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Modelling changes in Arctic Sea Ice Cover: an application of generalized and inflated beta and gamma densities
Journal of Statistical Distributions and Applications volume 1, Article number: 3 (2014)
Abstract
A modelling framework for changing Arctic sea ice extent is developed reflecting different trends and seasonal extremes in nine Arctic subregions. Core subregions retain partial ice cover throughout the year, and in winter show complete ice cover, while in peripheral subregions, winter coverage is not complete, and there is no ice cover at all in the summer. A generalized beta representation is developed for monthly ice extents in core subregions, with inflation to model maximum winter extents. For peripheral subregions, a gamma time series with excess zeroes (representing summer sea ice absence) is developed. Different trend representations (deterministic vs. stochastic) are compared for nonextreme observations. Other potential applications of the generalized beta density allowing zero or maximum inflation are discussed.
1 Introduction
In recent years, Arctic sea ice has been declining with wider climatic implications. The latter are multifaceted and subject to discussion and uncertainty; see, for example, (Screen et al. 2013). However, one implication follows from the role of sea ice in regulating the global temperature via its ability (compared to the ocean surface) to reflect the sun’s radiation. The albedo of snowcovered sea ice is 0.90, meaning it reflects 90% of the sun’s radiation, whereas the ocean surface reflects only 10%. Less sea ice and more ocean surface will lead to a warmer Arctic, and contribute to global warming. There is also evidence that decline in Arctic sea ice is influencing atmospheric circulation (including the jet stream) within and beyond the Arctic, with impacts including winter cold surges in Europe and North America (Liu et al. 2012).
Satellite records of Arctic sea ice extent (available with full annual coverage since 1979) show a downward trend, albeit with fluctuations such as unusually low points in 1990 (Serreze et al. 1995) and in 2007 (Comiso et al. 2008). For the Arctic as a whole, the rate of decline in summer ice extent has been greater (Stroeve et al. 2012), with an 8.2% linear rate of decline (per decade) in September ice extent between 1979 and 2011, as against a 3.9% linear rate of decline for March ice extent (see Table 1).
However, the Arctic extent totals in Table 1 aggregate over subregions with dissimilar trends and seasonal extremes; for example, some subregions show substantial falls in extent in both winter and summer months. The present study accordingly reviews issues in modelling sea ice trends for Arctic subregions, and includes short term projections three years beyond the range of the observed data. While a number of forecasts of sea ice cover have been developed (e.g. Stroeve et al. 2007), these generally consider the entire Arctic, and do not consider subregional differences. The observed data are monthly NASA satellite records from 1979 to 2011 (396 observations), disaggregated to nine Arctic subregions. These are the central Arctic Ocean, the Canadian Archipelago, Hudson Bay, Sea of Okhotsk, Bering Sea, Baffin Bay, Greenland Sea, KaraBarents Sea, and Gulf of St. Lawrence (see Figure 1). The observed data were extracted in June 2013 at http://neptune.gsfc.nasa.gov/csb/index.php?section=59.
There are important regional differences in trends and seasonal patterns according to location (e.g. Arctic core vs. periphery) that are relevant to statistical analysis and forecasts. These include modelling seasonal extremes. For central regions there is still some summer cover but a reversion to complete cover (maximum possible extent) in winter months. To represent this pattern, a generalized beta representation is applied, including "maximum inflation" to account for winter total coverage.
By contrast, in some noncore regions such as the Sea of Okhotsk and Bering Sea, winter extents do not cover the entire sea region, and there is no ice cover in the summer. For these regions, a gamma density with excess zeroes (representing summer sea ice absence) is developed. In both situations, regression models including trend and seasonal components are applied to represent the change in mean sea ice extent, while a separate logit regression is applied to the probability of inflation. Contrasting stochastic and deterministic trend representations are evaluated. Estimation is based on monthly observations in the first 31 years of satellite observations (19792009) with crossvalidation using the remaining two years of observed data as a test sample. Bayesian estimation and inference are implemented using the Winbugs package via Markov chain Monte Carlo (MCMC) methods (Lunn et al. 2009).
2 Generalized beta time series regression
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 m^{2} throughout January to March while the Canadian Archipelago had readings of 0.751 mn k m^{2} for January through to April. Figure 2 shows monthly extent totals for the central Arctic ocean during 200711 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 nonwinter 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 lognormal for extent readings y with ceiling d) but this precludes any analysis of the factors producing seasonal extremes.
