A Spatio-Temporal Model of Arctic Sea Ice

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.title[
# A Spatio-Temporal Model of Arctic Sea Ice
]
.subtitle[
## Comprehensive Exam
]
.author[
### Alison Kleffner
]
.date[
### Department of Statistics, University of Nebraska - Lincoln
]

---

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# Outline

<i class="fas fa-arrows-up-down-left-right" role="presentation" aria-label="arrows-up-down-left-right icon"></i> Motivation and Background

<i class="fas fa-list" role="presentation" aria-label="list icon"></i> Research Objectives

<i class="fas fa-info" role="presentation" aria-label="info icon"></i> Spatio-Temporal Clustering: Bounding Box

<i class="fas fa-info" role="presentation" aria-label="info icon"></i> Spatio-Temporal Interpolation: Intersection Model

<i class="fas fa-ruler" role="presentation" aria-label="ruler icon"></i> Simulation Study

<i class="fas fa-ruler" role="presentation" aria-label="ruler icon"></i> Results with Ice Data

<i class="fas fa-check-double" role="presentation" aria-label="check-double icon"></i> Discussion and Conclusion

<i class="fas fa-spinner" role="presentation" aria-label="spinner icon"></i> Steps for Program Completion

---
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# Importance of Arctic Sea Ice Crack Detection

+ Sea ice serves as a barrier between the atmosphere and the ocean
+ Cracks, or leads, may form in the ice pack due to dynamic processes
  - Allows for heat from the ocean to be transferred to the atmosphere (Schreyer, Sulsky, Munday, Coon, and Kwok, 2006). 
  - Accounts for half of the heat flux between the ocean and atmosphere (Badgley, 1961)
+ Previous ice crack detection methods can generally be split into two categories: thermal and deformation.

.center[

<img id='imgIceChunk' src='images/Ice Chunk.png' alt=' Artice Sea Ice with Crack' width='40%'>

<div id='modalIceChunk' class='modal'>
  
  <img class='modal-content' id='imgmodalIceChunk'>
  
  <div id='captionIceChunk' class='modal-caption'></div>
</div>
]

---
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# Thermal Ice Crack Detection Methods

+ Surface temperature differs between a crack and the surrounding sea ice.
+ Use thermal channels of the Advanced Very High Resolution Radiometer (AVHRR) (Key, Stone, Maslanik, and Ellefsen, 1993) 
  - Heavily dependent on clear skies and has issues with thin ice
+ Methods have been proposed to reduce the impact of clouds 
  - Moderate Resolution Imagery Spectroradiometer (MODIS)  (Willmes and Heinemann, 2015)
  - Fuzzy Cloud Artifact (FCAF) (Willmes and Heinemann, 2015)
  - Passive Microwave Data  (Rohrs, Kaleschke, Brohan, and Siligam, 2012)
.center[

<img id='imgthermal_example' src='images/thermal_example.png' alt=' Output from a Thermal Algorithm (Rohrs et al, 2012)' width='30%'>

<div id='modalthermal_example' class='modal'>
  
  <img class='modal-content' id='imgmodalthermal_example'>
  
  <div id='captionthermal_example' class='modal-caption'></div>
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]

---
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# Deformation Ice Crack Detection Methods

+ Deformation of a cell is determined by the motion of points (Peterson and Sulsky, 2011)
    - Finds the determinant of deformation gradient to measure accumulated area changes to show persistent diverged regions. 
    - Can find the size and orientation of the crack
+ Drawbacks
  - Need complete set of space-time observations to calculate deformation
  - The error in the deformation product may be strongly underestimated (Bouillon and Rampal, 2015)

.center[

<img id='imggrid_example' src='images/grid_example.png' alt='Example of how find deformation (Peterson & Sulsky, 2011)' width='40%'>

<img id='imgkinematic_crack_algorithm' src='images/kinematic_crack_algorithm.png' alt='Example of detected leads using a kinematic crack algorithm which uses the determinant of the deformation gradient to detect leads (Peterson & Sulsky, 2011)' width='30%'>

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  <img class='modal-content' id='imgmodalkinematic_crack_algorithm'>
  
  <div id='captionkinematic_crack_algorithm' class='modal-caption'></div>
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]

---
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# Data

.center[

<img id='imgrgps_grid' src='images/rgps_grid.jpg' alt='Example of initial grid used to track movement (Peterson & Sulsky, 2011)' width='25%'>

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+ Sea Ice can be tracked by NASA's RADARSTAT Geophysical Processor System (RGPS), which uses synthetic aperture radar (SAR) images to track the trajectory of points on an ice sheet.
+ Each grid cell vertex is assigned an identifier (cell `\(j=1,...,n\)`) which is used for tracking
+ Set of all trajectories:

.center[
`\(\mathcal{G} = \left\{g_1, ..., g_n\right\}\)` `\(\\\)`
where `\(g_{j} = \left\{s_{jt} : t \in \mathcal{T}_j\right\}\)`, `\(\mathcal{T}_j \subset \left\{t=1...T\right\}\)` a collection of time points where `\(cell_j\)` is observed `\(\\\)`
and `\({s_{jt}}\)` = `\((x_{jt}, y_{jt})\)`
]

+ For our study region, `\(n\)` = 8811, and `\(T\)` = 22

???

