# Define substitution and indel costs

### 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 |

### Arguments

`seqdata` |
a sequence object as returned by the seqdef function. |

`method` |
Character string. How to generate the costs. One of |

`cval` |
Scalar. For method |

`with.missing` |
Logical. Should an additional entry be added in the matrix for the missing states?
If |

`miss.cost` |
Scalar or vector. Cost for substituting the missing state. Default is |

`time.varying` |
Logical. If |

`weighted` |
Logical. Should weights in |

`transition` |
Character string. Only used if |

`lag` |
Integer. For methods |

`missing.trate` |
Logical. For methods |

`state.prop` |
??? |

`prop.weights` |
??? |

`prop.type` |
A list, ??? |

,

`proximities` |
Logical: should state proximities be returned instead of substitution costs? |

`...` |
Arguments passed to |

### 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 `cpal`

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

`sm` |
The substitution cost matrix of size |

`indel` |
The indel cost. Either a scalar or a vector of size |

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. Müller 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. Müller (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*. (Early view). 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

### See Also

`seqtrate`

, `seqdef`

, `seqdist`

.

### Examples

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
``` |

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