Draft:Time Series Trajectory Analysis

Introduction

Time Series Trajectory Analysis (TSTA) is a quantitative method used in land change science that quantifies and visualizes the trajectories of a binary variable during a time series. TSTA builds on existing land change analysis methods, particularly those developed by Robert Gil Pontius Jr and is presented as a method and analytic tool in Bilintoh, 2023. TSTA was first developed to better analyze trajectories in project data from Bahia, Brazil and Long Term Ecological Research sites in the United States (Figure 1). These projects received funding from NASA and the NSF, respectively.

Many existing methods of land change analysis focus on net change between two time points, while TSTA addresses the need in Geographic Information Science to capture the complexity of land change over multiple time points. By examining the land change of a time series over several time intervals, trends, and cyclical patterns can be identified. These patterns might be essential for understanding land degradation vegetation change or urban expansion.

Methods

Figure 1: Example Output of Time Series Trajectory Analysis for three study areas (From Bilintoh, 2023)

Figure 2: Example of stack bar plot outputs of Time Series Trajectory Analysis (From Bilintoh, 2023). This case only contains two time intervals, and thus only contains six of the trajectories. Segments above the time axis are gains, segments below the time axis are losses.

Time Series Trajectory Analysis provides a method to analyze a series of Boolean images and classify each pixel as having one of eight possible trajectories. Taken together these trajectories can describe key insights into trends of persistence and absence of a variable in an area. These trajectories include gain, loss and alternation. If both gain and loss occur during the time period of study, this is alternation.

TSTA is able to compare among sites and across inconsistent time intervals by expressing results as an annual percentage of each site’s unified size. A site’s unified size is the union of where the category exists at any of the site’s time points.

Key to the TSTA methods are visualizations of trajectory data. One visual output is a map showing the spatial distribution of different trajectories, as shown in Figure 1. Another, unique to TSTA, is stack bar plots, as shown in Figure 2. Stack bar plots show gain, loss, and stable trajectories, with or without alteration, over any degree of temporal resolution. Map e) in Figure 2 displays very different trajectories between the first and second intervals. This is an indicator of possible map errors/inconsistencies in data collection.

Example Analysis

Figure 3: Example data to show eight possible trajectories (from Bilintoh 2023).

Trajectory analysis identifies 8 different possible trajectories for a dataset with exactly two categories, typically absence and presence. Figure 3 shows a simplified data set of 12 pixels over 5 time points. Figure 4 names and defines the eight trajectories. TSTA also records 3 components of change.

The trajectories identified are:

(1) Loss without alternation: A pixel shows presence at the first time point and absence at the last and it only experiences change exactly once. Examples for this are labeled as Trajectory 1 and dark red in Figure 3.

(2) Gain without alternation: A pixel shows absence at the first time point and presence at the last and it only experiences change exactly once. Examples for this are labeled as trajectory 2 and dark blue in Figure 3.

(3) Loss with alternation: A pixel shows presence at the first time point and absence at the last. As a result, the pixel experiences change more than once and an uneven amount of times as the total net result changes (i.e. pixel changes at time point 2, 3, and 4 as seen in Figure 3). Examples for this are labeled as trajectory 3 and light red in Figure 3.

(4) Gain with alternation: A pixel shows absence at the first time point and presence at the last. As a result, the pixel experiences change more than once and an uneven number of times as the total net result changes (i.e. pixel changes at time point 1, 2, and 3 as seen in Figure 3). Examples of this are labeled as trajectory 4 and light blue in Figure 3.

Figure 4: Trajectory definitions (from Bilintoh 2023).

(5) All alternation loss first: A pixel of presence changes to absence and back with no net change between the first and the last time points. The pixel therefore experiences change at least twice and an even number of times at other timepoints (i.e. pixel changes at time point 2 and 3 as seen in Figure 3). Examples for this are labeled as trajectory 5 and dark yellow in Figure 3.

Figure 5: Bar graph of the trajectories and lines of gross gain and loss for a four-interval time series with unequal time intervals (from Bilintoh 2023).

