Optimizing human activity patterns using global sensitivity analysis
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Abstract
Implementing realistic activity patterns for a population is crucial for modeling, for example, disease spread, supply and demand, and disaster response. Using the dynamic activity simulation engine, DASim, we generate schedules for a population that capture regular (e.g., working, eating, and sleeping) and irregular activities (e.g., shopping or going to the doctor). We use the sample entropy (SampEn) statistic to quantify a schedule’s regularity for a population. We show how to tune an activity’s regularity by adjusting SampEn, thereby making it possible to realistically design activities when creating a schedule. The tuning process sets up a computationally intractable highdimensional optimization problem. To reduce the computational demand, we use Bayesian Gaussian process regression to compute global sensitivity indices and identify the parameters that have the greatest effect on the variance of SampEn. We use the harmony search (HS) global optimization algorithm to locate global optima. Our results show that HS combined with global sensitivity analysis can efficiently tune the SampEn statistic with few search iterations. We demonstrate how global sensitivity analysis can guide statistical emulation and global optimization algorithms to efficiently tune activities and generate realistic activity patterns. Though our tuning methods are applied to dynamic activity schedule generation, they are general and represent a significant step in the direction of automated tuning and optimization of highdimensional computer simulations.
Keywords
Global optimization Global sensitivity analysis Sample entropy Agentbased modeling Bayesian Gaussian process regression Harmony search1 Introduction
Highdimensional computer models for simulating real world phenomena have many variables and present a difficult challenge in understanding the relationship between input and output. Known as the curse of dimensionality, a full space analysis of the nature of inputoutput relationships is NPcomplete, scaling exponentially as s ^{ n }, where s is the number of sample values for each of the n input variables (Rabitz and Aliş 1999). This paper presents an efficient method for determining these inputoutput relationships in highdimensional models using a combination of global optimization and global sensitivity analysis. We demonstrate our method using a model of human activity and movement.
Human activity and movement patterns are complex and notoriously difficult to model (Berry et al. 2002). Large variations in movement patterns stem from demographic, geographic, and temporal differences. Quantifying the effects of these differences on human activity/schedules provides a difficult but important challenge (González et al. 2008). Realistic human activity and movement models are fundamental components for agentbased infrastructure simulations. These models use human activity patterns to simulate complex systems including epidemics (Eubank et al. 2004; Colizza et al. 2007; Mniszewski et al. 2008; Stroud et al. 2007), traffic (Kitamura et al. 2000), and natural disaster response (Pan et al. 2007). Despite their importance, models typically simplify the complexity of human movement and rely on estimates such as static activity patterns. The static approach results in a Groundhog Daylike effect, where every person performs the same activities day in and day out according to a fixed schedule. Since the schedule cannot be modified based on exogenous events, the schedule will inevitably repeat over some finite time scale.
The level of realism required in a model of natural phenomenon depends upon the scenario being modeled and the questions being addressed (Burton 2003; Burton and Obel 1995). In epidemic modeling, capturing emergent human behavior is crucial for accurately forecasting the spread of disease and the impact of mitigation strategies. Similarly, for modeling disaster response during a natural or manmade event, understanding people’s activities before and after the event will help emergency responders allocate resources. Finally, supply and demand modeling of various utilities (e.g., water, electricity, and communications) depends on the population’s activities as they move throughout the day. Therefore, capturing realistic activity patterns can help improve modeling efforts and save lives during emergencies.
We have built on the previous body of activity pattern research. Germann et al. presented a study analyzing mitigations for a pandemic influenza in the United States (Germann et al. 2006). In their study, 12hour schedules were cycled to direct the activities of seven different mixing groups consisting of work, school, day care, play group, neighborhood, neighborhood clusters, and communities. Paleshi et al. performed a similar study that featured stricter mixing patterns according to four coarselydefined demographic groups: preschool children (ages <1–4), school children (5–18), adults (19–64), and seniors (≥65) (Paleshi et al. 2011). In Stroud et al. (2007) and Mniszewski et al. (2008), epidemic simulations rely on static schedules with individuals cycling through nine different activities. Additionally, in contrast to other studies, individuals temporarily deviate from their schedules when ill, and parents stay home with sick children. Weekday schedules were further distinguished from weekends and holidays by replacement of work or school with home for a portion of the population in a study of social contact patterns and their effect on the spread of disease (Del Valle et al. 2008); it was found that the lack of weekday and weekend activities can greatly overestimate the impact of disease spread. Brockmann et al. moved beyond the realm of staticbased schedules by considering random walks as a proxy for human movement based on trajectories of almost 500 thousand dollar bills (Brockmann et al. 2006). González et al. studied the paths of 100 thousand mobile phone users and showed that humans do not behave randomly; rather, they follow simple reproducible spatial patterns (González et al. 2008). All of these models neglect basic human characteristics based on desire, need, and importance that can impact and change schedules accordingly (e.g., getting sick may force a person to go home early from work, or car maintenance may preclude shopping). A realistic human activity and movement model needs to dynamically take these basic human traits into account (Macy and Willer 2002).
