Dynamics from Multivariable Longitudinal Data

We introduce a method of analysing longitudinal data in variables and a population of observations. Longitudinal data of each observation is exactly coded to an orbit in a two-dimensional state space . At each time, information of each observation is coded to a point , where is the physical condition of the observation and is an ordering of variables. Orbit of each observation in is described by a map that dynamically rearranges order of variables at each time step, eventually placing the most stable, least frequently changing variable to the left and the most frequently changing variable to the right. By this operation, we are able to extract dynamics from data and visualise the orbit of each observation. In addition, clustering of data in the stable variables is revealed. All possible paths that any observation can take in are given by a subshift of finite type (SFT). We discuss mathematical properties of the transition matrix associated to this SFT. Dynamics of the population is a nonautonomous multivalued map equivalent to a nonstationary SFT. We illustrate the method using a longitudinal data of a population of households from Agincourt, South Africa.


Introduction
Analysis of multivariable longitudinal data involves either statistical or nonstatistical methods. Statistical methods include multivariate Markov chain model [1], regression model [2], and mixed models [3], while some nonstatistical methods involve extraction of dynamical system using state space reconstruction technique [4] or visual methods such as motion charts [5,6] and parallel coordinate plots [7,8]. A motion chart shows additional dimensions of the data at different time points, where the size and color of the bubble (among others) are used as variables. PCP represents variables as parallel axes, where a sequence of line segments intersects each axis at a point corresponding to the observation's value at the associated variable. Both methods aim to identify correlation among variables and identification of clusters and patterns among observations in the data. We present a novel nonstatistical method of analysis that is useful particularly for collecting information of change. State space reconstruction method mainly requires use of delay coordinates from data to come up with models for prediction. Our method does not rely on delay coordinates, and our aim is not prediction. Contrasting to motion charts, there is, in principle, no limitation to the number of variables studied in our method. Contrasting to PCP, for large and large number of observations, orbits over our two-dimensional state space, or over time, are easily visualized.
Here we consider dynamics of multivariable longitudinal data of a population of observations by applying a swap operation on the data of each observation. We suppose that values of ≥ 1 variables are discretised to > 1 bins so that each variable takes value from = {0, 1, . . . , − 1}. For a fixed order of variables (e.g., V 0 , V 1 , . . . , V −1 ) denote by the space of all -ary sequences of length and element ∈ by = 0 1 ⋅ ⋅ ⋅ −1 such that is the value of V . The -ary multivariable time series in ≥ 1 variables of an observation is defined by where each ∈ , . Our method can be used for -ary valued data, but for illustration, we use = 2. We denote our binary multivariable longitudinal data of a population of Table 1: Unfavourable = 0/favourable = 1 coding of four households to questions related to biological mother (BM), household head (HH), and death (AD).
