Chapter 3
Describing and Representing Trajectories
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1Samples of individual trajectories are often highly heterogeneous, making the task of describing all of them difficult, if not impossible. A typology is a simplified representation of a sample, which makes it easier to describe. However, given that each cluster in a typology itself contains some heterogeneity, the challenge of characterizing and presenting them in a simple and precise fashion is not a trivial one (see, for example, Halpin & Chan, 1998).1
I. Graphical Representations
2Looking at a graphical representation of a typology often offers a quick and relatively intuitive way to interpret a set of results. There are two main types of graphical representations that provide an overall view of the clusters in a typology of trajectories: state distribution plots and index plots.
1. State Distribution Plots
3State distribution plots consist of a succession of cross-sections, which show how the individuals in a cluster are distributed across the different states at each time step. In other words, at each moment in the trajectories, the graph presents the proportion of individuals in each of the situations.2 As an illustration, we will construct the trajectories of the school-to-work transition of 500 individuals, a sample chosen at random among the respondents to the Biographies et entourage survey (which is presented in Chapter 4). This trajectory is observed annually between the ages of 18 and 29 (inclusive), and at each observation, each individual in the sample is in one of the following situations: student, unemployed or inactive, or employed. The graph clearly shows that at age 18, individuals are divided approximately equally between school and employment (Figure 2). The proportion of students drops rapidly up to age 25 and then falls more slowly to near zero by age 29. At the end of the trajectory, the proportion who are unemployed or inactive—which increases mainly after age 23—reaches almost 20%, while almost 80% are employed.
Figure 2. State Distribution Plot of Activity Status, Ages 18–29

Coverage: Random sample of 500 individuals.
Source: Biographies et entourage survey (INED, 2001).
4The main advantage of this kind of graphical representation is the ease with which it can be read and used to compare clusters. However, by presenting a succession of separate cross-sections, state distribution plots obscure the individual character of the trajectories. They do not show us the series of situations that make up the life course of each person. For example, they do not give us any picture of what proportion of the individuals who are unemployed at the end of the trajectory were previously employed or students. Transitions—which are foundational to processes—are not represented.
5A variant of state distribution plots is modal sequence plots, which represent the sequence of modal states in each cluster (see Gabadinho, Ritschard, Müller et al., 2011). For each age, a bar is drawn in the color of the modal state, its height representing the frequency of that state at that age in the cluster.
2. Index Plots
6Unlike the state distribution plot, the second type of graphical representation—the index plot, proposed by Stefani Scherer (2001)—preserves the individual character of trajectories. Here again, the x-axis represents time. The trajectory of each individual is plotted on a single horizontal line as a succession of colored segments, each representing a period spent in one of the possible situations. The color, position, and length of each segment reflect the nature, timing, and duration of the situation, respectively. Figure 3 presents the trajectories of our sample of individuals in an index plot. Although it is not so easy to make out, we can see that most of the individuals who are unemployed or inactive at the end of the trajectory were previously employed. We can also see that the vast majority of individuals transition directly from education to employment without passing through a period of unemployment or inactivity. Finally, a significant proportion of individuals have a perfectly stable trajectory, remaining in employment from the start of the period to the end.
Figure 3. Index Plot of Activity Status, Ages 18–29

Coverage: Random sample of 500 individuals.
Source: Biographies et entourage survey (INED, 2001).
7Here, we can clearly see what makes index plots helpful. They retain the longitudinal aspect of individual trajectories and, thus, give us a better picture of the processes at work. But they are harder to interpret than state distribution plots—particularly, with a large number of trajectories. To make index plots more readable, it can be useful to sort trajectories by the situation at the start, at the end, or better, according to the first dimension in a multidimensional scaling (MDS) of the dissimilarities between trajectories (Piccarreta & Lior, 2010), as here.
8Piccarreta (2012) proposes smoothed MDS sequence plots, a way of making index plots easier to read by adding a step. After sequences are sorted by the first dimension in a multidimensional scaling, each one is replaced by the medoid of its neighbors.3 Fasang and Liao (2014) set out a similar approach: relative frequency sequence plots. Here, the trajectories are first sorted before being divided into K groups, with each group summarized by its medoid.
9Another option is to represent only the most frequent trajectories, potentially with thickness proportional to their frequency (sequence frequency plots: see Müller et al., 2008). Gabadinho, Ritschard, Studer et al. (2011) generalize this approach by selecting a small number of “representative” sequences. Their degree of representativeness can be calculated using different criteria (sequence frequency, neighborhood density, mean frequency of states, centrality, sequence likelihood).4
II. Composite Indicators
10Another way of describing the clusters in a typology of trajectories is to calculate some number of indicators for each cluster. This can clearly reveal significant differences, which may raise questions about the validity of the typology. Aside from the size of the cluster (the number of sequences it includes), several other types of indicators can be used.
