We introduce a method of analysing longitudinal data in n≥1 variables and a population of K≥1 observations. Longitudinal data of each observation is exactly coded to an orbit in a two-dimensional state space Sn. At each time, information of each observation is coded to a point (x,y)∈Sn, where x is the physical condition of the observation and y is an ordering of variables. Orbit of each observation in Sn 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 Sn 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.
1. 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 n 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 n≥1 variables are discretised to m>1 bins so that each variable takes value from V={0,1,…,m-1}. For a fixed order of variables (e.g., v0,v1,…,vn-1) denote by
(1)Mn,m=Vn
the space of all m-ary sequences of length n and element x∈M by x=x0x1⋯xn-1 such that xi is the value of vi. The m-ary multivariable time series in n≥1 variables of an observation k is defined by
(2)Xk={X0k,X2k,…,XTk},
where each Xtk∈Mn,m. Our method can be used for m-ary valued data, but for illustration, we use m=2. We denote our binary multivariable longitudinal data D of a population of K observations by the set of binary multivariable time series of observations
(3)D={X1,X2,…,XK}.
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 n=3 binary questions
(4)Q0:Is the biological mother present?(BM)Q1: Is the household headed by a minor?(HH)Q2: Is there an adult death in the household?(AD)
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 Q0 is favourable to child education, while ‘‘yes’’ to Q1 and Q0 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 t=0. To illustrate the answer for all time t 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 t to t+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
(5)M3={000,001,010,011,100,101,110,111}.
This can be used to generate probability matrices for Markov models [11, 12].
Unfavourable = 0/favourable = 1 coding of four households to questions related to biological mother (BM), household head (HH), and death (AD).
Time
h1
h2
h3
h4
BM, HH, AD
BM, HH, AD
BM, HH, AD
BM, HH, AD
0
0,1,1
0,1,1
0,0,1
0,1,1
1
1,1,0
0,0,1
0,0,1
1,0,1
2
0,1,1
1,1,1
1,0,1
1,1,1
3
0,1,1
1,0,0
0,0,0
0,0,1
4
0,1,1
0,1,0
0,0,1
0,1,1
5
0,1,1
0,1,0
0,0,1
1,0,1
6
1,1,1
0,0,0
1,0,1
0,0,1
Visual display of data of four subjects in Table 1 (a) at t=0 using PCP, where each parallel line is composed of answers 0 and 1 and (b) for all time t=0,1,…,6 using a Bratelli diagram, where each parallel line is composed of states from M3={0,1}3, the space of binary sequences of length 3.
Observe that, for question regarding Q1 (HH), household h1 has constant favourable answer, while h3 has constant unfavourable answer. On the other hand, h4 has constant favourable answer to Q3 (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 n-dimensional longitudinal data of an observation is represented as a 2-dimensional orbit in fitness-significance axes Sn composed of 2nn! points. Orbits in Sn 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.
2. 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 n variables v0,v2,…,vn-1 on Z.” Here we will only consider binary-valued 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 n≥1 questions (associated with variables) surveyed from a population P composed of K≥1 observations over T periods.
Denote the questionnaire by
(6)Q={Q0,Q1,…,Qn-1},
the set containing n questions. Let In={0,1,…,n-1} be an index set of n elements, and let time t=0,1,…,T. Let j, ij∈In and let xj∈{0,1}. For each observation k=1,2,…,K, denote a reordering of questions in Q at time t by
(7)Qtk={Qi0tk,Qi1tk,…,Qi(n-1)tk}:={Qi0,Qi1,…,Qin-1}tk,
the concatenation of answers to Qtk by
(8)xtk=x0tkx1tk⋯x(n-1)tk=(xjtk)j=0n-1:=(x0x1⋯x(n-1))tk,
and the concatenation of question indices of Qtk by
(9)ytk=i0tki1tk⋯i(n-1)tk=(ijtk)j=0n-1:=(i0i1⋯i(n-1))tk.
Example 1.
Coded data of observation k for t=0 and t=1 is shown in column 2 of Table 2. Suppose we (arbitrarily) reorder the questions at t=0 to {Q1,Q0,Q2}. Then x0k=010 and y0k=102. Similarly, if Q1k={Q2,Q0,Q1}, we have x1k=001 and y1k=201. As we are merely rewriting entries from the original data, all information is preserved.
Data of observation k (column 2), with questions reordered at t=0 and t=1.
Observation k
Coded answers to Q={Q0,Q1,Q2}
Concatenation of answers to Qtk
Concatenation of indices of Qtk
t=0
{0,1,0}
100
102
1
{0,0,1}
010
021
2.1. Fitness and Significance States
Consider questions in (4) and assign index i to Qi (i=0,1,2). Suppose we give more weight (significance) to Q0 and do this by positioning 0 in the left-end of question order, say, Q0, Q1, Q2, denoted by 012 (or Q0,Q2,Q1=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
(10)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 Q0, observation k has constant answer 0 while k′ has constant answer 1.
Concatenated coded answers of observations to question order Q0,Q1,Q2=(012).
t
k
k′
k′′
0
010
100
111
1
001
110
001
2
000
101
001
3
010
110
111
4
000
100
001
Suppose we arrange answers in (10) lexicographically along an x-axis. Then for question order 012, a one-dimensional dynamics on the x-axis composed of the eight states arises. Answers of k and k′ to Q0 seldom change (i.e., they are both constant in Q0) so the two households k and k′stay in the regions 0** and 1** of the x-axis, respectively. Recall that Q0 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 x=0**<x′=1** because the significant variable Q0 is unfavourable in x and favourable in x′. This holds true even if x has the same, or more, favourable values as x′. This argument can be extended to any two elements x,x′∈Mn with the same first ℓ entries.
