Inferring Changes in Arctic Sea Ice through a Spatio-Temporal Logistic Autoregression Fitted to Remote-Sensing Data

Abstract

Arctic sea ice extent (SIE) has drawn increasing attention from scientists in recent years because of its fast decline in the Boreal summer and early fall. The measurement of SIE is derived from remote sensing data and is both a lagged and leading indicator of climate change. To characterize at a local level the decline in SIE, we use remote-sensing data at 25 km resolution to fit a spatio-temporal logistic autoregressive model of the sea-ice evolution in the Arctic region. The model incorporates last year’s ice/water binary observations at nearby grid cells in an autoregressive manner with autoregressive coefficients that vary both in space and time. Using the model-based estimates of ice/water probabilities in the Arctic region, we propose several graphical summaries to visualize the spatio-temporal changes in Arctic sea ice beyond what can be visualized with the single time series of SIE. In ever-higher latitude bands, we observe a consistently declining temporal trend of sea ice in the early fall. We also observe a clear decline in and contraction of the sea ice’s distribution between ${70}^{\circ }\mathrm{N}$–${75}^{\circ }\mathrm{N}$, and of most concern is that this may reflect the future behavior of sea ice at ever-higher latitudes under climate change.

1. Introduction

Arctic sea ice cover has received increasing attention in recent years because of its fast decline in the Boreal summer and early fall, both a lagged and leading indicator of climate change. Geoscientists have observed a clear declining trend in Arctic sea ice by analyzing the aggregated sea ice data either spatially or temporally, e.g., [1,2,3,4,5,6,7,8]. Declining sea ice increases the amount of dark ocean water and results in an albedo–ice feedback effect, where a darker ocean absorbs more solar radiation, leading to further loss of sea ice [9,10,11]. Although the melting of sea ice in summer may provide new shipping routes, the Arctic is an important component of Earth’s energy system, and its loss of sea ice leads to climate change in other regions, e.g., [12,13,14,15,16]. Therefore, monitoring and modeling the spatio-temporal changes in Arctic sea ice is of critical importance to understanding climate change.

There are many geophysical factors related to the melting of sea ice, such as surface atmospheric temperature [17], the fragility of thinner sea ice [18], the occurrence of the polar anticyclone [19], and reflected solar radiation [20], to name a few, which cause difficulties in characterizing and predicting the presence of Arctic sea ice. Previous analyses have often focused on studying the temporal trend of the obvious summary, sea ice extent (SIE), which is defined as the total area of ice grid cells (grid cells with a sea-ice concentration ≥ 15%) in the Arctic region. Figure 1 shows the observed Arctic SIE in every September over the past two decades, which is derived from the NOAA/NSIDC Climate Data Record of sea ice concentration [21]. By summing up the areas of ice grid cells, we obtained the observed Arctic SIE in each year. An overall declining trend is easily apparent.

The ice/water status in zonal regions is studied here: a zonal band centered at ${\mathrm{lat}}_0$ is considered, and from binary ice–water observations on 25 km-resolution remote-sensing grid cells, the proportion of ice grid cells is calculated in each of the ninety $4^{\circ }\times 1^{\circ }$ longitude–latitude windows that straddle ${\mathrm{lat}}_0$. Then, visualization in the form of boxplots is made from these ninety ice/water proportions. Figure 2 shows a time series of such boxplots five-years apart in latitude bands at ${70}^{\circ }$N, ${72.5}^{\circ }$N, ${75}^{\circ }$N, …, ${85}^{\circ }$N. This summary adds a spatial dimension to the time series of SIE shown in Figure 1, illustrating the zonal behavior of sea-ice loss over time. By following the average ice proportions (dots) for each zone, we see that at ${70}^{\circ }$N, the lowest latitude band, there has been almost no September sea ice since 2000; at ${72.5}^{\circ }$N and ${75}^{\circ }$N, the ice proportions contracted towards zero around 2010; and at ${77.5}^{\circ }$N and ${80}^{\circ }$N, a pre-contraction pattern like that seen earlier at the lower latitudes is emerging.

These exploratory analyses of the observations are useful to detect changing patterns of Arctic sea ice; however, they are based on the aggregated binary data and typically do not come with proper quantification of uncertainty that is obtained by incorporating the spatio-temporal correlations of the data. Statistical models account for the spatio-temporal variability in the underlying ice–water process, as well as the uncertainty inherent in remote-sensing data, and hence they permit valid uncertainty quantification of the resulting summaries. Specifically, the model fitted in this paper allows an estimation of the probability of a grid cell being an ice grid cell in the past and present.

In this paper, we focus on the estimation problem, aiming to use last year’s ice/water statuses at the neighboring locations of a location $\mathbf{s}$ to estimate its ice/water status in the current year. The main goal is to obtain the estimated ice probability at each spatial location, which may contain more information than a binary outcome. The resulting surface of the estimated ice probabilities is generally smooth and less noisy. Then, various summaries can be obtained based on the estimated ice probabilities to infer the spatio-temporal changes in Arctic sea ice.

Our initial analysis is exploratory, focusing on Arctic binary sea ice data in the month of September, which is when the Arctic SIE typically reaches its minimum in any year and when decline is most likely to occur. Then, a Spatio-Temporal Logistic AutoRegressive (ST-LAR) statistical model is fitted, with spatio-temporally varying coefficients for the remotely sensed binary ice/water data taken in the previous September and defined on 25 km-resolution remote-sensing grid cells. Our model results in smooth estimates at the local level and allows for making inference on zonal and regional features of Arctic sea ice.

