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

^{1}

^{2}

^{3}

^{4}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Literature Review

## 3. Materials and Methods

#### 3.1. Data Description

**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 $25\phantom{\rule{3.33333pt}{0ex}}\mathrm{km}\times 25\phantom{\rule{3.33333pt}{0ex}}\mathrm{km}$ 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].

**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.

#### 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 $25\phantom{\rule{3.33333pt}{0ex}}\mathrm{km}\times 25\phantom{\rule{3.33333pt}{0ex}}\mathrm{km}$ 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:

**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:

**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

## 4. Analysis of the Arctic Sea Ice Extent Data

#### 4.1. Model Comparison Results

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

## 5. Discussion and Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Modeling Details

## Appendix B. Regularized Estimation Method

**Figure A1.**Left panel: A connected graph $\mathcal{G}$ of 30 nodes, where each node is connected with its 10 nearest (according to Euclidean distance) neighbors, and edge weights are equal to the Euclidean distances between the nodes. Middle panel: The minimum spanning tree of $\mathcal{G}$. Right panel: The three induced clusters by removing two edges of the spanning tree (the two dotted lines).

## Appendix C. Detailed Model Comparison Results

**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 | $\{{\beta}_{0,t}\left(\mathbf{s}\right),{\beta}_{rsr,t}\left(\mathbf{s}\right),{\eta}_{0,t}\left(\mathbf{s}\right),{\eta}_{1,t}\left(\mathbf{s}\right)\}$ |

**Initialization for Estimation**. Our proposed ST-LAR model given by (2) requires (centered) past observations at time $(t-1)$ as autoregressors to model the ice probability at time t. At the initial year ${t}_{0}=2000$, since the means of $\left\{{y}_{1999}\left({\mathbf{s}}_{i}\right)\right\}$ (i.e., $\left\{{\mu}_{1999}\left({\mathbf{s}}_{i}\right)\right\}$) are not available, we set $\eta $-coefficients in model (2) to zero and only use $\beta $-coefficients to fit the binary observations at ${t}_{0}$. After we obtain the estimates $\left\{{\widehat{\mu}}_{2000}\left({\mathbf{s}}_{i}\right)\right\}$, we can fully employ the ST-LAR models in Table A1 and obtain the estimated ice probabilities in a progressive manner from ${t}_{0}+1=2001$ onwards.

**Results of the Model-1 Class**. We first considered Model-1a, which uses a constant intercept ${\beta}_{0}$ for modeling the mean of $\mathrm{logit}\left({p}_{t}\left(\mathbf{s}\right|{\mathbf{y}}_{t-1})\right)$ and assumes only time-varying $\eta $-coefficients. We used all the data to estimate the intercept ${\beta}_{0}$ through a classical logistic regression, assuming no space-time dependence among the observations. Then, by fixing ${\beta}_{0}$ at its estimate, a simple logistic regression was used to estimate ${\eta}_{0,t}$ and ${\eta}_{1,t}$ at each time point.

**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 $t=2001,\phantom{\rule{4pt}{0ex}}\dots ,\phantom{\rule{4pt}{0ex}}2020$. 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 | $0.158$ | $0.306$ | $0.771$ | $0.042$ | $0.816$ | $0.945$ | $0.040$ | $0.826$ | $0.950$ |

**Figure A2.**For each of (

**a**) $t=2012$ and (

**b**) $t=2013$, shown are the observed ice/water statuses (

**upper-left panel**) and the estimated ice probabilities by Model-1a (

**upper-right panel**), Model-1b (

**lower-left panel**), and Model-1c (

**lower-right panel**).

**Figure A3.**The observed ice/water status and the estimated ice/water status (by Model-1a, Model-1b, and Model-1c) at $t=2012$.