In the general analysis of longitudinal data we see no recognition that data is taken with a purpose. Here we suppose that the value of a binary variable is either favourable or unfavourable to a given purpose. Suppose we would like to investigate the effect of change of households variables, namely, biological mother (BM), household head (HH), and adult death (AD), to a child's educational progress. Consider = 3 binary questions 0 : Is the biological mother present? (BM) 1 : Is the household headed by a minor? (HH) 2 : Is there an adult death in the household? (AD) (4) and suppose that answers are coded either favourable = 1 or unfavourable = 0 to our purpose. It is a reasonable hypothesis that the answer "yes" to 0 is favourable to child education, while "yes" to 1 and 0 is unfavourable. Table 1 shows coded data of 4 subjects (households) over 7 time steps. Using parallel coordinate plots (PCP), Figure 1(a) shows the answer of each subject at = 0.
To illustrate the answer for all time using PCP, the evolution of line segments, with time, will form surfaces that obscure each other. Figure 1(b) shows the Bratelli diagram [9,10] of transition of answers from time to + 1, to variable order BM, HH, and AD. Transitions per time interval can be associated with a state transition matrix, where states are analysed on the space 3 = {000, 001, 010, 011, 100, 101, 110, 111} .
This can be used to generate probability matrices for Markov models [11,12]. Observe that, for question regarding 1 (HH), household ℎ1 has constant favourable answer, while ℎ3 has constant unfavourable answer. On the other hand, ℎ4 has constant favourable answer to 3 (AD). Our aim is to extract clusters associated with stable variables. Underlying our method is our belief that the set of physical variables in which an observation spends the most time in is important (e.g., HH = 1 for ℎ1, HH = 0 for ℎ3, and AD = 1 for ℎ4) and that among the physical variables themselves the variables that are most often experienced (most probable) by the observations are important. We elegantly expose both most probable variable and value of variable, by a simple process of dynamically reordering variables.
There is no a priori indication of any absolute dynamics in data and here it is deterministically imposed. Because longitudinal data is fundamentally defined by change (if nothing changes, cross-sectional data is sufficient), frequency of answer change of variables then becomes a property of interest. A deterministic operation is applied to the multivariable data of each observation at each time step, dynamically reordering position of variables (and their corresponding values) by their stability; that is, the most stable is eventually positioned to the left, and the most frequently changing one is positioned to the right. All possible orders of variables are considered. From this, we introduce the significance state (order of variables) and fitness state (associated values with variables) of an observation. It is in a chosen ordering that the notion of fitness takes an objective and consistent meaning. The idea of fitness and significance is new in literature. The -dimensional longitudinal data of an observation is represented as a 2-dimensional orbit in fitness-significance axes composed of 2 ! points. Orbits in sufficiently encode the longitudinal data of each observation. Analysis of orbits at the individual and population levels in this space can then follow. This paper is organized as follows. In Section 2, we present the method of constructing orbits from multivariable longitudinal data. A detailed theory of the reordering operation applied to data is presented. In Section 3, a deterministic equation of motion that generates all observed orbits, as well as other possible orbits, is presented. In Section 4 we discuss transitions that occur in data. This is captured by a nonstationary SFT. We also discuss dynamics at the population level. An illustration of the method is presented in Section 5. We give concluding remarks in Section 6.