The rest of the paper is organized as follows. The related literature is reviewed in Section 2. We describe the data used in our spatio-temporal analysis of Arctic sea ice in Section 3.1. In Section 3.2, we introduce our proposed ST-LAR model, as well as the pseudo-likelihood parameter-estimation procedure used to fit the model. The model-evaluation scores are defined in Section 3.3. In Section 4.1, several models are fitted and compared, resulting in the best-performing model being chosen. Then, in Section 4.2, model-based estimates are presented in terms of summary statistics that describe the spatio-temporal changes of Arctic sea ice. Section 5 contains a resumé of our findings, as well as a brief discussion of future work that might be carried out. Technical details on the proposed model, the regularized estimation procedure, its fast computation, and the detailed model comparison results are contained in the Appendix A, Appendix B and Appendix C.

2. Literature Review

Recently, researchers have used statistical models to capture the spatio-temporal changing patterns of Arctic sea ice. Zhang and Cressie [22,23] proposed a spatio-temporal generalized linear model to model the binary Arctic SIE data, where an Expectation-Maximization (EM) algorithm, e.g., [24], and a fully Bayesian inference were developed, respectively. Horvath et al. [25] proposed a Bayesian logistic regression model to forecast September Arctic SIE, where several atmospheric, oceanic, and sea-ice variables at 1-month to 7-month lead times were used as covariates. Other related works on modeling ice sheet and sea ice belong to the paradigm of applying statistical models to calibrate outcomes from climate models so that the calibrated outputs better match the observed data. For example, Chang et al. [26] proposed a generalized principal component-based method to calibrate the ice sheet of West Antarctica, where principal components were modeled by Gaussian processes. Chang et al. [27] further combined information from both modern and paleo observations to calibrate the ice sheet model. On forecasting Arctic SIE, Director et al. [28] proposed a spatio-temporal statistical model to improve the forecasting of sea ice contours from physical sea ice models; Director et al. [29] further introduced a mixture contour forecasting, which is a weighted average of the recent climatology forecast and the forecast of a statistical model.

Here we adopt another popular route for modeling binary space-time data, a spatio-temporal logistic autoregressive model. The ST-LAR model can be considered as a special case of spatio-temporal autologistic models. Autologistic models, e.g., [30] (Ch.6) have been used extensively for modeling spatial or spatio-temporal binary data, since they have simple forms and directly specify the spatial/spatio-temporal dependence among binary data, e.g., [31,32,33]. Zhu et al. [34] proposed a spatio-temporal autologistic regresson model, which results in a valid spatio-temporal Markov random field for capturing data dependence. Zheng and Zhu [35] and Zhu et al. [36] proposed a Markov chain Monte Carlo method and a Monte Carlo maximum likelihood approach, respectively, to improve the traditional pseudo-likelihood-based estimation for autologistic models. Compared to other spatio-temporal autologistic models, the ST-LAR model does not include current year’s spatially neighboring observations as autoregressors at each time t. We use observations in the previous year $(t-1)$ as autoregressors, resulting in an ST-LAR model of order one.

Compared to the hierarchical spatio-temporal generalized linear models (GLMs) based on latent Gaussian processes, e.g., [23,37,38], the ST-LAR model directly models the data dependency and is relatively easy and fast to fit using likelihood-based inference. Since the model borrows information from the previous year’s spatially neighboring observations, it encourages estimated ice/water regions to be spatially contiguous. Following recent developments in high-dimensional spatial models (see, e.g., [33,39,40]), the new ST-LAR model we propose has autoregressive coefficients that vary in space and/or time. Spatially varying autoregressive coefficients are capable of capturing the nonhomogeneous autoregressive structure of real data. Our model extends the existing spatio-temporal autoregressive models that have autoregressive coefficients that do not depend on space or time. In addition, it takes into account the neighboring observations’ statuses being 1 (ice) or 0 (water), to allow more modeling flexibility. The proposed ST-LAR model along with regularized estimation of parameters offers a novel and flexible approach for modeling binary spatio-temporal data.

3. Materials and Methods

3.1. Data Description

The datasets upon which our analyses are based are described first, followed by the spatio-temporal statistical models that are fitted to the data. Definitions are then given for the model-evaluation criteria.

Arctic Sea Ice Data. The Arctic sea ice concentration data are produced by the National Oceanic and Atmospheric Administration (NOAA) as part of their National Snow and Ice Data Center’s (NSIDC) Climate Data Record (CDR). The data are obtained from a passive microwave instrument on the Nimbus 7 satellite, as well as from the F8, F11, and F13 satellites of the Defense Meteorological Satellite Program [1,4], which are projected onto $25\mathrm{km}\times 25\mathrm{km}$ grid cells. In this paper, we use Version 4 of the Arctic sea-ice-concentration data, which are available monthly starting from November 1978 [21].

Since the smallest Arctic sea ice cover in a year is typically attained in the month of September [4], we consider the September Arctic sea ice data for the time period from 2000 to 2020, which is a time series at times $t\in \{2000,\dots ,2020\}$. The original sea ice concentration data give the fraction of sea ice in each grid cell, taking values in $[0,1]$. The sea ice concentration data are from two well-established algorithms, the NASA Team algorithm [41] and the NASA Bootstrap algorithm [42]. To obtain the binary sea ice data, a $15\%$ cut-off value is often used to determine whether a grid cell is water (<15%) or ice (≥15%), e.g., [1,4,43,44]. The original NOAA data are on a $304\times 448$ grid in a polar stereographic projection but, here, we focus on the grid cells with latitude greater than ${60}^{\circ }$N, where sea ice is more prevalent. Figure 3, left panel, shows the ${60}^{\circ }$N cut-off along with the sea ice concentrations; the 25 km-resolution grid cells are small enough not to be distinguishable in the figure. Figure 3, right panel, shows the binary data, where recall that a sea ice concentration ≥ 15% in a grid cell defines it to be an ice grid cell.

