Brownian motion · A later measurement model

The particle.
The camera.
The estimate.

A camera averages motion during exposure and adds uncertainty to every position. Discover why those errors change what you can infer, even when the particle follows exactly the same path.

Start with ideal molecular-number inference → · Return to the argument

Noise-calibration assumption: stationary-feature clicks and the moving particle must have the same localization variance. The clicks cannot establish that assumption.

BM-08 · Observe, then infer

What did the camera change?

Static worked example

CurrentThese numbers match the current settings.

Model note

Primary output naiveD: Host calculation (inference.bm08). Owner inference.bm08.
Primary output centeredD: Host calculation (inference.bm08). Owner inference.bm08.
Primary output covarianceD: Host calculation (inference.bm08). Owner inference.bm08.
Primary output pairD: Host calculation (inference.bm08). Owner inference.bm08.
Primary output covarianceNoiseVariance: Host calculation (inference.bm08). Owner inference.bm08.
Primary output stationaryNoiseVariance: Host calculation (inference.bm08). Owner inference.bm08.
Primary output expectedVariance: Host calculation (inference.bm08). Owner inference.bm08.
Primary output expectedCovariance: Host calculation (inference.bm08). Owner inference.bm08.
Primary output sampleVariance: Host calculation (inference.bm08). Owner inference.bm08.
Primary output sampleCovariance: Host calculation (inference.bm08). Owner inference.bm08.
Primary output sdVariance: Host calculation (inference.bm08). Owner inference.bm08.
Primary output sdCovariance: Host calculation (inference.bm08). Owner inference.bm08.
Primary output naiveExpectation: Host calculation (inference.bm08). Owner inference.bm08.
Primary output modelDiffusion: Host calculation (inference.bm08). Owner inference.bm08.
Primary output pairDegrees: Host calculation (inference.bm08). Owner inference.bm08.
Primary output pairCount: Host calculation (inference.bm08). Owner inference.bm08.
Primary output emptyPairInterval: Host calculation (inference.bm08). Owner inference.bm08.
Primary output pairLowerClipped: Host calculation (inference.bm08). Owner inference.bm08.
Primary output recordingDraws: Host calculation (inference.bm08). Owner inference.bm08.
Primary output requestDraws: Host calculation (inference.bm08). Owner inference.bm08.
Primary output measurementDraws: Host calculation (inference.bm08). Owner inference.bm08.
Primary output reusedRecording: Host calculation (inference.bm08). Owner inference.bm08.
Primary output reusedObservation: Host calculation (inference.bm08). Owner inference.bm08.
Primary output retainedBytes: Host calculation (inference.bm08). Owner inference.bm08.
Primary output driftFit: Host calculation (inference.bm08). Owner inference.bm08.
Primary output noiseInterval: Host calculation (inference.bm08). Owner inference.bm08.
Primary output naiveInterval: Host calculation (inference.bm08). Owner inference.bm08.
Primary output centeredInterval: Host calculation (inference.bm08). Owner inference.bm08.
Primary output pairInterval: Host calculation (inference.bm08). Owner inference.bm08.
Primary output positions: Host calculation (inference.bm08). Owner inference.bm08.
Primary output idealPositions: Host calculation (inference.bm08). Owner inference.bm08.
Primary output blurredPositions: Host calculation (inference.bm08). Owner inference.bm08.
Primary output increments: Host calculation (inference.bm08). Owner inference.bm08.
Primary output times: Host calculation (inference.bm08). Owner inference.bm08.
Primary output stationaryClicks: Host calculation (inference.bm08). Owner inference.bm08.
Primary output latentWitness: Host calculation (inference.bm08). Owner inference.bm08.
Primary output speedTimes: Host calculation (inference.bm08). Owner inference.bm08.
Primary output idealSpeeds: Host calculation (inference.bm08). Owner inference.bm08.
Primary output cameraSpeeds: Host calculation (inference.bm08). Owner inference.bm08.
Primary output speedRatios: Host calculation (inference.bm08). Owner inference.bm08.
Primary output speedCrossover: Host calculation (inference.bm08). Owner inference.bm08.
Primary output coverageDraws: Host calculation (inference.bm08). Owner inference.bm08.
Primary output coverageIntervals: Host calculation (inference.bm08). Owner inference.bm08.
Primary output coverageNaiveCount: Host calculation (inference.bm08). Owner inference.bm08.
Primary output coveragePairCount: Host calculation (inference.bm08). Owner inference.bm08.
Primary output coverageEmptyCount: Host calculation (inference.bm08). Owner inference.bm08.
Seed 1905.
Accepted input revision 1.
Snapshot version 1.
Not modeled: Higher-order optical aberrations; only uniform exposure blur and Gaussian localization error.