In general, not all observations may be stable in the same variable; for example, k′′ in Table 3 is constant in Q0, not in Q2. Moreover, stability of an observation may change in time; it may be stable in variable i over one time interval and then stable in variable i′ over another time interval. We will not study orbits in a fixed question order alone. We construct a y-axis with states corresponding to question orders. The order of questions per observation becomes a new variable.
Definition 2.
Given n≥1 and Qkt, the fitness state and significance state of observation k at time t are the sequences
(11)xtk=(xjt)j=0n-1,ytk=(ijt)j=0n-1,
respectively. The set of fitness states of length n is called the fitness space of n variables defined by
(12)Xn:={(xj)j=0n-1:xj∈{0,1}}
and the set of significance states of length n is called the significance space of n variables defined by
(13)Yn:={(ij)j=0n-1∶ij∈In,ij′s
distinct}.
Elements of both Xn and Yn are arranged according to the lexicographic ordering (<) of sequences of length n.
Definition 3.
The space
(14)Sn={p=(x,y):x∈Xn,y∈Yn}=Xn×Yn
is called the change space for n variables.
Given n, we have the cardinalities |Xn|=2n, |Yn|=n!, and |Sn|=N=2nn!. A way of labeling state sj∈Sn is via the map
(15)ψ:Sn⟶{1,2,…,N},sj⟼ψ(sj)=j,ghffghffgjhlkj=1,2,…,N.
For convenience, we label states in Sn from left to right, and from top to bottom. If sj=(x,y) and ψ(sj)=j, we will refer to state sj as state j and write j=(x,y).
Remark 4.
In general, for multivariate data in n variables, with all variables m-ary valued, the space Sm,n is composed of mn×n! states.
2.2. The Method of Orbits
We define the dynamics of observations taken from a survey of n≥1 questions. Let ℕ0 be the set of nonnegative integers, let 𝒞n=𝒫(In) be the power set of In, and let xj* be such that
(16)xj*={1ifxj=00ifxj=1.
Definition 5.
Let Δ∈𝒞n and let j∈Δ.
The map ϕΔ:Sn→Sn is defined by
(17)ϕΔ(x0x1⋯xj⋯xn-1,y)=(x0x1⋯xj*⋯xn-1,y).
The map ϕJ:Sn→Sn is defined by
(18)ϕJ(x0x1⋯xj⋯xn-1,i0i1⋯ij⋯in-1)=(x0x1⋯xj-1xj+1⋯xn-1xj,i0i1⋯ij-1ij+1⋯in-1ij).
Let j,j′∈Δ. Then xj and xj′ both change values under ϕΔ. If j<j′, then ϕJ is first applied to j′; that is,
(19)ϕJ(x0x1⋯xj⋯xj′⋯xn-1,i0i1⋯ij⋯ij′⋯in-1)=(x0x1⋯xj-1xj+1⋯xj′-1xj′+1⋯xn-1xj′xj,wwwwwi0i1⋯ij-1ij+1⋯ij′-1ij′+1⋯in-1ij′ij).Definition 6.
Let ij∈In. Consider questionnaire Q with Qij∈Q. For each observation k, let fijk be the frequency of change in answer value of Qij over the observation period. Suppose
(20)0<fi0k<fi1k<⋯<fijk<⋯<fin-1k.
Inequality (20) is called the observation frequency relation and question order y0k=i0i1⋯in-1 is the initial significance state of observation k. If fijk=fij+1k and fijP<fij+1P at the population level, then choose question order ijij+1. If fijP=fij+1P, then choose question order as in the questionnaire (6). The initial fitness statex0k=x0x1⋯xn-1 is such that xj is the value of ij in y0k. The initial state of k is the ordered pair p0k=(x0k,y0k).
By choosing the initial significance y0k 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, fijP≠0. Otherwise, question ij may be deleted as any such property becomes an identifier of subpopulations of possible interest for analysis in its own right.
Definition 8.
For each observation k, define the change set at timet by
(21)Δtk(∈𝒞n)={j:Qij∈Qtkchanges answer value at timet+1}.
Let p0k=(y0k,x0k) be the initial state of k. Denote by ptk=(xtk,ytk) the state of k at time t≥0. The change mapφ:(ℕ0,𝒞n,Sn)→Sn is such that
(22)φ(t,Δtk,ptk):=φ[Δtk](ptk)=(ϕJ∘ϕΔtk)(ptk)=pt+1k.
The set Δtk∈𝒞n is given by the longitudinal data for each t≥0 and is a useful ordered listing of questions that change answer values from time t to t+1. For each k, the nonautonomous map φ[Δtk] 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 p0k of k, define the state of k at time t≥1 by
(23)pt=φ[Δt-1](⋯(φ[Δ0](p0))).
The forward orbit of k under φ is defined by
(24)𝒪φ(k)={ptk}t∈ℕ0.
2.3. Algorithm for Building the Orbit of an Observation in <inline-formula>
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<mml:mi>S</mml:mi></mml:mrow>
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We give a simple algorithm to determine the states pt>0k that comprise the orbit of an observation k from longitudinal data.
Step 1.
Determine initial question order y0k and initial state p0k=(x0k,y0k) of k and plot p0k in Sn.
Step 2.
Identify from the data of k the question that changes answer at time t=1, say Qij. Swap both ij (in y0k) and corresponding answer xj (in x0k) to the right of y0k and x0k, respectively, and change xj to xj*, where xj*=0 if xj=1 and 0 otherwise. This new question order and answer order give the next state p1k=(x1k,y1k) of k. Suppose both Qij and Qij change answers at t=1. If j<j′, then sequentially swap to the right ij and ij′ (resp., xj and xj′) of the question order (resp., answer order), starting with ij′ (resp., xj′). Change xj to xj* and xj′ to xj′*. Plot the point p1k in Sn and directed edge from p0k to p1k.