We further select the spatial grid cells that have experienced at least one ice–water or water–ice transition during the entire study period (21 years), leading to the final dataset with 8673 spatial locations (see Figure 4). This spatial domain consists of the union of ice–water boundary regions over two decades.

Reflected Solar Radiation Data. The monthly reflected solar radiation (RSR) data were obtained from NASA’s Clouds and the Earth’s Radiant Energy System (CERES) Energy Balanced and Filled (EBAF) top-of-atmosphere data product [45,46]. Here, we use Edition 4.1 of this data product, which contains monthly shortwave fluxes from March 2000 to January 2021 [47]. The CERES EBAF data product is on a $1^{\circ }\times 1^{\circ }$ longitude–latitude grid. There are two types of RSR data: one is clear-sky RSR, and the other is all-sky RSR. The clear-sky RSR (CS-RSR) data average the CERES footprints within a region that are cloud-free (defined to be regions where the cloud fraction is no greater than $0.1\%$), which is identified by a cloud-detection algorithm, e.g., [48,49,50]. The all-sky RSR (AS-RSR) data average all the CERES footprints within a region.

In our analyses, there was very little difference between the two and, henceforth, we show results involving CS-RSR. Spatially averaged CS-RSR, along with the Arctic SIE data for September, are shown as time series in Figure 5. The observed relationship leads us to consider using the June RSR data as an informative covariate in our proposed ST-LAR model.

3.2. Methodology

Notation. We first introduce some mathematical notation. Let $y_t\left(\mathbf{s}\right)$ denote a spatio-temporal binary datum for a $25\mathrm{km}\times 25\mathrm{km}$ grid cell at spatial location $\mathbf{s}\in \mathcal{D}$ and time point t, where $\mathcal{D}$ is the two-dimensional Arctic spatial domain and $t=1,2,\dots ,T$. Recall that $y_t\left(\mathbf{s}\right)$ is equal to one for an ice grid cell and equal to zero for a water grid cell. Let $\mathcal{S}\equiv \{{\mathbf{s}}_1,\dots ,{\mathbf{s}}_N\}$ denote the set of N grid-cell locations where there are data, and we assume $\mathbf{s}$ does not change over time. The data vector at time t is denoted as ${\mathbf{y}}_t\equiv {(y_t\left({\mathbf{s}}_1\right),\dots ,y_t\left({\mathbf{s}}_N\right))}^{\prime }$, for $t=1,2,\dots ,T$. We aim to model the probability of $y_t\left(\mathbf{s}\right)$ being one given all the previous ice/water observations at time $t-1$, denoted as $p_t\left(\mathbf{s}\right|{\mathbf{y}}_{t-1})\equiv \mathbb{P}(y_t\left(\mathbf{s}\right)=1|{\mathbf{y}}_{t-1})$, where $\mathbb{P}\left(A\right|B)$ denotes the conditional probability of A given B. We consider a logistic autoregressive model, where the key assumption is that $\mathrm{logit}\left(p_t\left(\mathbf{s}\right|{\mathbf{y}}_{t-1})\right)(\equiv log\{p_t\left(\mathbf{s}\right|{\mathbf{y}}_{t-1})/\left(1-p_t\left(\mathbf{s}\right|{\mathbf{y}}_{t-1})\right)\}$) depends on covariates at t with regression coefficients ${\mathit{\beta }}_t$, as well as the first-order autoregressive ice/water status of the spatially neighboring observations of $y_t\left(\mathbf{s}\right)$ at time $t-1$.

Standard ST-LAR Model. A standard spatio-temporal logistic (first-order) autoregressive model is as follows:

$$\begin{array}{c}\hfill \mathrm{logit}\left(p_t\left(\mathbf{s}\right|{\mathbf{y}}_{t-1})\right)={\mathbf{x}}_t{\left(\mathbf{s}\right)}^{\prime }{\mathit{\beta }}_t+\sum _{j\in \mathrm{Nr}\left(\mathbf{s}\right)}{\eta }_t(y_{t-1}\left({\mathbf{s}}_j\right)-{\mu }_{t-1}\left({\mathbf{s}}_j\right)),\end{array}$$

(1)

where ${\mathit{\beta }}_t$ are the spatially homogeneous regression coefficients associated with the covariate vector ${\mathbf{x}}_t\left(\mathbf{s}\right)$, ${\mu }_{t-1}\left(\mathbf{s}\right)={\mathrm{logit}}^{-1}\left({\mathbf{x}}_{t-1}{\left(\mathbf{s}\right)}^{\prime }{\mathit{\beta }}_{t-1}\right)$, ${\eta }_t$ is the coefficient of the (centered) autoregressor $y_{t-1}\left({\mathbf{s}}_j\right)$, $\mathrm{Nr}\left(\mathbf{s}\right)$ denotes the set of neighborhood locations of the location $\mathbf{s}$, and ${\mathrm{logit}}^{-1}\left(x\right)=exp\left(x\right)/(1+exp\left(x\right))$ is the logistic function for $x\in (-\infty ,\infty )$. The centered autoregressor directly models the effect of a neighboring water grid cell at $t-1$ (i.e., $y_{t-1}\left({\mathbf{s}}_j\right)=0$) on the presence of an ice grid cell at location $\mathbf{s}$ and time t, which helps to interpret ${\eta }_t$ [32]. In this paper, we focus on the case that ${\mathbf{x}}_t\left(\mathbf{s}\right)={(1,x_{rsr,t}\left(\mathbf{s}\right))}^{\prime }$, where $x_{rsr,t}\left(\mathbf{s}\right)$ is the RSR covariate at location $\mathbf{s}$ and time t.