A noisy image is not a new physical trajectory. Keep one wandering particle and change how it is observed. Then compare the estimates that ignore camera error with a procedure that accounts for it.

1 · Re-observe the same path

Displacements (3–1000)

Frame spacing (s)

Observed coordinates

Uniform exposure (s; multiples of 0.25)

Localization standard deviation (μm)

Stage drift in x (μm/s)

Camera-noise seed (unsigned 64-bit)

Exposure starts at each frame time and cannot exceed the spacing. Noise, stage drift, coordinates and frame selection leave the retained physical path unchanged.

2 · Declare what is known about noise

Pair-interval noise procedure

Stationary clicks (5–200)

Stationary-click seed (unsigned 64-bit)

Target interval coverage (%)

Changing the statistical procedure alone reuses the same observations. Stationary clicks have independent errors and an unknown constant feature location; they are not clicks on the moving particle.

3 · Change the physical setup

These changes start a new physical run

Generating diffusivity (μm²/s)

Fluid drift in x (μm/s)

Physical seed (unsigned 64-bit)

Fluid drift belongs to the physical path. Stage drift belongs to the measurement. Either produces apparent drift, but only the first changes this recording.

Comparisons use accepted settings, not unfinished drafts. Reset stage drift with a camera-error comparison; fluid drift remains an explicit physical input.

Static worked example. No calculation has started in this browser.

Accepted physical seed 1905; D = 0.42944 μm²/s; 101 frames in 2 coordinates; spacing 1 s; exposure 0.5 s; localization error 0.2 μm. Stage drift 0 μm/s; fluid drift 0 μm/s.

Dashed: retained position at frame start. Dotted: exposure average before stage drift or localization error. Solid: camera observation. Lines only connect the selected frames; the y coordinate, when selected, is included in all estimates and tables.

Four estimates, different assumptions

Accepted diffusivity estimates (μm²/s)
Generating value	0.42944
Ignore camera error and drift	0.40608
Fit drift, but ignore camera error	0.4067
Covariance estimate; known synthetic drift removed	0.51042
Disjoint pairs; noise and exposure corrected	0.47601

The covariance estimate uses the explicitly known synthetic drift, not a secretly fitted mean. It has no general confidence interval here. The pair procedure fits drift within its disjoint pairs and does not use generating D. Negative unconstrained estimates remain visible as diagnostics.

Which intervals are actually valid?

Ignoring the camera and drift

The zero-drift independent-increment interval does not apply to noisy, exposed, or drifting observations. Its point estimate remains visible as a deliberately naive comparison.

Fitting drift only

Fitting drift does not remove camera noise or the adjacent-increment correlation. The simple centered chi-square interval is not admitted.

Disjoint pairs, with a camera model

0.32569 – 0.69675

95% target coverage; μm²/s. Conservative procedure combining stationary-click noise uncertainty with disjoint-pair variance uncertainty.

50 disjoint pairs; 98 residual degrees of freedom.

Look for the camera’s fingerprint

Neighboring displacements share a localization error with opposite signs. Exposure averaging creates a different correlation. A variance alone cannot distinguish these effects.

Per-coordinate moments after subtracting known synthetic drift; μm²
Moment	Sample	Camera model	Asymptotic sampling SD
Increment variance	0.81215	0.79573	0.079699
Adjacent-increment covariance	0.10434	0.031573	0.0564

The last column is a large-sample MA(1) approximation for the moment estimates, not a confidence interval for D. It is particularly unreliable for very small samples. Expected naive D under this camera: 0.39787 μm²/s.

