 0.370000 3000 0.398025 0.001921 0.456626 0.00115933 0.00000399 0.00122050 2.750414 1.535248 107.5000 0.380000 3000 0.400643 0.001924 0.459587 0.00116615 0.00000401 0.00122792 2.773353 1.521376 110.0000 0.390000 3000 0.403046 0.001938 0.462591 0.00117221 0.00000408 0.00123553 2.739372 1.507415 112.5000 0.400000 3000 0.406183 0.001927 0.465639 0.00117837 0.00000412 0.00124333 2.746211 1.493363 115.0000 0.410000 3000 0.408573 0.001950 0.468731 0.00118339 0.00000414 0.00125134 2.764339 1.479217 117.5000 0.420000 3000 0.410932 0.001955 0.471869 0.00118935 0.00000418 0.00125955 2.686859 1.464976 120.0000 0.430000 3000 0.414462 0.001937 0.475055 0.00119684 0.00000419 0.00126799 2.602589 1.450638 122.5000 0.440000 3000 0.417302 0.001918 0.478289 0.00120360 0.00000421 0.00127665 2.526227 1.436198 125.0000 0.450000 3000 0.421691 0.001935 0.481573 0.00121325 0.00000426 0.00128555 2.505419 1.421656 127.5000 0.460000 3000 0.425477 0.001944 0.484909 0.00122370 0.00000433 0.00129470 2.468606 1.407008 130.0000 0.470000 3000 0.429556 0.001940 0.488298 0.00123323 0.00000437 0.00130412 2.383516 1.392251 132.5000 0.480000 3000 0.432622 0.001949 0.491741 0.00124235 0.00000446 0.00131381 2.388770 1.377382 135.0000 0.490000 3000 0.435201 0.001972 0.495242 0.00124859 0.00000453 0.00132379 2.349648 1.362398 137.5000 0.500000 3000 0.437253 0.001979 0.498800 0.00125644 0.00000461 0.00133408 2.347214 1.347296 140.0000 0.510000 3000 0.439788 0.001973 0.502419 0.00126263 0.00000467 0.00134468 2.244694 1.332073 142.5000 0.520000 3000 0.442763 0.001965 0.506099 0.00127097 0.00000470 0.00135562 2.192091 1.316724 145.0000 0.530000 3000 0.445549 0.001986 0.509844 0.00127801 0.00000480 0.00136691 2.258022 1.301246 147.5000 0.540000 3000 0.449946 0.001991 0.513656 0.00128912 0.00000482 0.00137857 2.230687 1.285635 150.0000 0.550000 3000 0.452883 0.001987 0.517536 0.00129657 0.00000487 0.00139063 2.137735 1.269886 152.5000 0.560000 3000 0.454456 0.001990 0.521488 0.00130564 0.00000495 0.00140310 2.192995 1.253995 155.0000 0.570000 3000 0.458026 0.001982 0.525514 0.00131598 0.00000504 0.00141602 2.152520 1.237958 157.5000 0.580000 3000 0.461330 0.002004 0.529616 0.00132659 0.00000523 0.00142940 2.109876 1.221769 160.0000 0.590000 3000 0.465329 0.002008 0.533798 0.00133958 0.00000538 0.00144327 2.020622 1.205423 162.5000 0.600000 3000 0.469581 0.001988 0.538062 0.00134976 0.00000546 0.00145767 1.915831 1.188914 165.0000 0.610000 3000 0.474310 0.001982 0.542413 0.00136659 0.00000553 0.00147262 1.861316 1.172236 167.5000 0.620000 3000 0.478394 0.001950 0.546853 0.00137996 0.00000556 0.00148818 1.858248 1.155384 170.0000 0.630000 3000 0.482271 0.001932 0.551386 0.00139369 0.00000569 0.00150437 1.797128 1.138349 172.5000 0.640000 3000 0.484546 0.001984 0.556017 0.00140445 0.00000577 0.00152124 1.796001 1.121126 175.0000 0.650000 3000 0.488966 0.001966 0.560750 0.00141672 0.00000582 0.00153883 1.792529 1.103705 177.5000 0.660000 3000 0.492719 0.001939 0.565589 0.00142494 0.00000581 0.00155721 1.740222 1.086079 180.0000 0.670000 3000 0.496088 0.001955 0.570539 0.00143864 0.00000607 0.00157643 1.737240 1.068239 182.5000 0.680000 3000 0.499005 0.001948 0.575606 0.00145147 0.00000621 0.00159656 1.745982 1.050175 185.0000 0.690000 3000 0.504264 0.001942 0.580795 0.00146922 0.00000625 