1 Since the increasing spread of COVID-19 in the U.S., with currently the highest number of con-2 ﬁrmed cases and deaths in the world, most states in the nation have enforced travel restrictions3 resulting in drastic reductions in mobility and travel. However, the overall impact and long-term4 implications of this crisis to mobility still remain uncertain. To this end, this study develops an5 analytical framework that determines the most signiﬁcant factors impacting human mobility and6 travel in the U.S. during the pandemic. In particular, we use Least Absolute Shrinkage and Selec-7 tion Operator (LASSO) to identify the signiﬁcant variables inﬂuencing human mobility and utilize8 linear regularization algorithms, including Ridge, LASSO, and Elastic Net modeling techniques to9 model and predict human mobility and travel. State-level data were obtained from various open- 10 access sources for the period from January 1, 2020 to June 13, 2020. The entire data set was 11 divided into a training data-set and a test data-set and the variables selected by LASSO were used 12 to train four different models by ordinary linear regression, Ridge regression, LASSO and Elas- 13 tic Net regression algorithms, using the training data-set. Finally, the prediction accuracy of the 14 developed models was examined on the test data. The results indicate that among all models, the 15 Ridge regression provides the most superior performance with the least error, while both LASSO 16 and Elastic Net performed better than the ordinary linear model. 17 18 Keywords : COVID-19, mobility, number of daily trips per person, regularization algorithms, 19 Ridge, LASSO, Elastic Net, linear regression 20 Chakraborty, Mahmud, Gates, and Sinha 3 INTRODUCTION1 The novel coronavirus (COVID-19) pandemic is delineating the global health crisis of our times2 and has had a tremendous impact on the way we understand our everyday lives as well as the world.3 Since its emergence in Asia late 2019, all continents in the world except Antarctica have been4 ﬁghting the virus in earnest. The World Health Organization (WHO) ( 1) has declared COVID-19 a 5 global pandemic on March 11, 2020 and the United States declared a national emergency on March6 13, 2020 ( 2). As of July 15, 2020, more than 13.6 million cases of COVID-19 were conﬁrmed in 7 215 countries around the globe ( 3). Among all countries, the U.S. has the highest number of 8 conﬁrmed cases and fatalities in the world due to COVID-19 ( 3). Several countries have closed 9 their borders, exercised lockdowns, curfews, stay-at-home orders, and social distancing protocols, 10 resulting in a sharp decrease in transport demand at local, regional, national, and also international 11 levels. By March 24, 2020, more than 20 percent of the world’s population has been ordered to 12 remain at home as governments, health, and administrative organizations take extreme measures 13 to protect their populations from the spread of the virus ( 4). 14 In the U.S., more than 40 states have already enforced stay-at-home order, the earliest 15 being in California effective from March 19, 2020 ( 5). Only in the U.S., the mobility restrictions 16 during the COVID-19 pandemic, in the form of travel bans, stay-at-home mandates, and lockdown 17 policies, have impacted millions of people. Overall, human mobility has been severely impacted 18 due to the travel bans and individual concerns to avoid public gatherings, resulting in tremendous 19 economic impacts in transportation sectors. However, the overall inﬂuence and the long-term 20 implications of this pandemic to mobility and transportation systems still remain uncertain at this 21 point in time. Against the background of this unprecedented global crisis, questions remain as to 22 how the different factors during the pandemic contribute to human mobility and travel. 23 With the increasing availability of high-quality data related to COVID-19, analyzing trans- 24 portation and mobility during and after this crisis is absolutely imperative. Although there are 25 still many unknowns, statistical models and analytical tools would help produce evidence-based 26 research and policy interventions after COVID-19 outbreak. At the national level, Zhang et al. 27 (2020) developed a COVID-19 impact analysis platform ( 6) that can inform users about the spread 28 of COVID-19 in the U.S., and the effects of the virus spread and government orders on mobil- 29 ity and social distancing in the country using privacy-protected smartphone device location data, 30 coupled with the census information. The platform gets updated daily and goes back to January 31 1, 2020, for benchmarking, and the results are scaled and aggregated to the entire population for 32 both state and county levels ( 7). Gao et al. (2020) reported on the interactive web-based map- 33 ping platform ( 8) developed by the GeoDS Lab at the University of Wisconsin, Madison, with the 34 support of the National Science Foundation RAPID program ( 9). This platform provides quan- 35 titative information on how people in different counties and states in the U.S. responded to the 36 social distancing directives and guidelines. The platform integrates geographic information sys- 37 tems (GIS) and the daily updated human mobility patterns obtained from large-scale anonymous 38 and aggregated smartphone data at the county-level ( 10). 