Inspect stationary clicks and inferred noise

Stationary-click variance: 0.027327 μm². Covariance-based noise estimate: -0.019274 μm². Noise bounds used by the pair procedure: 0.018697 – 0.04333 μm². In exact-noise mode the two bounds are the declared value; otherwise they consume half the allowed error probability.

Independent stationary-feature positions (μm)
Click	x	y
1	0.11424	-0.12339
2	0.013049	0.14156
3	0.30172	0.061022
4	-0.22497	-0.18133
5	-0.18734	0.17894
6	0.23688	-0.045071
7	-0.035658	-0.11956
8	-0.011732	0.010424
9	0.024987	0.13217
10	-0.12084	-0.11937
11	0.13412	0.16166
12	0.33155	-0.064944
13	-0.23434	0.17682
14	-0.043285	-0.14566
15	0.14079	-0.051643
16	-0.044961	-0.049523
17	0.038157	-0.042038
18	0.22326	-0.2329
19	0.2486	-0.27179
20	-0.074478	-0.41927
21	0.34235	0.39974
22	-0.13891	0.069009
23	-0.21279	-0.089314
24	-0.021301	-0.1282
25	0.011022	-0.07899
26	-0.18005	-0.23813
27	0.048898	0.060165
28	0.12458	-0.012509
29	-0.10319	-0.052014
30	-0.053618	-0.01365