**Results of the Model-2 Class**. Zhan and Davies [20] used the June RSR data to estimate the September Arctic sea ice with good success. This motivates us to include the June RSR as a covariate in our new model given by (2) to see whether the model evaluation scores can be improved further. At each time t, we standardized the RSR data by subtracting the sample means over space and dividing by the corresponding standard deviations. The CERES EBAF data product is on a ${1}^{\circ}\times {1}^{\circ}$ longitude–latitude grid, whose resolution is different from the 25 km resolution of the remotely sensed Arctic sea-ice data. When modeling ${p}_{t}\left(\mathbf{s}\right|{\mathbf{y}}_{t-1})$, we used the June RSR value at the location nearest to $\mathbf{s}$ as its RSR covariate.

**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 | $0.138$ | $0.390$ | $0.803$ | $0.048$ | $0.793$ | $0.938$ | $0.043$ | $0.812$ | $0.945$ | $0.044$ | $0.808$ | $0.944$ |

**Figure A4.**The observed Arctic SIE versus the estimated Arctic SIE using Model-1a and Model-2a (with the CS-RSR covariate). (

**a**) Observed (red solid line) and estimated (blue dashed line) Arctic SIE using Model-1a with coefficients $\{{\beta}_{0},{\eta}_{0,t},{\eta}_{1,t}\}$. (

**b**) Observed (red solid line) and estimated (blue dashed line) Arctic SIE using Model-2a with the CS-RSR covariate.

**More Validation Results for Model-1c**. We first check the number of distinct model parameters for Model-1c. Figure A5 shows the boxplots of the numbers of distinct values for the $\beta $ and $\eta $-coefficients. We can see that the numbers of distinct coefficients are small relative to the sample size (which is $N=8673$), with median values less than 100. The largest number of model parameters of the fitted ST-LAR models is about 300, which is much smaller than the sample size. Therefore, the model we choose (Model-1c) does not obviously suffer from over-fitting.

**Figure A6.**Years 2007 and 2012 are shown.

**Left panels**: The estimates $\left\{{\widehat{p}}_{t}\left({\mathbf{s}}_{i}\right|{\mathbf{y}}_{t-1})\right\}$.

**Middle panels**: The estimates $\left\{{\widehat{y}}_{t}\left({\mathbf{s}}_{i}\right|{\mathbf{y}}_{t-1})\right\}$ using a $0.5$ cut-off value to dichotomize $\left\{{\widehat{p}}_{t}\left({\mathbf{s}}_{i}\right|{\mathbf{y}}_{t-1})\right\}$.

**Right panels**: The binary Arctic sea ice data derived from satellite data.

**Figure A7.**Spatial maps of the observed ice/water statuses at $t-1$ and t and the autoregressive-coefficient estimates $\left\{{\widehat{\eta}}_{0,t}\left({\mathbf{s}}_{i}\right)\right\}$ and $\left\{{\widehat{\eta}}_{1,t}\left({\mathbf{s}}_{i}\right)\right\}$ of Model-1c, for $t=2007$ and $t=2012$.

$\mathit{N}=8673$ | ${\widehat{\mathit{y}}}_{\mathit{t}}\left(\mathbf{s}\right|{\mathbf{y}}_{\mathit{t}-1})=1$ | ${\widehat{\mathit{y}}}_{\mathit{t}}\left(\mathbf{s}\right|{\mathbf{y}}_{\mathit{t}-1})=0$ |
---|---|---|

${y}_{t}\left(\mathbf{s}\right)=1$ (actual ice) | $3653.35$ | $223.85$ |

${y}_{t}\left(\mathbf{s}\right)=0$ (actual water) | $213.20$ | $4582.60$ |

**Figure A8.**The time series of precision, recall, and F1-score parameters of the $2\times 2$ confusion matrix for Model-1c.

**Table A5.**Time-averaged model evaluation scores: The mean squared errors (MSEs), Nash–Sutcliffe model efficiency coefficients (NSEs), and the correct classification rates (CRs) for Model-1c.