We use the Dynamic Activity Simulator (DASim), previously known as ActivitySim (Galli et al. 2009), that incorporates activity utility and priority to develop schedules for a population of individuals. DASim generates schedules that give each individual close to the maximal utility that complements their priorities for activities. This allows one to design population schedules by specifying priorities and utilities of a variety of activities for any number of demographic groups. Moreover, new schedules can be generated dynamically during a simulation.
Once these schedules are dynamically generated, it is not immediately apparent if they are realistic for a population. In actual populations, we expect demand hours (i.e., the total number of people participating in an activity aggregated over one hour) for certain activities, such as grocery shopping or working, to be stable on any given weekday. For recreation activities or hospital visits, we expect daily demand hours to fluctuate, with possibly a more stable amount of demand hours on a monthly or quarterly timescale. In this way, regularity of demand hours can be required in population’s schedules to classify traits of certain activities, thus adding realism to the dynamic schedule generation. We propose to quantify an activity schedule’s regularity using the sample entropy (SampEn) statistic (Richman and Moorman 2000). That is, the SampEn of the time series associated with DASim output is used to dynamically adjust schedules to be consistent with regular and irregular activity patterns. By tuning SampEn, one can design schedules comprised of activities that occur with a desired level of regularity.
Tuning the SampEn statistic for a schedule can be posed as a highdimensional optimization problem. Global sensitivity analysis can be used to reduce the dimensionality of the optimization problem by targeting the input parameters in DASim that control the majority of variation in SampEn. The sensitivity analysis was carried out efficiently through the use of Bayesian Gaussian process regression. Once a lowdimensional set of influential parameters is discovered, a global optimization scheme, harmony search (HS) (Geem et al. 2001), is used to tune SampEn and therefore adjust the regularity of activities in a schedule. We demonstrate that reducing the search space for HS to only influential parameters results in a more efficient search.
2 Methods
2.1 Dynamic activities model
DASim is a dynamic parallel agentbased discrete event movement and activity simulator. DASim requires two components to generate schedules: (1) a population with demographic characteristics, and (2) locations with geographic coordinates. DASim can use any population and location data, but the synthetic population we use is based on U.S. census data^{1} and includes various demographic characteristics such as age, gender, income, and status (e.g., worker, student, and stay home). In addition, each person has a household consistent with the census data. Locations are derived from the Dun & Bradstreet business directory database,^{2} which include addresses and business type. Businesses can be aggregated in a geographic area and may include multiple business types such as a shopping mall. DASim integrates all this information to generate realistic schedules according to the person’s preferences and needs.
Example activity set. Each demographic group has a different set of activities, its own utility and priority functions, time and distance constraints, and weekend factor
Group  Home  Work  Shop  Leisure  Medical  Day Care  School 

Child (0–5)  ✓  ✓  ✓  ✓  
Youth (6–18)  ✓  ✓  ✓  ✓  
Worker (19–64)  ✓  ✓  ✓  ✓  ✓  
Senior (65+)  ✓  ✓  ✓  ✓ 
Parameters used in DASim’s utility and priority functions. Each parameter in DASim is shown with a brief description of what it influences and the ranges over which will allow it to vary in our optimization. Figure 1 shows how varying these parameters changes the sigmoid function in practice. Refer to Joh et al. (2001) for more details on these parameters
Parameter  Description  Range 

α _{{u,p}}  The α parameter determines the location of the sigmoid function along the Xaxis. In the case of utility, larger α values result in longerduration activities. Larger α parameters in priority functions result in less frequent activities.  [0,86400] 
β _{{u,p}}  The β parameter determines the slope of the sigmoid function. For utility functions, larger β values limit the range of desirable activity durations. For priority functions, larger β values increase how rapidly an activity becomes important once enough time has elapsed since last execution.  [0,1] 
γ _{{u,p}}  The γ parameter determines the inflection point in the curve. It defines how quickly the sigmoid’s concave region transitions to the convex region. For utility functions, larger γ values result in a more drastic concave to convex transformation, which in turn reduces activity duration range. Larger γ values in priority functions result in a larger range of time between activities.  [0,10] 
The two parameters in the objective function, C and D, weigh the importance of the priority function and travel time constraints, respectively. C and D are global parameters and apply equally to all activities for all demographic groups. The three parameters in each of the utility and priority functions are local parameters set on a peractivity basis.