Step 3.
Repeat Step 2, updating state pt-1k→ptk and t→t+1 for time t=1,2,…,T-1.
Visualization and analysis of orbits of observations in Sn allow capturing information of change in longitudinal data. We illustrate in Figure 2 an orbit in S3. The useful distance on Sn is given by the discrete metric; that is, d(p,p′)=1 if p≠p′ and zero otherwise. The visualized distance between points in Sn has no interpretation so we may represent Sn by a regularly spaced point.
State transitions of observation k (from Table 4) in S3.
Remark 10.
The set of all orderings of variables is captured in Sn. For fixed question order and x,x′∈Xn, x<x′ means that fitness state x′ is fitter than x. 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.
We now analyze possible state transitions which an observation k can take in Sn.
Definition 11.
Let p=(x,y) and p′=(x′,y′) be in Sn, and let Δ∈𝒞n.
Suppose Δ is such that φ[Δ](p)=p′. Then there is a transition from p to p′ under Δ, defined by p→[Δ]p′.
Suppose p→[Δ]p′. If y=y′ and x≠x′, then there is a horizontal transition from p to p′. If x=x′ and y≠y′, then there is a vertical transition from p to p′.
Suppose p→[Δ]p′. If, in addition, Δ is such that p′→Δp, then the transition from p to p′ is reversible, and one writes p↔[Δ]p′.
We have self-transitions if Δ=∅ (no change), the empty set. In general, horizontal transitions denote change in the right-most variable, while vertical transitions denote change in the last two variables.
Let p,p′∈Sn. Given an observation k, if ptk∈p and pt+1k∈p′, then there is Δt∈𝒞n such that p→[Δt]p′. We use the symbol “∈” to denote that there may be other observations in p or p′ at times t and t+1, respectively. State transitions of observation k in Sn are visualised as a sequence of directed edges.
Example 12.
Consider data of observation k in column 2 of Table 4. The asterisks denote changing answers in the next time step. The frequencies of change are f0k=2, f1k=3, and f2k=1, so the initial question order of k is y0k=201. And initial fitness state is x0k=010 (Definition 6). At t=0, we apply φ[Δ0k] to initial state p0k=(010,201). Questions i1=0 and i2=1 in Q0k change answers in the next time step so Δ0k={1,2} (not {0,1}). Applying ϕΔ0k to (010,201) changes x2=0 and x1=1 to x2*=1 and x1*=0, respectively. Next, ϕJ is first applied to both x2* and i2 by moving each to the right (they are already rightmost), followed by moving x1* and i1 to the right end. Hence, we have (x,y)1k=(010,210). For t=1 and t=2, verify that Δ1k={1} and Δ2k={0,1,2}, respectively. At each time t the bold numbers in the significance column are the question indices in Q that change answers at t+1. The orbit of k in S3 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.
States of observation k.
Unit k
Coded answers to
Fitness
Significance
Δtk
Q={Q0,Q1,Q2}
xtk
ytk
t=0
{1*,0*,0}
010
201
{1,2}
1
{0,1*,0}
010
210
{1}
2
{0*,0*,0*}
000
201
{0,1,2}
3
{1,1,1}
111
102
Example 13.
Figures 3(a) and 3(b) illustrate the associated orbits in S3 of the four households given in Table 1. Observe that orbit of h3 stays strictly on the left half of S3, particularly in subset where Q1(HH)=0, while h1 and h4 stay in the right half subset, where Q1(HH)=1 and Q2(AD)=1, respectively.
(a) Orbits of the four subjects 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.
Definition 14.
Let Δ∈𝒞n and let j∈Δ. The return mapφ~[Δ]:Sn→Sn is such that
(25)φ~[Δ](x0x1⋯xj⋯xn-1,i0i1⋯ij⋯in-1)=(x0x1⋯xj-1xn-1*xj⋯xn-2,wwi0i1⋯ij-1in-1ij⋯in-2).
That is, φ~[Δ] first inserts xn-1 and in-1 to the position, where xj and ij are located, followed by changing xn-1 to the new value xn-1*. If j,j′∈Δ and j<j′ then φ~[Δ] is first applied to xj and ij.
Trivially, for any n≥1, p∈Sn, and Δ∈𝒞n,
(26)φ~[Δ](φ[Δ](p))=φ[Δ](φ~[Δ](p))=p.
Theorem 15.
Let p∈Sn. Define the image set and preimage set of p over 𝒞n by
(27)𝒥p=⋃Δ∈𝒞n{p′∈Sn:φ[Δ](p)=p′},𝒥~p=⋃Δ∈𝒞n{p~∈Sn:φ[Δ](p~)=p},
respectively. For each Δ∈𝒞n,
there exists a unique p′∈Sn such that φ[Δ](p)=p′. Moreover, |𝒥p|=2n;
there exists a unique p~∈Sn such that φ[Δ](p~)=p. Moreover, |𝒥~p|=2n.
Proof.
(a) Let x=x0x1⋯xn-1 be the fitness state of p∈Sn. The image of x under both maps ϕΔ and ϕJ is unique so p′ is unique. Note that |𝒞n|=2n and the set of images of x under φ[Δ], Δ∈𝒞n, is the set of distinct binary numbers of length n, which can be associated with 2n distinct states in Sn. This is a bijection between elements of 𝒞n and the binary numbers of length n, so |𝒥p|=2n.
(b) For Δ∈𝒞n, choose p~=φ~[Δ](p), where φ~[Δ] is the return map. The proofs of the uniqueness of p~ and the cardinality of 𝒥~p follow a similar argument as in (a).
