May 18-19, 1995
Gus A. Koehler, PhD.
H. Richard Priesmeyer
Edward G. Cole
Before proceeding with a discussion of nonlinear tools and how they can be applied to disaster response data, it is important to provide some evidence that disaster response is a nonlinear process. Koehler, Comfort, Kreps and others have independently argued the case effectively from both theoretical and metaphorical positions. Additional mathematical evidence can now be supplied to suggest that nonlinear tools and the complexity theory view in general is an appropriate approach to studying, monitoring, and perhaps managing disaster response activities.
The most common nonlinear equation, known as the logistic equation is given as Xn= kXn-1(1-Xn-1). Although it most likely represents an entire family of similar nonlinear forms it has been shown able to model a wide range of natural phenomenon such as animal populations, cardiac rhythms, and corporate market share.
The logistic equation is most frequently associated with a "predator-prey" model of competition; it is for this reason that it models animal populations and market share so effectively. However, competition here can be taken in a larger context of balance between forces and it is from this view that it is seen as a legitimate model for disaster response.
The equation expresses each succeeding value as a function of the immediately preceding value and a constant. Unlike linear equations which provide a single solution, the logistic equation is iterated to produce a stream of solutions. That is, the first solution to the equation X0 is inserted as a variable into the right hand side to provide a second solution. The process continues to produce a series of solutions of any length. When the resulting values are adjusted for scale (using the mean and standard deviation of the original data) this simple equation can reproduce the original data with high precision. If the equation can model a given data set it provides strong evidence that the data set is nonlinear in nature. Further, the quality of the fit can be statistically tested for significance.
The Logistic Map
The logistic map describes all possible behaviors of a nonlinear system. It provides a picture of stability, oscillation, complex oscillation, and chaos. It can be used to better understand where stability ends and disorder begins.
Note that the logistic equation Xn= kXn-1(1-Xn-1) has only one variable (x) and one constant (k). The value of X describes the current state of the system. It might, for example, describe the current level of emergency medical service (EMS) activity. As the equation is iterated the current EMS activity level (Xn) is inserted into the right side of the equation to compute the next period's EMS activity level X(n+1). Continued iterations produce of stream of values describing the EMS actions over time. The values produced, however, depend almost entirely on the value of k. We can think of k as some kind of measure of the environment for the system ranging from stable and calm to highly energetic and excited. Knowing the value of k for a real system provides valuable insight into the potential behavior of that system.
Logistic Regression Fit To Total Involvement Data for First 24 Hours After Impact
The logistic map shown in Figure 1 provides a view of the solutions for the logistic equation over a wide range of k. It is produced by iterating the equation several times for each value of k and plotting the resulting values of X. As we increase k from 1.0 to 3.0 the values of X increase gradually. It is important to note that in this range the equation provides a single, stable solution as it is iterated. The result is a single line arching upward as higher values of X emerge at higher values of k. From a practical view this means that when the environment of a system is stable the performance coming from any system in that environment is also stable. In fact, it is resistant to being changed and tends to return to a point of stability if disturbed.
At k=3.0 the solutions bifurcate (split) and the equation begins to provide two oscillating solutions. Still higher values of k result in additional bifurcations and this simple equation begins to provide increasingly complex series of solutions. Above 3.7 the logistic equation provides random-appearing, non-repeating solutions which are considered chaotic. Again, from a practical view this means that as the environment of a system becomes more energetic the performance of that system becomes increasingly complex perhaps shifting from one state to another (oscillating), perhaps varying in non-repeating, random-appearing ways (chaos). It is useful to recall that there is no randomness here; all values of X, regardless of how complex, can be computed directly by simply iterating the equation.