An illustration of RGPS data is
shown in Fig. 7.1, where satellite views of a 50 km by 50 km region of Arctic ice
have a 5 km × 5 km RGPS grid superimposed. The time span between the first
and second observation is 18.5 h and the satellite images were recorded in mid May
2002.

---
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# Motivating Picture

+ Each line is a trajectory, `\(g_j\)`, plotted in an x-y coordinate

.center[

<img id='imgtraj_plot' src='images/traj_plot.png' alt='Plot of id trajectories to show movement and directiction of movement' width='90%'>

<div id='modaltraj_plot' class='modal'>
  
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  <div id='captiontraj_plot' class='modal-caption'></div>
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# Research Objectives

1. Develop a ice crack detection method
  - Using only the movement of the ice sheet
  - Cluster trajectories based on movement
  - The boundaries would be possible locations of ice cracks
2. Develop a spatio-temporal (ST) interpolation method to predict missing points along a trajectory 
  - Using the information gained from the clustering
  - Take into account nonstationarity of the data (estimate parameters separately for each cluster)

---
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# Existing ST Clustering Methods

+ Combination of geographic location with time introduces new challenges in clustering, where a cluster is now determined based on spatial and temporal similarity `\(\\\)` `\(\\\)`
+ Similarity Measures
  - One of the components of a clustering algorithm is to determine how to measure similarity
  - Similarity measures have been developed for spatio-temporal data, but many rely on having trajectories of the same length or are sensitive to noise
+ Density-Based Clustering
  - Objects that are densely packed in a region should be grouped together in a cluster
  - Can cluster objects into any shape and number of clusters do not need to be pre-defined
  - However, since our data is based on a grid, point density will be consistent across the domain
+ Model-Based Clustering
  - Can assume a model for each cluster, where the best fitting data for the model is found in order to determine cluster membership
  - Can be difficulty in finding the assume model

(Info from (Ansari, Ahmad, Khan, Bhushan, and others, 2020))

---
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# Existing ST Clustering Methods

+ Nonstationary Models (Kim, Mallick, and Holmes, 2005)
  - Data can be partitioned and a model can be developed for each partition
  - Partitioning can be accomplished through a Voronoi Tesselation and then use a Piecewise Gaussian Process to model the nonstationary process
  - Currently these methods are only for a spatial domain 
  
  .center[

<img id='imgvoronoi_tesselation' src='images/voronoi_tesselation.png' alt='Example of a Voronoi Tesselation' width='30%'>

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---
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# Find Ice Trajectory Features: Bounding Box

+ We create a bounding box around for each trajectory to represent it's movement
+ Bounding Box Features:
  - Length travel in x/y between the minimum and maximum location 
.center[
( `\(x_{max} - x_{min}\)` and `\(y_{max} - y_{min}\)`)]
  - Length travel in x/y between latest and earliest observation 
  .center[
  ( `\(x_{1} - x_{0}\)` and `\(y_{1} - y_{0}\)`)]
  - Angle of movement (direction)
  - Average x/y value
  - If clustering a sub-trajectory, can also include previous features

.center[

<img id='imgboundingbox' src='images/bounding-box.png' alt='Points used to Develop Bounding Box' width='25%'>

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  <img class='modal-content' id='imgmodalboundingbox'>
  
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---
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#Clustering with K-Means

+ Bounding Box features were used as input into K-Means clustering, which partitions n observations into k clusters.
+ K-Means Clustering (Steinley, 2006)
  - All observations must belong to a cluster and each cluster, k, must have at least one observation
  - Iterative Procedure 
  - Minimize squared Euclidean distance between an observation and the centroid vector of a cluster
  - Centroid vector found by averaging the features of each cluster member

---
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#Clustering with K-Means

+ **Drawback**: Number of Clusters must be specified prior to clustering
  - Number of Clusters determined using the Silhouette statistic
+ Silhouette Statistic (Kodinariya and Makwana, 2013)
  - Compares within cluster distances to between cluster distances
  
.center[
`\(s(i) = \frac{b(i) - a(i)}{max(a(i), b(i))}\)` `\(\\\)`
where `\(a(i)\)` is the average distance between i and observations in same cluster and `\(b(i)\)` is the minimum average distance between i and observations in other clusters
]

- Number of clusters determined by largest average silhoutte width

.center[

<img id='imgsilhouette_stat' src='images/silhouette_stat.jpeg' alt='Silhouette Statistic for Clustering using all of our Data' width='40%'>

<div id='modalsilhouette_stat' class='modal'>
  
  <img class='modal-content' id='imgmodalsilhouette_stat'>
  
  <div id='captionsilhouette_stat' class='modal-caption'></div>
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---
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#Missing Data

+ In general, data collection methods may fail, leaving positions in a trajectory unknown or may want to overcome sampling sparseness
+ In our case, missing data is due to the path of the satellite used to collect the data.