Theorem 16.
Let n≥2 and p∈Sn. If Δ={n-m,n-1}, then φ[Δ]m(p)=p for m=2,3,…,n.
Proof.
Fix n and m, where 2≤m≤n. Let a=x0x1⋯x(n-m-1) and b=i0i1⋯i(n-m-1) be sequences of length (n-m) so that
(28)p=(ax(n-m)x(n-m+1)⋯x(n-2)x(n-1),wwbi(n-m)i(n-m+1)⋯i(n-2)i(n-1)).
Let Δ={n-m,n-1}. Observe that, for any integer ℓ>0, a and b are fixed under φ[Δ]ℓ. Now
(29)φ[Δ](p)=(ax(n-m+1)⋯x(n-2)x(n-1)*x(n-m)*,wwax(n-m+1)⋯x(n-2)x(n-1)*x(n-m)*bi(n-m+1)⋯i(n-2)i(n-1)i(n-m))φ[Δ]2(p)=(ax(n-m+2)⋯x(n-2)x(n-1)*x(n-m)w×x(n-m+1)*,bi(n-m+2)⋯i(n-2)i(n-1)w×ax(n-m+2)⋯x(n-2)x(n-1)*x(n-m)i(n-m)i(n-m+1))⋮φ[Δ]ℓ(p)=(ax(n-m+ℓ)⋯x(n-2)x(n-1)*x(n-m)⋯x(n-m+ℓ-2)w×x(n-m+ℓ-1)*,bi(n-m+ℓ)⋯i(n-2)i(n-1)w×i(n-m)⋯i(n-m+ℓ-2)i(n-m+ℓ-1)).
For ℓ=m-1, φ[Δ]m-1(p)=(ax(n-1)*xn-m⋯xn-2*,bi(n-1)in-m⋯in-2). Hence, φ[Δ](φ[Δ]m-1(p))=φ[Δ]m(p)=p.
Corollary 17.
For any n≥1, p∈Sn, and Δ∈{{n-1},{n-2,n-1},In}, the transition from p to p′=φ[Δ](p) is reversible.
The next theorem states some nonallowable transitions in Sn.
Theorem 18.
Let
(30)y=i0i1⋯in-2in-1,y-=in-1i-1i-2⋯i-n-2i-n-1,i-j≠ijforj∈In∖{0},y′=i0′i1′⋯in-2′in-1,ij′≠ijforj∈In∖{n-1}.
Define
(31)Sy={p∈Sn:yisthesignificancestateofp},Sy-={p∈Sn:y-isthesignificancestateofp},Sy′={p∈Sn:y′isthesignificancestateofp}.
Let n≥2. The only transitions in Sy, aside from self-transitions, are horizontal transitions. Let n≥3.
There is no transition from Sy to Sy-.
There is no transition between Sy and Sy′.
Proof.
(a) It is easy to show that any transition between distinct points with the same significance state is under Δ={n-1}. In particular, horizontal transitions are between pairs p=(xp,y) and pˇ=(xpˇ,y), where xp=x0x1⋯xn-2xn-1 and xpˇ=x0x1⋯xn-2xn-1*.
(b) Let p∈Sy. The sets Δ=In and Δ={0,1,…,n-2} are the only elements of 𝒞n such that in-1 is moved to the leftmost position of the question order of φ[Δ](p), in which case question order must be in-1in-2⋯i0, different from y-.
(c) As in (b), Δ=∅ and Δ={n-1} are the only elements of 𝒞n such that in-1 remains in its position under φ[Δ]. The rest of the proof follows a similar argument as in (b).
2.5. Local Dynamics in <inline-formula>
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Consider the subset
(32)ℒm(⊂𝒮n)={(c0c1⋯c(n-m-1)x(n-m)⋯x(n-1),xj{0,1}∈ijInwwxj{0,1}∈ijInℓ0ℓ1⋯ℓn-m-1in-m⋯in-1)ww:cj,ℓjconstant,xj∈{0,1},ij∈In},
where n-m answers are constant. Since |ℒm|=m!2m states and |𝒮n|=n!2n states, then Sn is composed of 2n-mn!/m! subsets of the form ℒm. For n-1 constant answers (i.e., m=1), denote by ℒn-1(c,ℓ) the subset of Sn, where question ℓ has constant answer c. Then any p∈ℒn-1(c,ℓ) is given by p=(cx1x2⋯xn-1,ℓi1i2⋯in-1), xj∈{0,1}, ij∈In. Define ℒn-1(c) by
(33)ℒn-1(c)=⋃ℓ∈Inℒn-1(c,ℓ).
Using (33), we can express Sn as
(34)𝒮n=ℒn-1(0)∪ℒn-1(1),n≥1=ℒn.
We now define the set (as in 𝒞n) associated with transitions between points in ℒm, where m<n. Recall from Theorem 15 that transitions in Sn are under elements Δ of the set 𝒫(In)=𝒞n. For m=1, there are n-1 constant answer values so transitions in ℒ1 are under elements of the set
(35)𝒟1(n)=𝒫({n-1}),n≥1={∅,n-1}.
Similarly, for n-2 constant answer values (i.e., m=2), transitions in ℒ2 are under the set
(36)𝒟2(n)=𝒫({n-2,n-1}),n≥2={∅,n-1}∪{n-2,{n-2,n-1}}=𝒟1(n)∪(𝒫({n-2,n-1})∖𝒟1(n)).
In general, the dynamics in ℒm⊂𝒮n is described by transitions under the set
(37)𝒟m(n)=𝒫({n-m,n-m+1,…,n-1}),n≥m=𝒟m-1(n)∪(𝒫({n-m,n-m+1,…,n-1})∖𝒟m-1(n)).
