# seqcost: Define substitution and indel costs In seqdist2: Distance Between State Sequences

## Description

The function `seqcost` proposes different ways to generate substitution costs (supposed to reflect state dissimilarities) and possibly indel costs. Proposed methods are: `"CONSTANT"` (same cost for all substitutions), `"TRATE"` (derived from the observed transition rates), `"FUTURE"` (Chi-squared distance between conditional state distributions `lag` positions ahead), `"FEATURES"` (Gower distance between state features), `"INDELS"`, `"INDELSLOG"` (based on estimated indel costs). The substitution-cost matrix is intended to serve as `sm` argument in the `seqdist` function that computes distances between sequences. `seqsubm` is an alias that return only the substitution cost matrix, i.e., no indel.

## Usage

 ```1 2 3 4 5 6 7 8``` ```seqcost(seqdata, method, cval = NULL, with.missing = FALSE, miss.cost = NULL, time.varying = FALSE, weighted = TRUE, transition = "both", lag = 1, missing.trate = FALSE, state.prop = NULL, prop.weights = NULL, prop.type = list(), proximities = FALSE, miss.cost.fixed = !missing.trate, state.features = state.prop, feature.weights = prop.weights, feature.type = prop.type) seqsubm(...) ```

## Arguments

 `seqdata` a sequence object as returned by the seqdef function. `method` Character string. How to generate the costs. One of `"CONSTANT"` (same cost for all substitutions), `"TRATE"` (derived from the observed transition rates), `"FUTURE"` (Chi-squared distance between conditional state distributions `lag` positions ahead), `"FEATURES"` (Gower distance between state features), `"INDELS"`, `"INDELSLOG"` (based on estimated indel costs). `cval` Scalar. For method `CONSTANT`, the single substitution cost. For method `"TRATE"`, a base value from which transition probabilities are subtracted. If `NULL`, `cval=2` is used, unless `transition` is `"both"` and `time.varying` is `TRUE`, in which case `cval=4`. `with.missing` Logical. Should an additional entry be added in the matrix for the missing states? If `TRUE`, the ‘missing’ state is also added to state alphabet. Use this if you want to compute distances with missing values inside the sequences. See Gabadinho et al. (2010) for more details on the options for handling missing values when computing distances between sequences. `miss.cost` Scalar or vector. Cost for substituting the missing state. Default is `TRUE`. `miss.cost.fixed` Logical. Should the substitution cost for missing be set as the `miss.cost` value. Default is `cval`. `time.varying` Logical. If `TRUE` return an `array` with a distinct matrix for each time unit. Time is the third dimension (subscript) of the returned array. `weighted` Logical. Should weights in `seqdata` be used when applicable? `transition` Character string. Only used if `method="TRATE"` and `time.varying=TRUE`. On which transition are rates based? Should be one of `"previous"` (from previous state), `"next"` (to next state) or `"both"`. `lag` Integer. For methods `TRATE` and `FUTURE` only. Time ahead to which transition rates are computed (default is `lag=1`). `missing.trate` Logical. Deprecated, use `miss.cost.fixed` instead. `state.features` Data frame with features values for each state. `feature.weights` Vector of feature weights with length equal to the number of columns of `state.features` `feature.type` List of feature types. See `daisy` for details. `state.prop` ??? Deprecated. Use `state.features` instead. `prop.weights` ??? Deprecated. Use `feature.weights` instead `prop.type` ??? Deprecated. Use `feature.type` instead

,

 `proximities` Logical: should state proximities be returned instead of substitution costs? `...` Arguments passed to `seqcost`

## Details

The substitution-cost matrix has dimension ns*ns, where ns is the number of states in the alphabet of the sequence object. The element (i,j) of the matrix is the cost of substituting state i with state j. It defines the dissimilarity between

With method `CONSTANT`, the substitution costs are all set equal to the `cval` value, the default value being 2.

