"At the simplest level, an equation allows you to predict the results of an experiment without actually having to conduct it."

Brian Cox and Jeff Forshaw [106]

T-Model Formula

The following formula is used to estimate conservative (upper-bound) numbers of distal fingerprint look-alikes present in a given population group: 

L = 120 (R) / T (Log T)

Where,

L = Number of look-alikes or close matches 

R = Relevant fingerprint population

T = Total quantitative-qualitative discriminating value for the arrangement of ridge features 

The above formula is a derivative of the two foundational formula (described below under background and foundation for formula) and was developed by simple math in which the variable "P" (parts per fingerprint) was solved and incorporated into the formula that estimates numbers of look-alikes (L) as follows:

Background for Formula Development 

A rough, conservative estimate of the total discriminating value for the average amount of Level I and II [and Level III] ridge features present in the average flat fingerprint was needed in order to develop formula to estimate numbers of close matches or look-alikes in a given fingerprint population.  The average flat fingerprint was used because it is the best representation of a latent or evidence fingerprint.  This total discriminating value, or “T-Value”, was estimated as follows:

Based on the total number of ridge formation types found in the 39 fingerprints used in the Osterburg study, the number of occurrences for each ridge formation type per fingerprint was determined by dividing the frequency of occurrence for each by 39.  The discriminating value for each was raised to a power equivalent to the number of occurrences for that ridge feature type.  The discriminating values for each group of ridge feature types were multiplied, which established the aggregate discriminating value for the average flat fingerprint in terms of “type” (see below).   The same was done for ridge feature “position” (see below).  The product of these two values established the rough, conservative estimate for the total discriminating value or T-Value of ridge feature “types in position” for the average flat fingerprint.

Subsequently, this T-Value was applied to formula in order to estimate the numbers of close matches or look-alikes present in a given fingerprint population. 

Discriminating Value for Ridge Feature Shapes (Types)

Based on initial frequency of occurrence studies, i.e. the extrapolation of the Osterburg Study combined with experiment by the author (see below), the average flat fingerprint contains the following numbers of ridge features bearing the below discriminating values.

Experiment

The author examined 40 flat, distal fingerprints selected at random from ten-print files and counted the number of ending ridges in a funnel and the number of bifurcations in a funnel.  On average there were roughly 9.2 ending ridges in a funnel and 5.7 bifurcations in a funnel per fingerprint.  These numbers were subtracted from the total number of ending ridges and bifurcations found in the Osterburg study, i.e., 44.46 and 23.71 respectively, to estimate the number of ending ridges not in a funnel and number of bifurcations not in a funnel, i.e., 35.26 and 18.01 respectively.     

Based on the Osterburg study, 44.46 ending ridges were weighted at 13.34 each for an aggregate discriminating value of 13.34^44.46 and 23.71 bifurcations were weighted at 25.01 each for an aggregate discriminating value of 25.01^23.71.  These values were replaced with the below values for numbers of ending in/not in a funnel and bifurcations in/not in a funnel.

It is significant to note that regardless with values are used, there is little difference in the final aggregate weighting for each, and subsequently the final conservative discriminating value for Level I/II ridge feature types in position in the average flat fingerprint, i.e. 10^120, remains the same.

Discriminating Values for Ridge Feature Types 

The total discriminating value for ridge feature types in the average flat, distal fingerprint is estimated as follows:

44.46 ending ridges, i.e., 9.2 ending ridges in a funnel and 35.26 ending ridge not in a funnel for an aggregate discriminating value of 10^9.2 and 14.25^35.26 respectively.

23.71 bifurcations, i.e., 5.7 bifurcations in a funnel and 18.01 bifurcations not in a funnel, for an aggregate discriminating value of 18.75^5.7 and 26.75^18.01 respectively.

6.05 dots each weighted 45 for an aggregate weight of 45^6.05.*

.0256 trifurcations each weighted 23,136 for an aggregate weight of 23,136^.0256.

.9487 core areas each weighted 209.4 for an aggregate weight of 209.4^.9487.

.4358 delta areas each weighted 475 for an aggregate weight of 475^.4358.**

2.9 creases each weighted 634 for an aggregate weight of 634^2.9.

.41 scars each weighted 26,224 for an aggregate weight of 26,224^.41.

The combined aggregate weight for the above two (2) Level I ridge feature types (e.g., core and delta areas) and six (6) Level II" ridge features types is as follows:  1.84 x 10^106 

*  Based on extended experimentation on “dots”, the value for a single dot used by Osterburg is modified from 98, i.e., a match probability of 1/98, to 45, i.e., a match probability of 1/45 (see Validation Study).  