Consider the beta distribution on (0,1), with density function given by
with a > 0,b > 0. An alternative representation (Ospina and Ferrari 2010) involves mean and precision parameters (μ,ϕ), where a = μ ϕ,b = (1  μ)ϕ, namely
with ϕ > 0, and 0 < μ < 1. This form facilitates separate modelling of mean and variance trends (Huang and Oosterlee 2008). For values (a,b) apart from a = b = 1, the Beta(a,b) density has zero mass at the extreme values 0 and 1, and zeroinflated or oneinflated versions of the beta need to be applied (Ospina and Ferrari 2010). Let g = 0 or 1, then inflation at either boundary is achieved by the mechanism
where α_{ g } is an inflation probability.
The generalized beta is obtained by extending the support interval to an arbitrary bounded interval (c,d) (with d > 0) via a linear transformation y = c + z(d  c) (PhamGia and Duong 1989), so that
with equivalent representation
with mean c + (d  c)μ, and variance (d  c)^{2}μ(1  μ)/(ϕ + 1).
For the generalized beta in (1), inflation will need to be applied for values occurring at the boundary points, when y = c or y = d (this may be termed minimum and maximum inflation). The maximum inflated version of the generalized beta is particularly relevant to the sea ice application and has the form
In the generalized beta applied to core Arctic regions, c = 0 while d is the maximum winter extent (namely d_{1} = 7.158 mn k m^{2} for the central Arctic ocean, d_{2} = 0.751mn k m^{2} for the Canadian Archipelago, and d_{3} = 1.233 mn k m^{2} for Hudson Bay).
While summer minimum extents in the central Arctic and Canadian Archipelago remain well in excess of zero, those in Hudson Bay are becoming relatively small, e.g. y = 0.025 mn k m^{2} in September 2010. This raises the possibility of needing to represent both maximum and minimum inflation in the generalized beta. This can be handled by the mechanism
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 crossvalidatory and outofsample 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.
Two options for the trend are considered. One option is a stochastic trend, with random variation around a central linear trend. To allow for steeper declines in some months, a discrete mixture is implemented via
with ρ_{ m } ∼ Bern(π_{ m }) being binary indicators. With the constraint δ_{11} < δ_{01},φ_{1s} represents the stronger downward trend.
The other form of trend assumption (deterministic) involves a linear trend in years s_{ t } combined with a short term AR1 lag effect in extents y_{ t }. The latter represents carry over between successive months; for example, if September extent is relatively low in a particular year, then October extent may also be relatively low. The linear trend may vary between months and by broad subperiod. For example, (Comiso et al. 2008) report a stronger decline during 19962007 than 197996. Here we consider three subperiods p = 1,...,3 of 12 years, including outofsample years (201214), namely 19791990, 19912002, and 20032014. The trend parameter for a particular time point is chosen by a monthly specific discrete mixture between guide linear trend parameters, specific to broad period, Γ_{0p} and Γ_{1p}, with Γ_{1p} < Γ_{0p}. The AR1 lag effect is also taken to vary by month, with normal priors for each monthly lag parameter. Thus for month m = m_{ t }, and year s_{ t },
where the linear trend γ_{1m} for month m = m_{ t } in period p is chosen using a discrete mixture
Remaining aspects of the model are applicable across different representations of trend. Seasonal (monthly) effects are represented by a Fourier series (Höhle and Paul 2008),
where ω = 2π/M, with M = 12, and J_{1} is the number of harmonics. To allow for changing precision it is assumed that
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.
To represent extreme data (complete winter coverage), a logit regression is used to model the probabilities α_{ dt } of maximum inflation, with form
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 nonextreme 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 ydata), 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 seaice type (e.g. Fissel et al. 2011) between perennial multiyear sea ice and firstyear ice, so that subregion observations become (y_{1},y_{2},y_{3}) for area covered by multiyear ice, area covered by firstyear ice, and area without ice cover respectively.
3 Inflated gamma time series regression
In what are here denoted "peripheral" Arctic regions (Sea of Okhotsk, Bering Sea, Gulf of St. Lawrence), the winter ice extents do not provide complete ice cover of the sea region, while in at least one peak summer month (July, August, September) there is no ice cover at all. For example, Figure 3 shows the extent series for the Bering Sea for 200711. Let y_{ t } denote the observed extents (mn k m^{2}) for these regions following a gamma density,
with mean μ and variance μ^{2}/ϕ. A zero inflated version of the gamma is motivated by the need to represent absence of ice cover in summer months, as in the mechanism
Let {μ_{ t },ϕ_{ t },α_{0t}} denote the series of parameters underlying the observations y_{ t }. The link for μ_{ t } depends on which of two options for trend is applied.
