# Animal Movement Prediction Based on Predictive Recurrent Neural Network
## 4 Experiments
### Movement Prediction  
To evaluate the prediction performance of animal movements, we used three different metrics: mean square error (MSE), structural similarity index (SSIM), and Gaussian mixture centroid distance. MSE is a standard method for evaluating the prediction quality in statistics; it measures the average square difference between pixel values of ground truth image \(G\) and pixel values of predicted image \(P\) by using Equation (10). In the equation, \(N\times M\) is the total number of pixels and (i,j) represents a pixel of image. \[MSE(G,\ P)=\ \frac{1}{NM}\sum_{i=1}^{N}\sum_{j=1}^{M}\left[P(i,j)-G(i,j)\right]^{2} \tag{10}\] \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Interpolation Case \#**} & \multicolumn{5}{c}{**RMSE (latitude)**} \\ \cline{2-6}  & **Linear** & **Bezier** & **Cubic** & **Kinematic** & **Random Forest** \\ \hline Individual \#1 & **0.233** & 0.821 & 0.273 & 0.932 & 0.385 \\ Individual \#2 & **0.078** & 0.178 & 0.104 & 0.866 & 0.199 \\ Individual \#3 & 0.078 & 0.017 & 0.068 & 0.811 & **0.012** \\ Individual \#4 & 0.462 & 0.503 & 0.33 & 0.914 & **0.283** \\ Individual \#5 & 0.430 & 0.745 & 0.464 & 1.054 & **0.224** \\ Individual \#6 & **0.243** & 0.872 & 0.392 & 0.859 & 0.566 \\ Individual \#7 & **0.023** & 0.053 & 0.033 & 0.825 & 0.040 \\ Individual \#8 & **0.047** & 1.799 & 0.308 & 0.859 & 0.205 \\ Individual \#9 & **0.012** & 0.729 & 0.078 & 0.863 & 0.127 \\ Individual \#10 & 0.548 & 0.074 & 0.549 & 0.978 & **0.136** \\ Individual \#11 & 0.027 & 0.072 & 0.107 & 0.926 & **0.018** \\ Individual \#12 & **0.054** & 0.073 & 0.105 & 1.407 & 0.085 \\ Individual \#13 & 0.493 & 0.437 & 0.574 & 0.913 & **0.278** \\ Individual \#14 & 0.317 & 0.046 & 0.244 & 0.923 & **0.036** \\ Individual \#15 & **0.022** & 0.038 & 0.052 & 0.923 & 0.064 \\ \hline Avg. & 0.204 & 0.430 & 0.245 & 0.937 & **0.177** \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Comparison of RMSEs for missing latitude records interpolation. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Interpolation Case \#**} & \multicolumn{5}{c}{**RMSE (longitude)**} \\ \cline{2-5}  & **Linear** & **Bezier** & **Cubic** & **Kinematic** & **Random Forest** \\ \hline Individual \#1 & **0.067** & 0.230 & 0.087 & 0.880 & 0.106 \\ Individual \#2 & 0.115 & 0.081 & 0.156 & 0.854 & **0.094** \\ Individual \#3 & 0.024 & 0.016 & 0.024 & 0.815 & **0.003** \\ Individual \#4 & 0.188 & 0.455 & **0.144** & 0.828 & 0.465 \\ Individual \#5 & 0.270 & 0.511 & 0.271 & 0.916 & **0.022** \\ Individual \#6 & **0.158** & 0.280 & 0.192 & 0.818 & 0.352 \\ Individual \#7 & 0.028 &On the other hand, SSIM is a method for measuring the similarity between two images and is designed to improve on traditional methods such as peak signal-to-noise ratio (PSNR) and MSE. SSIM is calculated using Equation (11), where \(\mu\), \(\sigma\), and \(\sigma_{\text{GP}}\) are the average of pixels, variance, and covariance of \(G\) and \(P\), respectively. By setting the range of SSIM from \(-1\) and \(1\), we consider that, for the two images, there exists higher structural similarity, as their SSIM values are close to \(1\). \[SSIM(G,\ P)=\ \frac{(2\mu_{G}\mu_{P})(2\sigma_{GP}+c_{2})}{\Big{(}2\mu_{G}^{ \ 2}+\mu_{P}^{\ 2}+c_{1}\Big{)}\Big{(}\sigma_{G}^{\ 2}+\sigma_{P}^{\ 2}+c_{2}\Big{)}} \tag{11}\]  
In the ecology research field, centroid analysis is widely used for evaluating animal dynamics [45; 46; 47]. Since our movement prediction model produce a movement sequence as prediction, we can calculate the density of movements and their centroids. We measure the centroid distance between the input image, \(G\), and predicted image \(P\), which indicates how accurate our model is in the prediction of animal movements. For measuring centroid distance, we first apply Gaussian mixture fitting [48] to extract centroids from each image and calculate their real coordinates. Then, we calculate their distance using the real coordinates.