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  4. Elements of signal detection and estimation / Carl W. Helstrom. - Version details - Trove
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Course Catalogue for student exchange programmes. Cooperation with industry Research activities Educational activity. Recommended Links. Context Links.

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Specialised Academic Studies. Basic Information. Native organizations units.

Mastering the techniques of signal detection and estimation. Knowledge acquisition related to algorithms for signal detection and estimation.

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Elements of Signal Detection and Estimation

Detection of signals with known parameters. Signal detection with unknown parameters. They will miss more tumors, but they will be doing their part to reduce unnecessary surgeries. And they may feel that a tumor, if there really is one, will be picked up at the next check-up. These arguments are not about information. Two doctors, with equally good training, looking at the same CT scan, will have the same information. External noise: There are many possible sources of external noise. There can be noise factors that are part of the photographic process, a smudge, or a bad spot on the film.

Or something in the person's lung that is fine but just looks a bit like a tumor. All of these are to be examples of external noise. While the doctor makes every effort possible to reduce the external noise, there is little or nothing that they can do to reduce internal noise. Internal noise: Internal noise refers to the fact that neural responses are noisy.

These hypothetical tumor detectors will give noisy and variable responses. After one glance at a scan of a healthy lung, our hypothetical tumor detectors might fire 10 spikes per second. After a different glance at the same scan and under the same conditions, these neurons might fire 40 spikes per second. Internal response: Now I do not really believe that there are tumor detector neurons in a radiologist's brain. But there is some internal state, reflected by neural activity somewhere in the brain, that determines the doctor's impression about whether or not a tumor is present.

This is a fundamental issue; the state of your mind is reflected by neural activity somewhere in your brain. This neural activity might be concentrated in just a few neurons or it might be distributed across a large number of neurons. This internal response is inherently noisy. Even when there is no tumor present no-signal trials there will be some internal response sometimes more, sometimes less in the doctor's sensory system.


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Notice that we never lose the noise. The internal response for the signal-plus-noise case is generally greater but there is still a distribution a spread of possible responses. Notice also that the curves overlap, that is, the internal response for a noise-alone trial may exceed the internal response for a signal-plus-noise trial. Figure 1: Internal response probability of occurrence curves for noise-alone and for signal-plus-noise trials. Just to be really concrete, we could mark the horizontal axis in units of firing rate 10, 20, 30, This would mean that on a noise-alone no tumor trial, it is most likely that the internal response would be 10 spikes per second.

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It is also rather likely that the internal response would be 5 or 15 spikes per second. But it is very unlikely that the internal response would be 25 spikes per second when no tumor is present. Because I want to remain noncommittal about what and where in the brain the internal response is, I did not label the horizontal axis in terms of firing rates. The internal response is in some unknown, but quantifiable, units. The role of the criterion: Perhaps the simplest strategy that the doctor can adopt is to pick a criterion location along the internal response axis.

Whenever the internal response is greater than this criterion they respond "yes". Whenever the internal response is less than this criterion they respond "no". An example criterion is indicated by the vertical lines in Figure 2. The criterion line divides the graph into four sections that correspond to: hits, misses, false alarms, and correct rejections. On both hits and false alarms, the internal response is greater than the criterion, because the doctor is responding "yes''.

Hits correspond to signal-plus-noise trials when the internal response is greater than criterion, as indicated in the figure. False alarms correspond to noise-alone trials when the internal response is greater than criterion, as indicated in the figure. Figure 2: Internal response probability of occurrence curves for noise-alone and signal-plus-noise trials. Since the curves overlap, the internal response for a noise-alone trial may exceed the internal response for a signal-plus-noise trial. Vertical lines correspond to the criterion response. Suppose that the doctor chooses a low criterion Figure 3, top , so that they respond "yes'' to almost everything.

Then they will never miss a tumor when it is present and they will therefore have a very high hit rate.

Elements of signal detection and estimation / Carl W. Helstrom. - Version details - Trove

On the other hand, saying "yes'' to almost everything will greatly increase the number of false alarms potentially leading to unnecessary surgeries. Thus, there is a clear cost to increasing the number of hits, and that cost is paid in terms of false alarms.


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If the doctor chooses a high criterion Figure 3, bottom then they respond "no'' to almost everything. They will rarely make a false alarm, but they will also miss many real tumors. Notice that there is no way that the doctor can set their criterion to achieve only hits and no false alarms. The message that you should be taking home from this is that it is inevitable that some mistakes will be made. Because of the noise it is simply a true, undeniable, fact that the internal responses on noise-alone trials may exceed the internal responses on signal-plus-noise trials, in some instances.

Thus a doctor cannot always be right. They can adjust the kind of errors that they make by manipulating their criterion, the one part of this diagram that is under their control. ROC curves Figure 4 are plotted with the false alarm rate on the horizontal axis and the hit rate on the vertical axis.

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The figure shows several different ROC curves, each corresponding to a different signal strengths. We already know that if the criterion is very high, then both the false alarm rate and the hit rate will be very low, putting you somewhere near the lower left corner of the ROC graph. For an intermediate choice of criterion, the hit rate and false alarm rate will take on intermediate values. The ROC curve characterizes the choices available to the doctor.


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They may set the criterion anywhere, but any choice that they make will land them with a hit and false alarm rate somewhere on the ROC curve. Notice also that for any reasonable choice of criterion, the hit rate is always larger than the false alarm rate, so the ROC curve is bowed upward. Figure 4: Internal response probability of occurrence curves and ROC curves for different signal strengths.

When the signal is stronger there is less overlap in the probability of occurrence curves, and the ROC curve becomes more bowed. The role of information: Acquiring more information makes the decision easier. Unfortunately, the radiologist does not have much control over how much information is available.

20 Signal Detection Theory

In a controlled perception experiment the experimenter has complete control over how much information is provided. Having this control allows for quite a different sort of outcome. If the experimenter chooses to present a stronger stimulus, then the subject's internal response strength will, on the average, be stronger. Pictorially, this will have the effect of shifting the probability of occurrence curve for signal-plus-noise trials to the right, a bit further away from the noise-alone probability of occurrence curve.