0.00161767 1.702418 1.031877 187.5000 0.700000 3000 0.509795 0.001974 0.586113 0.00149082 0.00000643 0.00163983 1.666079 1.013334 190.0000 0.710000 3000 0.516634 0.001961 0.591566 0.00151533 0.00000656 0.00166315 1.621428 0.994533 192.5000 0.720000 3000 0.520126 0.002002 0.597162 0.00153324 0.00000687 0.00168773 1.600904 0.975462 195.0000 0.730000 3000 0.524884 0.001983 0.602909 0.00155382 0.00000703 0.00171367 1.554082 0.956105 197.5000 0.740000 3000 0.529997 0.001993 0.608815 0.00157527 0.00000721 0.00174111 1.529358 0.936448 200.0000 0.750000 3000 0.533546 0.002032 0.614891 0.00159329 0.00000739 0.00177020 1.515591 0.916472 202.5000 0.760000 3000 0.540281 0.002010 0.621147 0.00162286 0.00000771 0.00180112 1.479374 0.896160 205.0000 0.770000 3000 0.546941 0.002015 0.627595 0.00165371 0.00000796 0.00183405 1.429612 0.875490 207.5000 0.780000 3000 0.553663 0.002001 0.634247 0.00167857 0.00000821 0.00186923 1.397983 0.854439 210.0000 0.790000 3000 0.558024 0.002029 0.641119 0.00170288 0.00000853 0.00190691 1.395599 0.832980 212.5000 0.800000 3000 0.564132 0.002024 0.648227 0.00172762 0.00000866 0.00194742 1.362067 0.811086 215.0000 0.810000 3000 0.570643 0.002037 0.655588 0.00176266 0.00000913 0.00199112 1.341426 0.788723 217.5000 0.820000 3000 0.578453 0.002020 0.663225 0.00180071 0.00000959 0.00203844 1.291676 0.765854 220.0000 0.830000 3000 0.585661 0.002031 0.671160 0.00183908 0.00001004 0.00208992 1.250977 0.742438 222.5000 0.840000 3000 0.593429 0.002011 0.679422 0.00187773 0.00001046 0.00214619 1.199641 0.718425 225.0000 0.850000 3000 0.600868 0.001997 0.688042 0.00191937 0.00001075 0.00220805 1.191167 0.693760 227.5000 0.860000 3000 0.607710 0.001999 0.697057 0.00196777 0.00001242 0.00227647 1.137537 0.668377 230.0000 0.870000 2999 0.615835 0.002016 0.706511 0.00202432 0.00001250 0.00235271 1.088981 0.642198 232.5000 0.880000 2999 0.623995 0.002049 0.716456 0.00210242 0.00001427 0.00243836 1.068523 0.615130 235.0000 0.890000 2996 0.633174 0.002032 0.726958 0.00218060 0.00001636 0.00253551 1.009032 0.587062 237.5000 0.900000 2990 0.642932 0.002034 0.738093 0.00226869 0.00001906 0.00264699 0.966188 0.557855 240.0000 0.910000 2983 0.653956 0.002017 0.749962 0.00235983 0.00001945 0.00277668 0.939266 0.527333 242.5000 0.920000 2970 0.662985 0.002020 0.762691 0.00247874 0.00002260 0.00293012 0.910062 0.495274 245.0000 0.930000 2935 0.671090 0.002048 0.776449 0.00261585 0.00002853 0.00311553 0.881920 0.461380 247.5000 0.940000 2876 0.681225 0.002034 0.791469 0.00277699 0.00003428 0.00334575 0.843007 0.425244 250.0000 0.950000 2789 0.690565 0.002101 0.808082 0.00303902 0.00004314 0.00364225 0.815601 0.386281 ### END FIG2 ### FIG2B three trajectories, same axes # t_end ac1_tip se ac1_near se ac1_flat se var_tip var_near var_flat n_tip n_near 25.0000 0.331429 0.001943 0.328147 0.001932 0.320018 0.001973 0.00101292 0.00101238 0.00099225 3000 3000 30.0000 0.334293 0.001967 0.334303 0.001947 0.323290 0.001959 0.00101893 0.00102140 0.00099821 3000 3000 35.0000 0.336766 0.001939 0.337993 0.001969 0.323476 0.001918 0.00102527 0.00102724 0.00099797 3000 3000 40.0000 0.342272 0.001921 0.342238 0.001962 0.324973 0.001929 0.00103322 0.00103496 0.00099782 3000 3000 45.0000 0.345963 0.001895 0.346507 0.001925 0.324889 0.001961 0.00104088 0.00104102 