Background and Preliminaries: Method of Orbits
We suppose that longitudinal data is gathered by first specifying a purpose and then choosing questions that are of interest to purpose. The questionnaire may be designed to a purpose posed in the form "to investigate the effect of the variables V 0 , V 2 , . . . , V −1 on . " Here we will only consider binaryvalued questions with responses hypothesized as either favourable or unfavourable to the purpose. A favourable answer is coded 1 and 0 otherwise. Our longitudinal data is the response to a set of ≥ 1 questions (associated with variables) surveyed from a population composed of ≥ 1 observations over periods.

Journal of Nonlinear Dynamics
the concatenation of answers to by and the concatenation of question indices of by Example 1. Coded data of observation for = 0 and = 1 is shown in column 2 of Table 2. Suppose we (arbitrarily) reorder the questions at = 0 to { 1 , 0 , 2 }. Then 0 = 010 and 0 = 102. Similarly, if 1 = { 2 , 0 , 1 }, we have 1 = 001 and 1 = 201. As we are merely rewriting entries from the original data, all information is preserved.

Fitness and Significance
States. Consider questions in (4) and assign index to ( = 0, 1, 2). Suppose we give more weight (significance) to 0 and do this by positioning 0 in the left-end of question order, say, 0 , 1 , 2 , denoted by 012 (or 0 , 2 , 1 = 021). As in numbers or decimals, our weighting places the most significant number at the left-end position. All possible (concatenated) answers to 012 are given by 001, 000, 010, 111, 110, 101, 100, 011. Since 000 has all unfavourable answers we say that it is the least fit answer, while 111 is the fittest. Note that there are states with the same number of favourable values, for example, 001, 100, and 010. By a suitable weighting of questions, we show below that the lexicographic ordering of answers in (10) is an appropriate ordering of fitness. Table 3 illustrates concatenated coded answers to question order 012 of three observations from a population. To 0 , observation has constant answer 0 while has constant answer 1.
Suppose we arrange answers in (10) lexicographically along an -axis. Then for question order 012, a onedimensional dynamics on the -axis composed of the eight states arises. Answers of and to 0 seldom change (i.e., they are both constant in 0 ) so the two households and stay in the regions 0 * * and 1 * * of the -axis, respectively. Recall that 0 is the question associated with the significant (left) position of the question order so fitness is biased towards the left position. We can then write = 0 * * < = 1 * * because the significant variable 0 is unfavourable in and favourable in . This holds true even if has the same, or more, favourable values as . This argument can be extended to any two elements , ∈ with the same first ℓ entries.
In general, not all observations may be stable in the same variable; for example, in Table 3 is constant in 0 , not in 2 . Moreover, stability of an observation may change in time; it may be stable in variable over one time interval and then stable in variable over another time interval. We will not study orbits in a fixed question order alone. We construct a -axis with states corresponding to question orders. The order of questions per observation becomes a new variable.

Definition 2.
Given ≥ 1 and , the fitness state and significance state of observation at time are the sequences respectively. The set of fitness states of length is called the fitness space of variables defined by and the set of significance states of length is called the significance space of variables defined by Elements of both and are arranged according to the lexicographic ordering (<) of sequences of length .
For convenience, we label states in from left to right, and from top to bottom. If = ( , ) and ( ) = , we will refer to state as state and write = ( , ).

Remark 4.
In general, for multivariate data in variables, with all variables -ary valued, the space , is composed of × ! states.

The Method of Orbits.
We define the dynamics of observations taken from a survey of ≥ 1 questions. Let N 0 be the set of nonnegative integers, let C = P( ) be the power set of , and let * be such that Definition 5. Let Δ ∈ C and let ∈ Δ.
(a) The map Δ : → is defined by Let , ∈ Δ. Then and both change values under Δ . If < , then is first applied to ; that is, Definition 6. Let ∈ . Consider questionnaire with ∈ . For each observation , let be the frequency of change in answer value of over the observation period. Suppose Inequality (20) is called the observation frequency relation and question order 0 = 0 1 ⋅ ⋅ ⋅ −1 is the initial significance state of observation . If = +1 and < +1 at the population level, then choose question order +1 . If = +1 , then choose question order as in the questionnaire (6). The initial fitness state 0 = 0 1 ⋅ ⋅ ⋅ −1 is such that is the value of in 0 . The initial state of is the ordered pair 0 = ( 0 , 0 ).
By choosing the initial significance 0 given in Definition 6, we start the orbit in its most-likely significance state. This facilitates convergence to clusters (where they exist) and is useful for short data sets. Other strategies (e.g., using order of (6) or random choice) will nonetheless converge to question order according to (20).

Remark 7.
Longitudinal data is only of interest where change occurs; else cross-sectional surveys are adequate. We are interested in longitudinal data that give nontrivial information of change about the population; that is, Otherwise, question may be deleted as any such property becomes an identifier of subpopulations of possible interest for analysis in its own right.
The set Δ ∈ C is given by the longitudinal data for each ≥ 0 and is a useful ordered listing of questions that change answer values from time to + 1. For each , the nonautonomous map [Δ ] defines an evolutionary process that displaces the most frequently (resp., slowly) changing answers and corresponding questions to the right (resp., left).