(6) All alternation gain first: A pixel of absence changes to presence and back with no net change between the first and the last time points. The pixel therefore experiences change at least twice and an even number of times at other timepoints (i.e. pixel changes at time points 1, 2, 3, 4, as seen in Figure 3). Examples for this are labeled as trajectory 6 and light yellow in Figure 3.

(7) Stable Presence: Pixels of presence which never change. Examples for this are labeled as trajectory 7 and dark grey in Figure 3.

(8) Stable Absence: Pixels of absence which never change. Examples for this are labeled as trajectory 8 and light grey in Figure 3.

The components of change are:

Alternation: Pixels that experience at least one loss from a category and at least one gain to the same category. Pixels that change at least twice during the time series. It is computed using trajectories 5 and 6.

Quantity: The absolute net change in presence and absence between the beginning and end of the time series.

Exchange: The maximum value of an equal number of simultaneous gains and losses between the beginning and end of the time series. It is computed using trajectories 1, 2, 3, and 4.

Figure 2 displays stacked bar plots of these components of change in maps b), d), and f).

Figure 5 shows the actual trajectory analysis. The graphic shows the number of pixels for each trajectory (that can be identified by its color, as explained above) and time interval as a percentage of the annual change of the observed region. The dotted line above 0 shows the average annual gross gain observed over the total time period (in short just gross gain). The dotted line below 0 shows the average annual gross loss observed over the total time period (in short just gross loss).

Comparison with Existing Land Change Analysis Approaches

Trajectory Analysis is one of many methods to describe a time series. It is non-parametric and univariate. It can be used with unevenly spaced time series. While the examples presented here are related to Land Use Change, Trajectory Analysis can be used to describe any type of time series. The eight trajectories used in the analysis require a different number of time intervals in order to be produced:

With at least two time intervals, a component of Alternation can be described, and the trajectories of All Alternation Loss First and All Alternation Gain First can be described.

With at least three time intervals, Gain with Alternation and Loss with Alternation can be described. Thus, three time intervals or greater are needed in order to utilize all eight trajectories.

TSTA can be applied at different temporal resolutions and different spatial resolutions of a time series.

Trajectory Analysis can be used as a stepping-off point to deeper analysis. It is a package of trajectories and components that researchers can use for deeper understanding. It can be used to analyze different temporal or spatial resolutions of the same extent. A researcher may detect autocorrelation of the alternation trajectories for example, or an exponential trend in the gain trajectories.

Possible Applications

Trajectory Analysis was originally developed in the context of Remote sensing and Land Change Science. However, it can be used in a wide variety of applications involving a dataset with a binary variable of Presence of Absence.

Land Cover Change Detection: Helps to monitor changes in land cover over time. By analyzing the trajectories of specific land cover types, researchers can identify patterns of deforestation, urbanization and other land-use changes.

Politics: Trajectories of presence or absence of a political party's seats in a governing body.

Vegetation Dynamics: Monitoring the trajectories of vegetation derived from remote sensing data enables the assessment of ecosystem health, detecting disturbances, and understanding the impact of climate change of vegetation patterns.

Urban Growth and Expansion: Tracking trajectories of urban areas provides insights into urban growth and expansion. This is important for urban planning, resource management, and understanding the environmental consequences of rapid urbanization.

Natural disaster Assessment and Response: Can be used to understand patterns of severity and frequency in natural disasters such as floods, wildfires, and earthquakes. Monitoring the trajectory of changes before, during, and after a disaster helps in emergency response planning and post-disaster recovery efforts.

Medicine: Trajectories of presence or absence of cancerous tissue samples in a patient.

Land Management and Planning: Providing data-driven insights into the historical and current trends in land use. The information is valuable for making informed decisions about resource allocation, conservation efforts, and sustainable development.

R Package

Bilintoh has made available a package in the R programming language that can be adapted to perform time series trajectory analysis on any dataset of boolean images. It is available on GitHub.