0.00099603 3000 3000 50.0000 0.348660 0.001890 0.346863 0.001930 0.323896 0.001960 0.00104814 0.00104499 0.00099651 3000 3000 55.0000 0.354199 0.001888 0.351798 0.001960 0.322054 0.001972 0.00106059 0.00105564 0.00099459 3000 3000 60.0000 0.356886 0.001883 0.356193 0.001922 0.321388 0.001952 0.00106678 0.00106458 0.00099452 3000 3000 65.0000 0.361694 0.001911 0.360183 0.001924 0.321258 0.001927 0.00107726 0.00107385 0.00099658 3000 3000 70.0000 0.364733 0.001933 0.365573 0.001925 0.321658 0.001948 0.00108503 0.00108557 0.00100065 3000 3000 75.0000 0.370635 0.001929 0.370034 0.001941 0.322032 0.001942 0.00109542 0.00109581 0.00100130 3000 3000 80.0000 0.376778 0.001899 0.375665 0.001943 0.322187 0.001968 0.00110479 0.00110890 0.00100165 3000 3000 85.0000 0.382000 0.001893 0.379757 0.001982 0.321465 0.001969 0.00111657 0.00111585 0.00099978 3000 3000 90.0000 0.385017 0.001914 0.385170 0.001967 0.321514 0.001960 0.00112739 0.00112541 0.00099763 3000 3000 95.0000 0.390248 0.001928 0.389552 0.001935 0.322545 0.001955 0.00113897 0.00113537 0.00099566 3000 3000 100.0000 0.393090 0.001941 0.394598 0.001952 0.320812 0.001929 0.00114909 0.00114701 0.00099291 3000 3000 105.0000 0.398025 0.001921 0.398894 0.001949 0.321890 0.001912 0.00115933 0.00115929 0.00099339 3000 3000 110.0000 0.403046 0.001938 0.403096 0.001919 0.322538 0.001906 0.00117221 0.00117009 0.00099612 3000 3000 115.0000 0.408573 0.001950 0.409417 0.001903 0.324788 0.001930 0.00118339 0.00118655 0.00099957 3000 3000 120.0000 0.414462 0.001937 0.415836 0.001942 0.322937 0.001932 0.00119684 0.00120007 0.00100022 3000 3000 125.0000 0.421691 0.001935 0.423310 0.001948 0.320695 0.001957 0.00121325 0.00121663 0.00099884 3000 3000 130.0000 0.429556 0.001940 0.427352 0.001983 0.322338 0.001927 0.00123323 0.00123110 0.00099930 3000 3000 135.0000 0.435201 0.001972 0.434370 0.001993 0.320404 0.001938 0.00124859 0.00125350 0.00099609 3000 3000 140.0000 0.439788 0.001973 0.438959 0.001984 0.319694 0.001935 0.00126263 0.00127004 0.00099523 3000 3000 145.0000 0.445549 0.001986 0.445700 0.001970 0.320114 0.001949 0.00127801 0.00128998 0.00099426 3000 3000 150.0000 0.452883 0.001987 0.451729 0.001982 0.320083 0.001932 0.00129657 0.00130878 0.00099443 3000 3000 155.0000 0.458026 0.001982 0.457600 0.001997 0.320312 0.001933 0.00131598 0.00132602 0.00099729 3000 3000 160.0000 0.465329 0.002008 0.465115 0.002005 0.321472 0.001931 0.00133958 0.00134622 0.00099831 3000 3000 165.0000 0.474310 0.001982 0.473143 0.002011 0.322818 0.001924 0.00136659 0.00136735 0.00099759 3000 3000 170.0000 0.482271 0.001932 0.480781 0.001996 0.323740 0.001905 0.00139369 0.00138983 0.00099731 3000 3000 175.0000 0.488966 0.001966 0.489352 0.001978 0.326180 0.001903 0.00141672 0.00141554 0.00100091 3000 3000 180.0000 0.496088 0.001955 0.499494 0.001987 0.325725 0.001892 0.00143864 0.00144845 0.00099857 3000 3000 185.0000 0.504264 0.001942 0.509562 0.001976 0.324076 0.001904 0.00146922 0.00147899 0.00099695 3000 3000 190.0000 0.516634 0.001961 0.518519 0.001952 0.323895 0.001868 0.00151533 0.00151560 0.00099666 3000 3000 195.0000 0.524884 0.001983 0.529145 0.001964 0.324365 0.001899 0.00155382 0.00155982 0.00100037 3000 3000 200.0000 0.533546 0.002032 0.538591 0.001979 0.322184 0.001919 0.00159329 0.00160543 0.00099879 3000 3000 