Observations on Impostor Scores

Introduction

Based on a matrix representation of a fingerprint FAR examination certain conclusions on the nature of FAR determination are derived. Two experiments have been performed. In the first experiment every fingerprint reference has been matched against one sample of every request (query) fingerprint ID, excluding identical IDs. In the second case, every request sample of one impostor fingerprint ID has been matched against all reference IDs. These experiments shall give answers on questions such as:

• Is there an asymmetry between references and requests?
• What is the effect of several impostor trials with the same finger (ID)?

Experimental

• The first matrix (Table 1) comprises 68 columns for 68 successfully enrolled IDs used as reference (Ref) and 81 rows for all 81 IDs used as request (Req). The cells contain the match score for the corresponding IDs except for identical IDs.
• The second matrix (Table 2) comprises 68 columns for 68 successfully enrolled IDs used as reference (Ref) and 100 rows for all samples available for reference ID 10010 used as request (Req). The cells contain the score for the corresponding matches. The match with the identical ID is omitted in the calculations (marked red in the table).

Results

The following observations can be derived from Tables 1, 2, and 3:

1. The global mean and variance estimations of the "one sample" case and the "one ID" case deliver "significantly" different values which seem to favor the "one sample" case (Table 3). However this may be an artifact of the fact that the "one ID" case is ID specific whereas the "one sample" case is not.
2. In the "one sample" case the only "significant" deviation between the column and row distributions of the score values seems to be the measurement errors which are represented by "Sigma of Ref ID variance" and "Sigma of Req ID variance" and can be explained by the different number of IDs (Table 3)
3. The global score means within one case (Table 3) are equal to the means of the row and column means. (This is trivial.)
4. The sums of the variances of column / row means and the means of the column / row variances approaches the squared global score variances:

 

0.13 + 1.32 = 1.45 ~ 1.43		(one sample)
0.12 + 1.33 = 1.45 ~ 1.43		(one sample)
0.63 + 1.05 = 1.68 ~ 1.67		(one ID)
0.04 + 1.65 = 1.69 ~ 1.67		(one ID)

5. The variance of the means is small while the mean of the variances is near the global variance in both cases. (By theory, the variance of the means should approach zero and the mean of the variances approach the global variance for sufficiently high numbers of scores.)
6. In the "one ID case" there is an extreme difference between the Ref ID results and the Req sample results. Especially the variance of Ref ID means is unusually high (0.63).
7. In the "one ID" case the mean of the row standard deviations (1.29, estimated as the square root of the mean of Req sample variances) is significantly higher than the mean of the column standard deviations (1.03, estimated as the square root of the mean of Ref ID variance)
8. In the "one ID" case the variance of the Req sample variances (0.11) is considerably smaller than the variance of the Ref ID variances (0.40).
9. Suppose a test has been made with one request sample per ID (the first one in Table 2) and 68-1 reference IDs. Then the actual mean value would have been 1.22 (compared to 1.18 +- 0.20) and the actual standard deviation 1.14 (compared to 1.29).
10. Suppose a test has been made with only one request ID (the first one in Table 2) and one reference ID (the second one in Table 2) but 100 request samples. Then the actual mean value would have been 1.77 (compared to 1.18 +- 0.80) and the actual standard deviation 1.68 (compared to 1.03).
11. The second observation is that the measurement error (standard deviation) for the first test scenario is significantly smaller than for the second one:

 

Table 4: Results for different test scenarios

Calculation method		Mean of impostor score	Sigma of measurement error		Variance of impostor score	Sigma of measurement error
Global average: 6700 samples		1.18			1.67
ID average: 67 IDs, 100 samples		1.18	0.20		1.65	0.33
Test scenario 1: 67 IDs; 1 sample		1.22	(0.20)		1.30	(0.33)
Sample average: 67 IDs, 100 samples		1.18	0.80		1.05	0.63
Test scenario 2: 1 ID, 100 samples		1.77	(0.80)		2.83	(0.63)

• There are three kinds of impostor distributions which are not identical: one ID to one ID, one ID to many IDs, many IDs to many IDs (1, 6, 7, 8, 11)
• Request and reference prints seem to behave identical in this trial (2)
• The one-to-many impostor distribution (mean variance = 1.32) seems to be narrower than the many-to-many distribution (global variance = 1.43)
• The one-to-one impostor distributions (mean variance = 1.05) seems to be narrower than the one-to-many distribution (variance = 1.67)

Conclusions

• Although the number of IDs is smaller than the number of prints per ID, a test based on 67 IDs and 1 sample delivers more accurate results than a test based on 1 ID and 100 samples. For the planning of tests this means that it is more advisable to have a large number of participants than a large number of samples per finger (although this is easier to achieve).
• Due to the strong personal influences, each ID must be represented by the same number of samples when calculating global characteristics. Alternatively, the mean value of personal characteristics may be taken, provided it delivers the desired result. (Example: The average over all Ref ID means delivers an unbiased global mean value. This is not true for the average over all Ref ID variances which does not estimate the correct global variance. There should be no problem when calculating the impostor distribution or the FAR from the personal impostor distributions or the personal FARs by averaging.)

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