Modelling changes in Arctic Sea Ice Cover: an application of generalized and inflated beta and gamma densities

2.1 Application of the generalized beta to sea ice extents

As mentioned above, while there is decline in Arctic sea ice overall, there are considerable regional differences in trends and seasonal ice extent. What are here denoted "core" Arctic sea regions (namely the central Arctic Ocean, the Canadian Archipelago, and Hudson Bay) retain partial ice cover throughout the year, and in winter months show complete ice cover. For example, in 2011, the central Arctic had readings of 7.158mn k m² throughout January to March while the Canadian Archipelago had readings of 0.751 mn k m² for January through to April. Figure 2 shows monthly extent totals for the central Arctic ocean during 2007-11 and illustrates maximum inflation in winter. The trend in these three regions is similar to that in the Arctic ocean considered as an aggregate, namely stronger declines in summer extent.

A time series representation needs to express the annual reversion to complete cover (maximum recurrence) in winter months, together with the irregular trend for declining extent in non-winter months. The application of a generalized form of the beta density is motivated by the fact that the observed ice extents y can be seen as ratios r = y/d to a maximum possible extent d, though substantive interest is in trends in extents y. It is important that the bounded nature of the response is included in any model. Another possibility might be some form of truncated sampling mechanism (e.g. a log-normal for extent readings y with ceiling d) but this precludes any analysis of the factors producing seasonal extremes.

where α _g is an inflation probability.

with mean c + (d - c)μ, and variance (d - c)²μ(1 - μ)/(ϕ + 1).

In the generalized beta applied to core Arctic regions, c = 0 while d is the maximum winter extent (namely d₁ = 7.158 mn k m² for the central Arctic ocean, d₂ = 0.751mn k m² for the Canadian Archipelago, and d₃ = 1.233 mn k m² for Hudson Bay).

where the vector of probabilities (α _c ,α _c ,1 - α _c  - α _d ) should be modelled using a multinomial logistic.

2.2 Generalized beta time series regression with maximum inflation for sea ice extents

Let {μ _t ,ϕ _t ,α _{d t} } denote the series of parameters underlying the y _t series, m = m _t represent the month that observation t corresponds to, and s = s _t represent the year corresponding to observation t (e.g. s = 2 in 1980 for observations t = 13,..,24 and s = 31 for observations t = 361,..,372). A parsimonious time series model is sought (Ledolter and Abraham 1981), combining close fit with low predictive variability, especially for cross-validatory and out-of-sample predictions. As discussed in Section 4, these aspects of fit are assessed using a posterior predictive fit criterion (Laud and Ibrahim 1995). A parsimonious model for the level of the series is expressed by a logit regression in μ _t ,

logit(μ _t ) = Δ _ms  + ϑ _m , where Δ _ms represents trend for each combination of month m and year s, and ϑ _m represents seasonal effects.

with ρ _m  ∼ Bern(π _m ) being binary indicators. With the constraint δ₁₁ < δ₀₁,φ_1s represents the stronger downward trend.

namely a linear trend (varying by month) in year units. For example, (Stroeve et al. 2012) find evidence of increased variability in overall Arctic sea ice extent, especially in summer months.

A trend element in the inflation probability is not included as it would be confounded with the trend model in the mean.

2.3 Other generalized beta applications

While the application here focuses on sea ice extent and a time series application, the generalized inflated beta with mechanisms or regressions for both extreme and non-extreme observations has potential applications in other settings where the observations can be regarded as ratios r _i  = y _i /d _i of actual extents to a maximum extent d _i , but substantive interest is in the extents y _i . The extents may be, inter alia, expressed in spatial units (e.g. areas in millions of square kilometres) or time units (e.g. durations in hours). As an example with time extents, one might analyse hours with cloud cover y _i in relation to daylight hours d _i , while a spatial application might consider desertified extents y _i in relation to total area extents d _i .

Data of this form can be considered as a form of compositional data, and widely used methods (Aitchison and Egozcue 2005, Butler and Glasbey 2008) focus on the ratios r _i , or more specifically the log ratios. Spatial applications involving compositional data and zero inflation have been described in (Leininger et al. 2013), but also focus on the ratios.

