Sequence

# Alignment-based sequence methods

This notebook introduces the alignment-based sequence methods (operationalized by the Optimal Matching (OM) algorithm), which was originally developed for matching protein and DNA sequences in biology and used extensively for analyzing strings in computer science and recently widely applied to explore the neighborhood change.

It generally works by finding the minimum cost for aligning one sequence to match another using a combination of operations including substitution, insertion, deletion and transposition. The cost of each operation can be parameterized diferently and may be theory-driven or data-driven. The minimum cost is considered as the distance between the two sequences.

The sequence module in giddy provides a suite of alignment-based sequence methods.

Author: Wei Kang weikang9009@gmail.com

import numpy as np
import pandas as pd

import libpysal
import mapclassify as mc
f = libpysal.io.open(libpysal.examples.get_path("usjoin.csv"))
pci = np.array([f.by_col[str(y)] for y in range(1929,2010)])
q5 = np.array([mc.Quantiles(y,k=5).yb for y in pci]).transpose()
q5

/Users/weikang/anaconda3/lib/python3.6/site-packages/scipy/stats/stats.py:1713: FutureWarning: Using a non-tuple sequence for multidimensional indexing is deprecated; use arr[tuple(seq)] instead of arr[seq]. In the future this will be interpreted as an array index, arr[np.array(seq)], which will result either in an error or a different result.
return np.add.reduce(sorted[indexer] * weights, axis=axis) / sumval

array([[0, 0, 0, ..., 0, 0, 0],
[2, 2, 2, ..., 1, 1, 0],
[0, 0, 0, ..., 0, 0, 0],
...,
[1, 1, 1, ..., 0, 0, 0],
[3, 3, 2, ..., 2, 2, 2],
[3, 3, 3, ..., 4, 4, 4]])
q5.shape

(48, 81)

Import Sequence class from giddy.sequence:

from giddy.sequence import Sequence


### "hamming"

• substitution cost = 1
• insertion/deletion cost = $\infty$
seq_hamming = Sequence(q5, dist_type="hamming")
seq_hamming

<giddy.sequence.Sequence at 0x7ff498df1160>
seq_hamming.seq_dis_mat #pairwise sequence distance matrix

array([[ 0., 75.,  7., ..., 21., 81., 78.],
[75.,  0., 80., ..., 79., 57., 73.],
[ 7., 80.,  0., ..., 14., 81., 81.],
...,
[21., 79., 14., ...,  0., 81., 81.],
[81., 57., 81., ..., 81.,  0., 51.],
[78., 73., 81., ..., 81., 51.,  0.]])

### "interval"

Assuming there are $k$ states in the sequences and they are ordinal/continuous.

• substitution cost = differences between states
• insertion/deletion cost = $k-1$
seq_interval = Sequence(q5, dist_type="interval")
seq_interval

<giddy.sequence.Sequence at 0x7ff451d8d160>
seq_interval.seq_dis_mat

array([[  0., 123.,   7., ...,  21., 190., 225.],
[123.,   0., 130., ..., 116.,  69., 108.],
[  7., 130.,   0., ...,  14., 197., 232.],
...,
[ 21., 116.,  14., ...,   0., 183., 218.],
[190.,  69., 197., ..., 183.,   0.,  61.],
[225., 108., 232., ..., 218.,  61.,   0.]])

### "arbitrary"

• substitution cost = 0.5
• insertion/deletion cost = 1
seq_arbitrary = Sequence(q5, dist_type="arbitrary")
seq_arbitrary

<giddy.sequence.Sequence at 0x7ff451d8dc18>
seq_arbitrary.seq_dis_mat

array([[ 0. , 37.5,  3.5, ..., 10.5, 40.5, 39. ],
[37.5,  0. , 40. , ..., 39.5, 28.5, 36.5],
[ 3.5, 40. ,  0. , ...,  7. , 40.5, 40.5],
...,
[10.5, 39.5,  7. , ...,  0. , 40.5, 40.5],
[40.5, 28.5, 40.5, ..., 40.5,  0. , 25.5],
[39. , 36.5, 40.5, ..., 40.5, 25.5,  0. ]])

### "markov"

• substitution cost = $1-\frac{p_{ij}+p_{ji}}{2}$ where $p_{ij}$ is the empirical rate of transitioning from state $i$ to $j$
• insertion/deletion cost = 1
seq_markov = Sequence(q5, dist_type="markov")
seq_markov

<giddy.sequence.Sequence at 0x7ff451d8df28>
seq_markov.seq_dis_mat

array([[ 0.        , 72.31052406,  6.34073233, ..., 19.02219698,
80.2334688 , 77.48002783],
[72.31052406,  0.        , 77.05042347, ..., 74.77437281,
50.75696949, 65.9128181 ],
[ 6.34073233, 77.05042347,  0.        , ..., 12.68146465,
80.97128589, 80.51785856],
...,
[19.02219698, 74.77437281, 12.68146465, ...,  0.        ,
80.10306616, 80.46369148],
[80.2334688 , 50.75696949, 80.97128589, ..., 80.10306616,
0.        , 41.57088046],
[77.48002783, 65.9128181 , 80.51785856, ..., 80.46369148,
41.57088046,  0.        ]])

### "tran"

Biemann, T. (2011). A Transition-Oriented Approach to Optimal Matching. Sociological Methodology, 41(1), 195–221. https://doi.org/10.1111/j.1467-9531.2011.01235.x

seq_tran = Sequence(q5, dist_type="tran")
seq_tran

<giddy.sequence.Sequence at 0x7ff451d8d588>
seq_tran.seq_dis_mat

array([[ 0., 23.,  8., ..., 12., 24., 21.],
[23.,  0., 17., ..., 16., 28., 22.],
[ 8., 17.,  0., ...,  4., 18., 16.],
...,
[12., 16.,  4., ...,  0., 21., 15.],
[24., 28., 18., ..., 21.,  0., 23.],
[21., 22., 16., ..., 15., 23.,  0.]])
seq_tran.seq_dis_mat

array([[  0., 220.,  25., ...,  55., 220., 220.],
[220.,   0., 241., ..., 199.,  93., 123.],
[ 25., 241.,   0., ...,  44., 241., 241.],
...,
[ 55., 199.,  44., ...,   0., 207., 220.],
[220.,  93., 241., ..., 207.,   0.,  84.],
[220., 123., 241., ..., 220.,  84.,   0.]])