One option is a stochastic trend, with loglink
where Δ_{ ms } is as in (2) above. The other option, a deterministic trend, combines a linear trend in years s with a short term lag effect in the response. The linear trend may vary between months, and also between subperiod p. As above, the trend parameter for a particular time point is chosen according to a monthly specific discrete mixture between guide linear trends, specific to broad period, Γ_{0p} and Γ_{1p}, with Γ_{1p} < Γ_{0p}. Lag effects are also taken to vary by month. To avoid an explosive impact of the lag effect (Blundell et al. 2002, Jung et al. 2006), the following link scheme provides the deterministic trend option,
with the constraint ζ_{2m} ≥ 0. For example, one may take ζ_{2m} to have exponential or gamma priors.
Remaining aspects of the model are applicable across different representations of trend. Seasonal effects ϑ_{ m } are represented by a Fourier series, as in (3), and precision parameters modelled using the regression form (4). To represent summer extremes (no ice cover at all), a logit regression for the probability of zero inflation is used, with
For the remaining three Arctic subregions (Baffin Sea, Greenland Sea, KaraBarents), the data series demonstrate neither maximum recurrence in winter, nor complete summer ice disapperance as yet. The maximum values observed for these three series (in millions k m^{2}) are 1.766 (Baffin Sea, March 1993), 1.115 (Greenland Sea, January 1982) and 2.168 (KaraBarents, April 1979), whereas recent winter maxima are below these levels. Recent summer extents might be taken to indicate incipient ice disappearance in summer, such as a reading of 0.04 mn km^{2} in the Baffin Sea in August 2011. However, for these regions a gamma regression may be adopted, without any inflation mechanism. The time series regression therefore involves {μ_{ t },ϕ_{ t }}, and stochastic and deterministic trend options, as in (5) and (6) respectively, may be compared.
4 Model fit and comparison
4.1 Analysis framework
To illustrate a full comparative analysis, three subregions 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 insample crossvalidation for the two year period 20102011, namely for t = T + 1,...,T_{1} with T_{1} = 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 BrooksGelmanRubin 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,u_{1},γ_{0},γ_{1}}, 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 J_{1} = J_{2} = 3 harmonics, as insignificant regression effects {β,ξ} were obtained at higher numbers of harmonics.
Predictions are obtained by generating replicate inflation indicators D_{new,t} ∼ Bern(α_{ dt }), and replicate scaled beta values d z_{new,t}, where z_{new,t} ∼ Beta(μ_{ t }ϕ_{ t },ϕ_{ t }ϕ_{ t }μ_{ t }). Predictions are then
Let M_{new,t} and V_{new,t} be the posterior means and variances of y_{new,t}. Insample predictive fit (within the training dataset from January 1979 through to the end of 2009) is based on the Lcriterion (Laud and Ibrahim 1995), namely the square root of
Crossvalidatory predictions beyond T = 372 (for the 24 months in 2010 and 2011) are made by treating $\{{y}_{T+1},\mathrm{...},{y}_{{T}_{1}}\}$ as missing data, and generating predictions $\{{y}_{\mathit{\text{new}},T+1},\mathrm{...},{y}_{\mathit{\text{new}},{T}_{1}}\}.$ These can be compared with observed test data y_{obs,t} for 2010 and 2011. Cross validatory fit using the L criterion, denoted CVL, is based on
which will reflect the precision of future predictions (P_{1}) as well as the fit (P_{2}). 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).
Table 2 shows posterior means and variances during the crossvalidation period, while Table 3 summarises the Lcriteria for both crossvalidation periods (test data) and the insample estimation (training data). It is apparent that the deterministic trend has a better crossvalidation performance but a worse insample performance. In particular, the deterministic trend is less adaptable to anomalous observations such as occurred during the summers of 1990 and 2007. Both models are effective in reproducing the original data satisfactorily in terms of coverage obtained by y_{new,t}.
The better outofsample cross validation under the deterministic trend is due to lower predictive variability (P_{1} = 2.06 under the deterministic trend, whereas P_{1} = 6.13 under the stochastic trend); pure fit is broadly similar between the two options (P_{2} = 0.41 under deterministic trend, as against P_{2} = 0.34 under stochastic trend). Figure 5 shows observations and predictions for 201011, as well as outofsample 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 m^{2}. 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 197990 and for the partially observed third period (200314), with estimates based on observations for 200309. 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
The Sea of Okhotsk is one of three regions with recurrent summer disappearance of ice cover. Table 5 shows that unlike the trend for the whole Arctic as an aggregate (e.g. see Table 1), the most pronounced declines in average extent in the Sea of Okhotsk have been in the first quarter months (January, February, March). In applying gamma time series regression with zero inflation, priors are as in Section 4.2, except that exponential E(1) priors are adopted for the lag parameters ζ_{2m}, in order to ensure that μ_{ t } is positive  see equation (6). The Fourier seasonal effects series included in representations for μ_{ t } and the inflation probabilities α_{0t} have J_{1} = J_{2} = 3 harmonics. Predictions are obtained by generating replicate inflation indicators D_{new,t} ∼ Bern(α_{0t}), and replicate gamma values w_{new,t} ∼ Gamma(ϕ_{ t },ϕ_{ t }/μ_{ t }). Predictions are then obtained as
Table 6 summarises the Lcriteria for the crossvalidation period (when predictions are compared to test data) and for the insample 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 m^{2}) 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 subperiod (19791990). It may be noted that the actual data for the Sea of Okhotsk show a fall in annual average extent (mn k m^{2}) 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).
