2 nd Assignment ECG Signal


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1 Czech Technical University Prague Master in Biomedical Engineer 1 st Semester (2012/2013) Faculty of Mechanical Engineering Simulation of Biological Systems Teacher Ivo Bukovsky 2 nd Assignment ECG Signal Ana Filipa Vieira, nº Sandro Nunes, nº
2 Índex Page Introduction 3 Simulations, Results and Discussion 4 Conclusion 15 2
3 Introduction This assignment ought to describe a static model whose objective is to predict the future behavior of an ECG signal based on known data. To do so, we will use two samples: one is artificial, showing a perfect ECG behavior, while the other is taken from a real ECG exam, showing signs of arrhythmia and noise. In the first step, we will be using an interval of each of these samples to train our model. This consists in calculating the weights (parameters of the linear function) by measuring the error (difference between the modeled signal and the original data) and updating them accordingly, cycle by cycle. Once we have determined a linear function that can model the ECG signal in this interval, we will apply it to another interval of the samples and analyze its capacity to predict the future behavior of the ECG signal. In the second chapter, we will use our model to find novelties errors in the data generated. This is an important step as it indicates whether the information given by the model can be trusted or not. To do so we will make use of error calculation and weights updates to evaluate the model s response to addition of random perturbations. The model will be built in Matlab. Its schematic representation is shown below: Illustration 1 Scheme of the static model used. yr.  real data y  model output k  discrete index of time w  adapted weights 3
4 Simulations, Results and Discussion Training In order to build the static model needed to predict the future behavior of the ECG signal, we wrote the following code in Matlab: Illustration 2 Training code. To obtain this model, we used data from both a real and an artificial ECG signal. This model gathers information from the last 4 points to calculate the next one. Thus, we need a total of 5 weigths to model our function. These weights are being updated in each cycle based on the error calculated between the real value (obtained from the known data) and the value being calculated by the model. To obtain more reliable results for the weights, we added an external cycle, which goes through the data 30 times. Moreover, we added some normalization to the model in order to guarantee its stability. Finally, we stored the weights in a vector so that 4
5 they can be used in the next steps (testing and novelty analysis) and plotted both the original function and the one given by the model to compare them. Illustration 3 Original and modeled artificial ECG and Weights Illustration 4 Zoom in of the Artificial ECG signals Illustration 5 Original and modeled real ECG and Weights 5
6 Illustration 6 Zoom in of the Real ECG signals As we can see in both cases, the plot of the data and the function given by the model almost coincide, which shows that the weights calculated can model the signal with relatively high precision (this fact can clearly be seen in the zoomed in pictures). In addition, we can see that the weights are converging into a fixed value as their plot is almost horizontal for a high number of cycles. It is also worth mentioning that the artificial ECG shows a perfect periodic signal, with the same interval between each wave complex. In the contrary, there is no periodicity in the real ECG, showing some peaks characteristic of arrhythmia and some noise. Testing In order to evaluate the precision of our model, we applied it to another interval of the data samples (the purpose of this model is to predict different ECG signals based on the data collected from the one provided but, since we do not have other samples available, we decided to simply apply it to a different interval). The testing code is shown below: Illustration 7 Testing code 6
7 In this case, we are not updating the weights. Instead, we load the weight vector calculated in the training session to predict the future values of y. Since we want to predict the values from a different interval, we start the testing session at k=1001. Below are shown the plotting of both the original and modeled signals (artificial and real), as well as the difference between them: Illustration 8 Plotting of the original and modeled artificial ECG signals and its error Illustration 9  Zoom in of the original and modeled artificial ECG signals and its error 7
8 Illustration 10  Plotting of the original and modeled real ECG signals and its error Illustration 11  Zoom in of the original and modeled real ECG signals and error 8
9 Illustration 12 Zoomed in curve to evidence the following phenomenon If we look at the first graphs of each case either real or artificial  (with no zoom), the original and modeled signals almost overlap, which shows that, apparently, the model is capable of predicting the future values. However, if we look at the zoomed in graph of the real modeled signal, we can see that there is an increased difference comparing to the training plot. This is expected since, in the training session, the weights were being updated cycle by cycle, while, in testing, the same fixed weights are used throughout the entire interval. Regarding the artificial signal, the results are the same both in the training and testing: in spite of the fact that we are using a different interval, the signal is periodic throughout the whole the data and, thus, it does not matter which interval we are using as the weights calculated will be the same. Regarding the error (difference between the original and modeled signals) plots, we can see that it is periodical (as expected) in the artificial case, while in the real case there are some spikes. These are a consequence of the arrhythmia spikes and noise in the real data, which are not well predicted by the model. This was expected since the model was trained with data that only shows spikes occasionally, which does not reflect very heavily in the weights calculations. Finally, we need to mention the fact that this is a static linear model, which is not the ideal model to predict a sinuslike signal such as the ECG. Therefore, it seems that it is just following the original signal as evidenced in the last picture. Here we can see that, when the original plot goes up or down, the modeled signal responds by going up or down as well, only delayed. Novelty Analysis in Real ECG Instead of predicting the future values, in this chapter the aim is to find novelties (errors) in the generated data. To do so, we will be using the following Matlab code: 9