Definition 9.
Given initial state 0 of , define the state of at time ≥ 1 by The forward orbit of under is defined by

Algorithm for Building the Orbit of an Observation in .
We give a simple algorithm to determine the states >0 that comprise the orbit of an observation from longitudinal data.
Step 2. Identify from the data of the question that changes answer at time = 1, say . Swap both (in 0 ) and corresponding answer (in 0 ) to the right of 0 and 0 , respectively, and change to * , where * = 0 if = 1 and 0 otherwise. This new question order and answer order give the next state 1 = ( 1 , 1 ) of . Suppose both and change answers at = 1. If < , then sequentially swap to the right and (resp., and ) of the question order (resp., answer order), starting with (resp., ). Change to * and to * . Plot the point 1 in and directed edge from 0 to 1 .
Visualization and analysis of orbits of observations in allow capturing information of change in longitudinal data. We illustrate in Figure 2 an orbit in 3 . The useful distance on is given by the discrete metric; that is, ( , ) = 1 if ̸ = and zero otherwise. The visualized distance between points in has no interpretation so we may represent by a regularly spaced point.    Table 4) in 3 . Table 4: States of observation .

Unit
Coded answers to Remark 10. The set of all orderings of variables is captured in . For fixed question order and , ∈ , < means that fitness state is fitter than . Each level in the significance axis is question order under frequency ordering. The significance axis informs us which variables are weighted most strongly at each time, where significance is ordered from left to right. Clearly, reordering of variables is one among many families of operations; for example, swapping can be done by swapping changing variable to the left end. This operation however does not reveal clusters.

Transitions in .
We now analyze possible state transitions which an observation can take in .
Definition 11. Let = ( , ) and = ( , ) be in , and let → . If = and ̸ = , then there is a horizontal transition from to . If = and ̸ = , then there is a vertical transition from to .
We have self-transitions if Δ = 0 (no change), the empty set. In general, horizontal transitions denote change in  Table 1. A red edge denotes a transition to the left, a green edge is a transition to the right, and a blue edge is a self-transition. the right-most variable, while vertical transitions denote change in the last two variables.
Let , ∈ . Given an observation , if ∈ and +1 ∈ , then there is Δ ∈ C such that [Δ ] → . We use the symbol "∈" to denote that there may be other observations in or at times and + 1, respectively. State transitions of observation in are visualised as a sequence of directed edges.
Example 12. Consider data of observation in column 2 of Table 4. The asterisks denote changing answers in the next time step. The frequencies of change are 0 = 2, 1 = 3, and 2 = 1, so the initial question order of is 0 = 201. And initial fitness state is 0 = 010 (Definition 6). At = 0, we apply [Δ 0 ] to initial state 0 = (010, 201). Questions 1 = 0 and 2 = 1 in 0 change answers in the next time step so Δ 0 = {1, 2} (not {0, 1}). Applying Δ 0 to (010, 201) changes 2 = 0 and 1 = 1 to * 2 = 1 and * 1 = 0, respectively. Next, is first applied to both * 2 and 2 by moving each to the right (they are already rightmost), followed by moving * 1 and 1 to the right end. Hence, we have ( , ) 1 = (010, 210). For = 1 and = 2, verify that Δ 1 = {1} and Δ 2 = {0, 1, 2}, respectively. At each time the bold numbers in the significance column are the question indices in that change answers at + 1. The orbit of in 3 is shown in Figure 2. The vertical transition from state 11 to 3 denotes two changing answers, while a transition from state 9 to 32 denotes three changing answers. → is such that That is,̃[ Δ] first inserts −1 and −1 to the position, where and are located, followed by changing −1 to the new value * −1 . If , ∈ Δ and < theñ[ Δ] is first applied to and .
Trivially, for any ≥ 1, ∈ , and Δ ∈ C , respectively. For each Δ ∈ C , unique. Note that |C | = 2 and the set of images of under [Δ] , Δ ∈ C , is the set of distinct binary numbers of length , which can be associated with 2 distinct states in . This is a bijection between elements of C and the binary numbers of length , so |J | = 2 . ( is the return map. The proofs of the uniqueness of̃and the cardinality ofJ follow a similar argument as in (a).   The next theorem states some nonallowable transitions in .