With method `TRATE` (transition rates), the transition rates between all pairs of states is first computed (using the seqtrate function). Then, the substitution cost between states i and j is obtained with the formula

SC(i,j) = cval - P(i,j) -P(j,i)

where P(i,j) is the rate of transition from state i to j `lag` positions ahead.

With method `FUTURE`, the cost between i and j is the Chi-squared distance between the vector (d(alphabet | i)) of rates of transition from states i and j to all the states in the alphabet `lag` positions ahead:

SC(i,j) = ChiDist(d(alphabet | i), d(alphabet | j))

With method `FEATURES`, each state is characterized by the variables `state.prop`, and the cost between i and j is computed as the Gower distance between their vectors of `state.prop` values.

With methods `INDELS` and `INDELSLOG`, values of indels are first derived from the state relative frequencies f_i. For `INDELS`, indel_i = 1/f_i, and for `INDELSLOG`, indel_i = log[2/(1 + f_i)]. Substitution costs are then set as SC(i,j) = indel_i + indel_j.

For all methods but `INDELS` and `INDELSLOG`, the indel is set as max(sm)/2 when `time.varying=FALSE` and as 1 otherwise.

## Value

A list of two elements `indel` and `sm` with

 `indel` The indel cost. Either a scalar or a vector of size ns. `sm` The substitution cost matrix of size ns*ns, where ns is the number of states in the alphabet of the sequence object.

For `seqsubm`, the matrix `sm`.

## Author(s)

Matthias Studer and Alexis Gabadinho (first version) (with Gilbert Ritschard for the help page)

## References

Gabadinho, A., G. Ritschard, N. S. M\"uller and M. Studer (2011). Analyzing and Visualizing State Sequences in R with TraMineR. Journal of Statistical Software 40(4), 1-37.

Gabadinho, A., G. Ritschard, M. Studer and N. S. M\"uller (2010). Mining Sequence Data in `R` with the `TraMineR` package: A user's guide. Department of Econometrics and Laboratory of Demography, University of Geneva.

Studer, M. & Ritschard, G. (2015), "What matters in differences between life trajectories: A comparative review of sequence dissimilarity measures", Journal of the Royal Statistical Society, Series A. 179(2), 481-511. DOI: http://dx.doi.org/10.1111/rssa.12125

Studer, M. and G. Ritschard (2014). "A Comparative Review of Sequence Dissimilarity Measures". LIVES Working Papers, 33. NCCR LIVES, Switzerland, 2014. DOI: http://dx.doi.org/10.12682/lives.2296-1658.2014.33

`seqtrate`, `seqdef`, `seqdist`.
 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40``` ```## Defining a sequence object with columns 10 to 25 ## in the 'biofam' example data set data(biofam) biofam.seq <- seqdef(biofam,10:25) ## Optimal matching using transition rates based substitution-cost matrix ## and insertion/deletion costs of 3 trcost <- seqcost(biofam.seq, method="TRATE") biofam.om <- seqdist(biofam.seq, method="OM", indel=3, sm=trcost\$sm) ## Using the insertion/deletion cost returned by seqcost biofam.om <- seqdist(biofam.seq, method="OM", indel=trcost\$indel, sm=trcost\$sm) ## Using costs based on FUTURE with a forward lag of 4 fucost <- seqcost(biofam.seq, method="FUTURE", lag=4) biofam.om <- seqdist(biofam.seq, method="OM", indel=fucost\$indel, sm=fucost\$sm) ## Optimal matching using a unique substitution cost of 2 ## and an insertion/deletion cost of 3 ccost <- seqsubm(biofam.seq, method="CONSTANT", cval=2) biofam.om.c2 <- seqdist(biofam.seq, method="OM",indel=3, sm=ccost) ## Displaying the distance matrix for the first 10 sequences biofam.om.c2[1:10,1:10] ## ================================= ## Example with weights and missings ## ================================= data(ex1) ex1.seq <- seqdef(ex1,1:13, weights=ex1\$weights) ## Unweighted subm <- seqcost(ex1.seq, method="TRATE", with.missing=TRUE, weighted=FALSE) ex1.om <- seqdist(ex1.seq, method="OM", sm=subm\$sm, with.missing=TRUE) ## Weighted subm.w <- seqcost(ex1.seq, method="TRATE", with.missing=TRUE, weighted=TRUE) ex1.omw <- seqdist(ex1.seq, method="OM", sm=subm.w\$sm, with.missing=TRUE) ex1.om == ex1.omw ```