Model-1c | MSE | NSE | CR |
---|---|---|---|

Training | $0.036$ | $0.842$ | $0.954$ |

Testing | $0.050$ | $0.782$ | $0.936$ |

**Conclusion**. Our conclusion is that Model-1c, namely, the model given by (3) in the paper, is the best model. Its estimates give superior model evaluation scores, and hence those estimates are used to compute the summaries and their visualization of the spatio-temporal variability in Arctic sea ice.

**Uncertainty Estimation**. The parametric bootstrap method, e.g., [54] provides a feasible way for obtaining the standard errors of $\left\{{\widehat{p}}_{t}\left({\mathbf{s}}_{i}\right|{\mathbf{y}}_{t-1})\right\}$ for Model-1c. Let ${\widehat{\mathit{\theta}}}_{t}\equiv \{{\widehat{\beta}}_{0,t}\left({\mathbf{s}}_{i}\right),{\widehat{\eta}}_{0,t}\left({\mathbf{s}}_{i}\right),{\widehat{\eta}}_{1,t}\left({\mathbf{s}}_{i}\right),i=1,\dots ,N\}$ denote the parameter estimates of Model-1c at time t. The bootstrap samples can be generated in the following manner. At the initial time ${t}_{0}$, the estimates ${\widehat{\mathit{\theta}}}_{{t}_{0}}$ are plugged into equation (3) to obtain the estimated ice probabilities $\left\{{\widehat{p}}_{{t}_{0}}^{b}\left({\mathbf{s}}_{i}\right|{\mathbf{y}}_{{t}_{0}-1})\right\}$, and the bootstrap samples, $\left\{{\widehat{y}}_{{t}_{0}}^{b}\left({\mathbf{s}}_{i}\right)\right\}$, are independently generated from the Bernoulli distributions with parameters $\left\{{\widehat{p}}_{{t}_{0}}\left({\mathbf{s}}_{i}\right|{\mathbf{y}}_{{t}_{0}-1})\right\}$. Then, starting from $t={t}_{0}+1$, the bootstrap samples at time t are generated by (i) constructing the autoregressors using the bootstrap samples $\left\{{\widehat{y}}_{t-1}^{b}\left({\mathbf{s}}_{i}\right)\right\}$; (ii) plugging the autoregressors and the parameter estimates ${\widehat{\mathit{\theta}}}_{t}$ into (3) to obtain the estimated ice probabilities $\left\{{\widehat{p}}_{t}^{b}\left({\mathbf{s}}_{i}\right|{\mathbf{y}}_{t-1})\right\}$; and (iii) generating the bootstrap samples $\left\{{\widehat{y}}_{t}^{b}\left({\mathbf{s}}_{i}\right)\right\}$ independently from the Bernoulli distributions with parameters $\left\{{\widehat{p}}_{t}^{b}\left({\mathbf{s}}_{i}\right|{\mathbf{y}}_{t-1})\right\}$. This data-generating process is repeated B times to obtain B bootstrap samples.

**Figure A9.**Years 2007 and 2008 are shown. Left panels: The estimates $\left\{{\widehat{p}}_{t}\left({\mathbf{s}}_{i}\right|{\mathbf{y}}_{t-1})\right\}$ of Model-1c. Middle panels: The bootstrap means of the estimated ice probabilities from 100 bootstrap samples. Right panels: The bootstrap standard errors of the estimated ice probabilities.

**Figure A10.**The observed Arctic SIE (red solid dot), the estimated Arctic SIE by Model-1c (blue dashed line), and the bootstrap pointwise $95\%$ confidence interval of the Arctic SIE (shaded blue band).