Operators used by DASim when designing schedules. The local search metaheuristic randomly selects operators in order to create a new valid schedule from the current schedule. The objective function then compares the current schedule against this new schedule, as described in Fig. 2
Operator  Description 

DeleteIncreaseDuration  Delete an activity and increase the durations of the activities surrounding the deleted activity. 
AdjustDuration  Change the duration of two consecutive activities. 
Substitute  Replace an existing activity with another. 
DecreaseDurationInsert  Decrease the duration of two consecutive activities and insert a new activity. 
Append  Add a new activity. 
In this study, we concentrate on a randomlygenerated 10person test population. Each of the 10 people in the test population is allowed to create schedules from an activity set comprised of two activities. The first activity is allowed to be between 1 and 24 hours long (allowing for a variety of short or longduration activities, such as personal care, shopping, and medical appointments). The second activity is set to be between 4 hours and 10 hours (forcing longerduration activities, such as work, home, and sleep). The weekend factor for both activities is 1.0 (indicating that the activities are equally likely to occur during the weekend as they are during the week). The maximum travel time for each activity is fixed at 2 hours. Activities are allowed to start and end at any point during the day.
2.2 Sample entropy
Certain human activities occur with a high degree of regularity (e.g., working, going home), while others occur more erratically (e.g., medical treatment, social recreation) (Bhat et al. 2004; Kitamura and Hoorn 1987; Kitamura et al. 2006; Schlich and Axhausen 2003). Here, we develop a procedure to choose DASim parameters (α _{{u,p}},β _{{u,p}},γ _{{u,p}},C,D) that ensure spontaneity or regularity in an activity. We use the sample entropy (SampEn) statistic to detect regularity in a time series associated with a schedule.
SampEn was first introduced by Richman and Moorman (Richman et al. 2004; Richman and Moorman 2000) in response to Pincus’ seminal work on approximate entropy (ApEn) (Pincus 1991). Entropy quantifies the amount of order or disorder in a system. Ordered systems yield low entropy while disordered or chaotic systems yield high entropy. For a time series, this usually means that a low entropy system will have repeated changes or will remain constant, while a high entropy time series will have unpredictable changes that are highly variable. ApEn was originally developed to analyze regularity in medical and biological time series, specifically neonatal heart rates. It is still commonly used in medical literature (Goldberger et al. 2002; Hornero et al. 2005, 2006; Pincus and Goldberger 1994; Varela et al. 2003) and has also been applied to a variety of other fields including finance (Pincus and Kalman 2004) and human factors engineering (McKinley et al. 2011). SampEn improves on ApEn in several ways; most notably, it is a less biased statistic and requires about half the computing time (Richman and Moorman 2000).
SampEn computes the conditional probability that if a finite time series repeats itself within a tolerance r for m points, then it will also repeat itself for m+1 points, without allowing selfmatches (Lake et al. 2002). Small values of SampEn (values close to zero) indicate signal regularity (i.e., an ordered system), while relatively larger values indicate less regularity (i.e., a more disordered system). SampEn is still a comparative measure; there is no single threshold above which we may say that any arbitrary signal is irregular. It must be judged relative to the problem being addressed.
