Dynamic time-warping/warping/bending (WARPING,DTW)

Dynamic Time warping DTW

The distance we use most often in our daily life should undoubtedly be the European distance, but for some special cases, there are obvious flaws in Euclidean distance, such as time series, a simple example, sequence a:1,1,1,10,2,3, sequence b:1, 1,1,2,10,3, if the Euclidean distance, that is distance[i][j]= (B[j]-a[i]) * (B[j]-a[i]) to calculate, the total distance and should be 128, it should be said that the distance is very large, and actually the image of this sequence is very similar, In this case, someone began to consider the new method of calculating the distance of time series, then proposed the DTW algorithm, which plays an important role in speech recognition and machine learning convenience.
This algorithm is based on the idea of dynamic programming (DP), which solves the problem of template matching with different pronunciation length, and simply means to find the shortest path by constructing an adjacency matrix.
Also with the above 2 sequence as an example, a in 10 and B in the 2 corresponding to 2 in a and B in the 10 corresponding to the time, distance[3] and distance[4] must be very large, which leads directly to the last distance and expansion, this time, we need to adjust the time series, If we let the 10 in A and b in the corresponding 10, a in the 1 and B in the 2 corresponding, then the final distance and will be greatly shortened, this way can be seen as a time warp, see here, I believe it should be raised, why can not use the 2 in A and b in the corresponding problem, So the distance and must be 0, ah, the distance should be the smallest, but this is not allowed, because the 10 in a is in front of 2, and B 2 is in front of the 10, if the corresponding mode of intersection will lead to time confusion, does not conform to causality.
Next, to Output[6][6] (all the records subscript starting from 1, the beginning of all 0) to record the dtw distance between a, a simple introduction of the specific algorithm, the algorithm is actually a simple DP, the state transfer formula is output[i][j]=min (Min (Output[i-1][j],output[i][j-1]), output[i-1][j-1]) +distance[i][j] and the last output[5][5] is the DTW distance we need.
C Language implementation of DTW
#include <iostream>
#include <string.h>
using namespace Std;
#define NUM 5//The number of sample points in the sequence is simple, assuming that the sample points of 2 sequences are as many as
#define Min (A, B) (A&LT;B?A:B)
int main ()
{
int i,j,k;
int A[num],b[num];
int distance[num+1][num+1];
int output[num+1][num+1];
memset (distance,0,sizeof (distance));
memset (output,0,sizeof (output));
for (i=0;i<num;i++) cin>>a[i];
for (i=0;i<num;i++) cin>>b[i];
for (i=1;i<=num;i++)
for (j=1;j<=num;j++)
Distance[i][j]= (B[j-1]-a[i-1]) * (b[j-1]-a[i-1]); Calculates the Euclidean distance between a point and a point
for (i=1;i<=num;i++)
{
for (j=1;j<num;j++)
cout<<distance[i][j]<< ' \ t ';
cout<<endl;
}//output matrix of the entire Euclidean distance
cout<<endl;
for (i=1;i<=num;i++)
for (j=1;j<num;j++)
Output[i][j]=min (Min (output[i-1][j-1],output[i][j-1]), output[i-1][j]) +distance[i][j];
DP process, calculate DTW distance

for (i=0;i<=num;i++)
{
for (j=0;j<num;j++)
cout<<distance[i][j]<< ' \ t ';
cout<<endl;
}//Outputs the final dtw distance matrix, where Output[num][num] is the final dtw distance and

return 0;
}



Dynamic Time warping DTW

Dynamic time warping (DTW) is a mainstream method of speech recognition (warping).
The idea is: because the voice signal is a very large randomness of the signal, even if the same speaker on the same word, each pronunciation of the results are different, it is not possible to have exactly the same length of time. Therefore, when matched to the stored model, the timeline of the unknown word is distorted or bent evenly so that its characteristics are aligned with the template feature. It is a very powerful measure to improve the recognition accuracy of the system by using time-warping method.
Dynamic time warping (DTW) is a typical optimization problem, which uses the time-warping function W (n) which satisfies certain conditions to describe the time correspondence between the input template and the reference template, and solves the normalized function corresponding to the minimum accumulative distance of the two template matches.