For m=n, we see from (37) that 𝒟n(n)=𝒞n, as expected. Moreover, the transitions in Sn are given by the transitions in Sn-1, together with the additional transitions under the set difference 𝒞n∖𝒟n-1(n).
Since |ℒm|=|𝒮m| and |𝒟m(n)|=|𝒞m| for m≤n and that there is a one-to-one correspondence between transitions in ℒm and Sm (from (37)), we can write ℒm=𝒮m.
Example 19.
Figures 4(a) and 4(b) illustrate all possible transitions in S1 and S2, respectively. Consider horizontal transitions alternating between states 1 and 2. In Figure 4(a), this denotes alternating 0-1 answer to question 0. On the other hand, transitions alternating between states 1 and 2 in Figure 4(b) denote alternating 0-1 answer to question 0 with constant favourable answer to question 1. All transitions between states in the dashed boxes (containing ℒ1(cj,ℓj)) are under the map φ[{1}], where “1" refers to the index on the right end of question order i0i1. In each ℒ1(cj,ℓj), one answer value is constant and is positioned at the left end of the fitness state. The digraph associated with transitions in S1 is exactly the same as the digraph associated with transitions in each ℒ1(cj,ℓj). Transitions in these subsets ℒ1(cj,ℓj) are under 𝒟1(2). The set ℒ1(0,1) is composed of states 1 and 2, ℒ1(1,1) of states 3 and 4, and so on. Transitions under 𝒞2∖𝒟1(2)={{0},{0,1}} are shown as well.
Transitions in (a) S1 under 𝒞1 and (b) S2 under 𝒞2. Circles denote self-transition (Δ=∅).
Example 20.
Figure 5 illustrates orbits in S4 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 S4, we can remove Q3 and consider only the three questions Q0, Q1, and Q2 and analysing orbits in S3 characterised by Q3=0 and orbits characterised by Q3=1. Note that the frequency change of Q3 here is f3P=0.
Two clusters in S4 distinguished by the value of Q3. The left cluster comprises orbits with constant unfavourable value to Q3, while the right cluster is characterised by favourable value to Q3. Analysis of each cluster of orbits can be simplified in S3.
3. General Equation of Motion of Dynamics in <inline-formula>
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This section concerns properties of all possible paths in Sn.
Definition 21 (see [<xref ref-type="bibr" rid="B17">13</xref>]).
Let X and Y be arbitrary sets. A multivalued map F from X to Y, denoted by F:X⇉Y, is such that F(x) is assigned a set Yx⊂Y for all x∈X.
Consider our state space Sn. Denote by φ[Δ]:𝒮n→𝒮n the map in (22), where Δt is constant. Define
(38)G={φ[Δ]:Δ∈𝒞n}
and the multivalued map F:𝒮n⇉𝒮n by
(39)F(p)=⋃{g(p)∣g∈G}.|F(p)|=2n so F is a 1 to 2n map. Under F, the orbit {ptk}t∈ℕ0 is such that pt+1k∈F(ptk). The multivalued map F can be interpreted as a digraph whose N=n!2n vertices are the points in Sn and edge p→p′ if p′∈F(p).
Equivalently, F can be defined as a square matrix of size N that encodes all possible paths in Sn.
Definition 22.
Let V={1,2,…,N} and let i,j,∈V. For p,p′∈Sn, let ψ(p)=i and ψ(p′)=j. Let F be the multivalued map in (39). The theoretical transition matrix for F under n questions is denoted by Tn(th)=(Tn,ij(th)), where
(40)Tn,ij(th)={1ifp′∈F(p)0otherwise.Tn,ij(th) captures all theoretically admissible transitions between states in Sn. Each entry Tn,ij(th)=1 indicates that there is a transition from state i to j in one step. Hence, Tn,ij(th) gives all possible paths in Sn that any observation can take.
Example 23.
From Figure 4, we have Tn(th) for n=1 and n=2, as given below
(41)T1(th)=[1111]T2(th)=[1100010111000101001110100011101001011100010111001010001110100011].
Because the digraph in S1 is the same as the digraphs in ℒ1(cj,ℓj)⊂𝒮2 (Example 19), the matrix encoding transitions in S1 is the same as the matrix encoding transitions in ℒ1(cj,ℓj). The 2×2 submatrices of entries all equal to 1 (in bold numbers) in T2(th) correspond to transitions in ℒ1(cj,ℓj). The 2×2 zero submatrices in T2(th) (in bold) denote the nonallowable transitions between states with the same significance (Theorem 18(a)).
(a) The digraph 𝒢Tn(th) is 2n-regular; that is, the in- and out-degree of all its vertices is 2n.
(b) The largest eigenvalue of Tn(th) is λmax(Tn(th))=2n, with associated eigenvector vλmax=1.
(c) Consider trace (Tn(th))=N.
(d) Consider det(Tn(th))=0.
Proof.
(a) This is a consequence of Theorem 15.
(b) This is a consequence of (a) and the Perron-Frobenius theorem [14].
(c) Let ψ(p)=i. From the definition of self-transition, φ[∅](p)=p for any p∈Sn. Hence, Tn,ii(th)=1 for i=1,2,…,N.
(d) Let p=(x,y) and p′=(x′,y) be points in Sn such that
(42)x=x0x1⋯xn-2xn-1,y=i0i1⋯in-2in-1,x′=x0x1⋯xn-2xn-1*,wherexn-1*isgivenin(16).Claim. 𝒥p=𝒥p′, where 𝒥p and 𝒥p′ are given in (27).