.center[

<img id='imgdata_example' src='images/data_example.jpeg' alt='Missing Data within the Ice Sheet' width='80%'>

<div id='modaldata_example' class='modal'>
  
  <img class='modal-content' id='imgmodaldata_example'>
  
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#Linear Interpolation for Ice Trajectory

+ Popular due to ease in implementation
+ Performs best for linearly moving objects, or can potentially work well with non-linear data if sample in high enough frequency (Guo, Mou, Chen, and Chen, 2021)
  
<br>

.center[

<img id='imglinpic2' src='images/lin-pic2.png' alt='How Linear Interpolation Calculates a Missing Point' width='60%'>

<div id='modallinpic2' class='modal'>
  
  <img class='modal-content' id='imgmodallinpic2'>
  
  <div id='captionlinpic2' class='modal-caption'></div>
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#Finding Spatio-Temporal Neighbors

+ Using information gained from clusters to identify spatio-temporal neighbors
  - Know how other points in the cluster move at a time point
  - Would expect a missing point  at that time to move similarly to known points
+ Cluster by Weekly trajectories to find neighbors
  - Cluster by weeks because as smallest interval could detect movement and also see some continuity between weeks
  - Intersection of one week's clusters with the week before and week after would create groups
  - Each member of a group is then a spatio-temporal neighbor of the other members as they are in a similar geographic region over time. 
  
---
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#Finding Spatio-Temporal Neighbors - Steps

1. Create a polygon around each cluster for a week 
  - Finding boundary coordinates of the clusters
  - Done in a sequential manner
2. After a polygon is created, all `\(cell_j\)` located in that polygon are removed from the dataset.
  - This was done to decrease the amount of overlapping polygons
  - Make sure each `\(cell_j\)` is found within a polygon - do not want to lose data
3. Find the intersection of the weekly polygons
  - All of the points within that intersection are spatio-temporal neighbors.

.center[

<img id='imgintersection_ex' src='images/intersection_ex.jpeg' alt='Example of finding Intersection Polygons' width='50%'>

<div id='modalintersection_ex' class='modal'>
  
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---
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#Gaussian Process (GP)

**For Spatial Data**

`\(\left\{X(s): s \in D \subset R^2\right\}\)` is a Gaussian Process if all its finite-dimensional distributions are Gaussian

.center[
ie. `\(X(s) \sim GP(0,c(.|.))\)`
]

Meaning, for `\(\left\{s_1,...,s_n\right\}\)`, `\(x\)` = `\((x_1,...,x_n)^T \sim MVN(0, \Sigma_{\theta})\)`

Can define `\(\Sigma_{\theta}\)` as the Exponential Covariance Function (Guinness and Katzfuss, 2021)

.center[
`\(\Sigma_{\theta} = \sigma^2\exp(-||x-y||/\phi)\)` `\(\\\)`
where `\(\sigma^2\)` is the variance and `\(\phi\)` is the range
]

Joint density of the observations can be written as a product of conditional densities (Guinness, 2018)
.center[
`\(f(x_1,...,x_n) = f(x_1)\prod^n_{i=2} f(x_{i}|x_{1},...,x_{i-1})\)`

]

This can be a computationally complex process due to the inversion of `\(\Sigma_{\theta}\)`

---
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#Gaussian Process (GP)

**Extension to ST Data**

Now, the covariance function is an Exponential Space-Time, which is a separable covariance function (Wikle et al, 2019)

.center[
`\(\Sigma_{\theta}(s;t) = \Sigma_{\theta}^{(s)}*\Sigma_{\theta}^{(t)}\)` `\(\\\)` where 
`\(\Sigma_{\theta}^{(s)} = \sigma^2\exp\left\{-||x-y||/\phi\right\}\)`  and `\(\Sigma_{\theta}^{(t)} = \sigma^2\exp\left\{-|t|/\tau\right\}\)`

where `\(\sigma^2\)` =  variance, `\(\phi\)` = spatial range, `\(\tau\)` = temporal range
]

Separability can speed up computations if no missing data, but we do...