Logistic Fit of Disaster Response Data
A data set representing 257 key participants in 106 organized responses was obtained from Susan Bosworth and Gary Kreps at the College of William and Mary. It consists of survey data obtained from a wide range of disaster response personnel; it was collected in their research on role enactment. Included in the sample are EMS, fire, police, and a broad variety of support personnel involved in needs assessment, damage control, public information, and insurance. The events represented in the data set include a number of floods such as those in St. Paul, Colorado, and Texas, a number of tornados such as those in Jonesboro, Oaklawn, and Jackson, and other events such as the Alaska earthquake, Hurricane Camille, and the Mount St. Helens eruption.
The data set contains some rare time series variables which are essential for any nonlinear analysis. Specifically, for each respondent it contains an indication as to the time after impact when involvement began and the length of that involvement. These two measures make it possible to compute when involvement ended. Some subsequent analysis allows one to determine the level of overall activity for all participants at specific time intervals after impact. These activity levels can then be subjected to logistic regression analysis to determine the values of k and x which best fit the data.
Figure 1 also reveals the results of this analysis. When the equation is fitted to the first 24 hours of disaster response activity for this wide range of events it reveals a value for k of 3.66 with an initial value of X of .10. It also provides an F value of 6.75 which is significant at the 95 percent confidence level. Put simply, the nonlinear logistic equation can be fitted to this data and reveals the level of disorder in the resulting activity level during the first 24 hours after impact. These results add an important mathematical argument to the contention that disaster response is a nonlinear problem and can be addressed with nonlinear tools and techniques.
The value of k carries with it important insights into a system. Systems with low values of k and, therefore, high stability are not sensitive to initial conditions. They are quite resistant to any attempts to destabilize them. This can be either desirable or undesirable depending upon whether the system in question represents preferred behavior or not. Low values of k are desired when one wants little or no change in a system. The logistic map shows that the opportunity for true system change is built into the system, but is only possible when the system is operating in or near the chaotic region (when k exceeds 3.7). One will note that the derived value for k of 3.66 is very near but does not exceed the edge of the chaos domain at 3.7.
Only when the system is operating in "far from equilibrium conditions" does the opportunity for change exist. Chaos is therefore a necessary condition to permit substantial repositioning and adaptation. Put another way, it is likely that the chaos which results during these first 24 hours is a necessary and desirable condition which accommodates adaptation, cross-communication, the suspension of rules or policies, and other such emergent behavior essential to an efficient response.
Phase Plane Geometry
One nonlinear tool often used for analysis employs so-called phase planes which describe the evolving "state of the system." These phase planes can be extensively exploited to trace the level of activity and resource allocations during a disaster response. Specifically, a phase plane describes the changing condition of any system by identifying the "state of the system" on a common two-dimensional Cartesian plane. The two dimensions are simple two selected measures of the system. For this example consider the X Resource fire equipment and personnel and the Y Resource EMS equipment and personnel.
It is important to realize that phase planes plot changes in the two measures, rather than the actual values of the variables. That is, they plot marginal values between periods. Obviously, changes can be either positive or negative from period to period. The center of the phase plane is coordinate 0,0 representing no change in either measure. Quadrants are traditionally numbered as shown in Figure 2.
Descriptions of Resource Needs for Domains on a Phase Plane
At the onset of a disaster, the need for both resources (X and Y) would increase pushing the initial trajectory into Quadrant 1. As increasing quantities of equipment continued to arrive on the scene, the trajectory would stay in Quadrant 1, although its movement would be disordered rather than linear, and indicate subtle changes in the required equipment mix. As the disaster team begins to gain control of the situation, the necessity for equal resources would likely diminish. For example, a trajectory trip to Quadrant 2 would indicate the need for more EMS equipment and less need for fire equipment. Conversely a trip to Quadrant 4 would indicate a greater need for fire equipment and less need for EMS. The trajectory landing in Quadrant 3 would signal a reduction in the need for both resources.