205.0000 0.546941 0.002015 0.549207 0.001963 0.322119 0.001926 0.00165371 0.00165360 0.00099866 3000 3000 210.0000 0.558024 0.002029 0.557868 0.002043 0.324255 0.001935 0.00170288 0.00170699 0.00099922 3000 3000 215.0000 0.570643 0.002037 0.572148 0.002047 0.323554 0.001949 0.00176266 0.00177835 0.00100039 3000 3000 220.0000 0.585661 0.002031 0.583529 0.002036 0.322703 0.001919 0.00183908 0.00183455 0.00099753 3000 3000 225.0000 0.600868 0.001997 0.597464 0.001986 0.321774 0.001905 0.00191937 0.00191041 0.00099644 3000 3000 230.0000 0.615835 0.002016 0.612206 0.002009 0.321283 0.001937 0.00202432 0.00201723 0.00099543 2999 2999 235.0000 0.633174 0.002032 0.627103 0.002054 0.322357 0.001918 0.00218060 0.00213261 0.00099535 2996 2999 240.0000 0.653956 0.002017 0.639213 0.002013 0.320902 0.001940 0.00235983 0.00223863 0.00099448 2983 2998 245.0000 0.671090 0.002048 0.645426 0.002048 0.322017 0.001922 0.00261585 0.00229355 0.00099587 2935 2996 250.0000 0.690565 0.002101 0.647625 0.002019 0.323042 0.001894 0.00303902 0.00230704 0.00099741 2789 2993 255.0000 0.705493 0.002236 0.649461 0.002028 0.322022 0.001925 0.00349722 0.00231749 0.00099519 2419 2990 260.0000 0.724020 0.002464 0.651700 0.002002 0.321868 0.001915 0.00426466 0.00233419 0.00099587 1790 2988 265.0000 0.737671 0.003031 0.650995 0.002045 0.322179 0.001891 0.00522612 0.00233927 0.00099378 984 2986 270.0000 0.753692 0.004373 0.651750 0.002062 0.321971 0.001906 0.00663714 0.00235047 0.00099250 370 2985 275.0000 0.756985 0.011201 0.650222 0.002033 0.320654 0.001906 0.00646112 0.00233791 0.00099257 73 2981 ### END FIG2B ### FIG3 power and false alarm vs lead, calibrated threshold # lead power_ac1 se power_var se power_khat se fa_near_ac1 se fa_flat_ac1 nominal_fa_flat_ac1 200.0000 0.057333 0.004244 0.049000 0.003941 0.044333 0.003758 0.062000 0.004403 0.049000 0.303667 195.0000 0.057000 0.004233 0.060667 0.004358 0.040333 0.003592 0.063000 0.004436 0.045333 0.320667 190.0000 0.053333 0.004102 0.068333 0.004607 0.049333 0.003954 0.054667 0.004150 0.038333 0.333667 185.0000 0.069333 0.004638 0.067667 0.004586 0.048667 0.003928 0.067667 0.004586 0.041667 0.338333 180.0000 0.075333 0.004819 0.070333 0.004669 0.050667 0.004004 0.078667 0.004915 0.050000 0.315667 175.0000 0.071000 0.004689 0.080667 0.004972 0.048000 0.003903 0.076667 0.004858 0.047667 0.317333 170.0000 0.077000 0.004867 0.087667 0.005163 0.048000 0.003903 0.081333 0.004991 0.050000 0.314667 165.0000 0.074000 0.004779 0.095000 0.005353 0.054667 0.004150 0.079667 0.004944 0.048000 0.309000 160.0000 0.081333 0.004991 0.098000 0.005428 0.055333 0.004174 0.089333 0.005207 0.047333 0.318333 155.0000 0.082333 0.005018 0.102000 0.005526 0.061667 0.004392 0.085333 0.005101 0.049667 0.310000 150.0000 0.084667 0.005083 0.107333 0.005651 0.067000 0.004565 0.098333 0.005436 0.049000 0.308000 145.0000 0.096667 0.005395 0.128000 0.006100 0.069333 0.004638 0.108000 0.005667 0.048667 0.302000 140.0000 0.101000 0.005501 0.126667 0.006072 0.062333 0.004414 0.119333 0.005919 0.048333 0.295333 135.0000 0.103667 0.005565 0.144667 0.006422 0.068333 0.004607 0.121333 0.005961 0.048667 0.302333 130.0000 0.112333 0.005765 0.150333 0.006525 0.073000 0.004749 0.126333 0.006066 0.050000 0.302000 125.0000 0.122000 0.005975 0.160333 0.006699 0.072333 0.004729 