## References

- Parkinson, C.L.; Cavalieri, D.J.; Gloersen, P.; Zwally, H.J.; Comiso, J.C. Arctic sea ice extents, areas, and trends, 1978–1996. J. Geophys. Res. Ocean.
**1999**, 104, 20837–20856. [Google Scholar] [CrossRef] - Comiso, J.C.; Parkinson, C.L.; Gersten, R.; Stock, L. Accelerated decline in the Arctic sea ice cover. Geophys. Res. Lett.
**2008**, 35, L01703. [Google Scholar] [CrossRef] [Green Version] - Cavalieri, D.J.; Parkinson, C.L. Arctic sea ice variability and trends, 1979–2010. Cryosphere
**2012**, 6, 881–889. [Google Scholar] [CrossRef] [Green Version] - Parkinson, C.L. Global sea ice coverage from satellite data: Annual cycle and 35-yr trends. J. Clim.
**2014**, 27, 9377–9382. [Google Scholar] [CrossRef] [Green Version] - Parkinson, C.L. Spatially mapped reductions in the length of the Arctic sea ice season. Geophys. Res. Lett.
**2014**, 41, 4316–4322. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Parkinson, C.L.; DiGirolamo, N.E. New visualizations highlight new information on the contrasting Arctic and Antarctic sea-ice trends since the late 1970s. Remote Sens. Environ.
**2016**, 183, 198–204. [Google Scholar] [CrossRef] [Green Version] - Stroeve, J.; Notz, D. Changing state of Arctic sea ice across all seasons. Environ. Res. Lett.
**2018**, 13, 103001. [Google Scholar] [CrossRef] - Meier, W.N.; Stroeve, J. An updated assessment of the changing Arctic sea ice cover. Oceanography
**2022**, 35, 1–10. [Google Scholar] [CrossRef] - Curry, J.A.; Schramm, J.L.; Ebert, E.E. Sea ice-albedo climate feedback mechanism. J. Clim.
**1995**, 8, 240–247. [Google Scholar] [CrossRef] - Kumar, A.; Perlwitz, J.; Eischeid, J.; Quan, X.; Xu, T.; Tao, Z.; Hoerling, M.; Jha, B.; Wang, W. Contribution of sea ice loss to Arctic amplification. Geophys. Res. Lett.
**2010**, 37, 389–400. [Google Scholar] [CrossRef] - Pistone, K.; Eisenman, I.; Ramanathan, V. Observational determination of albedo decrease caused by vanishing Arctic sea ice. Proc. Natl. Acad. Sci. USA
**2014**, 111, 3322–3326. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Screen, J.A.; Simmonds, I. Exploring links between Arctic amplification and mid-latitude weather. Geophys. Res. Lett.
**2013**, 40, 959–964. [Google Scholar] [CrossRef] [Green Version] - Mori, M.; Watanabe, M.; Shiogama, H.; Inoue, J.; Kimoto, M. Robust Arctic sea-ice influence on the frequent Eurasian cold winters in past decades. Nat. Geosci.
**2014**, 7, 869–873. [Google Scholar] [CrossRef] - Cohen, J.; Screen, J.A.; Furtado, J.C.; Barlow, M.; Whittleston, D.; Coumou, D.; Francis, J.; Dethloff, K.; Entekhabi, D.; Overland, J.; et al. Recent Arctic amplification and extreme mid-latitude weather. Nat. Geosci.
**2014**, 7, 627–637. [Google Scholar] [CrossRef] [Green Version] - Cvijanovic, I.; Santer, B.D.; Bonfils, C.; Lucas, D.D.; Jch, C.; Zimmerman, S. Future loss of Arctic sea-ice cover could drive a substantial decrease in California’s rainfall. Nat. Commun.