The evolving "state of the system" is plotted as a trajectory on the phase plane. That is, as the values for each measure change over time, those changes are plotted on the phase plane. A trajectory is created by extending a line from each plotted position to the next position so one can see the evolution of the system. If several periods of data are available, the transitions in the data are presented as evolving trajectories on the phase plane. Rather than having to evaluate a series of one-dimensional variables, the phase plane image focuses attention on relative changes which might otherwise escape notice. It graphically depicts the changes in the two measures exposing subtle transitions in the data which are masked by the sheer size of the actual measures. Most often there is obvious structure and pattern to the changes and that structure is dramatically displayed in the trajectories of interactive changes as they are plotted. The patterns can be classified and they suggest a variety of other forms of analysis.
Interpretations, and Prescriptions
Because positions on the phase plane relate to specific changes in the system being monitored, it follows that there are unique interpretations for each quadrant and, often, appropriate generic prescriptions can be provided for any position on the plane. The ability to translate changes in data into written interpretations means that analysts applying this technique can generate a series of interpretative statements describing appropriate interventions for virtually any business activity. The authors have developed software with phase plane interpretations and appropriate corrective prescriptions for a variety of applications.
While Figure 2 provided a description of what was occurring, Figure 3 depicts possible interpretations. Movement of the trajectory into and within Quadrant 1 would alert the disaster team that the need for both EMS and fire equipment is continuing to grow and reserve capacity is declining. Further, the location of the trajectory within Quadrant 1 (closer to the X axis or the Y axis) would indicate subtle changes that are occurring in the required equipment mix. Movement into either Quadrant 2 or 4 would indicate the nature of the response is changing and one specific resource is needed more than the other. Finally movement into Quadrant 3 would signal a reduction in the need for both resources and an increase in reserve capacity.
The phase plane captures the interaction of the changes in the two measures thereby defining the "state of the system." That "state" or condition can then be reported and an appropriate prescription automatically generated. Again, it is important to realize that these conditions relate to the incremental evolution of the system hence the interpretations and prescriptions are on the margin. They provide insight into changes which would probably not otherwise be noted and therefore, they allow management to respond in a more timely manner.
Interpretation of Resource Allocation Activity
for Domains on a Phase Plane
The heart of phase plane analysis is the ability to generate appropriate prescriptions for each evolving condition (figure 4). As the trajectory moves into and around Quadrant 1, the following prescription might be appropriate: "As local resources continue to be committed and reserve capacity is being depleted, initiate Plan X (call in equipment from other communities and put the State Government on alert)."
Movement into either Quadrant 2 or 4 would advise the specific reallocation of resources from EMS support to fire support or vis-versa. And movement into Quadrant 3, signaling a reduction in the support needed, could generate the prescription to "dismiss personnel and equipment from other communities, reduce local commitment, resupply and standby."
Prescriptions for Disaster Response for Domains on a Phase Plane<
Figure 5 provides a trajectory of the first 12 hours for the variety of disaster responses represented in the William and Mary data set. In this example the activity levels of certain "active agents" were selected. These include personnel involved in evacuation, mobilization of emergency personnel, protective action (police), search and rescue personnel, and support personnel involved in providing basic needs to victims (it does not include EMS or fire personnel except those involved in search and rescue). The sample size which forms the basis of this trajectory is 140 personnel. The trajectory traces the number of these active agents relative to the total number of active agents.
Phase Plane of First 12 Hours After Impact for
Selected Active Agents to Total Agents
Note the initial increase in total involvement during the first two hours represented by the trajectory entering Quadrant 1. During the following two hours there is a substantial increase in the number of these selected agents and another increase in total involvement (recall that the origin of the phase plane represents "no change" and that any position to the right represents an increase even if that increase is less than in the previous period). During the period between four and six hours after impact the trajectory returns to a position near the origin reflecting a relative constant level (ie., no change) of activities both in terms of total and in terms of these selected agents. In the time period between six and eight hours after impact there is a slight reduction in activity as indicated by the trajectory venturing into the lower left quadrant (Quadrant 3). Between eight and 10 hours the trajectory returns to a low level in Quadrant 1 indicating a resumption of involvement; during the last two hours total involvement declines although there is a slight increase in the activity level of these selected agents (the 12 hour position resides above the origin indicating an increase in evacuation, mobilization, protective, search, and needs provision personnel).