39 With the availability of such large-scale data, this study applies an analytical framework 40 that helps to understand how different factors impacted the daily trips of the U.S. population dur- 41 ing the COVID-19 pandemic utilizing regularization regression techniques. Regression analysis 42 is widely used in data analysis and machine learning ( 11), and is applied extensively in various 43 applications, including transportation ( 12, 13 ). Among the various regression algorithms, linear 44 regression is computationally one of the most efﬁcient techniques and is often used as a starting 45 Chakraborty, Mahmud, Gates, and Sinha 4 point for many different problems, although in many cases the linear regression model may be1 suboptimal. In the analysis of time-series data coming from a dynamical system, linear regression2 arises naturally in data-driven analysis of dynamical systems using the transfer operators, namely3 Perron-Frobenius and Koopman operators ( 14–16 ). More recently ( 17, 18 ) used robust optimiza- 4 tion techniques to compute these operators from noisy data sets and the resulting optimization5 problem was a variation of the ordinary least squares (OLS), namely, least squares with regular-6 ization. Regularization is a standard technique used in data analysis to overcome some of the7 limitations of OLS, including the overﬁtting of training data and the susceptibility to noise in the8 data ( 11). 9 Literature on transportation analysis utilizing regularization is scant. Recently Polson and 10 Sokolov (2017) developed a deep learning model utilizing a linear model that is ﬁtted with `1 11 regularization to predict trafﬁc ﬂows. The study showed that the deep learning architecture was ca- 12 pable of capturing non-linear spatio-temporal effects in trafﬁc and providing short-term prediction 13 of trafﬁc ﬂow ( 19). Tan et al. (2011) proposed a semi-supervised Elastic Net regression method 14 for pedestrian counting by utilizing sequential information between unlabelled samples and their 15 temporally neighboring samples as a regularization term. The developed model was able to attain 16 superior prediction performance and select representative features from the original set of features 17 without losing their interpretability ( 20). Hasan et al. (2017) proposed statistical techniques to 18 identify spatial relationships among road links in an urban road network to select predictors for a 19 short-term trafﬁc prediction model for a given road link. The study uses a time-lagged multiple 20 linear regression method and utilizes two analytical methods, including the Elastic Net regular- 21 ization and Granger Causality test using one year of trafﬁc ﬂow and speed data from the selected 22 road network in Brisbane, Australia. For a given target link, the relevant predictors obtained by 23 the Granger Causality and Elastic Net are used separately to build the respective trafﬁc prediction 24 models. The results show that Granger Causality-based trafﬁc prediction model provides superior 25 prediction accuracy than that using the Elastic Net regression ( 21). More recently, Battifarano and 26 Qian (2019) explored the spatio-temporal correlations between the urban environment, trafﬁc ﬂow 27 characteristics, and surge multipliers and proposed a general framework for predicting the short- 28 term evolution of surge multipliers in real-time using a log-linear model with `1regularization, 29 integrated with pattern clustering. The modeling algorithm is validated by using Uber and Lyft 30 data from Pittsburgh ( 22). 31 While there is a plethora of information available related to mobility and travel during this 32 pandemic, it is absolutely critical to develop a robust methodological framework to accurately 33 identify the key factors that inﬂuence human mobility during such health crisis, so that it may help 34 direct the focus in policies and guidance in future. To this end, this study develops an analytical 35 framework that helps determine the most signiﬁcant factors affecting human mobility by utiliz- 36 ing linear regularization algorithms including the Ridge, Least Absolute Shrinkage and Selection 37 Operator (LASSO), and Elastic Net modeling techniques. 38 DATA DESCRIPTION 39 Data for this analysis were collected and combined from multiple web-based open-access sources. 40 The majority of data used in this analysis were requested and obtained from the COVID-19 Impact 41 Analysis Platform developed at the Maryland Transportation Institute of the University of Mary- 42 land (UMD) ( 6). This platform provides both state and county-based information for 50 states in 43 the U.S. and the District of Columbia. To match with the data available from other sources, for the 44 Chakraborty, Mahmud, Gates, and Sinha 5 purpose of this study, state-wise data were requested from this platform. The relevant statewide1 data obtained from this source include the daily number of new COVID-19 conﬁrmed cases per2 1000 people, daily COVID-19 death rate, the number of daily trips per person, social distancing3 index, percent of out of county or state trips, transit mode share, population density, percentage of4 African American or Hispanic Americans, income and employment, unemployment rates, popula-5 tion staying at home, and percent working from home, among others. Social distancing index in6 the data set indicates the increasing space between individuals and decreasing frequency of contact7 and is represented as an integer from 0 to 100, where 0 indicates no social distancing in the state8 and 100 indicates all residents are staying at home.9 Additional information was collected and appended with the data obtained from the UMD 10 platform. Statewide percentage of the elderly population of different age categories, percentage 11 of the population at different education levels, male to female ratio, and gross domestic product 12 (GDP) information were collected from the U.S. Census Bureau ( 23) and joined with the princi- 13 ple data set. Information on the percentage of population staying at home during this time was 14 obtained from the Bureau of Transportation Statistics ( 24). Moreover, the trends in mobility calcu- 15 lated from the number of trip directions requested by their user after comparing to the trip volume 16 of January 13, 2020, reported by Apple, were further joined with the study data set ( 25). Also, data 17 on community vulnerability index, which is a measure of poverty level estimated from seven in- 18 dicators including no health insurance coverage, education level, income, gross rent as percentage 19 of income, poverty, unemployment, and disability, were obtained from Surgo Foundation’s open- 20 access source ( 26). Furthermore, several states have exercised travel restrictions in the form of stay 21 at home order, limitations on gatherings, domestic travel limitations, or school closures. These in- 22 formation was obtained from the COVID-19 State and Territory Action Tracker ( 27) provided by 23 ESRI (Environmental Systems Research Institute) ( 28). 