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 m^{2}) fell from 0.98 in 197989 to 0.82 in 19902000 and 0.78 in 200111, while the average September extent fell from 0.37 (197989) to 0.34 (19902000) and 0.25 (200111). For this subregion, a deterministic trend model has better insample fit and crossvalidatory 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
It is relevant to assess how far aggregated predictions (totalling over models for each of the nine regions) compare with predictions from a single model applied to the extent time series for the entire Arctic sea (i.e. to the data summarised in Table 1). The aggregated predictions are based on training data for 19792009 and combine

a)
generalized beta regressions (with maximum inflation) applied to extent data for the Central Arctic, Canadian Archipelago, and Hudson Bay;

b)
zero inflated gamma regressions applied to extent data for the Sea of Okhotsk, Bering Sea, and Gulf of St.Lawrence;

c)
gamma regressions without inflation applied to extent data for the Baffin Sea, Greenland Sea, and KaraBarents.
The deterministic trend model is adopted, and MCMC iterations 25,00026,000 are cumulated over the nine models, providing posterior summary statistics (mean, variance, 2.5% percentile, 97.5% percentile) for crossvalidatory predicted extents in 201011 across the entire Arctic region. By contrast, the single region approach (applied to the extent series for 19792009 encompassing the entire Arctic region) involves a gamma regression with deterministic trend  since the stochastic trend option provided much less precise predictions to 201011.
The comparative Lcriteria for crossvalidatory fit (over the 24 months in 201011) are 2.43 (aggregated predictions over subregions) and 3.70 (single region approach). Figure 8 shows the crossvalidatory predictions (201011) and outofsample predictions (201214) for the combined forecast aggregating over subregion models. March extents (in mn km^{2}) 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 km^{2}) 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 subregion 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 subregional series at http://neptune.gsfc.nasa.gov/csb/index.php?section=59). In fact, the March 2012 figure of 15.21 mn k m^{2} (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 m^{2}.
5 Summary and conclusions
This paper has considered subregional aspects of the observed downward trend in Arctic ice sea cover. Arctic subregions differ in observed trends and also in seasonal extremes. Thus some regions still have total winter cover but partial summer cover, while other regions have partial winter cover and no ice cover in summer. While downward trends in sea ice extent across the Arctic as a whole show a stronger summer decline, this is not necessarily the case when subregions are considered.
These differences are important for choosing the appropriate distribution, and inflation mechanism. Subregional differences in ice loss may also be important for assessing relatively localized impacts on climate or economic activities (e.g. Fissel et al. 2011). There may also be benefits in prediction (see Section 4.4) through considering such subregional differences and in aggregating over regionspecific models.
Here generalized beta densities (with maximum inflation) and gamma regression (with zero inflation) have been used to represent recurrent winter maxima and summer ice disappearance respectively. Other applications of this methodology may be envisaged, outside time series applications. Possible applications of the generalized inflated beta density are discussed in Section 2.3. It may be noted that transformations of ice extent such as w_{ t } = log(1 + y_{ t }) might be envisaged as ways of avoiding beta or gamma regressions, and instead leading possibly to modelling using normal or skewnormal likelihoods. However, the problem of seasonal extreme inflation at particular values remains, and the methodology proposed in the paper provides a suitable representation for such extremes or for explaining them.
Another approach, as in compositional data analysis, is to focus on the ratios r_{ t } = y_{ t }/d_{ t } to the maximum, in the case when the data has two categories (e.g. area covered by sea ice y_{ t }, remaining area d_{ t }  y_{ t }). Generally compositional data analysis involves Gaussian likelihoods applied to logtransformed ratios. This method can adjust for zero inflation (e.g. Butler and Glasbey, 2008), but does not generate direct inferences or outofsample predictions for the extent data themselves.
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Keywords
 Sea ice extent
 Seasonal extremes
 Inflated generalized beta
 Inflated gamma
 Bayesian