It is useful to note the outer limits of the trajectory in Quadrant 1. Those activity levels at two and four hours likely represent capacity limits which will be addressed later.
Phase Plane of 12 to 48 Hours After Impact
for Selected Active Agents to Total Agents
Figure 6 traces the trajectory for the time period between 12 and 20 hours after impact. Note the close proximity to the origin (representing near constancy or stability) but also note that it resides for a considerable amount of time in Quadrant 3 reflecting the gradual subsidence of activity.
The interactive changes shown in Figures 5 and 6 are presented in a different format in Figure 7. The Marginal History Chart presents the changes in each measures independently for convenient examination of their independent behaviors. Along the top of the graph in Figure 7 is a list of the Quadrant positions (Quads) visited each hour for 48 hours. Note the first four Quadrant 1 positions representing the initial increase in activity. Also note the extended oscillations in activity for both the total and the selected (active) agents; an arrow at the 32nd hour indicates the approximate end of this variation and the onset of stability. This figure also presents a "velocity" measures which is computed as the product of the two other measures. A strong increase in both measures (associated with a trajectory entering Quadrant 1) will result in a sharp increase in velocity which quickly declines as activity levels stabilize.
Marginal History Chart of First 48 Hours After Impact In One-Hour Segment
The behavior of disaster response data which has been discussed in terms of interpretations and prescriptions and, above, as actual trajectories on a phase plane, can also be discussed conceptually as a resource allocation problem. Doing so provides insight into the technical and practical limits of the initial response and suggest some strategies for discussion.
Consider the diagram in Figure 8. For purposes of this discussion, again consider Resource X as fire fighting equipment and teams and Resource Y as EMS equipment and teams. The maximum values of these resources are given by Capacity Limits drawn into Quadrant 1. Any reserve resources such as state or federal assistance would extend the capacity boundaries outward. Any equipment already committed or disabled moves these capacity boundaries inward. Recall that the phase plane plots changes hence these boundaries are dynamic and changing reflecting the changing ability to increase response.
Capacity Limits and Coordination Barriers
A Coordination Barrier is included in Quadrant 1 to suggest that the ability to dispatch resources in a disaster is limited by technical problems. Any technical malfunctions, communication failures, physical restrictions, legal barriers, or personality conflicts can limit the ability of the system to respond to the limits of its resource capacity.
In Quadrant 3 there is a limit to the extent of downsizing as the disaster subsides. This can be considered a reserve limit which retains some capacity for normal or above normal activities. Figure 9 presents a phase plane depicting this Reserve Barrier. Note that downsize limits are also shown indicating the level at which all resources would be reserved.
Downsize Limits of Quadrant 1
There are also limits to which resources can be substituted and it is in this action where there may be potential for increasing the total response capacity. Figure 10 provides two Exchange Barriers which suggest the limits to which one resource can be exchanged for another. In practical terms this would suggest that to some degree one resource (fire personnel) can substitute for another resource (emergency medical personnel). On the phase plane this condition would be depicted as a point below the curve in Quadrant 2 (the upper left quadrant) because the substitution results in a decrease of Resource X (fire personnel) and an increase in Resource Y (emergency medical personnel). Similarly, a position above the curve in Quadrant 4 (lower right) suggests a substitution can be made reducing Resource Y and increasing Resource X.
Although this type of substitution ability may involve substantial practical barriers, costs, and coordination problems, it does provide one avenue for expanding the capacity limits of any resource. Although greater substitutability may not be practical or possible for this particular
example (fire and EMS), one is reminded that these concepts apply to any disaster response resources; it is highly likely that other, more readily accessible substitutions exist.
Substitution Limits in Quadrants 2 and 4
A Normal Need Domain can be said to exist on the phase plane which identifies the combination of resources most often applied. The size and shape of this Need Domain will vary depending on the resources in question and the specific environment in which they are applied. The specific character for any application could be identified from historical data.