24 Following the joining of the data from various aforementioned sources, a thorough screen- 25 ing and quality check of the data was performed for any missing values. The ﬁnal data set includes 26 daily data starting from January 1, 2020 to June 13, 2020 from 50 U.S. states and the District of 27 Columbia, and consists of a total of 8,415 observations for further analysis. 28 METHODOLOGY 29 In this section, we brieﬂy discuss ordinary linear regression and its different regularized versions 30 that are utilized in this study, including the Ridge, LASSO, and Elastic Net regression techniques. 31 Linear Regression 32 Consider the data set fyi;xi1;xi2;;xiNgn i=1where yiis the ithobservation of the dependent vari- 33 able yandxj,j=1;2;;Nare the Nindependent variables. In case of linear regression, the 34 model tries to ﬁt a straight line by minimizing the residuals. In particular, it assumes that the de- 35 pendent variable can be expressed as a linear combination of the independent variables, as given 36 by the following, 37 yi=a0+a1xi1+a2xi2++aNxiN+ei;i=1;2;;n; (1) 38 where eiis the residual. 39 In matrix form, the equation (1) can be written as 40 Y=Xa+e; (2) 41 Chakraborty, Mahmud, Gates, and Sinha 6 where1 Y= [y1;y2;;yn]>;X=0 BBB@1x11 x1N 1x21 x2N ............ 1xn1 xnN1 CCCA;a= [a0;a1;;aN]>;e= [e1;e2;;en]>: 2 The linear regression selects the parameters aj’s(j=1;;N)such that the norm of the 3 residual for every yi(i=1;;n)is minimized. Hence the optimal ais obtained as a solution of 4 the following optimization problem,5 min akYXak2: (3) 6 wherekk 2is the 2-norm of a vector. The optimization problem (3) is convex and can be solved 7 efﬁciently either using convex optimization techniques or analytically, such that the optimal a?is 8 given by9 a?=YX†; (4) 10 where X†is the Moore-Penrose inverse of X. 11 LASSO Regularization 12 Though linear regression is used extensively in many data-driven analyses and machine learning 13 applications, it suffers from some drawbacks. One of the main disadvantages of the linear re- 14 gression method is that it fails to identify the set of most important predictor variables. In many 15 applications, it may be computationally intensive, and in some cases, even redundant, to include 16 all explanatory variables in hand while ﬁtting a model. Hence, it is advisable to determine the key 17 predictors that have signiﬁcant associations with the dependent variable and in such cases, the lin- 18 ear regression is not ideal ( 29, 30 ). To circumvent this challenge, prior studies ( 29, 31 ) suggested 19 that the optimization problem (3) be modiﬁed as follows, 20 minakYXak2+l1kak1 subject tokak1t;(5) 21 wherekk 1is the 1-norm of a vector and the bound tis the tuning parameter. If tis large, it has no 22 effect on the regression coefﬁcients ais and in this case, the solution to the optimization problem 23 (5) approach the solution of normal linear regression optimization problem (3) in the limit of large 24 t. However, when the bound tis small, the parameters ais are constrained and hence are shrunk 25 and are smaller versions of the original least squares estimates. The 1-norm minimization puts 26 constraints on parameters that shrink coefﬁcients towards zero. This is the shrinkage property of 27 the LASSO regression that allows for a better interpretation of the model. By setting some of the 28 ais to zero, this technique identiﬁes the most important variables associated with the dependent 29 variable. 30 Ridge Regularization 31 Another major drawback of linear regression is that this algorithm has low bias and high variance 32 (11). This means that the linear regression may perform well on the train data, but it may not 33 generalize well to the test data set, thereby making the model performance unsatisfactory. In ma- 34 chine learning literature, this phenomenon is known as Bias-Variance trade-off ( 11). The intuition 35 of bias-variance trade-off is explained in Figure 1(a). Usually, with a highly complex model, it 36 is possible to ﬁt the training data as closely as possible. In this case, the training error is zero 37 Chakraborty, Mahmud, Gates, and Sinha 7 and the model is said to have a low bias. However, the highly complex model may not generalize1 well to the test data, thus making the test error large. This is due to the overﬁtting of the training2 data. The complex model, which overﬁts the training data and produces high test error, is said to3 have high variance. This situation is often reversed if the model considered is fairly simple. Ridge4 regression, which puts a 2-norm constraint on the set of coefﬁcients, is able to efﬁciently overcome5 this challenge.6 Error Optimal Model ComplexityTotal ErrorVarianceBias2Model Complexity (a) (b) Figure 1 (a) Bias-variance trade-off. (b) Effect of outlier on linear regression. Another drawback of ordinary linear regression is that the obtained model is highly inﬂu- 7 enced by the