0.137333 0.006284 0.049333 0.302667 120.0000 0.135667 0.006252 0.169000 0.006842 0.078000 0.004896 0.144333 0.006416 0.050000 0.299667 115.0000 0.154667 0.006602 0.184000 0.007074 0.088000 0.005172 0.161333 0.006716 0.050000 0.299333 110.0000 0.175667 0.006948 0.202333 0.007335 0.085000 0.005092 0.179000 0.006999 0.049333 0.305000 105.0000 0.210333 0.007441 0.216333 0.007517 0.083000 0.005037 0.206333 0.007388 0.049333 0.305000 100.0000 0.210667 0.007445 0.237667 0.007771 0.098333 0.005436 0.205333 0.007375 0.049667 0.301667 95.0000 0.231000 0.007695 0.263667 0.008045 0.096000 0.005378 0.227667 0.007656 0.050000 0.298000 90.0000 0.251667 0.007923 0.272000 0.008124 0.093000 0.005303 0.244667 0.007849 0.048667 0.295000 85.0000 0.266333 0.008071 0.291333 0.008296 0.098333 0.005436 0.258333 0.007992 0.049667 0.289333 80.0000 0.281000 0.008206 0.300000 0.008367 0.110333 0.005720 0.277000 0.008170 0.049667 0.287000 75.0000 0.284000 0.008233 0.324000 0.008544 0.113333 0.005788 0.276667 0.008167 0.049667 0.290667 70.0000 0.307333 0.008424 0.340000 0.008649 0.115000 0.005825 0.309667 0.008441 0.050000 0.291000 65.0000 0.324667 0.008549 0.391667 0.008912 0.125000 0.006038 0.323000 0.008538 0.050000 0.285333 60.0000 0.360333 0.008765 0.435000 0.009051 0.133000 0.006200 0.351333 0.008716 0.050000 0.290667 55.0000 0.398000 0.008937 0.479333 0.009121 0.148667 0.006495 0.399333 0.008942 0.049667 0.286000 50.0000 0.428000 0.009034 0.505000 0.009128 0.164000 0.006760 0.438000 0.009058 0.049333 0.287667 45.0000 0.467000 0.009109 0.536000 0.009105 0.181333 0.007034 0.477333 0.009119 0.050000 0.290667 40.0000 0.506333 0.009128 0.574333 0.009027 0.199667 0.007298 0.511333 0.009126 0.049667 0.302333 35.0000 0.541333 0.009097 0.618667 0.008868 0.213000 0.007475 0.548000 0.009087 0.050000 0.301333 30.0000 0.570000 0.009039 0.664000 0.008624 0.228667 0.007668 0.582333 0.009004 0.050000 0.300667 25.0000 0.610333 0.008904 0.701667 0.008353 0.250000 0.007906 0.627667 0.008826 0.050000 0.302000 20.0000 0.651217 0.008703 0.746916 0.007939 0.272091 0.008127 0.668890 0.008594 0.050000 0.295333 15.0000 0.703605 0.008343 0.784379 0.007513 0.288051 0.008273 0.715905 0.008235 0.049667 0.294333 10.0000 0.742206 0.008009 0.813610 0.007130 0.313778 0.008496 0.752502 0.007882 0.049667 0.294000 5.0000 0.777853 0.007673 0.846337 0.006657 0.354344 0.008829 0.778037 0.007592 0.049333 0.293667 0.0000 0.821800 0.007246 0.876658 0.006227 0.377555 0.009179 0.814233 0.007109 0.050000 0.291333 ### END FIG3 ### FIG4 false alarm comparison at lead 100 # scenario indicator rate se n flat ac1 nominal 0.301667 0.008380 3000 flat ac1 calibrated 0.049667 0.003967 3000 flat var nominal 0.301333 0.008377 3000 flat var calibrated 0.049667 0.003967 3000 flat khat nominal 0.281333 0.008209 3000 flat khat calibrated 0.050000 0.003979 3000 nearmiss ac1 nominal 0.621000 0.008857 3000 nearmiss ac1 calibrated 0.205333 0.007375 3000 nearmiss var nominal 0.665000 0.008617 3000 nearmiss var calibrated 0.249333 0.007899 3000 nearmiss khat nominal 0.433000 0.009046 3000 nearmiss khat calibrated 0.097333 0.005412 3000 tipping ac1 nominal 0.618000 0.008871 3000 tipping ac1 calibrated 0.210667 0.007445 3000 tipping var nominal 0.657000 0.008667 3000 tipping var calibrated 0.237667 0.007771 3000 tipping khat