However, for policy purposes, the interest may be in trends or patterns in the extents themselves (i.e. the y-data), rather than in the ratios, as is the case with the sea ice extents. Alternatively stated, "compositions provide information only about the relative magnitudes of the compositional components and so interpretations involving absolute values... cannot be justified" (Aitchison and Egozcue 2005) p. 839. Thus in the case of desertification (e.g. Zhao et al. 2010), the substantive focus may be on spreading desertification, implying analysis of desertified extents y _i . Some areas may be totally desertified with y _i  = d _i (maximum inflation). Regression modelling of desertification extents would then need to include a mechanism or regression describing maximum inflation, as would spatial forecasting (or interpolation) of desertified extents in situations where comprehensive assessment of desertification status is only available for some area units.

The generalized beta density with maximum or zero inflation might be potentially extended to Dirichlet density applications, and to generalized inflated Dirichlet densities parallel to equation (1), where there are more than two categories and where extreme observations can occur. For example, with three categories the observations would be (y_1i,y_2i,y_3i), with $\sum _k{y}_{\mathit{\text{ki}}}=d_i,$ and maximum inflation occurring when any y _ki  = d _i . The inflation probabilities can be modelled using a multiple logistic. For example, in the sea ice application, one may distinguish by sea-ice type (e.g. Fissel et al. 2011) between perennial multi-year sea ice and first-year ice, so that sub-region observations become (y₁,y₂,y₃) for area covered by multi-year ice, area covered by first-year ice, and area without ice cover respectively.

4.1 Analysis framework

To illustrate a full comparative analysis, three sub-regions are considered as representative of the three sets of regions described above: the central Arctic, the Sea of Okhotsk and the Greenland Sea. Different models are applied to monthly observations from January 1979 to December 2009 (namely for t = 1,..,T with T = 372), with in-sample cross-validation for the two year period 2010-2011, namely for t = T + 1,...,T₁ with T₁ = 396. Out of sample predictions are made for a further 36 months, January 2012 to December 2014. Posterior inferences are based on the second halves of two chain runs of 50,000 iterations, with convergence assessed using Brooks-Gelman-Rubin statistics (Brooks and Gelman 1998).

4.2 Generalized beta regression with maximum inflation

The central Arctic is one of three sea regions with recurrent winter maxima and partial summer ice cover, and Figure 4 shows the trend in yearly averages for the central Arctic, with the anomalous low cover years (1990, 2007) apparent. In applying the generalized beta regression, normal priors with mean zero and variance 1000 are adopted on fixed effect parameters {β,ξ,η,Γ,δ,u₁,γ₀,γ₁}, while the lag parameters γ_2m are assigned N(0,1) priors. A gamma prior with shape 1 and scale 0.001 is adopted for the precision $1/{\sigma }_u^2,$ while Beta(1,1) priors are adopted for the π _m . The Fourier seasonal representations for the means μ _t and inflation probabilities α _dt have J₁ = J₂ = 3 harmonics, as insignificant regression effects {β,ξ} were obtained at higher numbers of harmonics.

which will reflect the precision of future predictions (P₁) as well as the fit (P₂). To further assess predictive performance, a check is made whether observed extents are within 95% intervals of y _new . In a model that effectively reproduces the data, predictive coverage is at or above 95% (Gelfand 1996, p. 158).

The better out-of-sample cross validation under the deterministic trend is due to lower predictive variability (P₁ = 2.06 under the deterministic trend, whereas P₁ = 6.13 under the stochastic trend); pure fit is broadly similar between the two options (P₂ = 0.41 under deterministic trend, as against P₂ = 0.34 under stochastic trend). Figure 5 shows observations and predictions for 2010-11, as well as out-of-sample predictions through to December 2014, under the deterministic trend option. Posterior mean predictions for September extent in 2012, 2013 and 2014 are respectively 3.88, 3.78 and 3.68mn k m². Table 4 summarizes selected parameters from the deterministic trend model. The strongest downward trends in mean extent, as represented by the guide parameters Γ_0p and Γ_1p, are for 1979-90 and for the partially observed third period (2003-14), with estimates based on observations for 2003-09. Parameters η_2m for trends in precision are nonsignificant in summer months (i.e the 95% credible intervals straddle zero), though the 95% credible intervals for η_2m in July and August are biased to negative values, namely (-0.052,0.009) and (-0.058,0.009).