outliers in the training data set. For example, as in Figure 1(b), the outlier data point8 ykresults in the linear model ﬁt as represented by the red line. However, it is obvious by looking 9 at the overall data that the model ﬁt depicted by the black line is the more appropriate linear ﬁt to 10 the data. 11 Additionally, on many occasions, the real-life data is noisy or uncertain. When the ordi- 12 nary linear regression attempts to ﬁt that noise in the data, it eventually results in overﬁtting and 13 consequently degrading its performance for model prediction. As stated earlier, the data for this 14 study were obtained mostly from smartphone devices, and the chances of acquiring this data may 15 also be subject to individual user’s discretion, so it is reasonable to assume that the data utilized 16 in this study may contain some noise or uncertainty in it. To account for the noise in the data, 17 it is assumed that there is some uncertainty, DYandDX, in the dependent and independent vari- 18 ables, respectively. It is assumed that the uncertainties in both YandXare bounded, i.e. there 19 exists some positive real number r>0 such thatkDXk2randkDYk2r. With this, the 20 optimization problem (3) is modiﬁed to a min-max optimization problem ( 17, 18 ) given by, 21 minamax kDXk2r; kDYk2rk(Y+DY)(X+DX)ak2: (6) 22 Min-max optimization problems are generally hard to solve, but in this case, the optimiza- 23 tion problem (6) can be equivalently expressed as a convex optimization problem as follows, 24 Theorem 1 The optimization problem 25 Chakraborty, Mahmud, Gates, and Sinha 8 minamax kDXk2r; kDYk2rk(Y+DY)(X+DX)ak2: (7) 1 is equivalent to the following,2 minakYXak2+l2kak2 2; (8) 3 where l2is a positive real number, depending on the uncertainty bound r. 4 Proof. For proof, see ( 17, 18 ).  5 The optimization problem (8), known as Ridge regression, is a convex problem and can be 6 solved efﬁciently using any of the available convex optimization problem solvers. The parameter7 l2is called the regularization parameter and it acts as a trade-off between the ordinary least squares 8 cost and the cost on the coefﬁcients a. 9 The LASSO and Ridge regression techniques are the modiﬁed linear regression algorithms 10 that overcome the deﬁciencies of the linear regression algorithm and improves the performance of 11 a linear model. 12 Elastic Net Regularization 13 Although the LASSO regression is efﬁcient in feature selection from a large pool of variables, in 14 case of high-dimensional data with few observations, the LASSO algorithm saturates (it selects 15 up to nvariables, where nis the number of observations) ( 32). Also, if some of the independent 16 variables are highly correlated, the LASSO tends to select only one of those variables. The Elastic 17 Net modeling ( 32), which is a combination of both the LASSO and Ridge regressions, is capable 18 of overcoming this limitation. In particular, the Elastic Net uses the goods of both the worlds, the 19 LASSO and Ridge regressions. The optimization problem for the Elastic Net can be expressed as, 20 minakYXak2+l1kak1+l2kak2 2: (9) 21 The Ridge component of this technique, which corresponds to kak2 2, makes the cost 22 strongly convex and hence the optimization problem (9) has a unique minimum. But one prob- 23 lem with the Elastic Net modeling is the fact that often the optimization problem (9) is solved in 24 two steps, which leads to higher bias and poor variance ( 32). 25 In this study, since the number of observations is much higher compared to the number 26 of independent variables (dimension of the data), the LASSO regression is used to select the key 27 variables. This is followed by analyzing the prediction capabilities of the various regression algo- 28 rithms. 29 PRELIMINARY ANALYSIS 30 The preliminary analysis examines several explanatory variables of interest that were initially in- 31 cluded in the analysis. The distribution of the dependent variable i.e., the number of daily trips 32 per person during the analysis period (from January 1, 2020 to June 13, 2020) is shown in Figure 33 Figure 2. As can be clearly seen from Figure 2, the number of daily trips remains consistent from 34 January, 2020 to around mid-March, 2020. However, about the time when the U.S. started expe- 35 riencing rapid community outbreaks of the virus and the country declared a national emergency, 36 the number of daily trips drops substantially. Additionally, it is interesting to see that although the 37 Chakraborty, Mahmud, Gates, and Sinha 9 number of COVID-19 cases per day continues to increase considerably over time, people started1 making more trips from around early May, 2020.2 1.522.533.544.55 December 25, 2019 February 6, 2020 March 20, 2020 May 2, 2020 June 14, 2020Number of daily trips per person Date Figure 2 The number of daily trips per person during the analysis period. As stated earlier, the compiled data set includes a large number of variables of interest that 3 may impact the number of daily trips. Table 1 presents the descriptive statistics, including the4 minimums, maximums, means, and standard deviations of twenty-ﬁve variables that were initially5 considered as the set of potential explanatory predictors. For the purpose of scaling the data,6 population, median income, and GDP were included in their natural log forms.7 Chakraborty, Mahmud, Gates, and Sinha 10 TABLE 1 Descriptive Statistics of the Data Parameter Minimum Maximum Mean Std. Dev Number of daily trips per person 1.66 4.68 3.18 0.46 Social distancing index 10 83 32.62 16.38 Transit mode share 0.29 34.83 3.67 6.23 Percent of out of county trips 0.2 52.3 27.58 8.78 Number of new COVID-19 cases (per 1,000 people)0 0.72 0.04 0.06 Total population 577,737 39,557,045 6,415,047 7,271,390 Percent of population older