Here is an example from a small project that I did to split ecg data.
My approach was "switching autoregressive HMM" (google this if you haven’t heard it), where each datapoint is predicted from a previous datatot using a Bayesian regression model. I created 81 hidden states: an unwanted state to capture data between each hit and 80 separate hidden states corresponding to different positions in the heartbeat pattern. The state of pattern 80 was built directly from a selective single beat pattern and had two transitions — self-transition and transition to the next state in the pattern. The final state in the template goes either to itself or to an undesirable state.
I trained the model with Viterbi training , updating only the regression parameters.
In most cases, the results were adequate. A similar conditional random field structure will probably work better, but CRF training will require manual markings in the dataset if you have not already marked the data.
Edit:
Here is an example python code - it is not perfect, but it gives a general approach. He implements an EM training, not Viterbi, which can be a little more stable. Ecg dataset from http://www.cs.ucr.edu/~eamonn/discords/ECG_data.zip
import numpy as np import numpy.random as rnd import matplotlib.pyplot as plt import scipy.linalg as lin import re data=np.array(map(lambda l: map(float,filter(lambda x: len(x)>0,re.split('\\s+',l))),open('chfdb_chf01_275.txt'))).T dK=230 pattern=data[1,:dK] data=data[1,dK:] def create_mats(dat): ''' create A - an initial transition matrix pA - pseudocounts for A w - emission distribution regression weights K - number of hidden states ''' step=5 #adjust this to change the granularity of the pattern eps=.1 dat=dat[::step] K=len(dat)+1 A=np.zeros( (K,K) ) A[0,1]=1. pA=np.zeros( (K,K) ) pA[0,1]=1. for i in xrange(1,K-1): A[i,i]=(step-1.+eps)/(step+2*eps) A[i,i+1]=(1.+eps)/(step+2*eps) pA[i,i]=1. pA[i,i+1]=1. A[-1,-1]=(step-1.+eps)/(step+2*eps) A[-1,1]=(1.+eps)/(step+2*eps) pA[-1,-1]=1. pA[-1,1]=1. w=np.ones( (K,2) , dtype=np.float) w[0,1]=dat[0] w[1:-1,1]=(dat[:-1]-dat[1:])/step w[-1,1]=(dat[0]-dat[-1])/step return A,pA,w,K #initialize stuff A,pA,w,K=create_mats(pattern) eta=10. #precision parameter for the autoregressive portion of the model lam=.1 #precision parameter for the weights prior N=1 #number of sequences M=2 #number of dimensions - the second variable is for the bias term T=len(data) #length of sequences x=np.ones( (T+1,M) ) # sequence data (just one sequence) x[0,1]=1 x[1:,0]=data #emissions e=np.zeros( (T,K) ) #residuals v=np.zeros( (T,K) ) #store the forward and backward recurrences f=np.zeros( (T+1,K) ) fls=np.zeros( (T+1) ) f[0,0]=1 b=np.zeros( (T+1,K) ) bls=np.zeros( (T+1) ) b[-1,1:]=1./(K-1) #hidden states z=np.zeros( (T+1),dtype=np.int ) #expected hidden states ex_k=np.zeros( (T,K) ) # expected pairs of hidden states ex_kk=np.zeros( (K,K) ) nkk=np.zeros( (K,K) ) def fwd(xn): global f,e for t in xrange(T): f[t+1,:]=np.dot(f[t,:],A)*e[t,:] sm=np.sum(f[t+1,:]) fls[t+1]=fls[t]+np.log(sm) f[t+1,:]/=sm assert f[t+1,0]==0 def bck(xn): global b,e for t in xrange(T-1,-1,-1): b[t,:]=np.dot(A,b[t+1,:]*e[t,:]) sm=np.sum(b[t,:]) bls[t]=bls[t+1]+np.log(sm) b[t,:]/=sm def em_step(xn): global A,w,eta global f,b,e,v global ex_k,ex_kk,nkk x=xn[:-1] #current data vectors y=xn[1:,:1] #next data vectors predicted from current #compute residuals v=np.dot(x,wT) # (N,K) <- (N,1) (N,K) v-=y e=np.exp(-eta/2*v**2,e) fwd(xn) bck(xn) # compute expected hidden states for t in xrange(len(e)): ex_k[t,:]=f[t+1,:]*b[t+1,:] ex_k[t,:]/=np.sum(ex_k[t,:]) # compute expected pairs of hidden states for t in xrange(len(f)-1): ex_kk=A*f[t,:][:,np.newaxis]*e[t,:]*b[t+1,:] ex_kk/=np.sum(ex_kk) nkk+=ex_kk # max w/ respect to transition probabilities A=pA+nkk A/=np.sum(A,1)[:,np.newaxis] # solve the weighted regression problem for emissions weights # x and y are from above for k in xrange(K): ex=ex_k[:,k][:,np.newaxis] dx=np.dot(xT,ex*x) dy=np.dot(xT,ex*y) dy.shape=(2) w[k,:]=lin.solve(dx+lam*np.eye(x.shape[1]), dy) #return the probability of the sequence (computed by the forward algorithm) return fls[-1] if __name__=='__main__': #run the em algorithm for i in xrange(20): print em_step(x) #get rough boundaries by taking the maximum expected hidden state for each position r=np.arange(len(ex_k))[np.argmax(ex_k,1)<3] #plot plt.plot(range(T),x[1:,0]) yr=[np.min(x[:,0]),np.max(x[:,0])] for i in r: plt.plot([i,i],yr,'-r') plt.show()