**2017**, 8, 1947. [Google Scholar] [CrossRef] [Green Version] - Blackport, R.; Screen, J.A. Influence of Arctic sea ice loss in autumn compared to that in winter on the atmospheric circulation. Geophys. Res. Lett.
**2019**, 46, 2213–2221. [Google Scholar] [CrossRef] [Green Version] - Olonscheck, D.; Mauritsen, T.; Notz, D. Arctic sea-ice variability is primarily driven by atmospheric temperature fluctuations. Nat. Geosci.
**2019**, 12, 430–434. [Google Scholar] [CrossRef] - Labe, Z.; Peings, Y.; Magnusdottir, G. Contributions of ice thickness to the atmospheric response from projected Arctic sea ice loss. Geophys. Res. Lett.
**2018**, 45, 5635–5642. [Google Scholar] [CrossRef] [Green Version] - Wernli, H.; Papritz, L. Role of polar anticyclones and mid-latitude cyclones for Arctic summertime sea-ice melting. Nat. Geosci.
**2018**, 11, 108–113. [Google Scholar] [CrossRef] - Zhan, Y.; Davies, R. September Arctic sea ice extent indicated by June reflected solar radiation. J. Geophys. Res. Atmos.
**2017**, 122, 2194–2202. [Google Scholar] [CrossRef] - Meier, W.N.; Fetterer, F.; Windnagel, A.K.; Stewart, J.S. NOAA/NSIDC Climate Data Record of Passive Microwave Sea Ice Concentration, Version 4 [Data Set]. 2021. Available online: https://nsidc.org/data/g02202/versions/4 (accessed on 16 August 2022). [CrossRef]
- Zhang, B.; Cressie, N. Estimating spatial changes over time of Arctic sea ice using hidden 2 × 2 tables. J. Time Ser. Anal.
**2019**, 40, 288–311. [Google Scholar] [CrossRef] [Green Version] - Zhang, B.; Cressie, N. Bayesian inference of spatio-temporal changes of Arctic sea ice. Bayesian Anal.
**2020**, 15, 605–631. [Google Scholar] [CrossRef] - Dempster, A.P.; Laird, N.M.; Rubin, D.B. Maximum likelihood from incomplete data via the EM algorithm. J. R. Stat. Soc. Ser. B (Methodol.)
**1977**, 39, 1–22. [Google Scholar] - Horvath, S.; Stroeve, J.; Rajagopalan, B.; Kleiber, W. A Bayesian logistic regression for probabilistic forecasts of the minimum September Arctic sea ice cover. Earth Space Sci.
**2020**, 7, e2020EA001176. [Google Scholar] [CrossRef] - Chang, W.; Haran, M.; Applegate, P.; Pollard, D. Calibrating an ice sheet model using high-dimensional binary spatial data. J. Am. Stat. Assoc.
**2016**, 111, 57–72. [Google Scholar] [CrossRef] [Green Version] - Chang, W.; Haran, M.; Applegate, P.; Pollard, D. Improving ice sheet model calibration using paleoclimate and modern data. Ann. Appl. Stat.
**2016**, 10, 2274–2302. [Google Scholar] [CrossRef] - Director, H.M.; Raftery, A.E.; Bitz, C.M. Improved sea ice forecasting through spatiotemporal bias correction. J. Clim.
**2017**, 30, 9493–9510. [Google Scholar] [CrossRef] - Director, H.M.; Raftery, A.E.; Bitz, C.M. Probabilistic forecasting of the Arctic sea ice edge with contour modeling. Ann. Appl. Stat.
**2021**, 15, 711–726. [Google Scholar] [CrossRef] - Cressie, N. Statistics for Spatial Data, Revised Edition; John Wiley & Sons: Hoboken, NJ, USA, 1993. [Google Scholar]
- Besag, J. Spatial interaction and the statistical analysis of lattice systems. J. R. Stat. Soc. Ser. B (Stat. Methodol.)