than 60 years 15 27 21.88 2.32 Median income (US dollars) 44,445 84,342 61,786.9 10,522.4 Percent of Hispanic or African American population2.9 57 22.71 13.6 Unemployment rate 2.20 46.6 12.11 9.96 Percent of population working from home 2.30 55.7 16.04 11.62 Population staying at home 107,766 14,180,383 1,480,259 1,759,931 Gap between the ﬁrst case and the stay- at-home order (days)7 94 36.62 16.95 Socioeconomic status 0 1 0.51 0.29 State employee travel restriction 0 1 0.77 0.45 No stay-at-home order 0 1 0.25 0.44 Stay-at-home order 0 1 0.12 0.32 Stay-at-home guidance 0 1 0.63 0.48 School closures 0 1 0.96 0.19 Closure of some or all facilities 0 1 0.84 0.36 Recommended domestic travel limitation 0 1 0.26 0.44 Mandatory domestic travel limitation 0 1 0.2 0.40 Mandatory statewide mask policy 0 1 0.69 0.46 Driving mobility index 0 299.24 99.69 33.70 GDP in 2019 (109U.S. dollars) 30.5 2,792.03 370.04 482.7 Percent of population aged 25 years or over with bachelor degree or higher21.3 60.4 32.16 2.60 Chakraborty, Mahmud, Gates, and Sinha 11 RESULTS1 Model Selection2 One of the major advantages of the LASSO regression is that it is able to efﬁciently select the3 explanatory variables that are important for predicting the dependent variable. In other words,4 from a set of Nindependent variables, the LASSO can suitably select k<Nindependent variables. 5 In particular, the LASSO solves the optimization problem,6 minakYXak2+l1kak1 subject tokak1t(10) 7 and the kvariables corresponding to the klargest absolute values of the coefﬁcient vector aare 8 chosen as the new reduced set of important independent variables.9 ImportancePercent of population older than 60 years Social distancing index Percent of population working from home New COVID-19 cases per 1000 people Socioeconomic status Median income Driving mobility index Statewide mandatory mask policy Percent of African or Hispanic American population Percent of out of county trips Transit mode share Stay-at-home order Mandatory or recommended domestic travel restrictions Population staying at home Stay-at-home guidance Unemployment rate 0 1 2 3 Figure 3 Important independent variables identiﬁed by the LASSO regression. In this study, approximately 75 percent of the entire data were considered as training data 10 and the remaining 25 percent of the data were selected for testing the model results. As stated 11 before, the key variables were identiﬁed by the LASSO regression on the training data set with 12 the regularization parameter varying from zero to one. Hence, the identiﬁcation of the important 13 variables was achieved by solving the optimization problem, 14 Chakraborty, Mahmud, Gates, and Sinha 12 min akYXak2+l1kak1 subject to 0 <l11:(11) 1 The optimization problem (11) was solved for each l1with l12(0;1]and the optimal 2 coefﬁcients awere chosen for that particular l1for which the cost function of the optimization 3 problem (11) is the minimum. Ultimately, the LASSO regression problem (11) identiﬁed a total4 of sixteen predictors from the pool of twenty ﬁve features as the most important variables. The5 relative importance of the variables are shown in Figure 3.6 Model Training7 The ﬁnal data set used in this study included information from January 1, 2020 to June 13, 2020.8 After splitting the data set into two parts for training and testing, the train and test data sets ranged9 from January 1, 2020 to May 3, 2020 and May 4, 2020 to June 13, 2020, respectively. The ﬁrst 10 (train) part was utilized for training the regression models, while the latter part (test) was used to 11 test the efﬁciency of prediction from the obtained regression models. The sixteen key variables 12 selected by the LASSO regression from the training data set were then trained for prediction by 13 using all four modeling techniques, including the linear, Ridge, LASSO, and Elastic Net regres- 14 sion. This is accomplished by solving the optimization problems (3), (5), (8) and (9) to obtain the 15 linear, LASSO, Ridge, and Elastic Net models, respectively. It is important to note that except 16 for the linear model, the other optimization problems involve at least one regularization parameter 17 (l1andl2)and in all the optimization problems the lis were chosen such that, 0 <li1. 18 Interpretation and Discussion of the Model Results. 19 Table 2 compares the optimal coefﬁcients of the ﬁnal set of explanatory variables between all four 20 regression modeling techniques. As can be seen from Table 2, the coefﬁcients of the predictors are 21 fairly comparable across all modeling techniques. As expected, when the number of new cases per 22 1000 people increases, the number of daily trips per person decreases. Similarly, with the increase 23 in the social distancing index, the number of daily trips per person decreases. The percent of people 24 aged more than 60 years and the number of people staying at home are negatively associated with 25 the number of trips being made. Increasing concern regarding the higher health risk for the elderly 26 population may have escalated the decrease in their number of daily trips. 27 Moreover, the number of trips per person decreases with a higher median income and 28 socioeconomic status. Conversely, the number of daily trips per person increases with the increase 29 in the unemployment rate. Additionally, the increase in transit mode share, driving mobility index, 30 and percent of African and Hispanic American population are associated with the increase in the 31 number of daily trips being made. Interestingly, when the stay-at-home order is implemented in a 32 state, the count of daily trips per person goes lower. However, when the stay-at-home order is not 33 stringently implemented and rather is recommended in a state, people are less likely to follow the 34 guidance and tend to make more trips. Also, the number of trips drops when the states exercise 35 mandatory or recommended domestic travel restrictions. 