Mobile phone data reveals the importance of pre-disaster inter-city social ties for recovery after Hurricane Maria

Mobile phone location data were collected by Safegraph Inc.¹ during Hurricane Maria. GPS data were obtained from mobile phones of individuals who agreed to provide their location data for research purposes, and all information were anonymized to protect the identity and security of users. Each observation includes the spatial and temporal information (time, longitude, latitude) of mobile phones as well as a hashed unique identifier (ID), as shown in Table 1. The example ID is masked to protect privacy issues.

Figure 1 shows the probability density of the number of observations per day for a user in the location dataset. As shown in panel A), the probability density of the number of GPS logs per user follows a power law distribution. The average number of logs per user was 83 and the standard deviation was 262. For our analysis of population recovery on the county scale, it is important to account for the biases in population samples. To ensure that the data are not spatially biased, we plot the number of samples and sample rate (= observed IDs / population) proportional to census population in Fig. 1 for Puerto Rico. We can see that the number of mobile phone samples are highly correlated with census population for each county, with Pearson’s correlation coefficient of 0.985 Fig. 1b. Moreover, Fig. 1c shows that the sample rates have low correlations with census population. Thus, it is shown that the mobile phone data have little to no spatial bias in sample distributions.

However, after the Hurricane, the spatial bias in mobile phone user samples increases due to various reasons, including damages to mobile phone towers cutting off mobile phone users from the mobile network, and power outages causing people to not be able to recharge their phones. As explained more in depth in the “Methods” section, we overcome this issue by using data from only the temporal periods where the spatial bias of mobile phone user sampling rate was minimal.

In this study, population and income data of each county were used for regression analysis, in accordance with the findings from previous studies (Yabe et al. 2019). Population data were obtained from the US National Census², and median income data were obtained from the American Community Survey³.

Figure 2a and b show spatial plots of the population and income distribution of counties in Puerto Rico, respectively. We can observe that in Puerto Rico, the majority of the population is concentrated in the metro areas including San Juan County. The income disparity is less significant compared to the population disparity.

The tropical cyclone Hurricane Maria developed on September 16, 2017 in the Atlantic Ocean to the northeast of South America, and made landfall in southeastern Puerto Rico on September 20, 2017 with wind speeds of 155 miles per hour. Damage to Puerto Rico was severe and widespread following the Hurricane, with heavy rainfall, flooding, storm surge, and high winds causing considerable damage (Pasch et al. 2018). Various infrastructure systems were heavily damaged, causing power outages and water shortages for the entire island for months (United States Department of Energy and Restoration 2017). Fatalities as a consequence of Maria are still under investigation, however the most recent estimates suggest between 793 to 8,498 excess deaths occurred following the storm (Kishore et al. 2018). Total economic losses are estimated to be $ 92 billion.

Physical damage caused by the hurricane are measured by the housing damage rates in each county, which was provided through the “Housing Assistance Data” provided by the Federal Emergency Management Agency (FEMA). The raw data can be found through the link⁴. We defined “housing damage rate” for each county as the total number of houses that were inspected to have had more than $ 10,000 worth of damage due to the target hurricane, divided by the number of households in that county. Figure 2c shows the spatial distribution of the housing damages in Puerto Rico. They gray dotted lines show the trajectory of the hurricane, and we can observe higher housing damage rates near the trajectory. We can observe that many of the counties in Puerto Rico experienced high housing damage rates, between 20% and 60%.

To estimate the recovery time of each county, the population displacement dynamics were estimated from mobile phone observations. First, the home locations of individuals were estimated from the mobile phone data observed prior to the hurricane. It is well known that human mobility trajectories show a high degree of temporal and spatial regularity, each individual having a significant probability to return to a few highly frequented locations, including his/her home location (Gonzalez et al. 2008). Due to this characteristic, it has been shown that home locations of individuals can be estimated with high accuracy by clustering the individual’s stay point locations over night (Calabrese et al. 2011). Home locations of each individual were estimated by applying mean-shift clustering to the nighttime stay points (observed between 8PM and 6AM), weighted by the duration of stays in each location (Ashbrook and Starner 2003; Kanasugi et al. 2013). Mean shift clustering was implemented using the scikit-learn package on Python. An individual was detected to be displaced if the individual is estimated to be staying in a location outside the city of his/her estimated home location. Such home location estimates were used to check the representativeness of mobile phone user samples in Fig. 1.

Second, using the longitudinal observations during and after the hurricane, the displacement rate of each county was estimated. The night-time stay points of each individual were estimated from their mobile phone location data observed during night time, using mean-shift clustering of mobile phone observation points weighted by the length of stay at each point. We spatially aggregated the number of people estimated to have stayed the night in each county to obtain the daily population count for each county. However, we found that there was a significant decrease in the number of observed users during and after Hurricane Maria. The number of mobile phone users observed before, during and after Hurricane Maria are shown in Fig. 3a. We observe a significant decrease in the number of users over time, which is assumed to be due to various reasons including: 1) no signals due to damage to mobile phone towers, 2) dormant users due to lack of power to charge their phones, and 3) users who uninstalled their app which collected location data. However, since our interest is in the displacement rate of the users in each county, the absolute number of observed mobile phone users is less important in this study. Rather, whether or not the sampling rate of mobile phone users (# of mobile phone users estimated to be residents of county i / # of census population of county i) is uniform across all counties is important for our analysis. The Pearson Correlation between each county’s census population and observed mobile phone users for each day, are plotted in Fig. 3b. The red plots indicate days where the correlation is within 1% confidence interval from the mean correlation after December 15th, where the sample rates are stable. The blue plots indicate days where the correlation is significantly low, where sample rates of mobile phone user samples are biased across counties. We observe that until October 7th, the Pearson Correlation is significantly lower than usual, indicating that there is spatial inequality in the sampling rates of mobile phone users. However, after October 7th, which is 16 days from the Hurricane, the spatial bias becomes minimal (high correlation), and shows that we are able to construct the population recovery dynamics without significant sampling bias. Thus, we use population displacement data observed after October 7th to estimate the recovery time to avoid the effects of power outages and mobile phone tower outages.