The case where n=1 is given in Example 23. We prove our claim for n=2. By definition of self-transition, φ[∅](p)=p and φ[∅](p′)=p′. It is clear that φ[{1}](p)=p′ and φ[{1}](p′)=p. Let
(43)φ[{0}](p)=p1,φ[{0,1}](p)=p2.
Observe that
(44)φ[{0}](p′)=φ[{0}](φ[{1}]p)=φ[0,1](p)=p2,φ[{0,1}](p′)=φ[{0,1}](φ[{1}]p)=φ[{0}](φ[{1}]2(p))=φ[{0}](p)=p1.
Since p and p′ have the same image for all Δ∈𝒞2, 𝒥p=𝒥p′, as claimed.
Our claim implies that rows of T2(th) associated with p and p′ are the same. Because of these repeating rows, det(T2(th))=0. The proof can be readily extended to the general case and is treated in exactly the same manner.
3.2. Subshift of Finite Type
Let wi, j∈V={1,2,…,N}. If observation k is in state sj∈Sn after t time steps, we will write ptk∈sj to indicate that more than one observation may be in sj. We associate with the orbit 𝒪φ(k) a sequence of symbols w=(w0w1⋯wi⋯), where wi=j if pik∈sj. Denote the corresponding symbol space of one-sided sequences of Tn(th) by
(45)ΣTn(th)+={w=(w0w1⋯)∣Tn,wiwi+1(th)=1hjgTn,wiwi+1(th)∀i∈ℕ0,wi∈V}.
The equation of motion in Sn is given by the shift map ΣTn(th):ΣTn(th)+→ΣTn(th)+ defined by
(46)σTn(th)(w)=w′wherewi′=wi+1(∀i≥0).
Definition 25 (see [<xref ref-type="bibr" rid="B18">15</xref>]).
The pair (σTn(th),ΣTn(th)+) is called the subshift of finite type (SFT) determined by Tn(th).
The SFT determined by Tn(th) captures the exact detail of all possible itineraries of observations in Sn. Many dynamical properties of an SFT depend on the structure of its associated transition matrix or digraph.
Definition 26 (see [<xref ref-type="bibr" rid="B14">16</xref>]).
(i) A transition matrix T is irreducible if for each pair (i,j), there exists ℓ>0 so that Tℓ>0. Otherwise, T is reducible. (ii) The digraph GT=(V,E) associated with T is strongly connected if it is possible to get from any vertex to any other one by traversing a sequence of edges as directed by T. (iii) If there exists ℓ′>0 such that Tℓ>0 for all ℓ>ℓ′, then T is primitive. (iv) Let X be any set. A continuous map f:X→X is (topologically) transitive if, for all nonempty open sets U,U′⊂X, there exists ℓ≥0 such that fℓ(U)∩U′≠∅. (v) If there is ℓ′> 0 such that fℓ(U)∩U′≠∅ for all ℓ≥ℓ′, then f is (topologically) mixing.
Consider the SFT given by (σT,ΣT+). The following results [17, 18] concern dynamical properties of a transitive SFT and algebraic properties of its transition matrix. (i) The shift map ΣT is transitive (resp., mixing) if and only if T is irreducible (resp., primitive). (ii) T is irreducible if and only if the digraph GT=(V,E) is strongly connected. In that case we say that V is irreducible. (iii) A nonnegative, irreducible matrix with a positive element on the main diagonal is primitive.
Theorem 27.
Tn(th) is irreducible.
Proof.
We prove by induction on n. From Example 23, T1,ij(th)=1 for all (i,j) so Tn(th) is irreducible for n=1. Assume that Tn(th) is irreducible for n=m. Then the digraph GTm(th) is strongly connected, and Sm is irreducible. We prove that GTn(th) is strongly connected to n=m+1.
Recall that transitions in Sm+1 are under the set 𝒞m+1. From (37), this set can be expressed as 𝒞m+1=𝒟m(m+1)∪(𝒞m+1∖𝒟m(m+1)), where 𝒟m(m+1) is the set associated with transitions in the irreducible set ℒm(c,ij)(=𝒮m), c∈{0,1} and ij∈Im+1. With the additional transitions under 𝒞m+1∖𝒟m(m+1), we show that
the set ℒm(c)⊂𝒮m+1 (given in (33)) is irreducible;
there is a path to and from any pair p,p′, where p∈ℒm(0) and p′∈ℒm(1).
We prove (i) by showing that there is p∈ℒm(c,ij) such that, for all ij′≠ij, there is a transition from p to p′∈ℒm(c,ij′). By Theorem 16, φ[{0,m}]m+1(p)=p for any p∈𝒮m+1. Let
(47)p0=(cc⋯cc*,i0i1⋯im-1im)p1=φ[{0,m}](p0)=(cc⋯cc*,i1i2⋯imi0)p2=φ[{0,m}]2(p1)=(cc⋯cc*,i2i3⋯i0i1)⋮pm=φ[{0,m}]m(pm-1)=(cc⋯cc*,imi0⋯im-2im-1).
Observe that all pj’s have the same fitness states but distinct significance states. In particular, each pj is contained in a distinct subset ℒm(c,ij). This vertical transition allows all pj∈ℒm(c,ij) to visit a distinct ℒm(c,ij′) in at most m steps. Hence, ℒm(c) is irreducible. To show (ii), take
(48)p=(00⋯0,m(m-1)⋯10)∈ℒm(0),p′=(11⋯1,01⋯(m-1)m)∈ℒm(1).
Under Δ=Im+1, there is a reversible transition p↔[Δ]p′.
From (i), (ii), and the irreducibility of Tm(th), there is a path between any pair p,p′∈𝒮m+1. Hence, GTm+1(th) is strongly connected and Tm+1(th) is irreducible, as desired.
Corollary 28.
Tn(th) is primitive.