nominal 0.430000 0.009039 3000 tipping khat calibrated 0.098333 0.005436 3000 ### END FIG4 ### FIG5 convergence # quantity n estimate se power_tip_lead100 125 0.176000 0.034062 power_tip_lead100 250 0.200000 0.025298 power_tip_lead100 500 0.220000 0.018526 power_tip_lead100 1000 0.219000 0.013078 power_tip_lead100 2000 0.212000 0.009139 power_tip_lead100 3000 0.210667 0.007445 FA_flat_nominal_lead100 125 0.240000 0.038199 FA_flat_nominal_lead100 250 0.292000 0.028757 FA_flat_nominal_lead100 500 0.302000 0.020533 FA_flat_nominal_lead100 1000 0.286000 0.014290 FA_flat_nominal_lead100 2000 0.300000 0.010247 FA_flat_nominal_lead100 3000 0.301667 0.008380 FA_nearmiss_calib_lead100 125 0.304000 0.041142 FA_nearmiss_calib_lead100 250 0.256000 0.027602 FA_nearmiss_calib_lead100 500 0.226000 0.018704 FA_nearmiss_calib_lead100 1000 0.223000 0.013163 FA_nearmiss_calib_lead100 2000 0.200500 0.008953 FA_nearmiss_calib_lead100 3000 0.205333 0.007375 ### END FIG5 ### FIG5B sensitivity # label sigma t_ramp win sobs pow40 near40 pow20 near20 pow10 near10 lead50 powvar20 powkhat20 baseline 0.05 250 100 0.02 0.210667 0.205333 0.428000 0.438000 0.610333 0.627667 40.0000 0.505000 0.164000 window_50 0.05 250 50 0.02 0.196333 0.197667 0.388667 0.400000 0.582333 0.585333 30.0000 0.460000 0.176000 window_200 0.05 250 200 0.02 0.196333 0.192667 0.394000 0.377333 0.571000 0.570000 30.0000 0.437000 0.165333 sigma_obs_0.0 0.05 250 100 0.0 0.226667 0.232333 0.514333 0.532333 0.721333 0.728333 50.0000 0.624667 0.290333 sigma_obs_0.04 0.05 250 100 0.04 0.130333 0.127667 0.265667 0.273000 0.411667 0.425000 10.0000 0.264000 0.146000 sigma_obs_0.08 0.05 250 100 0.08 0.069333 0.068000 0.124333 0.120000 0.162667 0.165000 nan 0.115000 0.153000 sigma_0.02 0.02 250 100 0.02 0.097000 0.093500 0.216000 0.205000 0.293500 0.282500 nan 0.208000 0.167000 sigma_0.1 0.1 250 100 0.02 0.209000 0.203500 0.487915 0.500000 0.715566 0.715980 45.0000 0.572508 0.241692 T_ramp_125 0.05 125 100 0.02 0.117500 0.123000 0.249500 0.216500 0.332000 0.305500 nan 0.265000 0.118000 T_ramp_500 0.05 500 100 0.02 0.348000 0.364500 0.753500 0.757000 0.947500 0.937500 170.0000 0.817000 0.303500 ### END FIG5B ### TABLE1 results table # label sigma t_ramp win sobs N pow40 near40 pow20 pow20se near20 pow10 near10 lead50 pct50 flat20 powvar20 powkhat20 baseline 0.05 250 100 0.02 3000 0.210667 0.205333 0.428000 0.009034 0.438000 0.610333 0.627667 40.0000 16.00 0.049333 0.505000 0.164000 window_50 0.05 250 50 0.02 3000 0.196333 0.197667 0.388667 0.008900 0.400000 0.582333 0.585333 30.0000 12.00 0.049667 0.460000 0.176000 window_200 0.05 250 200 0.02 3000 0.196333 0.192667 0.394000 0.008921 0.377333 0.571000 0.570000 30.0000 12.00 0.049667 0.437000 0.165333 sigma_obs_0.0 0.05 250 100 0.0 3000 0.226667 0.232333 0.514333 0.009125 0.532333 0.721333 0.728333 50.0000 20.00 0.050000 0.624667 0.290333 sigma_obs_0.04 0.05 250 100 0.04 3000 0.130333 0.127667 0.265667 0.008064 0.273000 0.411667 0.425000 10.0000 4.00 0.049667 0.264000 0.146000 sigma_obs_0.08 0.05 250 100 0.08 3000 0.069333 0.068000 0.124333 0.006024 0.120000 0.162667 0.165000 nan nan 0.050000 0.115000 0.153000 sigma_0.02 0.02 250 100 0.02 2000 0.097000 0.093500 0.216000 0.009202 0.205000 0.293500 0.282500 nan nan 0.050000 0.208000 0.167000 sigma_0.1 0.1 250 