**1974**, 36, 192–225. [Google Scholar] [CrossRef] - Caragea, P.C.; Kaiser, M.S. Autologistic models with interpretable parameters. J. Agric. Biol. Environ. Stat.
**2009**, 14, 281–300. [Google Scholar] [CrossRef] - Shin, Y.E.; Sang, H.; Liu, D.; Ferguson, T.A.; Song, P.X. Autologistic network model on binary data for disease progression study. Biometrics
**2019**, 75, 1310–1320. [Google Scholar] [CrossRef] [PubMed] - Zhu, J.; Huang, H.C.; Wu, J. Modeling spatial-temporal binary data using Markov random model. J. Agric. Biol. Environ. Stat.
**2005**, 10, 212–225. [Google Scholar] [CrossRef] [Green Version] - Zheng, Y.; Zhu, J. Markov chain Monte Carlo for a spatial-temporal autologistic regression model. J. Comput. Graph. Stat.
**2008**, 17, 123–137. [Google Scholar] [CrossRef] - Zhu, J.; Zheng, Y.; Carroll, A.L.; Aukema, B.H. Autologistic regression analysis of spatial-temporal binary data via Monte Carlo maximum likelihood. J. Agric. Biol. Environ. Stat.
**2008**, 13, 84–98. [Google Scholar] [CrossRef] - Diggle, P.J.; Tawn, J.; Moyeed, R. Model-based geostatistics (with discussion). J. R. Stat. Soc. Ser. C (Appl. Stat.)
**1998**, 47, 299–350. [Google Scholar] [CrossRef] - Sengupta, A.; Cressie, N. Hierarchical statistical modeling of big spatial datasets using the exponential family of distributions. Spat. Stat.
**2013**, 4, 14–44. [Google Scholar] [CrossRef] [Green Version] - Chu, T.; Zhu, J.; Wang, H. Penalized maximum likelihood estimation and variable selection in geostatistics. Ann. Stat.
**2011**, 39, 2607–2625. [Google Scholar] [CrossRef] - Li, F.; Sang, H. Spatial homogeneity pursuit of regression coefficients for large datasets. J. Am. Stat. Assoc.
**2019**, 114, 1050–1062. [Google Scholar] [CrossRef] - Cavalieri, D.J.; Gloersen, P.; Campbell, W.J. Determination of sea ice parameters with the Nimbus 7 SMMR. J. Geophys. Res. Atmos.
**1984**, 89, 5355–5369. [Google Scholar] [CrossRef] - Comiso, J.C. Characteristics of Arctic winter sea ice from satellite multispectral microwave observations. J. Geophys. Res. Ocean.
**1986**, 91, 975–994. [Google Scholar] [CrossRef] - Zwally, H.J.; Comiso, J.C.; Parkinson, C.L.; Cavalieri, D.J.; Gloersen, P. Variability of Antarctic sea ice 1979–1998. J. Geophys. Res. Ocean.
**2002**, 107, 9-1–9-19. [Google Scholar] [CrossRef] [Green Version] - Meier, W.N.; Stroeve, J.; Fetterer, F. Whither Arctic sea ice? A clear signal of decline regionally, seasonally and extending beyond the satellite record. Ann. Glaciol.
**2007**, 46, 428–434. [Google Scholar] [CrossRef] [Green Version] - Kato, S.; Rose, F.G.; Rutan, D.A.; Thorsen, T.J.; Loeb, N.G.; Doelling, D.R.; Huang, X.; Smith, W.L.; Su, W.; Ham, S.H. Surface irradiances of edition 4.0 Clouds and the Earth’s Radiant Energy system (CERES) Energy Balanced and Filled (EBAF) data product. J. Clim.