In our simulations, SampEn is used to quantify regularity of demand hours for activities on an hourly basis (i.e., m=1). It is common practice to set r equal to some fraction of the standard deviation (σ) of the data being analyzed, allowing measurements on datasets with different amplitudes (Richman and Moorman 2000); thus, we set r=0.2σ, where σ is computed from DASim’s demand hours output. We use the SampEn implementation written in C provided by PhysioNet.^{3}
2.3 Global sensitivity analysis
A traditional Monte Carlo approach to compute the sensitivity indices is computationally expensive due to the repeated/iterated terms such as \(V(\mathbb{E}( \mathbf{Se}  \theta _{j}))\). A variety of approaches have been suggested to bring down the computational cost (Homma and Saltelli 1996; Marrel et al. 2009; Oakley and O’Hagan 2004; Saltelli 2002; Saltelli et al. 1999). We compute approximations to the sensitivity indices using a statistical surrogate model (Marrel et al. 2009; Neal 1997; Oakley and O’Hagan 2002, 2004), or emulator, for the function Se(θ). The emulator uses a Gaussian process regression (Higdon et al. 2008; Marrel et al. 2009; Neal 1997; Williams et al. 2006), which consists of fitting a Gaussian process Se _{ g }(θ;η) to samples of Se(θ) taken at different θ parameter sets specified by the rows of the M sample matrix Θ.
The Gaussian process emulator (MacKay 1998; Neal 1997) is constructed using Bayesian Gaussian process regression. For a more complete description of this process we refer the reader to Higdon et al. (2008), Marrel et al. (2009), Oakley and O’Hagan (2002, 2004), Williams et al. (2006). First, the emulator Se _{ g }(θ;η) is a stochastic process in the variable \(\theta\in\mathbb{R}^{14}\) with state variable η. It has the property that the evaluation at any finite number of θ samples (Se _{ g }(Θ _{1}),Se _{ g }(Θ _{2}),…,Se _{ g }(Θ _{ M }))^{ T } is a Gaussiandistributed Mdimensional random vector, having mean μ=μ(Θ _{1},Θ _{2},…,Θ _{ M }) and covariance Cov=Cov(Θ _{1},Θ _{2},…,Θ _{ M }).
In the Bayesian regression approach, Se _{ g } is constructed from samples of the output Se _{ i }=Se(Θ _{ i }), i=1,2,…,M. The mean and covariance of Se _{ g } are defined so that realizations of the simulated values have a maximized posterior probability given a prior distribution on the form of the covariance. The form for the covariance is specified so that when evaluating at a new parameter set, θ ^{∗}, the variance of Se _{ g }(θ ^{∗}) increases for θ ^{∗} further from the samples in the matrix Θ and goes to zero, if θ ^{∗} lies in this sample set. The mean of Se _{ g }(θ ^{∗}) is related to the sampled values so that it is equal to Se _{ i } for θ ^{∗}=Θ _{ i }. Thus, Se _{ g }(θ;η) is an interpolant of the sample values.
Sensitivity indices of \(\mathbb{E}_{\eta}(\mathbf{Se}_{g}(\theta; \eta))\) can be computed quickly once Se _{ g } is constructed from a sample set. We refer to Marrel et al. (2009), Oakley and O’Hagan (2004) for this computation. To construct the Gaussian process and to compute the sensitivity indices, we used the Los Alamos GPM/SA code^{4} (Higdon et al. 2008; Williams et al. 2006).
2.4 Global optimization
Our goal is to find values for each of the parameters in θ for which SampEn, for the given activities, is either minimized (for increased regularity in scheduling) or maximized (for increased spontaneity). Optimizing over the complete 14dimensional parameter space can be costly. Note that this 14dimensional space is only for two activities; each additional activity adds 6 new parameters. Therefore, analyzing five activities would require optimization over a 32dimensional space, which is computationally expensive for updating a schedule dynamically.
We use the global sensitivity indices to reduce the dimensionality of the optimization problem and identify parameters that contribute very little to the variance of SampEn. In an optimization step, these parameters are then fixed, and the remaining parameter space is searched using a global optimization procedure. If the number of parameters to which SampEn is sensitive is small, this can potentially result in a cheaper optimization procedure.
