cc [ flag... ] file... lmlib [ library... ] #include <mlib.h> mlib_status mlib_SignalDTWVectorPath_S16(mlib_d64 *dist, mlib_s32 *path, mlib_s32 *lpath, const mlib_s16 **dobs, mlib_s32 lobs, mlib_s32 sobs, void *state);
The mlib_SignalDTWVectorPath_S16() function performs dynamic time warping on vector data.
Assume the reference data are
r(y), y=1,2,...,N
and the observed data are
o(x), x=1,2,...,M
the dynamic time warping is to find a mapping function (a path)
p(i) = {px(i),py(i)}, i=1,2,...,Q
with the minimum distance.
In Kbest paths case, K paths with the K minimum distances are searched.
The distance of a path is defined as
Q dist = SUM d(r(py(i)),o(px(i))) * m(px(i),py(i)) i=1
where d(r,o) is the dissimilarity between data point/vector r and data point/vector o; m(x,y) is the path weighting coefficient associated with path point (x,y); N is the length of the reference data; M is the length of the observed data; Q is the length of the path.
Using L1 norm (sum of absolute differences)
L1 d(r,o) = SUM r(i)  o(i) i=0
Using L2 norm (Euclidean distance)
L1 d(r,o) = SQRT { SUM (r(i)  o(i))**2 } i=0
where L is the length of each data vector.
To scalar data where L=1, the two norms are the same.
d(r,o) = r  o = SQRT {(r  o)**2 }
The constraints of dynamic time warping are:
px(1) = 1 1 ≤ py(1) ≤ 1 + delta
and
px(Q) = M Ndelta ≤ py(Q) ≤ N
px(i) ≤ px(i+1) py(i) ≤ py(i+1)
See Table 4.5 on page 211 in Rabiner and Juang's book.
Itakura Type:
py  *** p4 p1 p0    ***  p2     *** px p3
Allowable paths are
p1>p0 (1,0) p2>p0 (1,1) p3>p0 (1,2)
Consecutive (1,0)(1,0) is disallowed. So path p4>p1>p0 is disallowed.
Due to local continuity constraints, certain portions of the (px,py) plane are excluded from the region the optimal warping path can traverse. This forms global path constraints.
See Equation 4.1503 on page 216 in Rabiner and Juang's book.
A path in (px,py) plane can be represented in chain code. The value of the chain code is defined as following.
============================ shift ( x , y )  chain code  ( 1 , 0 )  0 ( 0 , 1 )  1 ( 1 , 1 )  2 ( 2 , 1 )  3 ( 1 , 2 )  4 ( 3 , 1 )  5 ( 3 , 2 )  6 ( 1 , 3 )  7 ( 2 , 3 )  8 ============================ py  * 8 7 *  * 4 * 6  1 2 3 5  x0** px
where x marks the start point of a path segment, the numbers are the values of the chain code for the segment that ends at the point.
In following example, the observed data with 11 data points are mapped into the reference data with 9 data points
py  9  * * * * * * * * * **  /  * * * * * * * ** * *  /  * * * * * * * * * * *  /  * * * * ** * * * * *  /  * * * * * * * * * * *    * * * * * * * * * * *  /  * * * * * * * * * * *  /  * * * * * * * * * * *  / 1  * * * * * * * * * * *  + px 1 11
The chain code that represents the path is
(2 2 2 1 2 0 2 2 0 2 0)
See Fundamentals of Speech Recognition by Lawrence Rabiner and BiingHwang Juang, Prentice Hall, 1993.
The function takes the following arguments:
dist
path
lpath
dobs
lobs
sobs
state
The function returns MLIB_SUCCESS if successful. Otherwise it returns MLIB_FAILURE.
See attributes(5) for descriptions of the following attributes:

mlib_SignalDTWVectorInit_S16(3MLIB), mlib_SignalDTWVector_S16(3MLIB), mlib_SignalDTWVectorPath_S16(3MLIB), mlib_SignalDTWVectorFree_S16(3MLIB), attributes(5)