# Spatial Data Reconstruction via ADMM and Spatial Spline Regression +
## 5 Performance Evaluation
### Dataset Description  
The model in (13) is general and not attached to any particular empirical problem, and it does not contain many implicit assumptions. However, in order to measure its performance, we evaluate the model using real-world data. Due to generality of the proposed learning algorithm, the range of possible data sets is potentially big. For our empirical case study, we chose cell phone data, where there exists a problem of recovering a spatial field from coarse aggregations observed at sparse cell phone towers. We do not overestimate the problem, but rather see this particular data set suitable for an extensive case study. The Milan Call Description Records (CDR) dataset is a part of the Telecom Italia Big Data Challenge dataset provided by Telecom Italia Mobile. It contains the telecommunications activity records from 1 November 2013 to 31 December 2013 in the city of Milan [25]. The dataset divides Milan into a \(100\times 100\) square grid, where each square is size of about 235 m \(\times\) 235 m. In the dataset, each record consists of six entries: square ID, incoming call activity, outgoing call activity, incoming SMS activity, outgoing SMS activity, and time-stamp of 10-minute time slot. The values of the four types of activities are normalized to the same scale. Another dataset we utilized is the Milan geographical attribute dataset available from the Municipality of Milan's Open Data website [1]. This dataset consist of features of central 2726 squares among the whole 10,000 squares. The features of each square include: population, green area percentage, number of sport centers, number of universities, number of businesses, and number of bus stops. Figure 1 shows the area covered by these grid squares. The 2726 squares covers the central part of the Milan city and contains the majority of telecommunication activities in the dataset. We refer to [2] for more detailed description about this dataset. In our experiments, we compare the performance of different approaches on these squares. The general problem of recovering a spatial field from coarse aggregations observed at sparse points in the field in this particular case study is reformulated into the problem of recovering the distribution of cell phone activities over the whole 2726 square regions given that only aggregated activity observations in base stations are known. We need to further process the Milan CDR dataset to study this problem. First, we sum up the four types of activities during 1 November 2013 to 28 November 2013 and 1 December 2013 to 28 December 2013, respectively, to come up with the activity volume of each squares during November or December. These two datasets are served as the ground-truth datasets of Milan cell phone activity distributions. Figure 1a,b show the heat maps of activity volumes in each square during November and December. Second, after we aggregated the two months' activities for each square, we need to set the locations of base stations (BSs). According to [29], there are roughly 200 base stations in Milan. However, the exact locations are not available. Thus, we assume the 200 BSs are randomly distributed according to the following probability distribution  
\[\Pr(\text{Set square }i\text{ as BS})=f(\mathbf{p}_{i})/\sum_{j=1}^{N}f(\mathbf{p}_{j}), \tag{20}\]  
where \(f(\mathbf{p}_{i})\) is the cell phone activity volume in square \(i\), \(i=\{1,\ldots,N\}\), \(N=2726\) is the number of squares we are focusing on. Notice that, when we have 200 base station's aggregated observations, they only cover 7.34% of the whole 2726 squares region. This is extremely sparse and makes our problem highly challenging. In addition, we also assume \(n_{\text{BS}}=100\) and choose 100 squares as BSs accordingto the same probability distribution to stress-test our algorithm's capability under even sparser observations. Figure 2a,b show the base station distributions for \(n_{\text{BS}}=200\) and \(n_{\text{BS}}=100\), respectively.