We use the harmony search (HS) global optimization algorithm (Geem et al. 2001) to explore the parameter space. HS is a metaheuristic search algorithm, inspired from the improvisation process of jazz musicians, that optimizes (minimizes or maximizes) a certain objective function. Recently, HS has been successfully applied to a variety of problems including water distribution network design (Geem 2006b), parameter estimation (Kim et al. 2001), combined heat and power economic optimization (Vasebi et al. 2007), and even sudoku solving (Geem 2007). In many cases, it has been shown to outperform other commonly used search algorithms, such as simulated annealing (Kirkpatrick et al. 1983), tabu search (Glover 1989, 1990), and evolutionary algorithms (Bäck and Schwefel 1993).
In HS, sets of parameters (referred to as a harmonies) are randomly chosen (improvised) until the harmony memory is filled. A new harmony is improvised according to a set of rules: each parameter (note) may be chosen via random selection or memory consideration with an optional pitch adjustment (adjusting a parameter up or down slightly). The goodness of the new harmony is computed (in this case, the sum of the SampEn statistics for each activity), and if the harmony is better than the worst harmony stored in the harmony memory, the new harmony replaces the previously stored value.
HS features five main parameters: max_imp determines the maximum number of improvisations (iterations), hms is the harmony memory size (the number of best harmonies that should be remembered), hmcr is the harmony memory consideration rate (how often a note is chosen via memory consideration as opposed to random selection), par is the pitch adjusting rate (how often pitch adjustment is invoked), and mpap is the maximum pitch adjustment proportion (size of the perturbation).
A number of improvements and changes have been suggested since HS’ first introduction. One change added the notion of ensemble consideration, an operation that considers relationships between decision variables (Geem 2006a). Another modification, dubbed improved harmony search, dynamically modifies the par and mpap parameters as the search progresses (Mahdavi et al. 2007). Globalbest harmony search removes the mpap parameter altogether by altering the pitchadjustment step so that values are drawn from the best harmony in the harmony memory (Omran and Mahdavi 2008). Most recently, a parameter settingfree variation was introduced that dynamically modifies both hmcr and mpap as the search progresses (Geem and Sim 2010).
Harmony Search with Global Sensitivity Analysis Algorithm
 1.
Provide allowable intervals for each parameter \(\theta_{j} \in [\theta^{}_{j},\theta^{+}_{j}]\), j=1,2,…,14.
 2.
M samples of SampEn are taken at different parameter sets in the sample matrix Θ _{ M×14}.
 3.
Samples used to construct a Gaussian process emulator, Se _{ g }(θ;η).
 4.
Sensitivity indices, \(\{ S_{j}, S^{T}_{j} \}\), j=1,2,…,14, are computed from Se _{ g }(θ;η).
 5.A subset of parameters, \((\theta_{k_{1}}, \theta_{k_{2}}, \dots, \theta_{k_{d}})\), with high sensitivity values (see Fig. 4) are chosen on which to perform HS. The remaining parameters are fixed (note that we arbitrarily fix them at the mean value of their interval). Here, we use notation for an arbitrary subset of distinct parameter subscripts of size d≤14, {k _{1},k _{2},…,k _{ d }}⊂{1,2,…,14}.
 6.
HS is performed over the parameter subset to maximize a given functional of SampEn statistics for each activity.
Each 12week simulation of DASim for the 10person test population takes approximately 5 seconds wall time to complete. We initialize max_imp to 2000, hms to 50, hmcr to 0.75, par to 0.5, and mpap to 0.25. HS consistently converged to solutions of approximately the same fitness over many test runs, each with initial harmonies selected uniformly at random. As a result, we determined that a parameter sweep of the HS parameters was unnecessary.
3 Results
The global sensitivity analysis (Fig. 4) shows that the offset parameters, (α _{ u1},α _{ p1},α _{ u2},α _{ p2}), have the largest effect on the variation of the sample entropy for both activities. Thus, these α parameters have the most impact on regularity. Recall that, as seen in Table 2, α _{ u } and α _{ p } control the activity duration in the utility and activity frequency in the priority, respectively.
We performed random sampling over the entire 14dimensional parameter space and compared the variance in the SampEn for each activity against only varying the αparameters. Our results show that the variation in SampEn caused by only varying the αparameters was responsible for about 99 % of the variance in SampEn when the entire parameter set was allowed to vary. This result was consistent for each activity. This shows that our sensitivity analysis with the emulation gives realistic results and that optimization over the fourdimensional parameter space will suffice to approximate the minimum or maximum of the sample entropy or a functional thereof.