36 In situations where mandatory or recommended mask-wearing policy is in place, the num- 37 ber of trips increases, and this could partially be due to the fact that people may feel safer to go 38 and travel when they themselves or people around them wear masks. It may also seem counter- 39 intuitive that with the increase in the percent of the population working from home, the number 40 of daily trips escalates. However, this can be explained as people may make more trips with the 41 Chakraborty, Mahmud, Gates, and Sinha 13 ﬂexibility in work schedules while working from home (WFH), compared to the usual scenario1 when a signiﬁcant portion of the day is spent at ofﬁce locations. Lastly, although the percent of out2 of county trips is found to be negatively associated with the number of daily trips being made in3 the linear, LASSO, and Elastic Net regression models, the expected positive association between4 these two variables is rightly captured in the Ridge regression technique. This is also conﬁrmed by5 the error magnitudes, where the Ridge regression is providing the best model ﬁt with the least root6 mean square error (RMSE) among all modeling techniques explored in this analysis. Overall, the7 most important factors to inﬂuence the number of daily trips include the number of new COVID-8 19 cases per 1000 people, social distancing index, percent of the population working from home,9 socioeconomic status, percent of the population older than 60 years, and driving mobility index, 10 among others. 11 TABLE 2 Comparison of Model Coefﬁcients between the Linear, Ridge, LASSO, and Elastic Net Regressions Parameter Linear Ridge LASSO Elastic Net Intercept 7.1495 4.6603 6.7524 6.9383 New COVID-19 cases per 1000 people-0.2946 -0.2157 -0.2616 -0.2825 Social distancing index -2.7815 -2.0815 -2.7709 -2.7543 Transit mode share 0.0284 0.0217 0.0277 0.0280 Percent of out of county trips-0.0590 0.0918 -0.0476 -0.0504 Percent of people older than 60 years-3.1940 -2.5281 -3.0955 -3.1422 Percent of Hispanic and African American popu- lation0.0795 0.0441 0.0812 0.0791 Median income -0.2416 -0.0522 -0.2078 -0.2244 Unemployment rate 0.0044 0.0024 0.0043 0.0043 Percent of population working from home0.9885 0.3846 0.9531 0.9562 Population staying at home-0.0064 -0.0075 -0.0059 -0.0063 Socioeconomic status -0.2373 -0.1663 -.2238 -0.2307 Stay-at-home order -0.0255 -0.0250 -0.0259 -0.0259 Stay-at-home guidance 0.0060 0.0027 0.0050 0.0055 Mandatory or recom- mended domestic travel restrictions-0.0211 -0.0333 -0.0216 -0.0218 Mandatory statewide mask policy0.1111 0.0873 0.1075 0.1093 Driving mobility index 0.2073 0.3174 0.2004 0.2097 Furthermore, the log-lambda plots shown in Figure 4 depicts how the independent variables 12 Chakraborty, Mahmud, Gates, and Sinha 14 that enter in the model, vary across the Ridge, LASSO, and Elastic Net regression techniques as1 the regularization parameters change. When the regularization parameter lis small, the contribu- 2 tion of the regularization part to the cost functions in the optimization problems (3), (5), (8) and3 (9) is small and as such all these optimization problems are reduced to the ordinary linear regres-4 sion. However, as lis increased, the weight on the regularization component in the optimization 5 problems increases and as such, the coefﬁcients of the independent variables become smaller and6 approach zero. It is clearly seen from Figure 4 that all the coefﬁcients do not approach zero at7 the same time. In particular, the important variables remain non-zero for larger values of las 8 compared to the relatively non-important variables. For example, in Figure 4(a), the variables in-9 cluding social distancing index, percentage of people older than 60 years, and percentage of people 10 working from home remain the most important, because the coefﬁcients of these variables remain 11 non-zero as lis increased. Similarly, from Figures 4(b) and (c), the important variables corre- 12 sponding to the Ridge and Elastic Net regressions can be identiﬁed. However, based on the order 13 of importance, the coefﬁcients of the explanatory variables differ between the three regularization 14 techniques. For example, as can be seen in Figure 4, in case of the LASSO and Elastic Net re- 15 gressions, statewide mandatory mask policy is the third most important feature having a positive 16 association with the dependent variable, whereas, in the Ridge regression, percent of out of county 17 trips becomes the third in the order of importance among variables with a positive association with 18 the number of daily trips. This difference is due to the characteristic of the Ridge regularization 19 that reduces the norm of the coefﬁcients more uniformly, while the LASSO model attempts to set 20 as many coefﬁcients to zero as possible. 