**2018**, 31, 4501–4527. [Google Scholar] [CrossRef] - Loeb, N.G.; Doelling, D.R.; Wang, H.; Su, W.; Nguyen, C.; Corbett, J.G.; Liang, L.; Mitrescu, C.; Rose, F.G.; Kato, S. Clouds and the Earth’s Radiant Energy System (CERES) Energy Balanced and Filled (EBAF) top-of-atmosphere (TOA) edition-4.0 data product. J. Clim.
**2018**, 31, 895–918. [Google Scholar] [CrossRef] - NASA/LARC/SD/ASDC. CERES Time-Interpolated TOA Fluxes, Clouds and Aerosols Monthly Aqua Edition4A. 2015. Available online: https://asdc.larc.nasa.gov/project/CERES/CER_SSF1deg-Month_Aqua-MODIS_Edition4A (accessed on 7 August 2022). [CrossRef]
- Minnis, P.; Trepte, Q.Z.; Sun-Mack, S.; Chen, Y.; Doelling, D.R.; Young, D.F.; Spangenberg, D.A.; Miller, W.F.; Wielicki, B.A.; Brown, R.R.; et al. Cloud detection in nonpolar regions for CERES using TRMM VIRS and Terra and Aqua MODIS data. IEEE Trans. Geosci. Remote Sens.
**2008**, 46, 3857–3884. [Google Scholar] [CrossRef] - Minnis, P.; Sun-Mack, S.; Young, D.F.; Heck, P.W.; Garber, D.P.; Chen, Y.; Spangenberg, D.A.; Arduini, R.F.; Trepte, Q.Z.; Smith, W.L.; et al. CERES edition-2 cloud property retrievals using TRMM VIRS and Terra and Aqua MODIS data—Part I: Algorithms. IEEE Trans. Geosci. Remote Sens.
**2011**, 49, 4374–4400. [Google Scholar] [CrossRef] - Sun-Mack, S.; Minnis, P.; Chen, Y.; Doelling, D.R.; Scarino, B.R.; Haney, C.O.; Smith, W.L. Calibration changes to Terra MODIS Collection-5 radiances for CERES Edition 4 cloud retrievals. IEEE Trans. Geosci. Remote Sens.
**2018**, 56, 6016–6032. [Google Scholar] [CrossRef] [PubMed] - Tibshirani, R.; Saunders, M.; Rosset, S.; Zhu, J.; Knight, K. Sparsity and smoothness via the fused LASSO. J. R. Stat. Soc. Ser. B (Stat. Methodol.)
**2005**, 67, 91–108. [Google Scholar] [CrossRef] [Green Version] - Brier, G.W. Verification of forecasts expressed in terms of probability. Mon. Weather Rev.
**1950**, 78, 1–3. [Google Scholar] [CrossRef] - Nash, J.E.; Sutcliffe, J.V. River flow forecasting through conceptual models part I—A discussion of principles. J. Hydrol.
**1970**, 10, 282–290. [Google Scholar] [CrossRef] - Efron, B.; Tibshirani, R.J. An Introduction to the Bootstrap; Chapman & Hall/CRC Press: Boca Raton, FL, USA, 1994. [Google Scholar]
- Blanchard-Wrigglesworth, E.; Armour, K.C.; Bitz, C.M.; DeWeaver, E. Persistence and inherent predictability of Arctic sea ice in a GCM ensemble and observations. J. Clim.
**2011**, 24, 231–250. [Google Scholar] [CrossRef] - Besag, J. Statistical analysis of non-lattice data. J. R. Stat. Soc. Ser. D (Stat.)
**1975**, 24, 179–195. [Google Scholar] [CrossRef] [Green Version] - Lee, D.T.; Schachter, B.J. Two algorithms for constructing a Delaunay triangulation. Int. J. Comput. Inf. Sci.
**1980**, 9, 219–242. [Google Scholar] [CrossRef] - Akaike, H. A new look at the statistical model identification. IEEE Trans. Autom. Control
**1974**, 19, 716–723. [Google Scholar] [CrossRef] - Schwarz, G. Estimating the dimension of a model. Ann. Stat.
**1978**, 6, 461–464. [Google Scholar] [CrossRef]