4.3 Gamma regression time series

Table 6 summarises the L-criteria for the cross-validation period (when predictions are compared to test data) and for the in-sample estimation period (when predictions compared to training data). The deterministic trend has better performance (see Figure 6 for actual and fitted series). March extents (in mn k m²) are predicted to fall to 0.895 in 2012, 0.889 in 2013 and 0.879 in 2014 as compared to 0.964 in 2011. The strongest downward trend in mean extent as shown by the deterministic trend guide parameters {Γ_0p,Γ_1p} is actually in the first sub-period (1979-1990). It may be noted that the actual data for the Sea of Okhotsk show a fall in annual average extent (mn k m²) from 0.58 in 1979 to 0.37 in 1990. The posterior means (sd’s) for Γ_1p are respectively -0.016 (0.010), -0.0085 (0.0044), and -0.014 (0.005).

	Trend model
	Stochastic	Deterministic
In-sample fit (1979-2009)	2.95	2.43
	Stochastic	Deterministic
Cross validation fit (2010-11)	1.11	0.68
	% of observations within 95% credible intervals of posterior predictions
	Stochastic	Deterministic
In-sample period	97.6	97.6
Cross-validation period	100.0	95.8

For the three remaining sea regions (e.g. the Greenland Sea) with partial summer cover and no winter maximum recurrence, a gamma regression may be applied. In fact, like the Sea of Okhotsk, the Greenland Sea has declines in extent across both winter and summer months. For example, the average March extent (in mn k m²) fell from 0.98 in 1979-89 to 0.82 in 1990-2000 and 0.78 in 2001-11, while the average September extent fell from 0.37 (1979-89) to 0.34 (1990-2000) and 0.25 (2001-11). For this sub-region, a deterministic trend model has better in-sample fit and cross-validatory fit (see Figure 7). March extents are predicted as 0.767 in 2012, 0.766 in 2013 and 0.762 in 2014 as compared to 0.752 in 2011, and 0.764 in 2010. September extents are predicted as 0.294 in 2012, 0.290 in 2013 and 0.290 in 2014 as compared to 0.332 in 2011, and 0.264 in 2010.

4.4 Implications for prediction over entire Arctic

The deterministic trend model is adopted, and MCMC iterations 25,000-26,000 are cumulated over the nine models, providing posterior summary statistics (mean, variance, 2.5% percentile, 97.5% percentile) for cross-validatory predicted extents in 2010-11 across the entire Arctic region. By contrast, the single region approach (applied to the extent series for 1979-2009 encompassing the entire Arctic region) involves a gamma regression with deterministic trend - since the stochastic trend option provided much less precise predictions to 2010-11.

The comparative L-criteria for cross-validatory fit (over the 24 months in 2010-11) are 2.43 (aggregated predictions over sub-regions) and 3.70 (single region approach). Figure 8 shows the cross-validatory predictions (2010-11) and out-of-sample predictions (2012-14) for the combined forecast aggregating over sub-region models. March extents (in mn km²) are predicted as 14.42 in 2012, 14.35 in 2013 and 14.31 in 2014, as compared to actual extents of 14.34 in 2011, and 14.88 in 2010. September extents (in mn km²) are predicted as 4.97 in 2012, 4.84 in 2013 and 4.71 in 2014, as compared to actual totals of 4.60 in 2011, and 4.90 in 2010.

Although sub-region data are not (at the time of writing) available beyond 2011, totals for the entire Arctic are available from ftp://sidads.colorado.edu/DATASETS/NOAA/G02135/. (Note that the latter differ slightly before 2012 from entire Arctic extents based on totalling the sub-regional series at http://neptune.gsfc.nasa.gov/csb/index.php?section=59). In fact, the March 2012 figure of 15.21 mn k m² (NSIDC, 2012) was the highest March average ice extent since 2008. By contrast, the September 2012 figure was anomalously low at 3.61 mn k m².