Local Dynamics in . Consider the subset
where − answers are constant. Since |L | = !2 states and |S | = !2 states, then is composed of 2 − !/ ! subsets of the form L . For − 1 constant answers (i.e., = 1), denote by L ( ,ℓ) −1 the subset of , where question ℓ has constant answer . Then any ∈ L ( ,ℓ) −1 is given by Using (33), we can express as  Example 20. Figure 5 illustrates orbits in 4 that cluster into two. A red edge signifies a movement from right to left, a green edge a movement from left to right, and a blue edge a transition to the same state (self-transition). Instead of analysing in 4 , we can remove 3 and consider only the three questions 0 , 1 , and 2 and analysing orbits in 3 characterised by 3 = 0 and orbits characterised by 3 = 1. Note that the frequency change of 3 here is 3 = 0.

General Equation of Motion of Dynamics in
This section concerns properties of all possible paths in .
Definition 21 (see [13]). Let and be arbitrary sets. A multivalued map from to , denoted by : , is such that ( ) is assigned a set ⊂ for all ∈ . is such that +1 ∈ ( ). The multivalued map can be interpreted as a digraph whose = !2 vertices are the points in and edge → if ∈ ( ). Equivalently, can be defined as a square matrix of size that encodes all possible paths in . , gives all possible paths in that any observation can take.

Theorem 24. (a) The digraph G ( ℎ) is 2 -regular; that is, the in-and out-degree of all its vertices is
Proof. (a) This is a consequence of Theorem 15. (b) This is a consequence of (a) and the Perron-Frobenius theorem [14]. (43) Observe that Since and have the same image for all Δ ∈ C 2 , J = J , as claimed. Our claim implies that rows of (th) 2 associated with and are the same. Because of these repeating rows, det ( (th) 2 ) = 0. The proof can be readily extended to the general case and is treated in exactly the same manner.

Subshift of Finite
Type. Let , ∈ = {1, 2, . . . , }. If observation is in state ∈ after time steps, we will write ∈ to indicate that more than one observation may be in . We associate with the orbit O ( ) a sequence of symbols Journal of Nonlinear Dynamics Denote the corresponding symbol space of one-sided sequences of (th) by The equation of motion in is given by the shift map Σ (th) : Definition 25 (see [15]). The pair ( (th) , Σ + (th) ) is called the subshift of finite type (SFT) determined by (th) .
The SFT determined by (th) captures the exact detail of all possible itineraries of observations in . Many dynamical properties of an SFT depend on the structure of its associated transition matrix or digraph.
Definition 26 (see [16]). Consider the SFT given by ( , Σ + ). The following results [17,18] concern dynamical properties of a transitive SFT and algebraic properties of its transition matrix. (i) The shift map Σ is transitive (resp., mixing) if and only if is irreducible (resp., primitive). (ii) is irreducible if and only if the digraph = ( , ) is strongly connected. In that case we say that is irreducible. (iii) A nonnegative, irreducible matrix with a positive element on the main diagonal is primitive.
Proof. We prove by induction on . From Example 23, (th) 1, = 1 for all ( , ) so (th) is irreducible for = 1. Assume that (th) is irreducible for = . Then the digraph (th) is strongly connected, and is irreducible. We prove that (th) is strongly connected to = + 1.
We prove (i) by showing that there is ∈ L ( , ) such that, for all ̸ = , there is a transition from to ∈ L ( , ) . By . . .
Observe that all 's have the same fitness states but distinct significance states. In particular, each is contained in a distinct subset L Under Δ = +1 , there is a reversible transition ←→ . From (i), (ii), and the irreducibility of (th) , there is a path between any pair , ∈ S +1 . Hence, (th) +1 is strongly connected and (th) +1 is irreducible, as desired.
A continuous map on a compact metric space is chaotic if ℎ top ( ) > 0 [17]. The topological entropy for SFTs is given by the following theorem [15].

