# Robust Damage Estimation of Typhoon Goni on Coconut Crops with Sentinel-2 Imagery
## 4 Methods
### Active Learning for Density Estimation  
Based on our previous work on tree density retrieval from Sentinel-2 [15; 17], we derive a map of coconut canopy density, covering the entire region of interest in the Philippines. A Convolutional Neural Network (CNN), with 9 residual blocks each consisting of 3 convolutional layers, predicts the number of intact coconut canopies per pixel. For details of the network architecture we refer to [15].  
We rely on active learning [15] to keep the labelling effort as low as possible for the large area of interest. The fundamental idea of active learning is to start from some base model trained with only a small amount of reference data, and to iteratively grow the training set and improve the model, with additional help of an _acquisition function_\(g(x)\). The acquisition function scores unlabelled samples \(x\), so as to identify which samples are most informative and should be added to the training set. For our large regions of interest the set of candidate samples to choose from is very large, so \(g(x)\) should be cheap to compute and scale.  
As a rule of thumb, samples to be added to the training set should be diverse to avoid annotating redundant information (e.g., annotating several trees of highly similar characteristics) and they should have high prediction uncertainty under the old model, so that their correct label adds a lot of information to the training set. We use the acquisition function proposed in our previous work [15] to balance diversity and uncertainty:  
\[g(x_{i})=\frac{\sigma^{2}(\hat{g}_{i})}{\sum\limits_{\begin{subarray}{c}j\in \mathcal{X}\end{subarray}}\sigma^{2}(\hat{g}_{j})}+\frac{d^{x}(x_{i},\mu)^{2} }{\sum\limits_{\begin{subarray}{c}j\in\mathcal{X}\end{subarray}}d^{z}(x_{j},\mu)^{2}}}\, \tag{1}\]  
The first term in Equation (1) estimates the epistemic (model) uncertainty, i.e., the \"lack of knowledge\" of the model after seeing only a limited set of training examples. This uncertainty is estimated with an ensemble of models trained independently [66; 67], where  
Figure 2: Overview of our method. We use active learning for efficient labelling to train a model for coconut tree density estimation. Damage estimation is performed using images from before and after the typhoon pass. After a robustness and change analysis we predict the following change classes: damaged, new or unchanged coconut, and non-coconut pixels.  
the disagreement between ensemble members serves as a measure of how uncertain the ensemble is about a specific input. In particular, with \(M\) models \(f_{i}\) trained independently of each other, we can compute the epistemic uncertainty as the standard deviation of the model ensemble:  
\[\sigma_{\hat{y}}^{2}=\frac{1}{M-1}\sum_{i=1}^{M}(\hat{y}_{i}-\hat{y})^{2}\;, \tag{2}\]  
where \(\hat{y}_{i}=f_{i}(x)\) is the individual prediction from model \(f_{i}\) and \(\hat{y}=M^{-1}\sum_{i=1}^{M}(\hat{y}_{i})\) is the average over all individual predictions. We obtain this estimate for all available samples in our study area \(\mathcal{X}\).  
The second term in Equation (1) represents an estimate of the sample diversity, and aims to maximise the variety of the selected sample set. Diversity is measured via distances \(d^{z}\) from the mean feature value \(\mu\) in the latent feature space \(z\) of the CNN. Please refer to [15] for further details.