Optimal matching

Optimal matching is a sequence analysis method used in social science, to assess the dissimilarity of ordered arrays of tokens that usually represent a time-ordered sequence of socio-economic states two individuals have experienced. Once such distances have been calculated for a set of observations (e.g. individuals in a cohort) classical tools (such as cluster analysis) can be used. The method was tailored to social sciences from a technique originally introduced to study molecular biology (protein or genetic) sequences (see sequence alignment). Optimal matching uses the Needleman-Wunsch algorithm. Optimal matching is a sequence analysis method used in social science, to assess the dissimilarity of ordered arrays of tokens that usually represent a time-ordered sequence of socio-economic states two individuals have experienced. Once such distances have been calculated for a set of observations (e.g. individuals in a cohort) classical tools (such as cluster analysis) can be used. The method was tailored to social sciences from a technique originally introduced to study molecular biology (protein or genetic) sequences (see sequence alignment). Optimal matching uses the Needleman-Wunsch algorithm. Let S = ( s 1 , s 2 , s 3 , … s T ) {displaystyle S=(s_{1},s_{2},s_{3},ldots s_{T})} be a sequence of states s i {displaystyle s_{i}} belonging to a finite set of possible states. Let us denote S {displaystyle {mathbf {S} }} the sequence space, i.e. the set of all possible sequences of states. Optimal matching algorithms work by defining simple operator algebras that manipulate sequences, i.e. a set of operators a i : S → S {displaystyle a_{i}:{mathbf {S} } ightarrow {mathbf {S} }} . In the most simple approach, a set composed of only three basic operations to transform sequences is used: Imagine now that a cost c ( a i ) ∈ R 0 + {displaystyle c(a_{i})in {mathbf {R} }_{0}^{+}} is associatedto each operator. Given two sequences S 1 {displaystyle S_{1}} and S 2 {displaystyle S_{2}} ,the idea is to measure the cost of obtaining S 2 {displaystyle S_{2}} from S 1 {displaystyle S_{1}} using operators from the algebra. Let A = a 1 , a 2 , … a n {displaystyle A={a_{1},a_{2},ldots a_{n}}} be a sequence of operators such that the application of all the operators of this sequence A {displaystyle A} to the first sequence S 1 {displaystyle S_{1}} gives the second sequence S 2 {displaystyle S_{2}} : S 2 = a 1 ∘ a 2 ∘ … ∘ a n ( S 1 ) {displaystyle S_{2}=a_{1}circ a_{2}circ ldots circ a_{n}(S_{1})} where a 1 ∘ a 2 {displaystyle a_{1}circ a_{2}} denotes the compound operator. To this set we associate the cost c ( A ) = ∑ i = 1 n c ( a i ) {displaystyle c(A)=sum _{i=1}^{n}c(a_{i})} , thatrepresents the total cost of the transformation. One should consider at this point that there might exist different such sequences A {displaystyle A} that transform S 1 {displaystyle S_{1}} into S 2 {displaystyle S_{2}} ; a reasonable choice is to select the cheapest of such sequences. We thuscall distance d ( S 1 , S 2 ) = min A { c ( A )   s u c h   t h a t   S 2 = A ( S 1 ) } {displaystyle d(S_{1},S_{2})=min _{A}left{c(A)~{ m {such~that}}~S_{2}=A(S_{1}) ight}} that is, the cost of the least expensive set of transformations that turn S 1 {displaystyle S_{1}} into S 2 {displaystyle S_{2}} . Notice that d ( S 1 , S 2 ) {displaystyle d(S_{1},S_{2})} is by definition nonnegative since it is the sum of positive costs, and trivially d ( S 1 , S 2 ) = 0 {displaystyle d(S_{1},S_{2})=0} if and only if S 1 = S 2 {displaystyle S_{1}=S_{2}} , that is there is no cost. The distance function is symmetric if insertion and deletion costs are equal c ( a I n s ) = c ( a D e l ) {displaystyle c(a^{ m {Ins}})=c(a^{ m {Del}})} ; the term indel cost usually refers to the common cost of insertion and deletion. Considering a set composed of only the three basic operations described above, this proximity measure satisfies the triangular inequality. Transitivity however, depends on the definition of the set of elementary operations. Although optimal matching techniques are widely used in sociology and demography, such techniques also have their flaws. As was pointed out by several authors (for example L. L. Wu), the main problem in the application of optimal matching is to appropriately define the costs c ( a i ) {displaystyle c(a_{i})} . Optimal matching is also a term used in statistical modelling of causal effects. In this context it refers to matching 'cases' with 'controls', and is completely separate from the sequence-analytic sense.