In Figure 11 the Need Domain touches both the limit of the Y Resource and the limit of the X Resource. At these points the resource capacity is depleted (fully utilized). There are areas outside the Need Domain which provide excess capacity. At the top of the Y axis there is a domain were Y Resources are not normally used. Similarly, at the end of the X axis there are X Resources which normally used. Together, these additional resources provide the ability to expand further out into Quadrant 1 into the "Excess Capacity" domain.
Normal Need Domain on a Phase Plane
Although only a limited number phase plane trajectories have been presented here, it is clear that the myriad of agencies activated during a disaster is quite extensive and that phase plane images may be of limited value under those conditions. However, the real merit of these trajectories is to demonstrate the nonlinear character of disaster response and to provide a means for attaching interpretive and prescriptive information to the unfolding events.
There is, however, a nonlinear tool which may prove useful for combining several phase plane images into a readily accessible data matrix. The mutiquadrant matrix given in Figure 12 provides an example.
A Multiquadrant Matrix of Non-Critical, Critical, and Dead
During the Los Angeles Civil Disturbance of 1992
____________________________________________________________ Entity 1: LA EMS Agency Hospital Injury Vertical Horizontal Quadrant Matrix ____________________________________________________________ Non-crit / Total 11133313313233 Crit. / Total 41133212312232 Dead / Total 11133313213322 Time --------> Transition Codes . | .||:||.|. Observation Number 123456789*123456789*123456789*123456 ============================================================While phase plane trajectories can be generated for each interaction or combination of resources, a more convenient way to present the behavior of several phase planes over time is to produce a matrix of quadrant calls. The rows in such a matrix represent each of the several phase planes being examined while the columns represent the time evolution of the system. In this application each row represents one of three phase planes depicting non-critical, critical, and deceased patients during the Los Angeles Civil Disturbance of 1992. The columns represent time intervals of six hours for four days.
The multiquadrant matrix simplifies the behavior of each phase plane into a series of quadrant calls. This simplification allows one to depict in a single table the phase changes of several phase planes over time. One can examine the behavior of each phase plane by examining any selected row and can compare the changes in several phase planes at any given time by examining any column. Each matrix cell contains only the number of the quadrant visited. Recall that "1" indicates increased use of both measures, "2" indicates a decrease in the X Resource and an increase in the Y Resource, "3" indicates subsidence, and "4" indicates a increase in the use of the X Resource with a decrease in the Y Resource.
The consolidation of this information into a single matrix provides a compact way to depict the evolution of many dimensions of a system over time. For example, by referring to Figure 12, one can readily identify those time periods in which there was growth in all three of the phase plane trajectories; these are evident as columns of 1's in observation numbers 2 and 3. Similarly, examining any single row reveal the behavior of the resources in that phase plane.
Simplifying phase plane trajectories into a multiquadrant matrix does result in a loss of some information. Specifically, because only the quadrant number is provided, the matrix depicts only changes in direction; it does not indicate the magnitude of those changes. The phase plane trajectory can be produced or the original data may be examined to assess the magnitudes of the changes.
In a disaster situation, the need for timely information cannot be over emphasized. The advantage of using phase planes, and nonlinear methods in general, is the ability to process information and receive marginal analysis depicting changing conditions before they would otherwise become apparent. To effectively implement such a system one would need to acquire all the information at a common location, probably the disaster control center, in real-time mode. Each emergency call or incident report could be channeled automatically to a system that would provide control personnel immediate descriptions and interpretations of the evolving response. Prescription based on these automatic interpretation could be passed automatically to other disaster response agents or processed by personnel at the control center.
Much of the material presented here is based on analysis of actual disaster response data. The phase plane images, combined with well-established concepts of phase plane mapping and nonlinear system behavior, testify to the utility and merit of nonlinear systems for disaster response. As this line of inquire continues, this new science will undoubtedly supply other practical tools and techniques for practitioners.
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