The problem of detecting a major change point in a stochastic process is often of interest in applications, in particular when the effects of modifications of some external variables, on the process itself, must be identified. We here propose a modification of the classical Pearson \(\chi ^2\) test to detect the presence of such major change point in the transition probabilities of an inhomogeneous discrete time Markov Chain, taking values in a finite space. The test can be applied also in presence of big identically distributed samples of the Markov Chain under study, which might not be necessarily independent. The test is based on the maximum likelihood estimate of the size of the ’right’ experimental unit, i.e. the units that must be aggregated to filter out the small scale variability of the transition probabilities. We here apply our test both to simulated data and to a real dataset, to study the impact, on farmland uses, of the new Common Agricultural Policy, which entered into force in EU in 2015.
This is a preview of subscription content, access via your institution.
Buy single article
Instant access to the full article PDF.
Tax calculation will be finalised during checkout.
Subscribe to journal
Immediate online access to all issues from 2019. Subscription will auto renew annually.
Tax calculation will be finalised during checkout.
Anderson TW, Goodman LA (1957) Statistical Inference about Markov Chains. Ann Math Stat 28(1):89–110. https://doi.org/10.1214/aoms/1177707039
Bertoni D et al (2018) Farmland use transitions after the CAP greening: a preliminary analysis using Markov chains approach. Land Use Policy 79:789–800. https://doi.org/10.1016/j.landusepol.2018.09.012
Cortignani R, Severini S, Dono G (2017) Complying with greening practices in the new CAP direct payments: an application on Italian specialized arable farms. Land Use Policy 61:265–275. https://doi.org/10.1016/j.landusepol.2016.11.026
Eisinger RD, Chen Y (2017) Sampling for conditional inference on contingency tables. J Comput Graph Stat 26(1):79–87. https://doi.org/10.1080/10618600.2016.1153478
Fisz M (1963) Probability theory and mathematical statistics. Wiley, New York
Fu X, Wang X, Yang YJ (2018) Deriving suitability factors for CA–Markov land use simulation model based on local historical data. J Environ Manag 206:10–19. https://doi.org/10.1016/j.jenvman.2017.10.012
Gregorio AD, Iacus SM (2008) Least squares volatility change point estimation for partially observed diffusion processes. Commun Stat Theory Methods 37(15):2342–2357. https://doi.org/10.1080/03610920801919692
Iacus SM, Yoshida N (2012) Estimation for the change point of volatility in a stochastic differential equation. Stoch Process Their Appl 122(3):1068–1092. https://doi.org/10.1016/j.spa.2011.11.005
Knoke D, Bohrnstedt GW, Mee AP (2002) Statistics for social data analysis. F.E. Peacock Publishers, Itasca. ISBN: 0-87581-448-4
Micheletti A, Morale D, Rapati D, Nolli P (2010) A stochastic model for simulation and forecasting of emergencies in the area of Milano. In: 2010 IEEE Workshop on Health Care Management (WHCM). IEEE, Venice, Italy. https://doi.org/10.1109/WHCM.2010.5441259
Micheletti A, Nakagawa J, Alessi AA, Morale D, Villa E (2016) A germ-grain model applied to the morphological study of dual phase steel. J Math Ind 6:12. https://doi.org/10.1186/s13362-016-0033-5
Moreno E, Casella G, Garcia-Ferrer A (2005) An objective Bayesian analysis of the change point problem. Stoch Environ Res Risk Assess 19:191–204. https://doi.org/10.1007/s00477-004-0224-2
Olsen LR, Chaudhuri P, Godtliebsen F (2008) Multiscale spectral analysis for detecting short and long range change points in time series. Comput Stat Data Anal 52:3310–3330. https://doi.org/10.1016/j.csda.2007.10.027
Rancoita PMV, Giusti A, Micheletti A (2011) Intensity estimation of stationary fibre processes from digital images with a learned detector. Image Anal Stereology 30(3):167
Sang L et al (2011) Simulation of land use spatial pattern of towns and villages based on CA–Markov model. Math Comput Model 54(3):938–943. https://doi.org/10.1016/j.mcm.2010.11.019
Schütz N, Holschneider M (2011) Detection of trend changes in time series using Bayesian inference. Phys Rev E 84(2):021120. https://doi.org/10.1103/PhysRevE.84.021120
Solazzo R, Donati M, Arfini F (2015) Impact assessment of greening and the issue of nitrogen-fixing crops: evidence from northern Italy. Outlook Agric 44(3):215–222. https://doi.org/10.5367/oa.2015.0215
This study has been supported by Fondazione Cariplo, within the research project “Evaluation of CAP 2015–2020 and taking action—CAPTION” (Project Id: 2017-2513). Funding has also been received from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie project BIGMATH, Grant Agreement No 812912.
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
About this article
Cite this article
Micheletti, A., Aletti, G., Ferrandi, G. et al. A weighted \(\chi ^2\) test to detect the presence of a major change point in non-stationary Markov chains. Stat Methods Appl 29, 899–912 (2020). https://doi.org/10.1007/s10260-020-00510-0
- Weighted \(\chi ^2\) test
- Inhomogeneous discrete time Markov chains
- Nonparametric inference