Read all accepted frame positions

Every selected coordinate; exposure-start time in seconds, positions in μm
Start	Coordinate	Latent start	Exposure average	Camera position
0	x	0	0.059593	0.40616
0	y	0	0.49336	0.58492
1	x	0.76755	0.48381	0.45831
1	y	1.4664	1.6531	1.7576
2	x	1.3522	1.9374	1.7977
2	y	1.8208	2.3611	2.3077
3	x	1.6582	1.76	1.5403
3	y	4.7649	5.3406	5.6798
4	x	1.8259	2.2254	2.4785
4	y	6.106	6.4078	6.9054
5	x	2.6918	2.4524	2.6495
5	y	6.6218	7.1045	7.14
6	x	2.3015	2.2869	2.2653
6	y	6.8674	7.7694	7.9798
7	x	2.5376	3.3202	3.6063
7	y	7.815	8.1289	8.3653
8	x	3.9219	4.0269	4.0995
8	y	7.9772	8.0356	7.8886
9	x	3.662	4.0396	3.9277
9	y	8.7392	8.3412	8.4006
10	x	4.8784	5.035	5.1915
10	y	9.8519	9.6081	9.6218
11	x	3.2188	2.9586	3.114
11	y	10.096	9.8011	9.6936
12	x	1.7113	1.6358	1.6848
12	y	8.6935	8.6515	8.8669
13	x	1.5758	1.4006	1.2002
13	y	9.387	9.2404	9.443
14	x	2.0985	2.2149	2.3735
14	y	10.441	10.087	9.9655
15	x	2.3794	2.1728	2.162
15	y	9.6801	9.6963	10.071
16	x	3.0533	3.5569	3.6126
16	y	10.377	10.185	10.139
17	x	2.9836	3.0305	3.2455
17	y	10.019	10.529	10.491
18	x	3.9592	3.8842	4.1016
18	y	10.077	10.248	10.194
19	x	4.6413	4.7665	4.8111
19	y	10.528	10.525	10.182
20	x	6.0052	5.8067	5.9848
20	y	9.6698	9.6852	10.06
21	x	5.6737	4.5876	4.7685
21	y	9.5329	9.9584	9.6914
22	x	3.5548	3.4927	3.5383
22	y	9.2917	9.7142	9.7763
23	x	2.6572	2.3527	2.4738
23	y	9.8606	10.169	10.253
24	x	2.9066	2.6567	2.7187
24	y	11.145	11.579	11.513
25	x	3.0205	3.2533	3.3214
25	y	10.708	11.237	11.085
26	x	2.2934	2.5247	2.6839
26	y	10.448	10.97	10.838
27	x	1.6884	2.0406	1.7957
27	y	11.697	11.544	11.487
28	x	3.3604	3.3469	3.3858
28	y	10.959	10.63	10.509
29	x	4.0201	4.2709	4.3107
29	y	10.21	9.4371	8.9681
30	x	3.6998	3.6706	3.7236
30	y	8.8745	8.8583	9.2388
31	x	4.9056	4.5599	4.7906
31	y	9.4758	9.3429	9.443
32	x	5.1382	5.2829	5.1973
32	y	10.67	9.888	10.094
33	x	4.9866	4.4335	4.2462
33	y	7.6947	7.0434	7.0207
34	x	3.1814	2.6488	2.6366
34	y	4.9934	4.8296	4.8252
35	x	2.4513	2.1114	1.8492
35	y	5.6503	5.9031	5.8574
36	x	3.583	4.1399	4.2714
36	y	5.4076	6.4722	6.5851
37	x	5.0896	5.7585	5.5093
37	y	7.2942	6.9373	6.9627
38	x	4.9875	4.4822	4.757
38	y	7.5374	7.5797	7.5261
39	x	4.7811	4.6871	4.5648
39	y	6.8521	7.6846	7.4462
40	x	4.1958	3.7755	3.5298
40	y	7.795	7.9149	7.4695
41	x	4.0765	4.2618	4.2868
41	y	9.1455	8.5925	8.4906
42	x	3.0093	3.3736	3.2438
42	y	7.5917	7.8292	7.873
43	x	2.8804	3.1374	3.4481
43	y	7.9771	8.5259	8.2192
44	x	4.4756	4.5238	4.6473
44	y	8.3495	8.2293	8.186
45	x	5.2429	5.6274	5.5681
45	y	7.9999	7.9368	8.2303
46	x	5.633	5.3647	5.3908
46	y	8.5688	9.5611	9.5009
47	x	5.0369	5.0662	4.7665
47	y	10.82	11.466	11.533
48	x	4.0358	4.3932	4.2725
48	y	11.851	11.632	11.768
49	x	5.1979	5.1233	5.4984
49	y	10.331	10.257	10.197
50	x	4.7259	4.7283	4.9794
50	y	10.256	10.587	10.651
51	x	5.0725	4.9032	4.7221
51	y	9.7771	9.1211	9.1527
52	x	4.2456	4.1778	4.245
52	y	9.5051	9.5482	9.5819
53	x	4.9301	4.8117	5.0577
53	y	9.4363	9.3026	9.504
54	x	5.3437	5.1771	5.2083
54	y	8.8494	8.8335	8.5755
55	x	5.0866	5.346	5.7544
55	y	8.644	8.0718	7.9578
56	x	4.9806	4.3519	4.1889
56	y	7.6619	7.8874	7.9279
57	x	4.4448	3.9883	3.8757
57	y	8.5551	8.932	9.1778
58	x	2.6513	2.7795	2.5306
58	y	8.8701	9.2868	9.0655
59	x	2.5853	3.0001	3.0493
59	y	10.105	10.093	9.995
60	x	2.9557	3.4865	3.4012
60	y	10.321	10.228	10.189
61	x	3.4051	3.6652	3.7788
61	y	9.9667	10.256	10.348
62	x	2.7483	2.678	2.9185
62	y	10.964	11.261	11.034
63	x	1.9901	1.4699	1.5596
63	y	11.815	12.085	12.099
64	x	-0.51039	-1.0691	-0.79414
64	y	11.626	11.596	11.893
65	x	-0.55716	-0.60743	-0.55769
65	y	11.447	10.741	10.722
66	x	-0.63519	-0.76667	-0.67494
66	y	10.003	9.6375	9.8175
67	x	-0.013952	-0.28854	-0.098885
67	y	8.6998	8.708	9.1709
68	x	0.25495	0.098652	-0.1038
68	y	9.4474	9.9414	10.21
69	x	0.53966	-0.061082	0.10451
69	y	10.556	11.385	11.368
70	x	-0.9578	-1.1715	-0.94174
70	y	12.09	12.185	12.175
71	x	-2.5558	-2.2148	-1.9856
71	y	12.142	11.928	11.685
72	x	-1.1594	-1.9262	-1.9588
72	y	11.927	11.455	11.781
73	x	-2.6058	-3.5836	-3.1923
73	y	11.171	11.547	11.544
74	x	-3.8485	-3.1959	-3.0037
74	y	12.145	11.908	12.213
75	x	-4.0183	-4.5268	-4.7412
75	y	12.123	11.801	11.56
76	x	-4.5134	-4.4151	-4.6273
76	y	12.066	12.046	12.437
77	x	-5.3005	-5.1999	-5.1619
77	y	12.373	12.736	12.828
78	x	-5.7773	-6.3041	-6.1381
78	y	13.831	14.253	14.282
79	x	-6.6899	-6.6807	-6.4111
79	y	14	14.111	13.931
80	x	-5.5471	-5.4157	-5.5802
80	y	12.993	12.575	12.341
81	x	-4.4161	-4.292	-4.4776
81	y	12.939	12.538	12.542
82	x	-4.9302	-5.5939	-5.667
82	y	12.687	12.308	12.375
83	x	-5.7146	-5.5407	-5.5217
83	y	12.829	12.842	12.75
84	x	-5.8769	-5.8112	-5.6515
84	y	13.274	13.309	13.637
85	x	-6.6215	-7.0299	-7.1864
85	y	13.582	14.123	14.411
86	x	-7.3474	-7.5472	-7.5471
86	y	13.582	13.491	13.521
87	x	-7.1866	-7.0916	-7.1148
87	y	12.851	12.872	12.874
88	x	-6.8293	-6.9662	-6.9872
88	y	12.387	12.676	12.725
89	x	-7.1942	-7.1278	-7.1921
89	y	13.206	12.418	12.761
90	x	-7.4735	-6.8345	-7.1217
90	y	12.233	12.078	12.556
91	x	-7.2876	-7.418	-7.3593
91	y	12.508	12.152	11.962
92	x	-7.3322	-7.543	-7.5144
92	y	11.128	11.386	11.627
93	x	-7.9369	-7.7487	-7.9716
93	y	10.542	10.073	10.16
94	x	-7.2622	-7.0985	-7.1378
94	y	11.179	11.377	11.174
95	x	-6.4592	-6.7737	-6.6992
95	y	10.345	10.554	10.572
96	x	-7.2522	-7.6636	-7.6852
96	y	9.8799	10.11	9.9835
97	x	-7.7138	-7.5058	-7.5345
97	y	9.532	9.5214	9.658
98	x	-7.2317	-7.6763	-7.4877
98	y	8.9311	8.9259	8.7872
99	x	-7.3981	-7.4369	-7.5648
99	y	9.4625	9.0948	9.1115
100	x	-8.7425	-8.5387	-8.3366
100	y	8.0488	8.3731	8.4126

