Using Hidden Markov Models to analyse time series data

Transcription

1 Using Hidden Markov Models to analyse time series data September 9, 2011

2 Background Want to analyse time series data coming from accelerometer measurements. 19 different datasets corresponding to different individuals Figure 1: Typical dataset

3 Background Want to analyse time series data coming from accelerometer measurements. 19 different datasets corresponding to different individuals Aim: classify each datapoint as belonging to some state Figure 1: Typical dataset

4 Background Want to analyse time series data coming from accelerometer measurements. 19 different datasets corresponding to different individuals Aim: classify each datapoint as belonging to some state Compared models with 2-6 states Figure 1: Typical dataset

5 Exploratory data analysis

6 Exploratory data analysis Underlying process creating the jumps - use Hidden Markov Model (HMM)

7 Exploratory data analysis Underlying process creating the jumps - use Hidden Markov Model (HMM) PACF supports Markov assumption

8 Hidden Markov Models Assume data depends on a hidden Markov process

9 Hidden Markov Models Assume data depends on a hidden Markov process Each observation O t corresponds to a state q t.

10 Hidden Markov Models Assume data depends on a hidden Markov process Each observation O t corresponds to a state q t. Markov process transitions between states according to transition matrix A.

11 Hidden Markov Models Assume data depends on a hidden Markov process Each observation O t corresponds to a state q t. Markov process transitions between states according to transition matrix A. Each state has a different probability distribution

12 Transforming the data Model assumes normal distribution in each state

13 Transforming the data Model assumes normal distribution in each state Use Haar transform to decompose vector v = (v 1, v 2,..., v N ) into averages and differences:

14 Transforming the data Model assumes normal distribution in each state Use Haar transform to decompose vector v = (v 1, v 2,..., v N ) into averages and differences: Averages: am 1 = v 2m 1 + v 2m 2 Differences: d 1 m = v 2m 1 v 2m 2

15 Fisz transform After decomposing into averages and differences, want to normalize data

16 Fisz transform After decomposing into averages and differences, want to normalize data Use Fisz transform: f i m = d i m a i m

17 Fisz transform After decomposing into averages and differences, want to normalize data Use Fisz transform: f i m = d i m a i m Finally perform inverse Haar transform

18 Fitting the HMM Model is described by parameters λ = (A, {b i }, {π i })

19 Fitting the HMM Model is described by parameters λ = (A, {b i }, {π i }) Want to find the sequence of states that maximises the probability of the observations O 1, O 2,..., O t.

20 Fitting the HMM Model is described by parameters λ = (A, {b i }, {π i }) Want to find the sequence of states that maximises the probability of the observations O 1, O 2,..., O t. 2 problems in fitting the model: 1 How to choose the optimal sequence of states given O and λ. 2 How to adjust the parameters λ = (A, {b i }, {π i }) to maximise P(O λ).

21 Problem 1 Want to find the optimal path of states corresponding to observations

22 Problem 1 Want to find the optimal path of states corresponding to observations Ititialise: δ 1 (i) = π(i)b i (O 1 ).

23 Problem 1 Want to find the optimal path of states corresponding to observations Ititialise: δ 1 (i) = π(i)b i (O 1 ). General case: δ t+1 (j) = [max i δ t (i)a ij ] b j (O t+1 )

24 Problem 2 Probabilities of observations depend on the path through the parameters λ. Use these probabilities to estimate parameters by an iterative method:

25 Problem 2 Probabilities of observations depend on the path through the parameters λ. Use these probabilities to estimate parameters by an iterative method: 1 Set initial parameters λ = (A, {b i }, {π i })

26 Problem 2 Probabilities of observations depend on the path through the parameters λ. Use these probabilities to estimate parameters by an iterative method: 1 Set initial parameters λ = (A, {b i }, {π i }) 2 Calculate the probability of the observations and transitions given λ.

27 Problem 2 Probabilities of observations depend on the path through the parameters λ. Use these probabilities to estimate parameters by an iterative method: 1 Set initial parameters λ = (A, {b i }, {π i }) 2 Calculate the probability of the observations and transitions given λ. 3 Use these probabilities to reestimate the model parameters, e.g. E(# transitions from state i to state j) ā ij = E(# transitions from state i)

28 Problem 2 Probabilities of observations depend on the path through the parameters λ. Use these probabilities to estimate parameters by an iterative method: 1 Set initial parameters λ = (A, {b i }, {π i }) 2 Calculate the probability of the observations and transitions given λ. 3 Use these probabilities to reestimate the model parameters, e.g. E(# transitions from state i to state j) ā ij = E(# transitions from state i) 4 Repeat steps 2 and 3 until convergence.

29 Results Dataset 1 State Likelihood AIC

30 Results Dataset 2 State Likelihood AIC

31 Conclusions

32 Conclusions The HMM describes the shape of the data fairly well, although better for some of the datasets than others.

33 Conclusions The HMM describes the shape of the data fairly well, although better for some of the datasets than others. Using 5 states for the model gives the highest likelihood and lowest AIC for both datasets.

34 Conclusions The HMM describes the shape of the data fairly well, although better for some of the datasets than others. Using 5 states for the model gives the highest likelihood and lowest AIC for both datasets. The fitting procedure is very sensitive to the initial parameters

35 Conclusions The HMM describes the shape of the data fairly well, although better for some of the datasets than others. Using 5 states for the model gives the highest likelihood and lowest AIC for both datasets. The fitting procedure is very sensitive to the initial parameters Some datasets display very rapid fluctuations in the fitted states which do not seem consistent with the data. Could require a minimum time spent in each state.

36 Thank you for listening!