New Proposed ST-LAR Model. The standard ST-LAR model in (1) may be overly simplistic to characterize the complex dependence structure of $p_t\left(\mathbf{s}\right|{\mathbf{y}}_{t-1})$. To allow more modeling flexibility, we distinguish the effects of spatially neighboring observations at time $t-1$ according to their ice or water status, through using ${\eta }_0$ and ${\eta }_1$ to denote the corresponding two types of autoregressive coefficients, respectively (see Figure 6 for a conceptual illustration). Moreover, since the effect of covariates and autoregressors may change in space, we allow all the coefficients in the new ST-LAR model to vary with location $\mathbf{s}$. In summary, for any location ${\mathbf{s}}_i$ in the observed location set $\mathcal{S}$, our proposed model is:

$$\begin{array}{ccc}\hfill \mathrm{logit}\left(p_t\left({\mathbf{s}}_i\right|{\mathbf{y}}_{t-1})\right)& =& {\mathbf{x}}_t{\left({\mathbf{s}}_i\right)}^{\prime }{\beta }_t\left({\mathbf{s}}_i\right)+{\eta }_{0,t}\left({\mathbf{s}}_i\right)\sum _{j\in {\mathcal{I}}_{i,t-1}^0}(y_{t-1}\left({\mathbf{s}}_j\right)-{\mu }_{t-1}\left({\mathbf{s}}_j\right))\hfill \\ & & +{\eta }_{1,t}\left({\mathbf{s}}_i\right)\sum _{j\in {\mathcal{I}}_{i,t-1}^1}(y_{t-1}\left({\mathbf{s}}_j\right)-{\mu }_{t-1}\left({\mathbf{s}}_j\right)),\hfill \end{array}$$

(2)

where ${\mathcal{I}}_{i,t-1}^0$ and ${\mathcal{I}}_{i,t-1}^1$ denote the index sets of nearby water grid cells and ice grid cells at location ${\mathbf{s}}_i$ and time $(t-1)$, respectively. More detailed descriptions of the proposed model are given in Appendix A.

Inference Procedure. Although the proposed logistic autoregressive model with spatially varying coefficients is very flexible, the number of its parameters exceeds the number of observations. To overcome this over-fitting problem, we assume that the spatially varying coefficients are constant within clusters, which can be obtained in an unsupervised manner by imposing a spatially fused LASSO-type penalty [51] on the coefficients in the proposed model. Details on this regularized estimation method are given in Appendix B. We remark that due to the fact that we assume coefficients are clusterwise constant and we impose the fused LASSO penalty to penalize the difference of coefficients at two neighboring spatial locations, the resulting number of distinct coefficients is small relative to the sample size (see Appendix C).

3.3. Model Specification and Model Evaluation Criteria

We consider the ST-LAR model in (2) with different specifications of its coefficients (e.g., the coefficients are constant or spatially varying, and/or the covariate vector does or does not contain RSR). More details on these model specifications are given in Appendix C.

When evaluating the performances of different models, we use the following three criteria: (i) Mean squared error (MSE), which is essentially the Brier score for binary data [52]; (ii) Nash–Sutcliffe model efficiency coefficient (NSE, e.g., [53]); and (iii) correct classification rate (CR). When implementing CR, the $0.5$ probability threshold is used as a cut-off value to transform the estimated ice probabilities to dichotomized estimates taking values 1 (ice) or 0 (water).

Specifically, let $\left\{{\widehat{p}}_t\left({\mathbf{s}}_i\right|{\mathbf{y}}_{t-1})\right\}$ denote the fitted ice probabilities; then we have

$$\mathrm{MSE}\left(t\right)=\frac{1}{N}\sum _{i=1}^N{[{\widehat{p}}_t\left({\mathbf{s}}_i\right|{\mathbf{y}}_{t-1})-y_t\left({\mathbf{s}}_i\right)]}^2\mathrm{and}\mathrm{NSE}\left(t\right)=1-\frac{\mathrm{MSE}\left(t\right)}{{\mathrm{MSE}}_0\left(t\right)},$$

where ${\mathrm{MSE}}_0\left(t\right)\equiv \frac{1}{N}{\sum }_{i=1}^N{[y_t\left({\mathbf{s}}_i\right)-{\overline{y}}_t]}^2$, and ${\overline{y}}_t$ is the average of the binary data at time t. Note that the NSE score takes values in $(-\infty ,1]$, with a higher value indicating a greater estimation accuracy. The correct classification rate is

$$CR\left(t\right)=\frac{1}{N}\sum _{i=1}^N|y_t\left({\mathbf{s}}_i\right)-I({\widehat{p}}_t\left({\mathbf{s}}_i\right|{\mathbf{y}}_{t-1})\ge 0.5)|,$$

where $I\left(A\right)$ denotes an indicator function equal to one if A is true and equal to zero otherwise.

4. Analysis of the Arctic Sea Ice Extent Data

In this section, we consider the Arctic SIE data in each month of September from 2000 to 2020, focusing on grid cells in the region greater than ${60}^{\circ }$N that have experienced at least one ice–water/water–ice transition during the entire study period. The result is a sea ice dataset over 21 years at the 8673 spatial locations $\mathcal{S}$ illustrated in Figure 4.