21 Chakraborty, Mahmud, Gates, and Sinha 15 (a) (b) (c) Figure 4 Relative importance of the independent variables in terms of log-lambda for (a) LASSO regression, (b) Ridge regression and (c) Elastic-net regression. Prediction Performance1 Following the development of the models using training data, the prediction of the dependent vari-2 able (the number of daily trips per person) is tested and compared between the four regression3 techniques. Essentially, the performance of the four different models on the test data set is eval-4 uated, and the predicted values are compared with the observed values. Ultimately, the RMSEs5 from all models provide the measure of performance and efﬁciency of the models.6 TABLE 3 Root mean square error (RMSE) results of training and test data Model Train Test Linear 0.65890 0.2774 LASSO 0.65806 0.2734 Elastic Net 0.65787 0.2748 Ridge 0.63612 0.2413 The RMSEs of the different models utilizing both the train and test data set are presented in 7 Table 3. The comparison of the RMSEs between the four modeling techniques clearly shows that8 Chakraborty, Mahmud, Gates, and Sinha 16 the Ridge regression performs the best for both train and test data by having the least RMSE among1 all models. This is expected, as the Ridge regression provides superior prediction by overcoming2 the issue of overﬁtting with low variance and better generalization to the test data compared to3 other regularization methods (refer to Figure 1). Additionally, based on the RMSEs, both the4 LASSO and Elastic Net models provide better prediction performance compared to the ordinary5 linear regression.6 04 May 2014 May 2024 May 2003 Jun 2013 Jun 2022.22.42.62.833.23.43.63.84 (a) 04 May 2014 May 2024 May 2003 Jun 2013 Jun 2022.22.42.62.833.23.43.63.84 (b) Figure 5 Error in prediction of daily trips per person in the state of California by (a) Linear and Ridge regression, (b) LASSO and Elastic Net. 04 May 2014 May 2024 May 2003 Jun 2013 Jun 202.42.62.833.23.43.63.844.2 (a) 04 May 2014 May 2024 May 2003 Jun 2013 Jun 202.42.62.833.23.43.63.844.2 (b) Figure 6 Error in prediction of daily trips per person in the state of New York by (a) Linear and Ridge regression, (b) LASSO and Elastic Net. For the purpose of showing how the different modeling techniques perform at an individual 7 state level, graphical representations of RMSEs in the prediction of daily trips per person are given8 Chakraborty, Mahmud, Gates, and Sinha 17 separately for California and New York as examples in Figures 5 and 6, respectively. While the1 Figures 5(a) and 6(a) compare the errors in prediction by the ordinary linear regression and Ridge2 regression, the Figures 5(b) and 6(b) show the errors in prediction using the LASSO and Elastic3 Net modeling techniques. These ﬁgures clearly show that even at the individual state level, the4 Ridge model has the least error in prediction in both cases. Additionally, the LASSO and Elastic5 Net models perform almost similarly, which is expected because the RMSEs of the LASSO and6 Elastic Net over the entire test data set are found to be almost similar to each other (Table 3).7 04 May 2014 May 2024 May 2003 Jun 2013 Jun 200.180.20.220.240.260.280.30.320.340.360.38 (a) 0.220.230.240.250.260.270.28 LinearLASSORidgeElastic net (b) Figure 7 (a) Root mean square error (RMSE) in prediction of the different models (Eq. (13)) over the analysis period. (b) Average date-wise root mean square error (RMSE). Furthermore, the RMSEs in the prediction utilizing the four developed models over the 8 analysis period are presented in Figure 7. Let ydibe the observed value of daily trips per person 9 on the dthday for the state i. Hence for the test data set, d=1;2;;41 and i=1;2;;51. Let 10 ˆyp dibe the predicted daily trip per person on the dthday for ithstate when using the pthmodeling 11 technique. Here pis one of linear, Ridge, LASSO, or Elastic Net regression techniques. Therefore, 12 the error in the prediction using the pthmodel on the dthday for ithstate is, 13 ep di=ydiˆyp di: (12) 14 The root mean square error in the prediction of the pthmodel for the dthday over all the 15 states is, 16 rp d=s åns i=1(ep di)2 ns=s åns i=1(ydiˆyp di)2 ns; (13) 17 where ns=51 is the number of the states. 18 In Figure 7(a), rp dis plotted for all modeling techniques over the period ranging between 19 May 4, 2020 to June 13, 2020 (test period). From Figure 7(a), it can once again be seen that the 20 Ridge regression performs better for most of the test period, as its RMSE is usually lower compared 21 to the other methods. The average of rp dacross the test period, which can be expressed as, 22 Chakraborty, Mahmud, Gates, and Sinha 18 rp=ånd d=1rp d nd=0 @nd å d=1s åns i=1(ydiˆyp di)2 ns1 A, nd; (14) 1 where ndis the number of days (equals to 41 in this study), is plotted in Figure 7(b). This plot also 2 conﬁrms that the Ridge regression model has the least error in prediction.3 SUMMARY AND CONCLUSIONS4 Since the emergence and rapid growth of novel coronavirus (COVID-19), countries worldwide are5 taking extreme measures to help prevent the spread of the virus. The U.S. is greatly hit by the6 pandemic and currently has the highest number of conﬁrmed cases and deaths in the world due7 to COVID-19. Since the White House declared an emergency on March 13, 2020, most states in8 the U.S. implemented travel restrictions, stay-at-home orders, lockdowns, and social distancing9 protocols to combat the crisis, causing drastic reductions in travel demand at local, regional, and 10 national levels. However, the overall impact and the long-term implications of this crisis to mo- 11 bility still remain uncertain at this point in time. In order to understand these implications better, 12 statistical models and analytical tools utilizing the increasingly available open-access data is the 13 need of the hour. To this end, this study develops an analytical framework that helps to determine 14 the most signiﬁcant factors impacting human mobility and travel in the U.S. during the pandemic 15 by utilizing linear regularization algorithms, including the Ridge, Least Absolute Shrinkage and 16 Selection Operator (LASSO), and Elastic Net modeling techniques. 