The observed population dynamics are fitted with an exponential function. We use the exponential model instead of the linear recovery function due to better fit to data (see “Appendix” section). We define “time until recovery”, denoted by \(\tilde {T}_{i}^{\alpha }\), as the timing of the first time the city’s population recovers to \(M_{i}^{\alpha } = M_{i}^{\infty } \cdot \alpha \), which is characterized by the long term normalized population after the disaster \(\left (M_{i}^{\infty }\right)\) and a constant parameter 0<α<1. In cases where the long term population exceeds the original population \(\left (M_{i}^{bef} = 1 < M_{i}^{\infty }\right)\), the amount of time after exceeding the original population should not be considered as time used for recover (rather, the population is growing after the disaster). In these cases, \(\tilde {T}_{i}^{\alpha }\) is defined as the timing of the first time the city’s population recovers to α, which is the population of pre-disaster level multiplied by α. In sum, time until recovery is defined as the following:

where min(x,y) returns the smaller value between x and y. α=0.95 is used as the threshold parameter in the further results, since time until recovery are not affected by the parameter value significantly (see “Appendix” section). For simple notations, we write \(\tilde {T}_{i}^{0.95}\) as T_i and call the variable “Time Until Recovery” in the following analysis. Figure 4 shows the estimated recovery times T_i of each county. We observe a general trend where the counties around San Juan recover quickly (yellow), and recovery spreads to coastal cities, then interior regions (red).

First, ordinary least squares (OLS) method is used to estimate the parameters of the general regression model specified below.

where, y is an n×1 vector representing the objective variable (recovery time), x is an n×k matrix of the independent variables, and β is a k×1 vector of the coefficients. Here, the error term ε is assumed to be an i.i.d. normal. When there is spatial dependence in the error term, the i.i.d. normal assumption is violated. Two approaches are taken to deal with spatial correlation (Ward and Gleditsch 2018). First is the spatial lag model, where the error term is decomposed into a spatially lagged term for the dependent variable and an independent error term, ε=ρWy+e, where W is the matrix that reflects the spatial proximity between areas which is commonly defined by encoding the k-nearest neighbors. This gives us the spatial lag model, described by the following equation:

where ε∼N(0,σ²I). The parameters can be estimated using maximum likelihood estimation.

The other approach is to assume that the error is spatially correlated, instead of the objective variables affecting the objective variables of neighboring areas. We can write the spatial error model as follows:

where ε∼N(0,σ²I). In the following regression analysis, we first test the general regression model and also test for significance in spatial lag ρ and spatial error λ terms. Then, we apply the appropriate spatial regression analysis accordingly.

The normalized population data were fitted using two different functions to denoise the observations. We test two functions: 1) linear model and 2) exponential model.

The exponential model is defined as follows:

where, τ_i is the recovery speed of city i, \(M_{i}^{0}\) is the initial population rate of city i, and \(M_{i}^{\infty }\) is the long term population rate of city i. t=T₀ denotes the timing of initial population rate. Figure 7a shows an illustration of the exponential model.

The alternative is the linear model, defined as the following:

where, β_i is the coefficient of the linear recovery speed of city i, \(M_{i}^{0}\) is the initial population rate of city i, and \(M_{i}^{\infty }\) is the long term population rate of city i. t=T_c denotes the timing where the linear recovery line reaches \(M_{i}^{\infty }\). Figure 7b shows an illustration of the linear model.

We chose the model by testing which model is able to fit the empirical data obtain from Puerto Rico after Hurricane Maria. Parameters (τ_i, β_i and \(M_{i}^{\infty }\)) of the models were fitted using least squares method. \(M_{i}^{0}\), which is the initial population rate, is defined as \(M_{i}^{0} = \min _{t} M_{i}(t)\). Squared error (SE) and Pearson’s correlation coefficient ρ between the observed and fitted time series were used as evaluation metrics. SE is defined as:

where M(t) and \(\hat {M(t)}\) are observed and fitted values of the time series, and T is the total number of time steps. Pearson’s correlation coefficient ρ is defined as:

where Cov(x,y) is the covariance between x and y, σ_x is the standard deviation of x.

Figure 8 shows the histograms of the two metrics for the two models, to evaluate the goodness of fit of the two models. We can observe that the exponential model has lower SE and higher correlation, meaning that it is able to better approximate the observations. Thus, in our further analysis, we use the results obtained by fitting the observations to the exponential model.

As shown in Fig. 9, Pearson’s correlation coefficients between \(\tilde {T}_{i}^{0.80}\), \(\tilde {T}_{i}^{0.85}\), \(\tilde {T}_{i}^{0.90}\), \(\tilde {T}_{i}^{0.95}\) are all very high (ρ>0.96). This implies that the \(\tilde {T}_{i}^{\alpha }\) values do not depend highly on the values of α. Thus, we use α=0.95 as a parameter for our further analysis. For notational simplicity, we write \(\tilde {T}_{i}^{0.95}\) as \(\tilde {T}_{i}\) and call the variable “time until recovery” in the analysis.