4.1. Model Comparison Results

We fitted the ST-LAR model given by (2) to the SIE data in the 2000s and 2010s, with different choices of its coefficients and covariates (see

Table A1. The ST-LAR models for fitting the September Arctic SIE data. The mean of the latent $y_t\left(\mathbf{s}\right)$ is either modeled by an intercept term or by using the RSR as a covariate.

Model-1a	$\{{\beta }_0,{\eta }_{0,t},{\eta }_{1,t}\}$
Model-1b	$\{{\beta }_{0,t},{\eta }_{0,t}\left(\mathbf{s}\right),{\eta }_{1,t}\left(\mathbf{s}\right)\}$
Model-1c	$\{{\beta }_{0,t}\left(\mathbf{s}\right),{\eta }_{0,t}\left(\mathbf{s}\right),{\eta }_{1,t}\left(\mathbf{s}\right)\}$
Model-2a	$\{{\beta }_{0,t},{\beta }_{rsr,t},{\eta }_{0,t},{\eta }_{1,t}\}$
Model-2b	$\{{\beta }_{0,t},{\beta }_{rsr,t},{\eta }_{0,t}\left(\mathbf{s}\right),{\eta }_{1,t}\left(\mathbf{s}\right)\}$
Model-2c	$\{{\beta }_{0,t},{\beta }_{rsr,t}\left(\mathbf{s}\right),{\eta }_{0,t}\left(\mathbf{s}\right),{\eta }_{1,t}\left(\mathbf{s}\right)\}$
Model-2d	$\{{\beta }_{0,t}\left(\mathbf{s}\right),{\beta }_{rsr,t}\left(\mathbf{s}\right),{\eta }_{0,t}\left(\mathbf{s}\right),{\eta }_{1,t}\left(\mathbf{s}\right)\}$

in Appendix C). Since the RSR observations $x_{rsr,t}(·)$ are only available from $t=2000$, the data $\left\{y_{2000}\left({\mathbf{s}}_i\right)\right\}$ were used as autoregressors to initialize the fitting of the ST-LAR models, and estimates $\left\{{\widehat{p}}_t\left({\mathbf{s}}_i\right|{\mathbf{y}}_{t-1})\right\}$ (and $\left\{{\widehat{y}}_t\left({\mathbf{s}}_i\right|{\mathbf{y}}_{t-1})\right\}$) were obtained from $t=2001$. That is, we fitted the ST-LAR models to the SIE data at $N=8673$ spatial locations in the “donut " region, as shown in Figure 4, from which we obtained the corresponding estimated ice probabilities and model evaluation scores from $t=2001$ to $t=2020$.

From the model comparison results given in Appendix C, the best model takes the following form:

$$\begin{array}{ccc}\hfill \mathrm{logit}\left(p_t\left({\mathbf{s}}_i\right|{\mathbf{y}}_{t-1})\right)& =& {\beta }_{0,t}\left({\mathbf{s}}_i\right)+{\eta }_{0,t}\left({\mathbf{s}}_i\right)\sum _{j\in {\mathcal{I}}_{i,t-1}^0}(y_{t-1}\left({\mathbf{s}}_j\right)-{\mu }_{t-1}\left({\mathbf{s}}_j\right))\hfill \\ & & +{\eta }_{1,t}\left({\mathbf{s}}_i\right)\sum _{j\in {\mathcal{I}}_{i,t-1}^1}(y_{t-1}\left({\mathbf{s}}_j\right)-{\mu }_{t-1}\left({\mathbf{s}}_j\right)),\hfill \end{array}$$

(3)

where ${\mathcal{I}}_{i,t-1}^0$ and ${\mathcal{I}}_{i,t-1}^1$ are the index sets consisting of the neighboring water grid cells and ice grid cells for location ${\mathbf{s}}_i$ at time $t-1$, respectively. That is, the best model has spatially varying coefficients for the intercept and the autoregressive coefficients, and it does not have any covariates.

Our numerical results in Appendix C show that the standard ST-LAR model with spatially constant coefficients but without the RSR covariate cannot capture the spatial behaviors of the observed ice–water/water–ice transitions, resulting in low model evaluation scores (see

Table A2. Model evaluation scores: The mean squared errors (MSEs), Nash–Sutcliffe model efficiency coefficients (NSEs), and the correct classification rates (CRs) for the Model-1 class based on their estimates at $t=2001,\dots ,2020$. The last row gives the time-averaged model evaluation scores.