Definition 29 (see [<xref ref-type="bibr" rid="B18">15</xref>]).
The topological entropy of the shift map ΣT:ΣT+→ΣT+ is defined by
(49)htop(σT)=limm→∞ln(|𝒲(m)|)m,
where 𝒲(m) is the set of allowable sequences of length m≥1.
A continuous map f on a compact metric space X is chaotic if htop(f)>0 [17]. The topological entropy for SFTs is given by the following theorem [15].
Theorem 30.
Let T be a transition matrix and let ΣT:ΣT+→ΣT+ be the associated SFT. Then
(50)htop(σT)=ln(λmax),
where λmax is the maximum eigenvalue of T.
Since (σTn(th),ΣTn(th)+) is the associated subshift of the multivalued map F in (39), one has
(51)htop(F)=htop(σTn(th))=ln(2n)=nln2>0.
4. Dynamics from Data
Given multivariable longitudinal binary data of dimension n, every observed orbit is an orbit of an SFT determined by Tn(th). Dynamics of real-world longitudinal data however is not often defined by an SFT. Data usually selects certain paths given by Tn(th) and may sometimes stay in a particular subspace of Sn.
4.1. Nonstationary SFT
Let p,p′∈Sn, ψ(p)=i, and ψ(p′)=j. For each t≥0, denote by Tn,t(data) the matrix that records observed transitions that occur in the longitudinal data from time t to t+1, where
(52)(Tn,t(data))ij={1if∃k∈K,suchthatp→[Δtk]p′0otherwise.
We note that Tn,t(data) is defined over an interval of time and that Tn,t(data) may vary with time. Some allowable transitions between states given by Tn(th) might not occur in the observed data so Tn,t(data)≠Tn(th). For the case where Tn,t(data) is constant we write Tn,t(data)≡Tn(data) and we can define the SFT given by the pair (σTn(data),ΣTn(data)+).
If Tn,t(data) is not constant, then we have a sequence of matrices
(53)A={Tn,t(data)}t≥0.
Given A, define
(54)ΣA(data)={(Tn,t(data))wtwt+1w=(w0w1⋯w𝒯⋯)∈ΣTn(th)+bbj:(Tn,t(data))wtwt+1=1}.
We call ΣA(data) the nonstationary symbolic space restricted by the sequence of matrices A. The shift takes place as usual in ΣA(data) and is denoted by σA:ΣA(data)→ΣA(data). We refer the reader to [19–21] for a discussion on nonstationary SFT.
Definition 31.
The pair (σA,ΣA(data)) is the nonstationary SFT (NSFT) determined by the sequence of matrices A.
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 Sn over time. Figure 6(a) illustrates the orbits of a population in a subset of S3 over time.
(a) Orbits of Agincourt households in S3, over time. (b) Number of orbit visits to each state in S3, where majority are in ℒ~, having the favourable property of being headed by an adult (i.e., Q1=1).
4.2. Population Dynamics
We discuss the longitudinal data of a population of K>1 observations. Aside from all possible paths that an observation can take in Sn, 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 Sn at a given time, we can consider the number of observations that follow the same transition in Sn.
In general, given a transition matrix T (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 Sn can then follow as usual. In what follows, let V denote a finite index set with |V|≥2 and i,j∈V.
Definition 32.
Let Ht(i) be the number of observations in state i at time t. The density matrix at time t is defined by
(55)Dt=(dt,ij),
where dt,ij is the number of observations in state i at time t that go to state j at time t+1. The (net) flux of state i at time t is defined by
(56)Ft(i)=Ht+1(i)-Ht(i).
Let ℓt,ij=dt,ji-dt,ij, i,j∈V. The flux in (56) can also be expressed as
(57)Ft(i)=∑jℓt,ij.
From (56) and (57), we have
(58)Ht(i)=∑jdt,ij,Ht+1(i)=∑jdt,ji.
Let Ht be a row vector whose ith entry is Ht(i). We will refer to Ht as the observed capacity vector at time t. Given the initial observed capacity vector H0, there are two methods that we can use to determine Ht+1.
(i) Nonhomogenous Case. From data, we are encouraged to construct a probability matrix based directly on the density matrices Dt. For each Dt, we construct a time-dependent probability matrix Pt=(Pt,ij), where
(59)Pt,ij={dt,ij∑jdt,ij,if∑jdt,ij≠00otherwise.
The capacity vector Ht+1 is given by the product
(60)Ht+1=Ht·Pt=H0·∏m=0tPm.
We show that the capacity Ht+1(i) in (56) agrees with the ith entry of (60). It is trivial if ∑jdt,ij=0 for all t. Otherwise we have
(61)Ht+1(i)=Ht(i)+Ft(i)=Ht(i)+∑j(dt,ji-dt,ij)=Ht(i)+∑j(Pt,ji∑idt,ji-Pt,ij∑jdt,ij)=Ht(i)+∑jPt,jiHt(j)-∑jPt,ijHt(i)=Ht(i)(1-∑jPt,ij)+∑jPt,jiHt(j)=∑jPt,jiHt(j).
(ii) Homogenous Case. Let Dt be the density matrix at time t and let
(62)D(data)=∑t=0T-1Dt
be the accumulated density matrix of the data over the observation period. Define the mean density matrix by
(63)D=1TD(data).
Suppose D is irreducible. Define the mean probability matrix from D by PD=(PD,ij), where
(64)PD,ij=Dij∑jDij.
The capacity vector at t+1 is given by the product
(65)Ht+1=Ht·(PD)=H0·(PD)t.
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 Sn are then estimated. Markov modeling on the bigger space Sn may give better results in generating data as Sn is a more detailed space than a space where order of variable is fixed (e.g., Mn).
Definition 33.