17 Data for this study were obtained for 50 states and the District of Columbia from vari- 18 ous databases created and maintained to analyze the impacts of this pandemic. Relevant data on 19 COVID-19 spread and its impact on mobility, social distancing, health, economy, and vulnerabil- 20 ity of different groups were obtained from COVID-19 Impact Analysis Platform developed at the 21 University of Maryland. Statewide percentages of the elderly population, population at different 22 education levels, and GDP related information were obtained from the U.S. Census Bureau, while 23 information on percentage of population staying at home was obtained from the Bureau of Trans- 24 portation Statistics. Also, Data on travel restrictions, school closure, limitation on gatherings, stay 25 at home order were obtained from COVID-19 State and Territory Action Tracker. Additionally, 26 data were also collected from other sources on the mobility index and community vulnerability in- 27 dex. The compiled data set includes daily observations from 50 states and the District of Columbia 28 starting from January 1, 2020 to June 13, 2020. 29 Evaluating an extensive data set requires advanced analysis techniques to identify the most 30 important factors in explaining the response variable. Commensurate with analyzing such a rich 31 data, this study employs different linear regularization techniques, including the Ridge, LASSO, 32 and Elastic Net models, along with ordinary linear regression. The entire data set was split into 33 two parts, where approximately 75 percent of the data (from January 1, 2020 to May 3, 2020) 34 were used for training the models, and the remaining 25 percent (from May 4, 2020 to June 13, 35 2020) was used for testing the prediction performance. Determining the set of most important fac- 36 tors impacting the number of daily trips per person from the pool of several independent variables 37 to increase the prediction accuracy was accomplished using the LASSO regression. The sixteen 38 selected variables were further analyzed employing the linear, Ridge, Lasso, and Elastic Net re- 39 gression techniques using the training data. Finally, the performance of the prediction is tested by 40 feeding the test data into the models for all regression techniques. 41 Chakraborty, Mahmud, Gates, and Sinha 19 The results of this study reveal that the coefﬁcients of the predictors are fairly comparable 1 across all modeling techniques. When factors including the number of new cases, social distanc-2 ing index, median income, socioeconomic status, percent of people aged more than 60 years, the3 number of people staying at home, transit mode share, driving mobility index, and the percent4 of African and Hispanic American population increase, the number of daily trips per person de-5 creases. Conversely, the number of daily trips per person increases with the increase in the unem-6 ployment rate and the percent of population working from home. Although, when the stay-at-home7 order is implemented, the number of daily trips drops, when the stay-at-home order is only rec-8 ommended but not stringently enforced, people are less likely to follow the guidance and tend to9 make more trips. Also, the number of daily trips drops when the states exercise either mandatory 10 or recommended domestic travel restrictions. The count of daily trips also increases with a manda- 11 tory or recommended mask-wearing policy in place. Lastly, although the percent of out of county 12 trips is found to be negatively associated with the number of daily trips being made in the linear, 13 LASSO, and Elastic Net regression models, the expected positive association between these two 14 variables is correctly captured in the Ridge regression technique. 15 Furthermore, the developed models were used to predict the number of daily trips per 16 person for all the states for a period of 41 days (from May 4, 2020 to June 13, 2020). Although all 17 the developed models, namely ordinary linear, LASSO, Ridge and Elastic Net models, compare 18 favorably, the Ridge regression model performed the best by having the least root mean square 19 error (RMSE) in prediction among the different models. This result makes sense because the 20 Ridge regression is robust in overcoming the issue of overﬁtting and thus generalizes better to the 21 test data set, resulting in lesser prediction error. Also, other regularization techniques, including 22 the LASSO and Elastic Net models, as well, performed superior to the ordinary linear regression. 23 The study is only the starting point to help understand the associations between different 24 factors and human mobility during the COVID-19 pandemic. The authors of this study intend 25 to expand this study to utilize county-based data to understand these associations from a more 26 granular level. Moreover, it would be insightful to include additional variables into the analysis 27 as potential independent variables. Also, additional factors such as vehicle miles traveled or the 28 number of miles traveled per person would also be required to be explored in explaining mobility. 29 From the modeling perspective, it is reasonable to argue that the available data is subjected to 30 some uncertainties and future research should be carried out to explicitly take the uncertainties into 31 account to derive at more precise models. Furthermore, as the crisis is moving on to the greater 32 peaks in terms of the number of conﬁrmed cases and deaths over time in the U.S., subsequent 33 analysis is warranted with data from the following months (post June 13, 2020). 34 AUTHOR CONTRIBUTIONS 35 The authors conﬁrm contribution to the paper as follows: study conception and design: Subhra- 36 jit Sinha, Meghna Chakraborty, Md Shakir Mahmud; data collection and preparation: Meghna 37 Chakraborty, Subhrajit Sinha, Md Shakir Mahmud; analysis and interpretation of results: Subhrajit 38 Sinha, Meghna Chakraborty, Md Shakir Mahmud; draft manuscript preparation: Meghna Chakraborty, 39 Subhrajit Sinha, Md Shakir Mahmud, and Timothy Gates. All authors reviewed the results and ap- 40 proved the ﬁnal version of the manuscript. 41 Chakraborty, Mahmud, Gates, and Sinha 20