™ combines time-warping with distance measurement, using dynamic programming technology to compare two different sizes of patterns to solve the problem of variable speed in speech recognition;
™ A nonlinear time-structured pattern matching algorithm;

DTW (Dynamic time warping), which is either "dynamically temporal distortion" or "dynamic time warping." This is a set of methods that are rooted in "dynamic programming" (dynamically programming, referred to as DP), which effectively reduces the time to search.
The goal of DTW is to find the shortest distance between two vectors. In general, for vectors x and y in two n-dimensional spaces, the distance between them can be defined as a straight-line distance between two points, called the Ulla Distance (Euclidean Distance).
Dist (x, y) = |x–y| ，
But if the lengths of the vectors are different, the distances between them cannot be calculated using the above mathematical formula. In general, the assumption that the element position of both vectors represents time, because we have to tolerate deviations in the timeline, so we don't know the element correspondence between the two vectors, so we have to rely on a set of efficient algorithms to find the best correspondence.

DTW is used for speech recognition in relation to the Speaker (Speaker Dependent), which is recorded by the user in his own voice and then compared to the previous audio recording. This method is more suitable for the voice of the same speaker, so the scope of application is narrower, such as the current phone Name dialing and so on.

DTW's question:
Large number of ™ operations;
™ Recognition performance relies too much on endpoint detection;
™ relies too much on the speaker's original pronunciation;
™ cannot dynamically train the sample;
™ does not make full use of the temporal dynamic characteristics of speech signals;
DTW is suitable for the small case of the specific human primitive, and it is used to identify the isolated words more.

The general idea of dynamic programming algorithm
The basic idea of dynamic programming algorithm is to decompose the problem to be solved into several sub-problems
But the sub-problems that are decomposed are often not independent of each other. The number of different sub-problems is often only polynomial magnitude. When solving, some sub-problems are repeatedly computed many times.
If you can save the answers to the solved sub-problems, and then find out the answers that have been obtained when needed, you can avoid a large number of repeated computations to get the polynomial time algorithm.

Dynamic Planning Basic Steps
The property of the optimal solution is found out and its structural characteristics are engraved.
Defines the optimal value recursively.
Calculates the optimal value in the bottom-up way.
The optimal solution is constructed according to the information obtained when the optimal value is calculated.


First, dynamic time regularization of the proposed
The speech signal has the very strong randomness, the different pronunciation habit, the pronunciation is in the environment different, the mood different will cause the pronunciation duration to differ the phenomenon. such as the last sound of the word with some drag, or take a little breath sound, at this time, because the drag or breath sound will be mistaken for a phoneme, resulting in the word endpoint detection is not allowed, resulting in the change of characteristic parameters, thus affecting the measure estimation, reduce the recognition rate, so in the speech recognition, first of all, it is necessary to time
Second, the definition of dynamic time regularization
A correct pronunciation should contain all the sounds that make up the sound and the correct order of the phoneme connection. The duration of each phoneme is related to the phoneme itself and the state of the speaker. In order to improve the recognition rate and overcome the difference of pronunciation time, the input speech signal is stretched or shortened until it is consistent with the length of the standard mode. This process is called time warping.
Three, dynamic time-warping principle description
In the 60 's, it was suggested by Japanese scholars that the idea of an algorithm was to lengthen or shorten the unknown (compression) until it was consistent with the length of the reference template, in which the time axis of an unknown word would distort or bend so that its characteristic volume corresponded to the standard pattern.
Principle Description
DTW is a combination of time-warping and distance measurement calculations. The test speech parameters have an I-frame vector, and the reference template has J-frame vectors, I and J, looking for a time-warping function j=w (i), which maps the time axis I of the test vector to the template's timeline J, and causes the function w (i) to satisfy:
Dynamic time-warping/warping/bending (<wbr>time <wbr>warping,dtw)
Formula
The distance measure d between the I frame Test vector T (i) and the J frame template vector R (j).
The distance between all vector frames in the case of optimal time normalization is also called the cost function.
Dynamic time-warping/warping/bending (<wbr>time <wbr>warping,dtw)
Equation 2
The Euclidean distance between two cepstrum vector frames (i and J) is computed, and the two vector frames have P-cepstrum parameters respectively.
In order to make the first sample of T (test) and R (Reference) J Sample pair positive, the corresponding point is not on the diagonal line, a curved curve j=w (i) is obtained. J=w (i) is called a regular function.
The basis of time normalization
Set T={A1, A2, ...., Ai, ....., ai} i=1~i r={b1, B2, ...., BJ, ... j=1~j i≠j time-warping the problem to be solved is to match the element A and element B so that the difference between each pair of matched samples Minimum, to reach the least Euclidean distance.





