# The Living Thing / Notebooks : Change of time and probability measure

TBD. Various notes on a.e. continuous monotonic changes of index in order to render a process “simple” in some sense.

In Warping and registration problems you try to align two or more processes. But here, the target is some “null”, basic process. This special case is often a lot more computationally tractable and statistically well behaved.

To explore:

Lamperti representation for continuous state branching processes, Girsanov transformation (may not be a change of measure in the sense that I want it to be; must look up whether I can shoehorn it.) Mass transport problems - do these relate? Maximum spacing estimators. Surely this pops up in copula modelling too? Ogata’s time rescaling: Intensity estimation for point processes uses this as a statistical test.

to understand:

• relation to martingale transform.
• how this multivariate version works
• estimating the time transform itself

A summary in VeSc04 of the point process flavour:

Knight (Knig70) showed that for any orthogonal sequence of continuous local martingales, by rescaling time for each via its associated predictable process, we form a multivariate sequence of independent standard Brownian motions. Then Meyer (Meye71) extended Knight’s theorem to the case of point processes, showing that given a simple multivariate point process ${N_i ; i = 1, 2, . . . , n}$, the multivariate point process obtained by rescaling each $N_i$ according to its compensator is a sequence of independent Poisson processes, each having intensity 1. Since then, alternative proofs and variations of this result have been given by Brem72, Papa72, AaHo78, Kurt80 and BrNa88. Papangelou (Papa72) gave the following interpretation in the univariate case:

Roughly, moving in $[0, \infty)$ so as to meet expected future points at a rate of one per time unit (given at each instant complete knowledge of the past), we meet them at the times of a Poisson process.

[…]

Generalizations of Meyer’s result to point processes on $\mathbb{R}^d$ have been established by MeNu86, Nair90 and Scho99. In each case, the method used has been to focus on one dimension of the point process, and rescale each point along that dimension according to the conditional intensity.

## Refs

AaHo78
Aalen, O. O., & Hoem, J. M.(1978) Random time changes for multivariate counting processes. Scandinavian Actuarial Journal, 1978(2), 81–101. DOI.
AnKo05
Anatolyev, S., & Kosenok, G. (2005) An Alternative to Maximum Likelihood Based on Spacings. Econometric Theory, 21(2). DOI.
BTMH05
Baddeley, A., Turner, R., Møller, J., & Hazelton, M. (2005) Residual analysis for spatial point processes (with discussion). Journal of the Royal Statistical Society: Series B (Statistical Methodology), 67(5), 617–666. DOI.
Brem72
Bremaud, P. (1972) A martingale approach to point processes (No. Memorandum ERL-M345). . Electronics Research Laboratory
BBVK02
Brown, E., Barbieri, R., Ventura, V., Kass, R., & Frank, L. (2002) The time-rescaling theorem and its application to neural spike train data analysis. Neural Computation, 14(2), 325–346. DOI.
BrNa88
Brown, T. C., & Nair, M. G.(1988) A Simple Proof of the Multivariate Random Time Change Theorem for Point Processes. Journal of Applied Probability, 25(1), 210–214. DOI.
ChPR13
Chaumont, L., Pantí, H., & Rivero, V. (2013) The Lamperti representation of real-valued self-similar Markov processes. Bernoulli, 19(5B), 2494–2523. DOI.
ChAm83
Cheng, R. C. H., & Amin, N. A. K.(1983) Estimating Parameters in Continuous Univariate Distributions with a Shifted Origin. Journal of the Royal Statistical Society. Series B (Methodological), 45(3), 394–403.
ChSt89
Cheng, R. C. H., & Stephens, M. A.(1989) A goodness-of-fit test using Moran’s statistic with estimated parameters. Biometrika, 76(2), 385–392. DOI.
Cox55
Cox, D. R.(1955) Some Statistical Methods Connected with Series of Events. Journal of the Royal Statistical Society. Series B (Methodological), 17(2), 129–164.
GiKM08
Giesecke, K., Kakavand, H., & Mousavi, M. (2008) Simulating point processes by intensity projection. In Simulation Conference, 2008. WSC 2008. Winter (pp. 560–568). DOI.
HaPB10
Haslinger, R., Pipa, G., & Brown, E. (2010) Discrete Time Rescaling Theorem: Determining Goodness of Fit for Discrete Time Statistical Models of Neural Spiking. Neural Computation, 22(10), 2477–2506. DOI.
Knig70
Knight, F. B.(1970) An Infinitesimal Decomposition for a Class of Markov Processes. The Annals of Mathematical Statistics, 41(5), 1510–1529. DOI.
Kurt80
Kurtz, T. G.(1980) Representations of Markov Processes as Multiparameter Time Changes. The Annals of Probability, 8(4), 682–715. DOI.
Lamp67
Lamperti, J. (1967) Continuous-state branching processes. Bull. Amer. Math. Soc, 73(3), 382–386.
Merzbach, E., & Nualart, D. (1986) A Characterization of the Spatial Poisson Process and Changing Time. The Annals of Probability, 14(4), 1380–1390. DOI.
Meye71
Meyer, P. A.(1971) Demonstration simplifiee d’un theoreme de Knight. In Séminaire de Probabilités V Université de Strasbourg (pp. 191–195). Springer Berlin Heidelberg
Nair90
Nair, M. G.(1990) Random Space Change for Multiparameter Point Processes. The Annals of Probability, 18(3), 1222–1231. DOI.
Papa72
Papangelou, F. (1972) Integrability of expected increments of point processes and a related random change of scale. Transactions of the American Mathematical Society, 165, 483–506. DOI.
PrMW09
Priesemann, V., Munk, M. H., & Wibral, M. (2009) Subsampling effects in neuronal avalanche distributions recorded in vivo. BMC Neuroscience, 10(1), 40. DOI.
Rann84
Ranneby, B. (1984) The Maximum Spacing Method An Estimation Method Related to the Maximum Likelihood Method. Scandinavian Journal of Statistics, 11(2), 93–112.
RaRT05
Ranneby, B., Rao Jammalamadaka, S., & Teterukovskiy, A. (2005) The maximum spacing estimation for multivariate observations. Journal of Statistical Planning and Inference, 129(1–2), 427–446. DOI.
RaWu01
Rao, C. R., & Wu, Y. (2001) On model selection. In Institute of Mathematical Statistics Lecture Notes - Monograph Series (Vol. 38, pp. 1–57). Beachwood, OH: Institute of Mathematical Statistics
Scho99
Schoenberg, F. (1999) Transforming spatial point processes into Poisson processes. Stochastic Processes and Their Applications, 81(2), 155–164. DOI.
Scho02
Schoenberg, F. P.(2002) On Rescaled Poisson Processes and the Brownian Bridge. Annals of the Institute of Statistical Mathematics, 54(2), 445–457. DOI.
VeSc04
Vere-Jones, D., & Schoenberg, F. P.(2004) Rescaling Marked Point Processes. Australian & New Zealand Journal of Statistics, 46(1), 133–143. DOI.
WoLi06
Wong, T. S. T., & Li, W. K.(2006) A note on the estimation of extreme value distributions using maximum product of spacings. In Institute of Mathematical Statistics Lecture Notes - Monograph Series (pp. 272–283). Beachwood, Ohio, USA: Institute of Mathematical Statistics