10 Illustration 13 Code for Novelty Analysis Finding novelties require keeping track of the error (difference between original and modeled signals) and the weights increment. Thus, we created two new vectors which store the module of both the error and the weights increments: e and absdwall. The novelties are, thus, found by multiplying these two vectors: if one or both of these values are high enough, we are able to detect a spike in the novelty plot, thus allowing us to find where the model fails to predict correctly. Moreover, since we need the previous 4 original values of y, we had to create an auxiliary vector which stores this information. At the end of the cycle this vector is substituted by the current 4 values of y and so on. Apart from these additions, once more we had to add some normalization to the model to guarantee its stability. Below we present the plotting of the original and modeled signal, the error, the weights increments and the product of these last two (novelty plot). First, we present the unchanged real signal and then we will present 2 examples where we introduce some random perturbation to the data. To allow for a better comparison, we also present a zoom in of the intervals where the perturbations will be added. 10
11 Illustration 14  Original and modeled signals (Green), Error (Red), Absolute value of Weights increments and Novelty plots In this first case (unchanged), we can see that the model still detects some anomalies around the peaks, which was expected. As discussed before, the model was trained with data with only occasional spikes, which makes it impossible for the model to correctly predict them. Illustration 15  Zoom in of the interval [7440:7445] 11
12 Illustration 16  Zoom in around point k=10000 These two graphs show a close up of the interval where the perturbations will be introduced next. As we can see, both the error and the weight increments are very small around these intervals and no novelty is detected. Now, we will introduce a random perturbation around the interval 7440 to 7445 and analyse what happens. This is done by introducing the following line of code in the beginning of the program: In this first graph, the noise introduced is not very noticeable in the signal plotting. However, even zoomed out, we can see a big spike in the error and weights incremements, which naturally reflects on the novelty plot. Illustration 17  Original and modeled signal with introduced perturbation on an interval (Green), Error (Red), Absolute value of Weights increments and Novelty plots 12
13 In this zoomed in picture, the difference is even more noticeable. Now we can see a change in the signal plot, as well as an increase in the value of the other plots. Illustration 18 Zoom in around interval [7440; 7445] In the next example, the perturbation is introduced, not in an interval, but in a single point (k=10000). Once more, the difference is not noticeable in the signal plot but the change is apparent in the error, weight increments and obviously in the novelty plot. Illustration 19  Original and modeled signal with perturbation added on a point (Green), Error (Red), Absolute value of Weights increments and Novelty plots 13
14 The following zoomed in plots evidence this difference: Illustration 20  Zoom in around point k=
15 Conclusion As we have seen in the first part of this assignment, we tested the capacity of a static model to predict the behavior of an ECG signal. In one hand, we concluded that our model could predict with great accuracy the future behavior of the artificial ECG. This was somewhat expected since this is a periodic signal and, thus, it is sufficient to gather information about a small interval to calculate suitable weights to model the rest of the interval. However, this was not our main concern since the principal goal of this kind of models is to predict real ECG signals. Regarding this point, the results were a little bit behind. Since we obtained the weights based in data in which peaks are only occasional, the model struggled to predict the ECG behavior when an arrhythmia peak happened (the error increased at these points). However, it could still detect them. Looking closer to the signal plot, we could see that the model seemed to be only following the original data based on the previous values rather than actually predicting it. This shows one of the main weaknesses of this kind of signals: the ECG is a sinustype signal, which needs a more sophisticated model than a static linear model to describe it with high accuracy. In spite of its simplicity, our model still showed very useful, especially in the second chapter of this assignment. In the second chapter, we used the same model (with some modifications as explained before) to detect anomalies in the data generated. In this field, our model presented very good results as it was able to detect very small random perturbations in the signal. In spite of not being visible in the overview plotting of the signals, it still could detect novelties based on error and weight increments calculation. To summarize, we demonstrated that, in spite of their simplicity, static models can prove to be a useful tool in prediction of signals such as ECG and especially on novelty detection, which makes them more reliable. 15
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