Veeraraghavan and others use dynamic time warping (WARPING,DTW) to match motion sequences. DTW is a time-varying data sequence matching method, which is commonly used for DNA matching, string and symbol comparison, and speech analysis in microbiology [77]. The idea of DTW algorithm is to find an optimal time-warping function for a given reference template feature vector sequence and an input feature vector sequence, so that the total cumulative distortion of the time axis of the input sequence is mapped to the reference template. For DTW, even if the time-scale of the test sequence pattern and the reference sequence pattern is not exactly the same, the time order constraint exists, it can still well complete the pattern matching between the test sequence and the reference sequence. DTW has the advantages of simple concept and robust algorithm, and can classify image sequences. The paper [35] calculates the distance between two shape sequences by dynamic time regularization method in the shape space to identify the movement and gait, and obtains a good classification result. However, the DTW algorithm is computationally large and lacks the dynamic characteristics between adjacent sequential sequences, while in practice, the adjacent sequence in the motion sequence is highly correlated in time and space.
1) Dynamic Time War ping (DTW)
Dynamic time axis bending or dynamic regulation Dhamma (DTW)
2) DTW
Dynamic time warping (DTW)
3) Dynamic Time warping
Dynamic Time Bending
1.Time series similar pattern matching based on wavelet and dynamic time warping;
Similarity matching of time series based on wavelet and dynamic time bending
2.Similarity matching algorithm for interval-valued time series based on dynamic time warping;
Interval value time series matching algorithm based on dynamic time bending
3.Through Dynamic time warping analysis to the Hot-fire test data and simulated fault data of a certain liquid rocket engi Ne,the warped Path sets were obtained.
In this paper, the thermal test data of a large liquid rocket engine and the fault data obtained by the engine model simulation are analyzed dynamically, the curved path set is obtained, and then the fault detection and diagnosis is carried out by the decision tree method.
More examples >>
4) Dynamic time warping (DTW)
Dynamic Time Bending
1.An efficient lower bounding technique is proposed based on Dynamic time warping (DTW) for time series similarity search,w Hich measures the distance between original sequence reduced dimensionality by piecewise Aggregate approximation (PAA) appr Oximation method and query sequence reduced dimensionality by Grid Minimum bounding Rectangle (gmbr) representation APPROAC H.
Aiming at time series data, a new nether technique based on dynamic time bending is proposed, which is based on the linear representation of piecewise aggregation approximation to reduce the dimension of the original sequence, and the minimum boundary rectangle of the grid of the query sequence is generated, and then the lower bound distance is measured based on the dynamic time bending distance.
More examples >>
5) Dynamic Time alignment
Time Dynamic Regulation Dhamma
6) Dynamic Time warping
Dynamic time Regulation Dhamma
1.Gait contour by using Fourier descriptors is described,to make quasi-periodic analysis on height and width ratio of gai T Image,and to solve the problems resulting from image sequence of different gait cycle by using dynamic time warping.
By using Fourier descriptors to describe the gait contour image, the quasi-periodicity of gait is analyzed by using the aspect ratio of gait image, and the dynamic time regulation Dhamma algorithm is used to solve the problem of the comparison between the different gait cycles of image sequences.
2.In the system, RelAtive sepetral (RASTA) filter is used to reduce the convolution channel noise mixed in speech signal, And the improved dynamic Time warping (DTW) algorithm is adopted for recognizing speech command.
The system uses Rasta filtering method to remove the convolution channel noise in speech signal, and uses the improved dynamic time regulation Dhamma (DTW) algorithm to recognize the Voice command.

