Sequential recognition procedures

If, in the previously considered recognition methods, the decision on whether an object was in one form or another was carried out immediately along the entire set of features, then in this section we will discuss the case of their consistent measurement and use.

Let be . First, the object is measured and on the basis of this information, the question of attributing this object to one of the images is decided. If this can be done with a sufficient degree of confidence, then other signs are not measured and the recognition procedure ends. If there is no such certainty, then the sign is measured. and the decision is made in two ways: and . Further, the procedure is either terminated or the sign is measured. and so on until either a decision is made to classify the object to any image, or all signs.

Such procedures are extremely important in cases where the measurement of each of the signs requires a significant expenditure of resources (material, temporary, etc.).

Let be and known Where Note that if the distribution is known then all distributions of smaller dimension are known (the so-called marginal distributions). For example,

Let measured signs. Build likelihood ratio If a then the object is attributed to the image , if a then to the image . If then the sign is measured and calculate the likelihood ratio etc.

It is clear that the thresholds and associated with the permissible probability of recognition errors. Achieving inequality we strive to ensure that the probability of correctly assigning the object of the first image to Was in times more than the erroneous assignment of the object of the second image to , i.e or . Insofar as then (upper threshold). Similar reasoning is carried out to determine . Achieving inequality we strive to ensure that the probability of correctly assigning the object of the second image to Was in times more than the incorrect assignment of the object of the first image to , i.e

,

,

(lower threshold).

In a consistent procedure for measuring signs, a very useful property of these signs is their statistical independence. Then and there is no need to store (and most importantly, build) multidimensional distributions. In addition, it is possible to optimize the sequence of the measured signs. If they are ranked in descending order of classification informativity (the amount of discriminating information) and a consistent procedure is organized in accordance with this ranking, the number of measured signs can be reduced on average.

We have reviewed the case of (two images). If a , then likelihood relationships can be built , for example of this type: Stopping boundary (threshold) for th image is chosen equal If a then th image is discarded and built likelihood ratios and thresholds . The procedure continues until only one image remains unreleased or all of them have been exhausted. signs. If in the latter case more than one image remained unreleased, the decision is made in favor of the one for which the likelihood ratio as much as possible.

If there are two images ( ) and the number of signs is not limited, then a sequential procedure with probability 1 ends in a finite number of steps. It is also proved that given and considered procedure with the same informativeness of various signs will give a minimum of the average number of steps. For Vald introduced a sequential procedure and called it a successive criterion for the ratio of probabilities (c.to.v.).

For optimal procedure is not proven.

With known prior probabilities, you can implement a Bayesian sequential procedure, and if you know the costs of character measurements and the matrix of penalties for incorrect recognition, then the sequential procedure can be stopped to minimize the average risk. The point here is to compare the losses caused by recognition errors when the procedure is terminated, and the expected losses after the next measurement plus the cost of this measurement. Such a problem is solved by the dynamic programming method if successive measurements are statistically independent. More detailed information on the optimization of the Bayes sequential procedure can be found in the recommended literature [8].