	Model-1a			Model-1b			Model-1c
Year	MSE	NSE	CR	MSE	NSE	CR	MSE	NSE	CR
2001	$0.145$	$0.248$	$0.791$	$0.037$	$0.810$	$0.952$	$0.038$	$0.803$	$0.954$
2002	$0.210$	$0.124$	$0.691$	$0.036$	$0.849$	$0.955$	$0.041$	$0.829$	$0.948$
2003	$0.100$	$0.570$	$0.868$	$0.031$	$0.868$	$0.959$	$0.043$	$0.813$	$0.946$
2004	$0.173$	$0.267$	$0.763$	$0.062$	$0.738$	$0.908$	$0.040$	$0.833$	$0.952$
2005	$0.204$	$0.181$	$0.699$	$0.035$	$0.858$	$0.956$	$0.042$	$0.830$	$0.946$
2006	$0.150$	$0.377$	$0.795$	$0.037$	$0.846$	$0.955$	$0.041$	$0.828$	$0.948$
2007	$0.176$	$0.162$	$0.701$	$0.020$	$0.906$	$0.975$	$0.030$	$0.856$	$0.960$
2008	$0.195$	$0.168$	$0.723$	$0.064$	$0.725$	$0.924$	$0.048$	$0.795$	$0.935$
2009	$0.156$	$0.376$	$0.785$	$0.039$	$0.846$	$0.952$	$0.045$	$0.822$	$0.943$
2010	$0.127$	$0.479$	$0.828$	$0.037$	$0.850$	$0.955$	$0.046$	$0.810$	$0.938$
2011	$0.118$	$0.484$	$0.850$	$0.038$	$0.832$	$0.952$	$0.043$	$0.810$	$0.949$
2012	$0.102$	$0.288$	$0.821$	$0.024$	$0.833$	$0.970$	$0.027$	$0.812$	$0.969$
2013	$0.235$	$0.058$	$0.590$	$0.086$	$0.657$	$0.893$	$0.049$	$0.804$	$0.936$
2014	$0.181$	$0.273$	$0.747$	$0.038$	$0.847$	$0.952$	$0.032$	$0.872$	$0.961$
2015	$0.190$	$0.193$	$0.715$	$0.045$	$0.808$	$0.941$	$0.051$	$0.784$	$0.932$
2016	$0.142$	$0.410$	$0.815$	$0.042$	$0.824$	$0.946$	$0.029$	$0.880$	$0.964$
2017	$0.119$	$0.501$	$0.839$	$0.053$	$0.777$	$0.924$	$0.031$	$0.872$	$0.962$
2018	$0.164$	$0.314$	$0.778$	$0.045$	$0.814$	$0.936$	$0.055$	$0.769$	$0.926$
2019	$0.132$	$0.394$	$0.814$	$0.026$	$0.880$	$0.966$	$0.030$	$0.862$	$0.962$
2020	$0.143$	$0.260$	$0.804$	$0.047$	$0.755$	$0.935$	$0.033$	$0.831$	$0.961$
Average	$0.158$	$0.306$	$0.771$	$0.042$	$0.816$	$0.945$	$0.040$	$0.826$	$0.950$

). When allowing the $\eta$-coefficients to vary over the spatial domain $\mathcal{D}$, the estimates $\left\{{\widehat{p}}_t\left({\mathbf{s}}_i\right|{\mathbf{y}}_{t-1})\right\}$ match the observed ice/water status very well (see Figure A2), and the inferred dichotomized ice/water estimates using the $0.5$ cut-off value result in high correct classification rate (CR) scores. Furthermore, allowing the intercept to also vary in space slightly improves the CR scores, where significant improvements occur in years $t=2004$ and $t=2013$. Figure 7 shows the estimation results of the model fitted according to (3), compared to the SIE observations, for $t=2008$ and 2013. Clearly, $\{{\widehat{p}}_t\left({\mathbf{s}}_i\right|{\mathbf{y}}_{t-1}):i=1,\dots ,N\}$ match the spatial patterns of the binary Arctic SIE data for those years very well. The excellent performance of our proposed model (3) can be attributed to its spatially varying coefficients, whose values can adapt to the observed ice–water/water–ice transitions in regions that lose or gain substantial sea ice from $t-1$ to t (see Figure 8).

We also considered the model given by (2) with spatially constant autoregressive coefficients and covariate given by the June reflected solar radiation (RSR) in the same year, but the model evaluation scores were not competitive (

Table A3. The MSEs, NSEs, and CRs for the Model-2 class with clear-sky RSR (CS-RSR) as a covariate. The last row gives the time-averaged model evaluation scores.