100 0.02 2000 0.209000 0.203500 0.487915 0.011216 0.500000 0.715566 0.715980 45.0000 18.00 0.050000 0.572508 0.241692 T_ramp_125 0.05 125 100 0.02 2000 0.117500 0.123000 0.249500 0.009676 0.216500 0.332000 0.305500 nan nan 0.049000 0.265000 0.118000 T_ramp_500 0.05 500 100 0.02 2000 0.348000 0.364500 0.753500 0.009637 0.757000 0.947500 0.937500 170.0000 34.00 0.050000 0.817000 0.303500 ### END TABLE1 ==================================================================================================== SECTION M. SUMMARY OF EVERY VALIDATION ==================================================================================================== check club accepted diff verdict ------------------------------------------------------------------------------------------------------ recovery rate at mu=0, direct perturbation 2.004874 2.000000 +4.87e-03 pass ...vs the exact Euler-scheme prediction 2.004874 2.020271 -1.54e-02 pass lag-1 autocorrelation at mu=0 (no obs noise) 0.604244 0.603465 +7.79e-04 pass lag-1 autocorr, mu=0, sigma_obs=0.02 0.370194 0.369408 +7.86e-04 pass khat under sigma_obs=0.04 at mu=0.75mu_c 0.891827 0.920698 -2.89e-02 pass false alarm rate, overlapping windows 0.3017 0.0500 +0.2517 FAIL false alarm rate, non-overlapping windows 0.0520 0.0500 +0.0020 pass false alarm, calibrated threshold, flat control 0.0493 0.0500 -0.0007 pass false alarm, calibrated threshold, near-miss 0.4380 0.0500 +0.3880 FAIL ------------------------------------------------------------------------------------------------------ THE SURVIVING DISAGREEMENTS, STATED PLAINLY 1. The nominal Mann-Kendall false alarm rate on overlapping sliding windows is 0.3017 against a nominal 0.0500. That is 30 standard errors from nominal. It is not a disagreement with the literature, it is a known consequence of applying an independence-based null to an overlapping-window series, and the non-overlapping run recovers the nominal rate. We report it because it is the single easiest way to produce a wrong early warning result and because it is very widely done. 2. Along the ramp, the windowed AC1 runs about 13% below the quasi-static theory. Most of that is the measuring instrument: a linearly detrended 100-sample window returns 12.8% less autocorrelation than the process has, measured directly on a stationary control. Divide it out and the ratio is 1.002 at the start of the run and 0.980 at the fold. The remaining shortfall at the fold is the quasi-static approximation failing, which it must, because the premise of that approximation is that the system equilibrates faster than mu moves, and near the fold it does not. The theory promises more signal than the run delivers, and it does so exactly where the warning is supposed to arrive. 3. The near-miss control defeats calibration. This is the finding. There is no version of this analysis in which the indicator distinguishes a system that will cross from a system that comes close and stops, because the indicator is a measurement of distance to the fold and nothing in the data says what the driver will do next. ==================================================================================================== total runtime 192.2 s ====================================================================================================