---
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#Vecchia's Approximation for a GP

+ **Goal:** To speed up calculation of a GP
+ Writes the joint density as a product of conditional distributions, where only a subset of the data is used to create the conditional distributions (Guinness, 2018)

.center[
`\(f(x_1,...,x_n) = f(x_1)\prod^n_{i=2} f(x_i|x_{n(i)})\)` `\(\\\)`
where `\(n(i)\)` are the neighbors of observation `\(i\)`
]

+ Neighbors are obtained from the order of points (Vecchia, 1988)

+ Vecchia's Approximation is implemented in the GpGp package, where updates to the ordering method and a grouping method were introduced to speed up calculations (Guinness and Katzfuss, 2021)

---
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#Spatio-Temporal Interpolation

+ Individual model developed for both x and y using the GpGp package in R (Guinness and Katzfuss, 2021)
  - Within each intersection
  - Due this to take into account the nonstationarity
+ Use Exponential Space-Time covariance function (as previously defined)
+ Output is the maximum likelihood estimates for the mean and covariance parameters
+ Use model to determine estimates of missing locations
  - From the conditional expectation of the model
  - Create a grid encompassing our ice sheet to give a starting value of the missing locations. 
  - The model will then adjust this location using its known neighbors

---
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#Simulation Study

+ Create Underlying Process Grid 
  - Simulating movement of ocean that causes observations to move
  - Initial grid was created and shifted seven times to represent seven days of movement in the underlying process.
  - This data is then used to create the covariance matrix, `\(C_{d,c}(\theta)\)`
  - Covariance parameters and defined mean trend ( `\(\mu_{d,c}\)` ) is different for each cluster `\(\\\)`

.center[
`\(U_{d,c}(s,t) \sim GP(\mu_{d,c}, C_{d,c}(\theta))\)` `\(\\\)`

where `\(U_{d,c}(s,t)\)` is the displacement at location `\(s\)` & time `\(t\)` `\(\\\)` for `\(d\)` = x or y, cluster `\(c\)`

]

**Note**: Only 2 clusters for simplicity

---
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#Simulation Study

+ Create Observed Grid
  - Movement of an observed point is determined by the value of the nearest point of the underlying process for that day ( `\(g\)` )
  - Obtained a week's worth of simulated data
  
.center[
  `\((x_{t,j}, y_{t,j}) = (U^{X}_{t-1,c,g}, U^{Y}_{t-1,c,g}) + (x_{t-1,j}, y_{t-1,j})\)` `\(\\\)`
  where t=1,...,7 (time), j = 1,...,121 (id), `\(\\\)` U is the underlying process value at `\(t-1\)` for cluster ( `\(c\)` ) and grid id ( `\(g\)` )
]

.center[

<img id='imgboth_grid2' src='images/both_grid2.jpeg' alt='Underlying and Observed Grid Plotted Together' width='40%'>

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#Simulated Data

Created 3 different scenarios, each with different parameter values.

.center[

<img id='imgsim_traj' src='images/sim_traj.png' alt='Simulated Trajectories for Each Simulation' width='90%'>

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#Simulated Clustering Results

+ Results are shown at two different time points
  - On initial grid ( `\(t=0\)` )
  - On last day of the week ( `\(t=7\)` )
<br>
<br>
<br>

.center[

<img id='imgsim_init_clust' src='images/sim_init_clust.png' alt='Clusters for each Simulation at t=0' width='40%'>

<img id='imgsim_last_clust' src='images/sim_last_clust.png' alt='Clusters for each Simulation at t=7' width='40%'>

<div id='modalsim_init_clust' class='modal'>
  
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---
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#Simulated Interpolation Results

+ Simulated and clustered another week of data.
+ 10% of the data for the first week are randomly assigned to be missing.
+ Other methods for comparison:
  - Linear Interpolation
  - Instead of running model inside each intersection, a model was developed using all known points (essentially ignoring the nonstationarity aspect of our data)

<br>
  
<table style="width:80%;">
<caption>RMSE for Interpolation Methods</caption>
 <thead>
<tr>
<th style="empty-cells: hide;border-bottom:hidden;" colspan="1"></th>
<th style="border-bottom:hidden;padding-bottom:0; padding-left:3px;padding-right:3px;text-align: center; " colspan="2"><div style="border-bottom: 1px solid #ddd; padding-bottom: 5px; ">Intersection Model</div></th>
<th style="border-bottom:hidden;padding-bottom:0; padding-left:3px;padding-right:3px;text-align: center; " colspan="2"><div style="border-bottom: 1px solid #ddd; padding-bottom: 5px; ">Linear</div></th>
<th style="border-bottom:hidden;padding-bottom:0; padding-left:3px;padding-right:3px;text-align: center; " colspan="2"><div style="border-bottom: 1px solid #ddd; padding-bottom: 5px; ">No Intersection Model</div></th>
</tr>
  <tr>
   <th style="text-align:center;"> Simulation </th>
   <th style="text-align:center;"> X </th>
   <th style="text-align:center;"> Y </th>
   <th style="text-align:center;"> X </th>
   <th style="text-align:center;"> Y </th>
   <th style="text-align:center;"> X </th>
   <th style="text-align:center;"> Y </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:center;"> 1 </td>
   <td style="text-align:center;"> 1.496 </td>
   <td style="text-align:center;"> 1.517 </td>
   <td style="text-align:center;"> 1.042 </td>
   <td style="text-align:center;"> 1.226 </td>
   <td style="text-align:center;"> 1.438 </td>
   <td style="text-align:center;"> 1.295 </td>
  </tr>
  <tr>
   <td style="text-align:center;"> 2 </td>
   <td style="text-align:center;"> 1.628 </td>
   <td style="text-align:center;"> 1.580 </td>
   <td style="text-align:center;"> 1.455 </td>
   <td style="text-align:center;"> 1.540 </td>
   <td style="text-align:center;"> 1.474 </td>
   <td style="text-align:center;"> 1.488 </td>
  </tr>
  <tr>
   <td style="text-align:center;"> 3 </td>
   <td style="text-align:center;"> 1.342 </td>
   <td style="text-align:center;"> 1.338 </td>
   <td style="text-align:center;"> 0.950 </td>
   <td style="text-align:center;"> 0.920 </td>
   <td style="text-align:center;"> 1.458 </td>
   <td style="text-align:center;"> 1.489 </td>
  </tr>
</tbody>
</table>