**Figure 1.**The observed Arctic SIE from 2000 to 2020, with a fitted linear regression line (blue dashed line) using year t as a single covariate.

**Figure 2.**Boxplots of ice proportions in different latitude bands centered at ${\mathrm{lat}}_{0}={70}^{\circ}\mathrm{N},\phantom{\rule{4pt}{0ex}}72.{5}^{\circ}\mathrm{N},\dots ,\phantom{\rule{4pt}{0ex}}{85}^{\circ}\mathrm{N}$. The average of ice proportions is represented by a dot (•) and the horizontal bars in the boxplots show the five-number summaries of a distribution (namely, minimum, first quartile, median, third quartile, and maximum). The five-year-interval boxplots are shaded from blue (2000) through green (2010) to red (2020).

**Figure 3.**

**Left panel**: The Arctic sea ice concentrations in September of 2001.

**Right panel**: The corresponding binary Arctic SIE data in the same month. The spatial grid cells are displayed in a polar stereographic projection.

**Figure 4.**The spatial grid cells (in red) with at least one observed ice–water or water–ice transition from 2000 to 2020.

**Figure 5.**The time series of spatially averaged CS-RSR data for June (blue dashed line) versus the Arctic SIE data for September (red solid line). The averaged values of RSR are taken over all RSR observations in the Arctic region (latitudes $\ge {60}^{\circ}$N).

**Figure 6.**Spatio-temporal representation of the autoregressive structure, where the black square indicates the spatial domain, the colored circles indicate spatial locations, the solid black lines in the black square define the neighborhood of those spatial locations, and the dashed colored lines define the autoregressive dependence at location $\mathbf{s}$. Circles corresponding to water (ice) grid cells are colored blue (red).

**Figure 7.**Years 2008 and 2013 are shown.

**Left panels**: The estimates $\left\{{\widehat{p}}_{t}\left({\mathbf{s}}_{i}\right|{\mathbf{y}}_{t-1})\right\}$.

**Middle panels**: The estimates $\left\{{\widehat{y}}_{t}\left({\mathbf{s}}_{i}\right|{\mathbf{y}}_{t-1})\right\}$ using a $0.5$ cut-off value to dichotomize $\left\{{\widehat{p}}_{t}\left({\mathbf{s}}_{i}\right|{\mathbf{y}}_{t-1})\right\}$.

**Right panels**: The binary Arctic sea ice data derived from satellite data.

**Figure 8.**Spatial maps of the autoregressive-coefficient estimates $\left\{{\widehat{\eta}}_{0,t}\left({\mathbf{s}}_{i}\right)\right\}$ and $\left\{{\widehat{\eta}}_{1,t}\left({\mathbf{s}}_{i}\right)\right\}$ for the best model at $t=2008$ and $t=2013$.

**Figure 9.**Observed Arctic SIE (red solid line) versus the estimated Arctic SIE based on the model given by (3) (blue dashed line).

**Figure 10.**Boxplots of the estimates $\{{\widehat{p}}_{t}\left({\mathbf{s}}_{i}\right|{\mathbf{y}}_{t-1}):i=1,\phantom{\rule{4pt}{0ex}}\dots ,\phantom{\rule{4pt}{0ex}}N\}$ from the model given by (3) for different latitude bands with half-bandwidth $\delta =0.{5}^{\circ}$. The horizontal dashed line indicates a $0.5$ probability, and the solid dots indicate the averaged values of $\{{\widehat{p}}_{t}\left({\mathbf{s}}_{i}\right|{\mathbf{y}}_{t-1}):i=1,\phantom{\rule{4pt}{0ex}}\dots ,\phantom{\rule{4pt}{0ex}}N\}$. The horizontal bars in the boxplots show the five-number summaries of a distribution (namely, minimum, first quartile, median, third quartile, and maximum). The five-year-interval boxplots are shaded from blue (2000) through green (2010) to red (2020).

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Zhang, B.; Li, F.; Sang, H.; Cressie, N.
Inferring Changes in Arctic Sea Ice through a Spatio-Temporal Logistic Autoregression Fitted to Remote-Sensing Data. *Remote Sens.* **2022**, *14*, 5995.
https://doi.org/10.3390/rs14235995

**AMA Style**

Zhang B, Li F, Sang H, Cressie N.
Inferring Changes in Arctic Sea Ice through a Spatio-Temporal Logistic Autoregression Fitted to Remote-Sensing Data. *Remote Sensing*. 2022; 14(23):5995.
https://doi.org/10.3390/rs14235995

**Chicago/Turabian Style**

Zhang, Bohai, Furong Li, Huiyan Sang, and Noel Cressie.
2022. "Inferring Changes in Arctic Sea Ice through a Spatio-Temporal Logistic Autoregression Fitted to Remote-Sensing Data" *Remote Sensing* 14, no. 23: 5995.
https://doi.org/10.3390/rs14235995