A set of observations K′ cluster in Sn if for all k∈K′, there exists R⊂Sn and time interval I such that the orbit O(k)⊂R for all t∈I.
An advantage in using the bigger space Sn to analyse multivariable longitudinal data is that all possible clusterings in stable variables are revealed. We can also visualize in Sn the shift in clusters from one region R to another region R′. There is much interesting information that we can gather from a cluster in a subset R of Sn. Clustering in ℒm (m≠n) defined in (32) involves observations with the same stable n-m variables. Subsets ℒm are important as they automatically determine which variables are trivial in the analysis. We either reduce analysis from Sn to ℒm or replace the n-m 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 Lm have strong correlation among the n-m variables.
5. Application
We illustrate our method using the n=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.
Agincourt questionnaire with corresponding coded answer and frequency of answer change in the population.
Question
Answer
Code
fiP
Q0: Is the biological mother present (>6 months)? (BM)
Yes
1
760
Q1: Is the household headed by a minor? (HH)
Yes
0
4
Q2: Is there an adult death in the household? (AD)
Yes
0
119
The longitudinal data consists of K=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 P is f1P≪f2P≪f0P. The number of unfavourable = 0 answers (over 1692 answers) is Q0:844(49.9%), Q1:2(0.1%), and Q2:71(4.2%). We expect that this, together with the very small value of f1P, will give clustering in the question order 1**, where Q1=1.
Figure 6(a) illustrates orbits of households in S3 over time, where 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. Most of the orbits are in
(66)ℒ~={21,22,23,24,29,30,31,32}={(x,y):x=1**,y=1**}
and are characterised by the favourable property of being headed by an adult (i.e., Q1=1). Of the 179 households that stay in ℒ~, 159 distinct orbits are found. However, we see from Figure 6(b) that the number of state visits in S3 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(a) are then split into defaulting and nondefaulting populations, as shown in Figures 7(a) and 7(b), respectively.
Orbits in S3 of (a) defaulting and (b) nondefaulting households.
Note that if we include a fourth question
(67)Q3:isthehouseholddefaulting(=0)ornondefaulting(=1)?
then orbits will be plotted in S4, as seen in Figure 5. Orbits that cluster on the left are the defaulting households (Q4=0), while those on the right are nondefaulting (Q4=1).
Figure 8 gives Ht(i), the number of observed defaulting and nondefaulting households at each state i=23,24 from t=1999 to t=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 S4, respectively.
Number of defaulting and nondefaulting households in states 23 and 24 for t=1999 to t=2007.
(a) Orbits of defaulting (left cluster) and nondefaulting (right cluster) households in S4, over time. (b) Accumulated number of visits to each state of S4.
Table 6 gives the (accumulated) number of state transitions i→j, i,j∈{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 P values and these support our hypothesis that mother out-migration 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.
Total transitions from state i→j(i,j=23,24) in defaulting and nondefaulting household, together with the odds ratios (95% confidence interval and P value).
State i→j
Defaulting
Nondefaulting
Odds ratio (95% CI; P value)
23→23
256
63
1.59 (1.17 to 2.17; P = 0.0031)
23→24
238
84
1.00 (0.75 to 1.33; P = 0.9925)
24→23
260
96
0.94 (0.72 to 1.24; P = 0.6653)
24→24
214
99
0.70 (0.53 to 0.92; P = 0.0110)
6. 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 Sn 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 binary-valued variables but the method may be generalized to m-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 Sn is that information of change can be extracted directly from the visualized orbits. In addition, orbits in Sn 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.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgment
Maria Vivien Visaya is supported by a Claude Leon Fellowship.
FungE. S.ChingW. K.ChuS.NgM. K.ZangW.Multivariate Markov chain models3Proceedings of the IEEE International Conference on Systems, Man and CyberneticsOctober 2002Hammamet, Tunisia2983022-s2.0-0037670110BijleveldC.van der KampL. J. T.MooijaartA.van der KlootW.van der LeedenR.van der BurgE.VerbekeH.FiewsS.MolenberhghsG.DavidianM.The analysis of multivariate longitudinal data: a reviewVlachosI.KugiumtzisD.State space reconstruction for multivariate time series predictionAl-AzizJ.ChristouN.DinovI. D.SOCR motion charts: an efficient, open-source, interactive and dynamic applet for visualizing longitudinal multivariate dataGrossenbacherA.The globalisation of statistical contentInselbergA.ZhouH.YuanX.QuH.CuiW.ChenB.VilanovaA.TeleaA.ScheuermannG.MöllerT.Visual clustering in parallel coordinates27Proceedings of the Eurographics/IEEE-VGTC Symposium on Visualization2008BasiosV.FortiG.NicolisG.Symbolic dynamics generated by a combination of graphsDurandF.BerthV.RigoM.Combinatorics on Bratteli diagrams and dynamical systemsBeckettL.DiaconisP.Spectral analysis for discrete longitudinal dataGottschauA.Markov chain models for multivariate binary panel dataMrozekM.AlefeldG.FrommerA.LangB.Inheritable properties and computer assisted proofs in dynamicsMincH.RobinsonC.KitchensB.LindD.MarcusB.VargaR.ArnouxP.FisherA. M.Anosov families, renormalization and non-stationary subshiftsFanA. H.PollicottM.Non-homogeneous equilibrium states and convergence speeds of averaging operatorsFisherA. M.Nonstationary mixing and the unique ergodicity of adic transformationsKahnK.TollmanS. M.CollinsonM. A.ClarkS. J.TwineR.ClarkB. D.ShabanguM.Gómez-OlivéF. X.MokoenaO.GarenneM. L.Research into health, population and social transitions in rural South Africa: data and methods of the Agincourt health and demographic surveillance system