Dynamics from Data
Given multivariable longitudinal binary data of dimension , every observed orbit is an orbit of an SFT determined by (th) . Dynamics of real-world longitudinal data however is not often defined by an SFT. Data usually selects certain paths given by (th) and may sometimes stay in a particular subspace of . , the matrix that records observed transitions that occur in the longitudinal data from time to + 1, where We note that (data) , is defined over an interval of time and that (data) , may vary with time. Some allowable transitions between states given by (th) might not occur in the observed data so (data) , ̸ = (th) . For the case where (data) , is constant we write (data) , ≡ (data) and we can define the SFT given by the pair ( (data) , Σ + (data) ).
is not constant, then we have a sequence of matrices Given , define We call Σ (data) the nonstationary symbolic space restricted by the sequence of matrices . The shift takes place as usual in Σ (data) and is denoted by : Σ (data) → Σ (data) . We refer the reader to [19][20][21] for a discussion on nonstationary SFT.
Visualization of dynamics of longitudinal data defined by an NSFT is illustrated by a sequence of directed graphs called a Bratteli diagram [9,10]. Equivalently we may plot orbits of observations in over time. Figure 6(a) illustrates the orbits of a population in a subset of 3 over time.

Population Dynamics.
We discuss the longitudinal data of a population of > 1 observations. Aside from all possible paths that an observation can take in , we are also interested in the number of observations on paths. Because it is possible for more than one observation to occupy a state in at a given time, we can consider the number of observations that follow the same transition in .
In general, given a transition matrix (e.g., those in (40) or (52)), standard analysis is to accumulate number of transitions between states, and from this construct the associated stochastic transition matrix. Construction of associated transition and stochastic matrices on states of can then follow as usual. In what follows, let denote a finite index set with | | ≥ 2 and , ∈ .
Definition 32. Let ( ) be the number of observations in state at time . The density matrix at time is defined by where , is the number of observations in state at time that go to state at time + 1. The (net) flux of state at time is defined by Let ℓ , = , − , , , ∈ . The flux in (56) can also be expressed as From (56) and (57), we have Let be a row vector whose th entry is ( ). We will refer to as the observed capacity vector at time . Given the initial observed capacity vector 0 , there are two methods that we can use to determine +1 .
(i) Nonhomogenous Case. From data, we are encouraged to construct a probability matrix based directly on the density matrices . For each , we construct a time-dependent probability matrix = ( , ), where The capacity vector +1 is given by the product We show that the capacity +1 ( ) in (56) agrees with the th entry of (60). It is trivial if ∑ , = 0 for all . Otherwise we have The capacity vector at + 1 is given by the product The probabilities in (64) are based on observed frequencies at which one state follows another. The transition probabilities of a Markov model using states in are then estimated. Markov modeling on the bigger space may give better results in generating data as is a more detailed space than a space where order of variable is fixed (e.g., ).
Definition 33. A set of observations cluster in if for all ∈ , there exists ⊂ and time interval such that the orbit ( ) ⊂ for all ∈ .
An advantage in using the bigger space to analyse multivariable longitudinal data is that all possible clusterings in stable variables are revealed. We can also visualize in the shift in clusters from one region to another region . There is much interesting information that we can gather from a cluster in a subset of . Clustering in L ( ̸ = ) defined in (32) involves observations with the same stable − variables. Subsets L are important as they automatically determine which variables are trivial in the analysis. We either reduce analysis from to L or replace the − constant variables with other nonconstant variables that are of interest to purpose. In addition, orbits that spend most (if not all) of the time in have strong correlation among the − variables.