---
class:primary

#Simulated Interpolation Results

<table>
<caption>RMSE for Interpolation Methods by cluster</caption>
 <thead>
<tr>
<th style="empty-cells: hide;border-bottom:hidden;" colspan="2"></th>
<th style="border-bottom:hidden;padding-bottom:0; padding-left:3px;padding-right:3px;text-align: center; " colspan="2"><div style="border-bottom: 1px solid #ddd; padding-bottom: 5px; ">Intersection</div></th>
<th style="border-bottom:hidden;padding-bottom:0; padding-left:3px;padding-right:3px;text-align: center; " colspan="2"><div style="border-bottom: 1px solid #ddd; padding-bottom: 5px; ">Linear</div></th>
<th style="border-bottom:hidden;padding-bottom:0; padding-left:3px;padding-right:3px;text-align: center; " colspan="2"><div style="border-bottom: 1px solid #ddd; padding-bottom: 5px; ">No Intersection</div></th>
</tr>
  <tr>
   <th style="text-align:center;"> Simulation </th>
   <th style="text-align:center;"> Cluster </th>
   <th style="text-align:center;"> X </th>
   <th style="text-align:center;"> Y </th>
   <th style="text-align:center;"> X </th>
   <th style="text-align:center;"> Y </th>
   <th style="text-align:center;"> X </th>
   <th style="text-align:center;"> Y </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:center;"> 1 </td>
   <td style="text-align:center;"> 1 </td>
   <td style="text-align:center;"> 1.383 </td>
   <td style="text-align:center;"> 1.555 </td>
   <td style="text-align:center;"> 0.815 </td>
   <td style="text-align:center;"> 1.163 </td>
   <td style="text-align:center;"> 1.562 </td>
   <td style="text-align:center;"> 1.290 </td>
  </tr>
  <tr>
   <td style="text-align:center;">  </td>
   <td style="text-align:center;"> 2 </td>
   <td style="text-align:center;"> 1.573 </td>
   <td style="text-align:center;"> 1.488 </td>
   <td style="text-align:center;"> 1.188 </td>
   <td style="text-align:center;"> 1.272 </td>
   <td style="text-align:center;"> 1.337 </td>
   <td style="text-align:center;"> 1.298 </td>
  </tr>
  <tr>
   <td style="text-align:center;"> 2 </td>
   <td style="text-align:center;"> 1 </td>
   <td style="text-align:center;"> 1.700 </td>
   <td style="text-align:center;"> 1.653 </td>
   <td style="text-align:center;"> 1.316 </td>
   <td style="text-align:center;"> 1.329 </td>
   <td style="text-align:center;"> 1.658 </td>
   <td style="text-align:center;"> 1.596 </td>
  </tr>
  <tr>
   <td style="text-align:center;color: black !important;background-color: yellow !important;">  </td>
   <td style="text-align:center;color: black !important;background-color: yellow !important;"> 2 </td>
   <td style="text-align:center;color: black !important;background-color: yellow !important;"> 1.584 </td>
   <td style="text-align:center;color: black !important;background-color: yellow !important;"> 1.534 </td>
   <td style="text-align:center;color: black !important;background-color: yellow !important;"> 1.503 </td>
   <td style="text-align:center;color: black !important;background-color: yellow !important;"> 1.612 </td>
   <td style="text-align:center;color: black !important;background-color: yellow !important;"> 1.407 </td>
   <td style="text-align:center;color: black !important;background-color: yellow !important;"> 1.451 </td>
  </tr>
  <tr>
   <td style="text-align:center;"> 3 </td>
   <td style="text-align:center;"> 1 </td>
   <td style="text-align:center;"> 1.318 </td>
   <td style="text-align:center;"> 1.346 </td>
   <td style="text-align:center;"> 1.156 </td>
   <td style="text-align:center;"> 1.096 </td>
   <td style="text-align:center;"> 1.434 </td>
   <td style="text-align:center;"> 1.405 </td>
  </tr>
  <tr>
   <td style="text-align:center;">  </td>
   <td style="text-align:center;"> 2 </td>
   <td style="text-align:center;"> 1.353 </td>
   <td style="text-align:center;"> 1.334 </td>
   <td style="text-align:center;"> 0.647 </td>
   <td style="text-align:center;"> 0.673 </td>
   <td style="text-align:center;"> 1.484 </td>
   <td style="text-align:center;"> 1.581 </td>
  </tr>
</tbody>
</table>