Faster pictures need not reveal a physical speed

This separate analytical comparison holds the accepted diffusivity and localization error fixed, sets exposure to zero, and varies a hypothetical observation interval. It does not create extra frames on the retained quarter-second grid. Neither curve is an instantaneous velocity.

Dashed: ideal spread divided by interval. Solid: the same model with localization noise and zero exposure. These are analytical comparisons, not extra observed frames or instantaneous particle velocities.

Noise-only crossover interval: 0.093145 s. At positive exposure, blur also changes the spread; that is not part of this zero-exposure comparison.

Analytical zero-exposure comparison, not additional sampled data
Spacing (s)	Ideal (μm/s)	With noise (μm/s)	Ratio
4	0.46338	0.46874	1.0116
1	0.92676	0.96896	1.0455
0.25	1.8535	2.1715	1.1716
0.0625	3.707	5.85	1.5781
0.01	9.2676	29.764	3.2116
0.0025	18.535	114.65	6.1853

Does the interval procedure cover the generating value?

Compare 100 other hypothetical paths under the same physical and camera settings. The dashed comparison deliberately applies the naive formula even where it is invalid; a pleasing interval is not evidence that its assumptions hold.

Run hypothetical experiments explicitly to compare procedures.

Every trial, including misses and empty sets, is retained. Changing the noise procedure reuses the same hypothetical identities. The primary path and camera observations are not replaced. A realized coverage fraction need not equal the target.

