Analysis of Stochastic Patterns of Daily Minimum Extreme Temperature of Karachi in Global Climate Change Perspective

Analysis of Stochastic pattern of extreme temperature


  • Muhammad Atif Idrees DHA SUFFA University
  • Syed Ahmed Hassan University of Karachi, Karachi, Pakistan
  • Muhammad Arif Hussain DHA SUFFA University, Karachi, Pakistan



Temperature, Markov Chain, Transition Probability Matrix, Karachi


Effects of climate change are a critical and globally accepted phenomenon and gradually becoming inevitable and catching the attention of policymakers around the world. Temperature is a principal climatic factor and is defined as the degree or intensity of heat causing huge consequences on human beings’ lives. This paper suggests some stochastic approaches to do an analysis of the Karachi region’s daily minimum extreme temperature from Jan 1, 2010, to Dec 31, 2014. It is observed that the average daily minimum temperature fits the Markov chain and its limiting probability has reached steady-state conditions after 20 to 87 steps or transitions. The results indicate that after 20 to 87 days the distribution becomes stationary. The smaller steady state time represents the stationary of the data series, whereas long-term behavior shows non-stationarity in trend behavior in the respective seasonal time series. Furthermore, the overall annual dormancy of 24 oC to 31oC daily minimum temperature was analyzed early part of the
summer season. This study can be useful for weather variability forecasting.


M.A. Idrees, and H.A. Baig. Simulation and Modelling of Maximum and Minimum Temperature of Karachi. Journal of Natural Sciences Research 8: 19 (2018).

Y.Q. Tawfeek, F.H. Jasim, and M.H. Al-Jiboori. A Study of Canopy Urban Heat Island of Baghdad, Iraq. Asian Journal of Atmospheric Environment 14: (2020).

R.H. Daren, C.W. Richardson, C.L. Hanson, and G.L. Johnson. Simulating maximum and minimum daily temperature with the normal distribution. ASAE Annual Meeting, American Society of Agricultural and Biological Engineers (1998).

E.K. Mustafa, Y. Co, G. Liu, M.R. Kaloop, A.A. Beshr, F. Zaraoura, and M. Sadek. Study for Predicting Land Surface Temperature (LST) Using Landsat Data: A Comparison of Four Algorithms. Advances in Civil Engineering 2020: 16 (2020).

A.A. Squintu, G. Schrier, E. Besselaar, E. Linden, D. Putrasahan, C. Roberts, M. Roberts, E. Scoccimarro, R. Senan, and A.K. Tank. Evaluation of trends in extreme temperatures simulated by HighResMIP models across Europe. Climate Dynamics 56: 2389–2412 (2020).

J. Sillmann, M.G. Donat, J.C. Fyfe, and F.W. Zwiers. Observed and simulated temperature extremes during the recent warming hiatus. Environmental Research Letters 9 (2014).

N. Christidis, P.A. Stott, and S.J. Brown. The Role of Human Activity in the Recent Warming of Extremely Warm Daytime Temperatures. Journal of Climate 24: 1922-1930 (2011).

J. Turner, J.C. King, T.A. Lachlan-Cope, and P.D. Jones. Recent temperature trends in the Antarctic. Nature 418: 291-292 (2002).

P. You, and A. Jézéquel, Simulation of extreme heat waves with empirical importance sampling. Geoscientific Model Development 13: 763–781 (2020).

A.D. Stone, and A.J. Weaver. Daily maximum and minimum temperature trends in a climate model. Geophysical Research Letters 29: 70-1 (2002).

H. P. Dasari, S. Rui, J. Perdigao, and V.S. Challa. A regional climate simulation study using WRF-ARW model over Europe and evaluation for extreme temperature weather events. International Journal of Atmospheric Sciences 2014: 704079 (2014).

H.R. Shumway, and S.D. Stoffer. ARIMA models. Time series analysis and its applications 75-163 (2017).

S.A. Hassan, and M.R.K. Ansari. Nonlinear analysis of seasonality and stochasticity of the Indus River. Hydrological Sciences Journal Journal des Sciences Hydrologiques 55: 250-265 (2010).

H. Khan, and S.A. Hassan. Stochastic River Flow Modelling and Forecasting of Upper Indus Basin. Journal of Basic and Applied Sciences 11: 630-636 (2015).

W.K. Ching, S.F. Eric, and K.N. Michael. Higher‐order Markov chain models for categorical data sequences. Naval Research Logistics (NRL) 51: 557-574 (2004).

W.K. Ching, K.N. Michael, and E.S. Fung Higherorder multivariate Markov chains and their applications. Linear Algebra and its Applications 428: 492-507 (2008).

T. Liu. Application of Markov chains to analyze and predict the time series. Modern Applied Science 4: 162 (2010).

J. T. Chu, J. Xia, C-Y. Xu, and V. P. Singh Statistical downscaling of daily mean temperature, pan evaporation and precipitation for climate change scenarios in Haihe River, China. Theoretical and Applied Climatology 99: 149-161 (2010).

J. A. Buzacott, and J. G. Shanthikumar. Stochastic models of manufacturing systems. Pearson (1993).

D.J. Bartholomew. Recent developments in nonlinear stochastic modelling of social processes. Canadian Journal of Statistics 12: 39-52 (1984).

W. Feller 2008. An introduction to probability theory and its applications. John Wiley & Sons 2: 704 (1991).

J.G. Kemeny, and J.L. Snell. Mathematical Models in the Social Sciences. Introduction to Higher Mathematics. New York, Toronto, London, Blaisdell Publishing Company, A Division of Ginn and Company 145 (1963).

W.K. Ching. Iterative methods for queuing and manufacturing systems. Springer Science & Business Media (2001).

A. Berger. Linear Algebra Application~ Markov Chains. Mathematics (2007).

W.T. Anderson, and A.G. Leo. Statistical inference about Markov chains. The Annals of Mathematical Statistics 89-110 (1957).




How to Cite

Muhammad Atif Idrees, Syed Ahmed Hassan, & Muhammad Arif Hussain. (2023). Analysis of Stochastic Patterns of Daily Minimum Extreme Temperature of Karachi in Global Climate Change Perspective: Analysis of Stochastic pattern of extreme temperature. Proceedings of the Pakistan Academy of Sciences: A. Physical and Computational Sciences, 59(4), 37–44.



Research Articles