---
class:primary

#Simulated Interpolation Results

+ A benefit of using a model-based approach is that are able to determine the uncertainty of the estimate. 
+ Conducted 30 simulations of predictions, which are used to calculate the standard deviation
+ Found Intervals by
.center[
`\(\hat{x} \pm (2*\sigma_x)\)` and `\(\hat{y} \pm (2*\sigma_y)\)` 
]
+ Then found the proportion of intervals that contained the true value

<table style='width:80%;'>
<caption>Coverage</caption>
 <thead>
  <tr>
   <th style="text-align:center;"> Simulation </th>
   <th style="text-align:center;"> X </th>
   <th style="text-align:center;"> Y </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:center;"> 1 </td>
   <td style="text-align:center;"> 0.281 </td>
   <td style="text-align:center;"> 0.298 </td>
  </tr>
  <tr>
   <td style="text-align:center;"> 2 </td>
   <td style="text-align:center;"> 0.188 </td>
   <td style="text-align:center;"> 0.250 </td>
  </tr>
  <tr>
   <td style="text-align:center;"> 3 </td>
   <td style="text-align:center;"> 0.291 </td>
   <td style="text-align:center;"> 0.375 </td>
  </tr>
</tbody>
</table>

---
class:primary

#Ice Data Results: Clustering Using All Data

.center[

<img id='imgall_clust' src='images/all_clust.jpeg' alt='Clustering of Ice Trajectories using All Data in Bounding Box' width='90%'>

<div id='modalall_clust' class='modal'>
  
  <img class='modal-content' id='imgmodalall_clust'>
  
  <div id='captionall_clust' class='modal-caption'></div>
</div>
]

---
class:primary

#Clustering Using All Data - Comparison

+  Can compare our results with deformation data found using a kinematic crack algorithm calculated using the RGPS data (Peterson and Sulsky, 2011)
  - Note that this image does not represent the true ice cracks, just the cracks determined by this method.

.center[

<img id='imgallweekscomp' src='images/all-weeks-comp.png' alt='Comparison of Our Results to a Kinematic Crack Algorithm' width='80%'>

<div id='modalallweekscomp' class='modal'>
  
  <img class='modal-content' id='imgmodalallweekscomp'>
  
  <div id='captionallweekscomp' class='modal-caption'></div>
</div>
]

---
class:primary

#Ice Data Results: Clustering by Week

.center[

<img id='imgclust_by_week' src='images/clust_by_week.png' alt='Results of Clustering by Week' width='90%'>

<div id='modalclust_by_week' class='modal'>
  
  <img class='modal-content' id='imgmodalclust_by_week'>
  
  <div id='captionclust_by_week' class='modal-caption'></div>
</div>

]

---
class:primary

#Overall Interpolation Results along Border

+ Took a random hold-out of points along the border, as the border may be locations of irregular movement
  
<table style="width:80%;">
<caption>RMSE for Interpolation Methods along Border</caption>
 <thead>
<tr>
<th style="empty-cells: hide;border-bottom:hidden;" colspan="1"></th>
<th style="border-bottom:hidden;padding-bottom:0; padding-left:3px;padding-right:3px;text-align: center; " colspan="2"><div style="border-bottom: 1px solid #ddd; padding-bottom: 5px; ">Intersection</div></th>
<th style="border-bottom:hidden;padding-bottom:0; padding-left:3px;padding-right:3px;text-align: center; " colspan="2"><div style="border-bottom: 1px solid #ddd; padding-bottom: 5px; ">Linear</div></th>
</tr>
  <tr>
   <th style="text-align:right;"> Week </th>
   <th style="text-align:right;"> X </th>
   <th style="text-align:right;"> Y </th>
   <th style="text-align:right;"> X </th>
   <th style="text-align:right;"> Y </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:right;color: black !important;background-color: yellow !important;"> 1 </td>
   <td style="text-align:right;color: black !important;background-color: yellow !important;"> 3.215 </td>
   <td style="text-align:right;color: black !important;background-color: yellow !important;"> 3.408 </td>
   <td style="text-align:right;color: black !important;background-color: yellow !important;"> 1.606 </td>
   <td style="text-align:right;color: black !important;background-color: yellow !important;"> 4.494 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 2 </td>
   <td style="text-align:right;"> 3.539 </td>
   <td style="text-align:right;"> 3.256 </td>
   <td style="text-align:right;"> 2.034 </td>
   <td style="text-align:right;"> 1.832 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 3 </td>
   <td style="text-align:right;"> 3.031 </td>
   <td style="text-align:right;"> 2.958 </td>
   <td style="text-align:right;"> 0.991 </td>
   <td style="text-align:right;"> 1.240 </td>
  </tr>
</tbody>
</table>

