The stochastic view used in climate sciences: (some) perspectives from (some of) mathematical statistics

TL;DR

This chapter surveys the stochastic view in climate sciences by connecting climate statistics with statistical methodology. It highlights themes such as robust time-series modelling, barrier-crossing prediction with uncertainty via confidence distributions and curves, monitoring parameter stability, fusing information from diverse sources, and assessing extreme-event probabilities. Concrete illustrations include skiing-days regression, Oslo temperature anomalies with segmented regression, the Hjort Liver Index linked to Kola temperatures, and extreme-value analyses of sprint records; the work emphasizes practical uncertainty communication and cross-disciplinary collaboration. The findings underscore the value of combining methodological rigor with climate science insights to improve inference, prediction, and decision-relevant understanding of climate-related phenomena.

Abstract

Climate statistics is of course a very broad field, along with the many connections and impacts for yet other areas, with a history as long as mankind has been recording temperatures, describing drastic weather events, etc. The important work of Klaus Hasselmann, with crucial contributions to the field, along with various other connected strands of work,is being reviewed and discussed in other chapters. The aim of the present chapter is to point to a few statistical methodology themes of relevance for and joint interest with climate statistics. These themes, presented from a statistical methods perspective, include (i) more careful modelling and model selection strategies for meteorological type time series; (ii) methods for prediction, not only for future values of a time series, but for assessing when a trend might be crossing a barrier, along with relevant measures of uncertainty for these; (iii) climatic influence on marine biology; (iv) monitoring processes to assess whether and then to what extent models and their parameters have stayed reasonably constant over time; (v) combination of outputs from different information sources; and (vi) analysing probabilities and their uncertainties related to extreme events.

Figures (9)

• Figure 2.1: Left panel: the number of skiing days per year, at the location Bjø rnholt close to Oslo, from 1896 to 2022, though with a gap in the series, with no records from 1938 to 1954. The dashed line is the estimated regression from the four-parameter autoregressive model, with 90 percent confidence band. Right panel: The confidence curve ${\rm cc}(\rho)$ for the autocorrelation parameter of the residuals $y-a-bx$, with point estimate $0.303$ and 95 percent interval $[0.116,0.490]$.
• Figure 3.1: Average monthly temperature anomalies, land and sea, at Oslo position, from 1901 to 2024. Left panel: for month October, with pointwise 90 percent confidence interval for the regression line, along with 90 percent prediction intervals for the values of $y$ themselves, for 2030 and 2050. Right panel: for month September, using connected segmented regression. The estimated break-point, from small derivative to strong derivative, is at 1986.
• Figure 3.2: At which year $x_0$ in the future will the mean level $a+b(x_0-\bar{x})$ reach the threshold $y_0$, defined as 1.5$^\circ$ C above the average value over 1901--2000, for the month in consideration? Here are confidence curves ${\rm cc}(x_0)$, for October temperatures (left panel) and January temperatures (right panel). For October, the point estimate is 2056, with 90 percent interval from 2024 to 2139, skewed to the right. For January, the skewness is stronger, and the confidence curve never goes above 0.864; confidence intervals at levels higher than this will include infinity, i.e. the threshold will never be reached. The climate increase is stronger for the summer months than for the winter months.
• Figure 3.3: Left panel: log-likelihood profile for the segmented regression model, for the September series, with maximum very significantly exceeding both the log-likelihood maxima $\ell_{\max,1}$ of linear regression and $\ell_{\max,2}$ of quadratic regression. Right panel: monitoring plot $B_n$ for the regression coefficient $b$ in $y_i=a+bx_i+\varepsilon_i$, for months January, February, June, September; the plot for September reaches above the 0.95 null distribution quantile 2.856, indicates that $b$ for that month has not remained constant.
• Figure 4.1: Left panel: the Hjort liver index, 1859--2013 (percentage of liver in the skrei, the Northeastern Atlantic cod) with the annual Kola temperature, 1921--2013 (in degrees Celsius). Right panel: running t tests plot for Kola temperatures 1921 to 2013, one curve for each month.