	Model-2a			Model-2b			Model-2c			Model-2d
Year	MSE	NSE	CR	MSE	NSE	CR	MSE	NSE	CR	MSE	NSE	CR
2001	$0.126$	$0.345$	$0.820$	$0.037$	$0.808$	$0.952$	$0.047$	$0.756$	$0.940$	$0.035$	$0.816$	$0.956$
2002	$0.153$	$0.363$	$0.766$	$0.039$	$0.837$	$0.952$	$0.053$	$0.779$	$0.929$	$0.048$	$0.800$	$0.939$
2003	$0.108$	$0.535$	$0.858$	$0.046$	$0.800$	$0.939$	$0.039$	$0.834$	$0.947$	$0.038$	$0.837$	$0.955$
2004	$0.130$	$0.450$	$0.818$	$0.057$	$0.758$	$0.925$	$0.043$	$0.819$	$0.938$	$0.047$	$0.801$	$0.936$
2005	$0.164$	$0.342$	$0.762$	$0.059$	$0.762$	$0.922$	$0.052$	$0.793$	$0.932$	$0.047$	$0.813$	$0.941$
2006	$0.135$	$0.439$	$0.824$	$0.043$	$0.820$	$0.945$	$0.045$	$0.815$	$0.949$	$0.050$	$0.792$	$0.943$
2007	$0.142$	$0.322$	$0.834$	$0.032$	$0.846$	$0.956$	$0.036$	$0.829$	$0.956$	$0.033$	$0.842$	$0.958$
2008	$0.189$	$0.192$	$0.725$	$0.043$	$0.814$	$0.946$	$0.056$	$0.762$	$0.931$	$0.058$	$0.753$	$0.927$
2009	$0.140$	$0.438$	$0.788$	$0.029$	$0.883$	$0.963$	$0.041$	$0.836$	$0.948$	$0.066$	$0.737$	$0.920$
2010	$0.128$	$0.476$	$0.830$	$0.042$	$0.828$	$0.945$	$0.041$	$0.832$	$0.949$	$0.044$	$0.820$	$0.944$
2011	$0.108$	$0.525$	$0.855$	$0.048$	$0.787$	$0.943$	$0.032$	$0.859$	$0.960$	$0.050$	$0.783$	$0.935$
2012	$0.102$	$0.286$	$0.812$	$0.026$	$0.819$	$0.967$	$0.022$	$0.845$	$0.974$	$0.028$	$0.805$	$0.966$
2013	$0.180$	$0.280$	$0.741$	$0.114$	$0.543$	$0.838$	$0.060$	$0.758$	$0.921$	$0.041$	$0.835$	$0.945$
2014	$0.162$	$0.348$	$0.764$	$0.084$	$0.664$	$0.889$	$0.061$	$0.755$	$0.934$	$0.046$	$0.817$	$0.940$
2015	$0.171$	$0.273$	$0.746$	$0.045$	$0.811$	$0.945$	$0.051$	$0.781$	$0.937$	$0.060$	$0.745$	$0.922$
2016	$0.135$	$0.438$	$0.818$	$0.043$	$0.822$	$0.946$	$0.031$	$0.869$	$0.961$	$0.039$	$0.835$	$0.948$
2017	$0.119$	$0.503$	$0.841$	$0.047$	$0.803$	$0.942$	$0.040$	$0.832$	$0.947$	$0.036$	$0.849$	$0.957$
2018	$0.131$	$0.454$	$0.816$	$0.053$	$0.781$	$0.932$	$0.054$	$0.774$	$0.926$	$0.050$	$0.793$	$0.935$
2019	$0.101$	$0.536$	$0.856$	$0.031$	$0.858$	$0.960$	$0.027$	$0.874$	$0.965$	$0.033$	$0.850$	$0.961$
2020	$0.144$	$0.257$	$0.778$	$0.036$	$0.814$	$0.953$	$0.034$	$0.826$	$0.955$	$0.030$	$0.846$	$0.961$
Overall	$0.138$	$0.390$	$0.803$	$0.048$	$0.793$	$0.938$	$0.043$	$0.812$	$0.945$	$0.044$	$0.808$	$0.944$

in Appendix C). Furthermore, when using spatially varying autoregressive coefficients and the RSR covariate, it did not improve on average the model evaluation scores of the model without the RSR covariate.

Henceforth, we focus on the estimates from the model given by (3) and propose several summaries of them to reflect the spatio-temporal changes in Arctic sea ice over the two decades from 2000 to 2020. These summaries are intended to visually capture the decline in Arctic sea ice in the 2000s and the 2010s.

4.2. Summaries Based on the Model Given by (3)

We first consider the Arctic SIE, which is defined as the sum of areas of all ice grid cells in the Arctic region. By setting ${\widehat{y}}_t\left(\mathbf{s}\right|{\mathbf{y}}_{t-1})=I({\widehat{p}}_t\left(\mathbf{s}\right|{\mathbf{y}}_{t-1})\ge 0.5)$ using the estimated ice probability, ${\widehat{p}}_t\left(\mathbf{s}\right|{\mathbf{y}}_{t-1})$, from the best model, we obtain dichotomized ice/water estimates at spatial locations that experienced at least one ice–water transition in the past two decades (i.e., 8673 locations in the “donut” region). For the remaining locations $\mathbf{s}$ that were always ice or always water during these two decades, we simply let

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{\widehat{y}}_t\left(\mathbf{s}\right|{\mathbf{y}}_{t-1})={\widehat{p}}_t\left(\mathbf{s}\right|{\mathbf{y}}_{t-1})=1,\hfill & \mathrm{if} y_t\left(\mathbf{s}\right)\mathrm{was} \mathrm{always} \mathrm{ice};\hfill \\ {\widehat{y}}_t\left(\mathbf{s}\right|{\mathbf{y}}_{t-1})={\widehat{p}}_t\left(\mathbf{s}\right|{\mathbf{y}}_{t-1})=0,\hfill & \mathrm{if} y_t\left(\mathbf{s}\right)\mathrm{was} \mathrm{always} \mathrm{water}.\hfill \end{array}\right.\end{array}$$

(4)

Then, by summing up the areas of the estimated ice grid cells, we obtain the estimated Arctic SIE. From Figure 9, we can see that the time series of estimated SIEs agrees with the time series of observed SIEs very well.

Next, similarly to the exploratory analysis shown in Figure 2, we can describe the zonal changes in sea ice at different latitude bands by summarizing the estimates $\left\{{\widehat{p}}_t\left({\mathbf{s}}_i\right|{\mathbf{y}}_{t-1})\right\}$, obtained from (3) and (4). Then, for a latitude band centered at ${\mathrm{lat}}_0$ with half-bandwidth equal to $0.5^{\circ }$, we can summarize the estimates $\left\{{\widehat{p}}_t\left({\mathbf{s}}_i\right|{\mathbf{y}}_{t-1})\right\}$ within this latitude band over $t\in \{2000,\dots ,2020\}$.