---
class:primary

#Interpolation Results along Border for Week 2

<table style="width:80%;">
<caption>RMSE for Interpolation Methods by cluster for Week 2</caption>
 <thead>
<tr>
<th style="empty-cells: hide;border-bottom:hidden;" colspan="1"></th>
<th style="border-bottom:hidden;padding-bottom:0; padding-left:3px;padding-right:3px;text-align: center; " colspan="2"><div style="border-bottom: 1px solid #ddd; padding-bottom: 5px; ">Intersection</div></th>
<th style="border-bottom:hidden;padding-bottom:0; padding-left:3px;padding-right:3px;text-align: center; " colspan="2"><div style="border-bottom: 1px solid #ddd; padding-bottom: 5px; ">Linear</div></th>
</tr>
  <tr>
   <th style="text-align:center;"> Cluster </th>
   <th style="text-align:center;"> X </th>
   <th style="text-align:center;"> Y </th>
   <th style="text-align:center;"> X </th>
   <th style="text-align:center;"> Y </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:center;color: black !important;background-color: yellow !important;"> 1 </td>
   <td style="text-align:center;color: black !important;background-color: yellow !important;"> 2.81 </td>
   <td style="text-align:center;color: black !important;background-color: yellow !important;"> 3.04 </td>
   <td style="text-align:center;color: black !important;background-color: yellow !important;"> 3.82 </td>
   <td style="text-align:center;color: black !important;background-color: yellow !important;"> 5.19 </td>
  </tr>
  <tr>
   <td style="text-align:center;"> 2 </td>
   <td style="text-align:center;"> 2.94 </td>
   <td style="text-align:center;"> 2.96 </td>
   <td style="text-align:center;"> 2.39 </td>
   <td style="text-align:center;"> 0.99 </td>
  </tr>
  <tr>
   <td style="text-align:center;"> 3 </td>
   <td style="text-align:center;"> 2.94 </td>
   <td style="text-align:center;"> 3.07 </td>
   <td style="text-align:center;"> 1.46 </td>
   <td style="text-align:center;"> 1.97 </td>
  </tr>
  <tr>
   <td style="text-align:center;"> 4 </td>
   <td style="text-align:center;"> 3.26 </td>
   <td style="text-align:center;"> 3.20 </td>
   <td style="text-align:center;"> 1.32 </td>
   <td style="text-align:center;"> 0.54 </td>
  </tr>
  <tr>
   <td style="text-align:center;"> 5 </td>
   <td style="text-align:center;"> 4.14 </td>
   <td style="text-align:center;"> 3.88 </td>
   <td style="text-align:center;"> 0.43 </td>
   <td style="text-align:center;"> 0.34 </td>
  </tr>
</tbody>
</table>

---
class:primary

#Interpolation Results along Border by Week

+ Cluster 1: Data is More Spread out and not linear
+ Cluster 5: Data does not move much

.center[

<img id='imgw1_c1_traj' src='images/w1_c1_traj.jpeg' alt='Cluster Trajectories where our method performs the best' width='40%'>

<img id='imgw2_c5_traj' src='images/w2_c5_traj.jpeg' alt='CCluster Trajectories where Linear Interpolation performs the best' width='40%'>

<div id='modalw1_c1_traj' class='modal'>
  
  <img class='modal-content' id='imgmodalw1_c1_traj'>
  
  <div id='captionw1_c1_traj' class='modal-caption'></div>
</div>

<div id='modalw2_c5_traj' class='modal'>
  
  <img class='modal-content' id='imgmodalw2_c5_traj'>
  
  <div id='captionw2_c5_traj' class='modal-caption'></div>
</div>

]

---
class:primary

#Coverage Proportion by Cluster

<br>

<table style="width:75%;">
<caption><b>Coverage for Week 2</b></caption>
 <thead>
  <tr>
   <th style="text-align:center;"> Cluster </th>
   <th style="text-align:center;"> X </th>
   <th style="text-align:center;"> Y </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:center;color: black !important;background-color: yellow !important;"> 1 </td>
   <td style="text-align:center;color: black !important;background-color: yellow !important;"> 0.467 </td>
   <td style="text-align:center;color: black !important;background-color: yellow !important;"> 0.413 </td>
  </tr>
  <tr>
   <td style="text-align:center;"> 2 </td>
   <td style="text-align:center;"> 0.280 </td>
   <td style="text-align:center;"> 0.278 </td>
  </tr>
  <tr>
   <td style="text-align:center;"> 3 </td>
   <td style="text-align:center;"> 0.337 </td>
   <td style="text-align:center;"> 0.351 </td>
  </tr>
  <tr>
   <td style="text-align:center;"> 4 </td>
   <td style="text-align:center;"> 0.260 </td>
   <td style="text-align:center;"> 0.251 </td>
  </tr>
  <tr>
   <td style="text-align:center;"> 5 </td>
   <td style="text-align:center;"> 0.149 </td>
   <td style="text-align:center;"> 0.163 </td>
  </tr>
</tbody>
</table>

---
class:primary

# Discussion of Methods

**Clustering with Bounding Box:**

+ Advantages
  - Provides information about how the data is moving
  - Shown through simulations and the ice data to provide a reasonable estimation of the locations of cracks
+ Drawbacks
  - Relies on a pre-defined number of clusters, which generally is not known
  - Have a set number of cluster, so a limit on how many cluster boundaries, thus cracks.
  