1. Composite Indicators to Describe Trajectories
11Clusters can be characterized by calculating the mean (and, more generally, the distribution) of the values of indicators that describe the trajectories they contain: for example, the amount of time spent in the different states and the number of episodes in them, or “turbulence.” A more extensive list of these indicators is presented in Chapter 2.
2. Homogeneity of Clusters
12Clustering procedures are aimed at producing categories of sequences that are at once internally homogeneous and distinct from each other. But both internal homogeneity and interclass heterogeneity vary between clusters. These can be measured using a range of indicators. Using the matrix of distances between trajectories (calculated through optimal matching or another dissimilarity measure), we can calculate the average distance between the trajectories in a given cluster (intracluster distance), the average distance between the trajectories in a given cluster and those in other clusters (intercluster distance), and the maximum or mean distance to a typical trajectory that characterizes the cluster (Aassve et al., 2007). There are other indicators of homogeneity, such as age-specific entropy (Fussell, 2005) and the Gini index.
III. Typical Trajectories
13Characterizing the clusters in a typology can also respond to the question of what trajectories are typical of a given cluster. In this case, we seek to reduce each cluster to one or more particularly representative trajectories. In theory, we might be able to identify the most common individual trajectory in the cluster. In general, though, while the trajectories in a cluster may be similar, they are more often complex and too diverse to identify a single trajectory that is significantly more common on its own.
14Another way of identifying a typical trajectory is to construct an average trajectory, consisting of the sequence of modal situations at each time step. But this “average” trajectory will not necessarily correspond to any actually observed trajectory. This approach can even produce nonsensical results, particularly in cases where some transitions are irreversible. Taking a study of parental trajectories as an example, in such an average trajectory, the state “has had at least one child” might precede a situation “never had children.”
15A better solution is to use a real, observed trajectory rather than an artificially constructed one. Taking the medoid of the cluster is a good way to do this. This is the trajectory in a cluster that is closest to the cluster’s center of gravity—in other words, the one with the smallest mean distance from the other trajectories in the cluster. It is, thus, a trajectory experienced by an actual individual. This is a particularly good way of illustrating the character of a cluster as a whole—“embodying” it—and is sometimes used in research that uses typologies of trajectories (Aassve et al., 2007; Wiggins et al., 2007). But when a cluster is too heterogeneous, it can be better to present a set of “typical” trajectories rather than just one, to represent the diversity of pathways included in a cluster.
IV. We Have a Typology. Now What?
16After constructing and interpreting our typology of trajectories, it is generally interesting to look at the relationship between the typology and variables that were not used to code the trajectories. Using a categorical variable representing the cluster that each sequence has been assigned to, we can conduct classical bivariate or multivariate analyses.
17If we analyze the relationship between the typology and a categorical variable (e.g., type of employment trajectory by gender), we might use cross tabulation, overall measures of association (e.g., Cramér’s V), or local measures of association (phi, Pearson residuals or Percentages of Maximum Deviation from Independence [Cibois, 1993], to measure the degree of mutual attraction or repulsion between cells in the table), mosaic plots (see Friendly & Meyer, 2015; Le Guen, 2003), etc.
18If we want to analyze the relationship between the typology and a continuous variable (e.g., type of employment trajectory by year of birth), we may want to compare distributions (e.g., central tendency, dispersion), use an overall measure of association (eta squared) or a local one (point biserial correlation—i.e., Pearson’s correlation for each category in the typology coded as a binary variable), boxplots, etc.
19If we are investigating how the typology is related to multiple variables, it can be used as an independent or supplementary variable in a correspondence analysis, or in a regression as a dependent variable (what factors determine whether individuals have one type of trajectory rather than another?) or an explanatory variable (does the type of trajectory contribute to explaining some other phenomenon?).
Notes de bas de page
1 The graphical representations and indicators presented in this chapter are also useful even before we begin constructing a typology because they offer some broad outlines for the description of the trajectories found in the population as a whole.
2 The same proportions can also be represented in other ways: in unstacked form, for example, with bar charts (histograms) or with simple curves.
3As a reminder, the medoid is the trajectory with the lowest mean distance with respect to the others in a set.
4 All graphical representations in this section are available in the R packages TraMineR or TraMineRextras.
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