DTW (Dynamic time warping), which is based on the idea of dynamic programming (DP), solves the problem of template matching with different pronunciation length, and is an early and more classical algorithm in speech recognition. For the isolated word recognition, the DTW algorithm and the HMM algorithm need to provide a lot of speech data in the training stage, and the model parameters can be obtained by repeated computation, while the DTW algorithm seldom needs extra computation in training. So in the speech recognition of isolated words, the DTW algorithm is still widely used.


Dynamic bending distance of parallel computing time series on cluster system
Mo Zhong Cheng
The similarity measure of time series is the basis to measure the similarity of two sequences. The dynamic time bending distance calculation method has strong robustness and can measure the similarity between time series of different lengths, but it is time-consuming. The dynamic time bending distance is calculated by using the wave-front propulsion method and the local sub-results are transmitted in parallel with the pipeline. A parallel algorithm for measuring the similarity of two time series in a cluster system is proposed. Experimental results on PC cluster system show that the parallel algorithm is efficient and has good acceleration and expandability.
Keywords: time series; dynamic time bending; parallel computing; cluster system

Research on time-series clustering algorithm based on piecewise linear dynamic time bending
Ong Yingjun Zhu Zhongying
Time series is a kind of important complex type data, and time series Knowledge discovery is becoming one of the hotspots of knowledge discovery. Euclidean distance and its extension are widely used in the comparison of time series, but the distance measure has no good robustness to the data. Dynamic time bending technology is a pattern matching algorithm based on nonlinear dynamic programming, but its computational complexity is quite high. In this paper, a dynamic time-bending algorithm based on piecewise linear representation of time series is proposed, and the matching of sequences is obtained by calculating the shortest curved path between linear piecewise series data. The comparison results of cluster analysis based on different distance measure for integrated control time series data show that the proposed algorithm has high accuracy and strong robustness to amplitude difference, noise and linear drift, greatly reduces computational complexity and has good application value.

1) Segmented Dynamic time warping Distance
piecewise Dynamic Bending Distance
1.After finding similar segmented subsequence by segmented Dynamic time warping Distance,this method then search this subs Equence point by Point,by which the subsequence is acquired accurately.
The algorithm first uses the median distance threshold and the extremum point two constrained piecewise linear fitting time series, obtains the similar sub-series by the piecewise dynamic bending distance measure, and retrieves the exact query of the subsequence sequence by point.
2) Time warping Distance
Dynamic bending distance
3) Segmented time warping distance
Segment Time Bend distance
4) Dynamic Time warping distance
Dynamic Time bending distance
5) Range Curvature Distortion
Distance Bend
1.In This dissertation, the presence of range curvature distortion in conventional PFA are analyzed from data format Perspe Ctive.
In this paper, we discuss the polar format algorithm of the classical algorithm of the bunching pattern (polar algorithm, PFA), and analyze the limitation of the distance bend in the PFA from the angle of signal format to the size of the high resolution imaging area, and give the general method of compensating distance bending. However, the space variability of the distance curvature makes the compensation complex and computationally large.
More examples >>
6) Range curvature distinction
Distance Bend Difference
1.Effect caused by range curvature distinction for deception jamming under SAR imaging
Effect of distance bend difference on SAR deception jamming imaging

Dynamic time-warping algorithm http://www.doc88.com/p-91694157837.html
Dynamic time-warping algorithm http://www.docin.com/p-86210991.html
Dynamic bending distance http://wenku.baidu.com/view/c40dabba1a37f111f1855b60.html of parallel computing time series on cluster system


Dynamic Time Warping
The confused with the time Warp mechanism for discrete event simulation, or the time Warp Operating System that used This mechanism.