Accepted identity, random streams and limits

Run bm08-_R_bav5uiv5b_/run/1; snapshot 1; revisions {"input":1,"observer":0,"measurement":0,"estimator":0}. Owner inference.bm08.

Retained path and bridge draws: 32896. New latent work for this request: 32896. Camera/click stream evaluations: 524. Hypothetical experiment draws: 0. Retained latent bytes: 131600.

Source identity: source:sha256:c7ff276529014c8c0b7bc51fdeef19fe4faf6888459839a8196da9b6d99cffe0. Two latent coordinates and exact Brownian subinterval averages are retained on 4,112 quarter-second intervals. Frame errors have separate timestamp-indexed streams. Remeasurement may evaluate the same noise draws again; it does not create a new latent trajectory. No irregular timing, overlapping exposure, censoring, empirical import or arbitrary camera likelihood is admitted. This is host arithmetic, not audited WASM or historical evidence.

Each placement has its own accepted settings, worker and observations.

Why these procedures differ

An exposure is an average, not a point

The camera model uses a uniform exposure beginning at each frame time. Its position is the average latent position during that exposure, plus stage drift and independent Gaussian localization error. At zero exposure it uses the instantaneous frame-start position. The generated bridge integrals make these averages consistent with the same Brownian path.

Y_i=\frac{1}{T_e}\int_{t_i}^{t_i+T_e}X(s)\,ds+v_{\mathrm{stage}}(t_i+T_e/2)+\varepsilon_i,\quad \varepsilon_i\sim\mathcal N(0,\sigma^2)

Neighbors are correlated

For uniform, non-overlapping exposures and equally spaced frames, the per-coordinate variance and adjacent covariance of drift-subtracted increments are:

\gamma_0=2D(\Delta t-T_e/3)+2\sigma^2,\qquad \gamma_1=DT_e/3-\sigma^2

Localization error enters two neighboring displacements with opposite signs. Motion blur contributes the other sign. Ignoring both can understate or overstate diffusion. Subtracting fitted drift alone does not remove either camera effect.

Use covariance, or choose independent pairs

For known drift, the combination of the second moment and neighboring product cancels the localization and blur terms. The covariance point estimate can still be negative in a finite sample; clipping it would conceal a diagnostic.

\widehat D_{\mathrm{CVE}}=\frac{\widehat\gamma_0/2+\widehat\gamma_1}{\Delta t}

A different route takes disjoint frame pairs (0,1), (2,3), and so on. No pair shares a position error, and exposure is no longer than the spacing. These pair displacements are independent Gaussian vectors. Fitting one drift per coordinate leaves q = d(K − 1) degrees of freedom for K pairs. A chi-square interval for their variance can therefore be inverted using the known exposure and localization variance.

D=\frac{V-2\sigma^2}{2(\Delta t-T_e/3)}

When localization variance comes from independent stationary-feature clicks, half the allowed error probability goes to its variance interval and half to the pair-variance interval. Combining their endpoints gives a conservative confidence set. The set is intersected with D ≥ 0. An empty intersection is a possible outcome and is retained as a miss, not turned into a positive answer.

Shorter intervals can magnify camera error

With zero exposure and mean drift removed, dividing the coordinate spread by the observation interval gives an apparent speed. Localization error dominates below the noise-only crossover interval σ²/D. This does not reveal a finite instantaneous Brownian velocity.

v_{\mathrm{app,ideal}}=\sqrt{2D/\Delta t},\qquad v_{\mathrm{app,camera}}=\frac{\sqrt{2D\Delta t+2\sigma^2}}{\Delta t}

Keep this later model separate from the paper

These are original draft explanations of modern camera and inference models, not a transcription of Einstein’s argument. Synthetic camera observations are not experimental evidence for a physical suspension. The interval procedures assume constant drift, constant diffusivity, exact timing and calibration, uniform exposure, and independent Gaussian position errors. No irregular or censored data importer is supplied.

Berglund (2010): camera-based single-particle tracking · Vestergaard, Blainey and Flyvbjerg (2014): covariance-based diffusion estimation · Read the uncertainty foundation

Return to ideal inference →Return to the tracer ensemble