Application
We illustrate our method using the = 3 demographic binary questions taken from Agincourt [22]. With purpose P: to investigate the effect of household change on a child's educational progress, the three questions in (4) are associated with household change and are all of interest to P. Table 5 gives the question (indexed by 0, 1, and 2, resp.) with associated variables (BM, HH, and AD), favourable/unfavourable coding, and frequency of change.
The longitudinal data consists of = 188 observations with no missing data. Each household is surveyed each year, for nine years [22], and the population is constant throughout the observation period; that is, no households enter nor leave the population. Using (20), the frequency relation in the population is 1 ≪ 2 ≪ 0 . The number of unfavourable = 0 answers (over 1692 answers) is 0 : 844(49.9%), 1 : 2(0.1%), and 2 : 71(4.2%). We expect that this, together with the very small value of 1 , will give clustering in the question order 1 * * , where 1 = 1. and are characterised by the favourable property of being headed by an adult (i.e., 1 = 1). Of the 179 households that stay inL, 159 distinct orbits are found. However, we see from Figure 6(b) that the number of state visits in 3 is dominant in states 23 = (110, 120) and 24 = (111, 120) so dynamics principally takes place in these two states.
State 23 is characterised by the most stable variable HH = 1, followed by AD = 1 and then by BM = 0. Similarly, state 24 is characterised by the most stable variable HH = 1, followed by AD = 1 and then by BM = 1. Orbits that stay strictly in states 23 and 24 are associated with households headed by an adult and with no adult death during the survey period. Those idle in 23 (resp., 24) have mothers away (resp., present) during the survey period. Households that transition from 23 to 24 are characterized by mother out-to in-migration, while transitions from 24 to 23 denote in-to out-migration.
With regard to purpose, we define educational default by the number of years that a child has failed in his school life. A household with a child who has failed 4 years or more in his school life is classified as defaulting. Orbits in Figure 6 Note that if we include a fourth question Q 3 : is the household defaulting (=0) or nondefaulting (=1)?
then orbits will be plotted in 4 , as seen in Figure 5. Orbits that cluster on the left are the defaulting households ( 4 = 0), while those on the right are nondefaulting ( 4 = 1). Figure 8 gives ( ), the number of observed defaulting and nondefaulting households at each state = 23, 24 from = 1999 to = 2007. It is clear that, for both subpopulations, there is an exchange of numbers between states 23 (BM-out) and 24 (BM-in). Figures 9(a) and 9(b) illustrate the orbits of defaulting and nondefaulting households over time, and the accumulated number of visits in 4 , respectively. Table 6 gives the (accumulated) number of state transitions → , , ∈ {23, 24} for both subpopulations. Using odds ratio, we have the value 1.59 (>1) for transition 23 → 23, which means that once mothers from defaulting households out-migrate, they take long to return to their households. Out-migrating mothers are then more likely to be away from home in the defaulting households. For 24 → 24 we have odds ratio of 0.70 (<1) indicating that in-migrating mothers are more likely to stay at home in the nondefaulting households. Both odds ratios have significant values and these support our hypothesis that mother outmigration places child educational progression at risk. We have no strong evidence that mother out-migration (24 → 23) is more likely in the defaulting households nor that in-migration (23 → 24) is more likely in nondefaulting households.

Concluding Remarks
By defining a specific swap operation on multivariable longitudinal data, we have imposed dynamics on data. No approximations are made and orbit of an observation in the 2-dimensional space can be decoded back to the observation's original data. By including the order of variables in our analysis, we have introduced new states, along with an objective criterion of relative fitness and identification of subshift. We have only considered binaryvalued variables but the method may be generalized to -ary valued multivariable longitudinal data. Although the method of orbits is nonstatistical, it can however be used to aid in statistical analysis. The advantage in plotting orbits in is that information of change can be extracted directly from the visualized orbits. In addition, orbits in facilitate the identification of cause and effect, of preceding events that might uniquely and usefully associate with change. Frequency of change is a property taken directly from the data so that, under our reordering operation, any practitioner with the same favourable and unfavourable coding will find the same ordering of questions and arrive at the same analysis of longitudinal data.