**ST Interpolation with Intersection Models:**

+ Advantages
  - Takes into account the nonstationarity of the data
  - Showed some improvement, in terms of RMSE, over linear interpolation for curved data that is not highly sampled
  - Able to estimate data on first and last day of a dataset, which linear interpolation is not able to do. 
  - Able to calculate uncertainty
  
---
class:primary

#Comments on Model Development

+ Log-likelihood does not converge 
  - Temporal range value is exceptionally large especially in comparison to the spatial range
+ Impractical Covariance Parameter estimates - large temporal range
  - This may be due to only working with small amount of data that is not moving much over time
  - With the simulated data, as more days were added to the dataset, the temporal range does slowly decrease
+ Potentially not a lot of data in the intersection
  - Some intersections are small
  - Need variation in time
  
  
---
class:primary

# Steps for Program Completion 
+ Modeling Arctic Sea Ice
  - Bivariate Interpolation
  - Visualization of results
  - Investigate model further (why covariance parameter estimates impractical)
+ Visualization and comparison of spatial field data
+ Explaining Machine Learning Output for Spatial Data (Next Step)
  - To explain what the optimal management decisions are for profitability for a farmer's field, accounting for a range of factors. 
  - Develop a user-interface which will be designed around explaining machine learning output to non-experts, 
  - Build trust in the model predictions without requiring farmers to learn the details of statistical modeling.

---
class:primary
# References

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(1961).
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(2015).
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<p><cite><a id='bib-gpgp_pkg'></a><a href="#cite-gpgp_pkg">Guinness, J. and M. Katzfuss</a>
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<p><cite><a id='bib-guo_improved_2021'></a><a href="#cite-guo_improved_2021">Guo, S., J. Mou, L. Chen, et al.</a>
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(1993).
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<p><cite><a id='bib-kim_analyzing_2005'></a><a href="#cite-kim_analyzing_2005">Kim, H., B. K. Mallick, and C. Holmes</a>
(2005).
&ldquo;Analyzing Nonstationary Spatial Data Using Piecewise Gaussian Processes&rdquo;.
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<p><cite><a id='bib-kodinariya_2013'></a><a href="#cite-kodinariya_2013">Kodinariya, T. and P. Makwana</a>
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<p><cite><a id='bib-peterson_evaluating_2011'></a><a href="#cite-peterson_evaluating_2011">Peterson, K. and D. Sulsky</a>
(2011).
&ldquo;Evaluating Sea Ice Deformation in the Beaufort Sea Using a Kinematic Crack Algorithm with RGPS Data&rdquo;.
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<em>Remote Sensing of the Changing Oceans</em>.
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<p><cite><a id='bib-rohrs_algorithm_2012'></a><a href="#cite-rohrs_algorithm_2012">Rohrs, J., L. Kaleschke, D. Brohan, et al.</a>
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</font>

---
class:primary
# References
<font size="2">
<p><cite><a id='bib-steinley_kmeans_2006'></a><a href="#cite-steinley_kmeans_2006">Steinley, D.</a>
(2006).
&ldquo;K-means clustering: A half-century synthesis&rdquo;.
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DOI: <a href="https://doi.org/10.1348/000711005X48266">10.1348/000711005X48266</a>.</cite></p>

<p><cite><a id='bib-vecchia1988estimation'></a><a href="#cite-vecchia1988estimation">Vecchia, A. V.</a>
(1988).
&ldquo;Estimation and model identification for continuous spatial processes&rdquo;.
In: <em>Journal of the Royal Statistical Society: Series B (Methodological)</em> 50.2, pp. 297&ndash;312.</cite></p>

<p><cite><a id='bib-willmes_pan-arctic_2015'></a><a href="#cite-willmes_pan-arctic_2015">Willmes, S. and G. Heinemann</a>
(2015).
&ldquo;Pan-Arctic lead detection from MODIS thermal infrared imagery&rdquo;.
In: <em>Annals of Glaciology</em> 56, pp. 29&ndash;37.
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Wikle, C.K., Zammit-Mangion, A., and Cressie, N. (2019). Spatio-Temporal Statistics with R. Chapman & Hall/CRC, Boca Raton, FL.

</font>

---
class:inverse
<br>
<br>
<br>
.center[
# Questions?
<br>
<br>

]