Figure 10 shows time series of such boxplots for latitude bands ${70}^{\circ }$N, ${72.5}^{\circ }$N, ${75}^{\circ }$N, ${77.5}^{\circ }$N, ${80}^{\circ }$N, ${82.5}^{\circ }$N, and ${85}^{\circ }$N at five-year intervals from 2000 to 2020. For each latitude band ${\mathrm{lat}}_0$ with bandwidth $2\delta$, the estimates $\{{\widehat{p}}_t\left({\mathbf{s}}_i\right|{\mathbf{y}}_{t-1}):i=1,\dots ,N\}$ with latitudes in $[{\mathrm{lat}}_0-\delta ,{\mathrm{lat}}_0+\delta )$ are summarized; we chose $\delta ={0.5}^{\circ }$, resulting in a latitude bandwidth of $1^{\circ }$. Since these boxplots are made at a finer resolution than the exploratory results in Figure 2, the changes in sea ice both in level and variability are more apparent. We can see that for the mid-high latitude bands ${\mathrm{lat}}_0={72.5}^{\circ }\mathrm{N},{75}^{\circ }\mathrm{N},{77.5}^{\circ }\mathrm{N},{80}^{\circ }\mathrm{N}$, there is a clear deceasing temporal trend of $\left\{{\widehat{p}}_t\left({\mathbf{s}}_i\right|{\mathbf{y}}_{t-1})\right\}$ over the past two decades. Even for the higher latitude band ${\mathrm{lat}}_0={82.5}^{\circ }\mathrm{N}$, the estimated ice probabilities have large variations at $t=2020$ compared to those concentrated near one with a small interquartile range in the previous years. Critically for ${\mathrm{lat}}_0={72.5}^{\circ }\mathrm{N},{75}^{\circ }\mathrm{N}$, once the third quartile of $\{{\widehat{p}}_t\left({\mathbf{s}}_i\right|{\mathbf{y}}_{t-1}):i=1,\dots ,N\}$ drops below $0.5$, the sea ice almost disappears in later years and never rebounds afterwards. This behavior may reflect the future trend of the ice probabilities for the higher latitude bands such as ${\mathrm{lat}}_0={77.5}^{\circ }\mathrm{N},{80}^{\circ }\mathrm{N}$ (and even ${82.5}^{\circ }\mathrm{N}$), under climate change.

5. Discussion and Conclusions

In summary, we consider ST-LAR models with spatially varying coefficients to characterize the changes in sea ice in the Arctic, where the probability of a grid cell at location $\mathbf{s}$ and time t being ice (i.e., $p_t\left(\mathbf{s}\right|{\mathbf{y}}_{t-1})$) depends on the observations in ${\mathbf{y}}_{t-1}$ at the neighboring grid cells of $\mathbf{s}$. Compared to standard ST-LAR models, we allow the autoregressive coefficients ($\eta$-parameters) to vary in space and time to capture local spatial variations in the binary Arctic sea ice data. The effects of autoregressors are distinguished according to their ice/water statuses to allow more modeling flexibility. To reduce the number of spatially varying parameters, we employed a regularized estimation approach using spatially fused LASSO penalties to yield clusterwise constant parameters. The proposed model, along with regularized estimation, offers a novel and flexible approach for modeling binary spatio-temporal data. The spatially varying autoregressive coefficients can adapt to the nonhomogeneous autoregressive structure of the Arctic sea ice data, thus leading to superior fitting results. Our results show that with spatially varying autoregressive coefficients, the estimated ice probabilities can capture the spatial patterns of observed ice–water/water–ice transitions very well.

We found that with spatially varying autoregressive coefficients, further adding the RSR covariate to the ST-LAR models did not significantly improve the estimation results. The best model form has spatially varying autoregressive coefficients and a spatially varying intercept, as given in (3). This model results in correct classification rates generally above $94\%$ for each year. Based on the estimated ice probabilities $\{{\widehat{p}}_t\left({\mathbf{s}}_i\right|{\mathbf{y}}_{t-1}):i=1,\dots ,N\}$ of the best model, we computed several summary statistics to describe the spatio-temporal changes in Arctic sea ice. For the time series of Arctic SIE, the estimated SIEs match the observed ones very well and show a clear declining temporal trend over the past two decades. Then, the estimated ice probabilities were summarized zonally in selected latitude bands through a time series of boxplots for each latitude band over different years. We observed that for the mid-high latitude bands, there is a clear declining trend of the estimated ice probabilities over the past two decades. For latitude bands at ${72.5}^{\circ }\mathrm{N}$ and ${75}^{\circ }\mathrm{N}$, the sea ice almost disappeared after 2010 and did not rebound afterwards. A similar zonal contraction towards zero may now be underway for the higher latitudes ${77.5}^{\circ }\mathrm{N}$ and ${80}^{\circ }\mathrm{N}$, since the estimated ice probabilities already show decreasing levels and/or increasing variabilities from 2000 to 2020.

By assuming $\left\{y_t\left(\mathbf{s}\right)\right\}$ are conditionally independent given ${\mathbf{y}}_{t-1}$ and model parameters at each time t, the proposed ST-LAR model leads to a valid joint distribution for $\left\{y_t\left(\mathbf{s}\right)\right\}$. Hence, the parametric bootstrap approach [54] can be applied to obtain the standard errors of parameter estimates and the estimated ice probabilities (see Appendix C). While it can be time-consuming to run the model on multiple bootstrap samples, importantly, it results in valid statistical inferences.

In this paper, we focus on obtaining the estimated ice probabilities in the Arctic region. Since the spatial-specific coefficients in the proposed ST-LAR model are estimated using data at time t, our model cannot be readily used to predict the ice/water status in future years. To enable forecasting, we need to assume that the autoregressive relationship does not change in a short time period. The choice of a suitable time window in which the autoregressive structure approximately stays the same will need further investigation. We leave investigating the forecasting aspect of ST-LAR models for future research. In addition, the Arctic SIE data at targeted months in the same year can also be considered for constructing the autoregressors in an ST-LAR model, e.g., [25,55].