4 Discussion
This study focuses on schedule realism in a human activity model, but the methods presented here are generic and can be applied to a variety of other problems where a specific property in a highdimensional model is desired. These types of highdimensional tuning/optimization problems are ubiquitous in modern complex computer simulations. Thus, there is a significant need for methods of automatic tuning that incorporate systematic dimension reduction. Our combination of global sensitivity analysis and a global optimization method is effective for the application presented here. Additionally, it is sufficiently general to warrant application in many other areas.
Dynamic scheduling for synthetic populations is necessary to make simulations of human behavior phenomena more realistic. The dynamic scheduling program DASim was designed to aid in largescale agentbased infrastructure simulation (e.g., transportation and epidemic modeling). DASim can generate schedules that are different over demographics and change in response to events, such as disease outbreaks and nonpharmaceutical interventions.
To evaluate the realism of a dynamicallygenerated schedule, we must select metrics on which it should be evaluated. We presented a method for tuning a dynamic scheduling model for schedule regularity, which we quantify using the sample entropy (SampEn) statistic applied to population demand hours. Adjusting the SampEn statistic requires working with a highdimensional optimization problem. We used global sensitivity analysis and statistical surrogate models to significantly lower the dimensionality of the search space. A global optimization algorithm, harmony search (HS), was used to efficiently tune the degree of regularity of a schedule.

Demand hour regularity of activities over a population can be controlled by tuning the SampEn statistic.

DASim parameters that most influence the SampEn statistic can be identified using global sensitivity analysis combined with a statistical surrogate model. We determined that the α parameters in the utility and priority functions have the largest effect on the variation of the sample entropy of an activity.

DASim parameters that result in close to optimal (i.e., minimized/maximized) SampEn values can be discovered using HS. Furthermore, this can be done efficiently with many fewer iterations by searching a parameter subspace determined by global sensitivity analysis first (just the α parameters in this study).
While we have shown how to reduce the search space and computation time when analyzing parameter importance under a particular metric, this process still takes a significant amount of compute time. We tuned our parameters in a reduced problem environment, using a 10person population. Although this approach works for the measure of regularity discussed in this paper, it may not work for more complex measures of interest. Our initial search space of 14 dimensions is still relatively small; some simulations may have many tens, hundreds, or even thousands of dimensions. Understanding parameter importance and interactions in such highdimensional spaces may prove difficult or even impossible in some instances using our methods.
Our analysis is based on hourly regularity for demand hours of schedules. Many other granularities may be desirable. For example, work may be regular every 12 hours. Some studies suggest that the size of the dataset be at least 10^{ m } and preferably at least 30^{ m } in the approximate entropy (ApEn) algorithm (Pincus and Goldberger 1994). While this is certainly possible for small values of m (recall that m=1 in this paper), larger values of m quickly become problematic (e.g., work would require at least 10^{12} demand hour data points). Alternative measures of regularity may be considered for larger values of m.
We are considering other evaluative measures to quantify additional properties, beyond regularity, of a schedule’s realism. Here, we analyzed measures of regularity of time usage and found that it is controlled by a small set of the defining parameters in the model. Another possibility would be to look at quantifying the efficiency of a dynamicallygenerated schedule in terms of location usage, whether an individual’s schedule is geographically arranged in a sensible way given his or her current location. One could also look at the total percentage of time spent on an activity. Evaluation of a measure of each of these effects would lend testable realism to a generated schedule for a population. The use of statistical emulation, global sensitivity analysis, and optimization as demonstrated here would then allow for efficient tuning of these measures.
For models that rely on human activity patterns and movement, such as disease and infrastructure models, capturing realistic activity patterns is crucial for decision support. Therefore, new techniques such as the ones proposed here are needed for analyzing highdimensional problems. However, more research still needs to be done related to efficiently solving these problems computationally and understanding human activity patterns and behavior.
Footnotes
Notes
Acknowledgements
We would like to acknowledge the Institutional Computing Program at Los Alamos National Laboratory for use of their HPC cluster resources. This research has been supported at Los Alamos National Laboratory under the Department of Energy contract DEAC5206NA25396 and a grant from the NIH/NIGMS in the Models of Infectious Disease Agent Study (MIDAS) program U01GM09765801.
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