Dynamic time Warping (DTW) is a algorithm for measuring similarity between, sequences which may vary in time or speed. For instance, similarities in walking patterns would is detected, even if in one video the person is walking slowly and If in another he or she were walking more quickly, or even if there were accelerations and decelerations during the course of one observation. DTW have been applied to video, audio, and graphics-indeed, any data which can is turned into a linear representation can is analyzed with DTW. A well known application have been automatic speech recognition, to cope with different speaking speeds.
In general, DTW are a method that allows a computer to find a optimal match between both given sequences (e.g. time series) With certain restrictions. The sequences is "warped" non-linearly in the Time dimension to determine a measure of their similarity independent of CE Rtain non-linear variations in the Time dimension. Thissequence alignment method is often used in the context of hidden Markov models.
One example of the restrictions imposed on the matching of the sequences are on the monotonicity of the mapping in the time Dimension. Continuity is less important in DTW than on other pattern matching algorithms; DTW is a algorithm particularly suited to matching sequences with missing information, provided there was long enough seg ments for matching to occur.
The extension of the problem for two-dimensional ' series ' like images (planar warping) are np-complete, while the problem F Or one-dimensional signals like time series can is solved in polynomial time.
Example of one of the many forms of the algorithm
This example illustrates the implementation of a dynamic time warping when the sequences strings is discrete of symbols . D (x, y) is a distance between symbols, i.e. D (x, y) = | X-y |.
int dtwdistance (char S[1..N], char T[1..M]) {Declare int DTW[0..N, 0..M] Declare int i, J, cost for I: = 1 to M dtw[0, I] : = Infinity for I: = 1 to n dtw[i, 0]: = Infinity dtw[0, 0]: = 0 for I: = 1 to N for j: = 1 to M cost:= D (S[i], t[j]) DTW [I, j]: = Cost + Minimum (Dtw[i-1, j],//Insertion dtw[i, j-1],//deletion dtw[i-1, j-1])//Match return dtw[n, M]}
We sometimes want to add a locality constraint. That's, we require that if s[i] are matched with t[j], then |i-j | is no larger than W, a window parameter.
We can easily modify the above algorithm to add a locality constraint (differences marked in bold italic). However, the above given modification works only if |n-m| is no larger than W, i.e. the end point is within the window length from diagonal. The algorithm work, the window parameter w must is adapted so, |n-m≤w| (See the line marked with (*) in the code).
int dtwdistance (char S[1..N], char t[1..m], int w) {declare int DTW[0..N, 0..M] Declare int i, J, cost W: = Max (W, ABS (n m)//ADAPT window size (*) for I: = 0 to N for j:= 0 to M Dtw[i, j]: = Infinity dtw[0, 0]: = 0 for I: = 1 to N for j: = Max (1, i-w) to Min (M, i+w) Cost: = D (S[i], t[j]) Dtw[i, j]: = Cost + Minimum (Dtw[i-1, j],//Insertion dtw[i, j-1],//D Eletion dtw[i-1, j-1])//Match return dtw[n, M]}
Open Source Software
The lbimproved C + + library implements Fast Nearest-neighbor retrieval algorithms under the Dynamic Time warping (GPL). It also provides a C + + implementation of Dynamic time warping as well as various lower bounds.
The R package DTW implements most known variants of the DTW algorithm family, including a variety of recursion rules (also Called step patterns), constraints, and substring matching.
The Mlpy Python Library implements DTW.
References
Sakoe, H. and Chiba, S., Dynamic programming algorithm optimization for spoken word recognition, IEEE transactions on Acou Stics, Speech and Signal processing, (1) pp. 43-49, 1978, issn:0096-3518
C. S. Myers and L. R. Rabiner. A Comparative Study of several dynamic time-warping algorithms for connected word recognition. The Bell System Technical Journal, (7): 1389-1409, September 1981.
L. R. Rabiner and B. Juang. Fundamentals of speech recognition. Prentice-hall, Inc., 1993 (Chapter 4)
See Also
Levenshtein distance
Elastic Matching
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Dynamic time-warping/warping/bending (WARPING,DTW)