**  For purposes of simplicity the average value for a “Y-Shaped Delta” and “Non-Y Shaped Delta” was used, i.e., (190 + 570) / 2 = 475





Discriminating Value for Ridge Feature Positions

The total discriminating value for ridge feature positions in the average flat, distal fingerprint is estimated as follows:

Values for intervening ridge count to the nearest neighbor were used to estimate the total discriminating value for the average flat fingerprint by the following experiment:

39 flat fingerprints were selected at random from ten-print files.  A total of 3,738 minutiae were counted (i.e., ending ridges, bifurcations and dots).  Number of continuous ridge units, pores, scars and creases, were not included in this count. 

Based on the above data, there is an average of 96 minutiae per flat fingerprint.  As a result there are 48 pairs of nearest neighbor ridge features bearing intervening ridge counts.  Based on a previous intervening ridge count frequency of occurrence study from ridge features to its nearest neighbor, 64.52% minutiae pairs have an intervening ridge count equal to 0 or 1, i.e., these features possess no additional discriminating value to the value assigned to ridge feature type. 

However, the remaining minutiae pairs have intervening ridge counts greater than 1 and as a result possess additional discriminatory value, which are defined based on their frequency distribution.  These values are multiplied against the number of times they occur in the average flat fingerprint as follows:

11.8128 ridge feature types had an average of 2 ridge counts with a discriminating value of 4 each (24.61% of 48 = 11.8128).  As a result the total discriminating value for these events in the average flat fingerprint equals 4 ^ 11.8128, or 12,942,383.

3.1968 ridge feature types had an average of 3 ridge counts with a discriminating value of 15 each (6.66% of 48 = 3.1968).  As a result the total discriminating value for these events in the average flat fingerprint equals 15 ^ 3.1968, or  5,750.80

1.1472 ridge feature types had an average of 4 ridge counts with a discriminating value of 41 each (2.39% of 48 = 1.1472).  As a result the total discriminating value for these events in the average flat fingerprint equals 41 ^ 1.1472, or 70.82.

.6144 ridge feature types had an average of 5 ridge counts with a discriminating value of 78 each (1.28% of 48 = .6144).  As a result the total discriminating value for these events in the average flat fingerprint equals 78  ^ .6144, or 14.53.

.1632 ridge feature types had an average of 6 ridge counts with a discriminating value of 294 each (.34% of 48 = .1632).  As a result the total discriminating value for these events in the average flat fingerprint equals 294 ^ .1632, or 2.52

.0816 ridge feature types had an average of 7 ridge counts with a discriminating value of 588 each (.17% of 48 = .0816).  As a result the total discriminating value for these events in the average flat fingerprint equals 588 ^ .0816, or 1.68.

Subsequently, the aggregate discriminating value for all of the above events in the average flat fingerprint is estimated as follows:

(12,942,383) x (5,750.80) x (70.82) x (14.53) x (2.52) x (1.68) =  324,245,435,357,544.66

Note:  The aggregate discriminating value for Level III ridge features, i.e., continuous ridge units and pores, in the average flat fingerprint is estimated as follows: 

1.     The discriminating value for a single continuous ridge unit is estimated to be 1.1577.  There are approximately 512.41 continuous ridge units in the average flat fingerprint.  As a result the total discriminating value for these events in the average flat fingerprint is defined as 1.1577^512.41.

2.     The discriminating value for a single pore is estimated to be 5.398.  There are approximately 119.89 pores in the average flat fingerprint.  As a result the total discriminating value for these events in the average flat fingerprint is defined as 5.398^119.89. 

Subsequently the aggregate discriminating value for the above events in the average flat fingerprint is estimated as follows:  1.1577^512.41 x 5.398^119.89 = 2.37 x 10^120.  For purposes of conservatism the value 2.37 x 10^120 was rounded down to 10^120.

Total Discriminating Value for Level I/II Ridge Feature "Types in Positions":

The total discriminating value for the average number of ridge feature types in position in the average flat distal fingerprint was estimated by multiplying the values for each as follows:

(1.84 x 10^106) x (324,245,435,357,544.66) = 5.9 x 10^120 ≈ 10^120.

The discriminating value 10^120, i.e., the reciprocal of the fingerprint match probability, e.g., 1/10^120, for the average number of Level I and Level II ridge features types in position” present in the average flat fingerprint, e.g., not Level III. 

NOTE

The discriminating value for Level III detail in the average flat fingerprint is estimated using the total value for “continuous ridge units” and “pores” as follows:  512.41 continuous ridge units each weighted 1.1577 for an aggregate weight of 1.1577^512.41; 119.89 pores each weighted 5.398 for an aggregate weight of 5.398^119.89. 

The product of the total values for Level I, II and II ridge detail in the average flat fingerprint is estimated as follows:

(5.14 x 10^117) x (1.1577^512.44) x (5.398^119.89) = 1.21 x 10^238

For purposes of conservatism the value 1.21 x 10^238 is rounded up to 10^240.

The value 10^240 equates to the total discriminating value for Level I, II and III ridge features in the average flat fingerprint.  However, for purposes of performing latent print analysis during the course of routine criminal casework, Level III ridge features, e.g., matching pores and matching continuous ridge unit edge contours and widths, are not generally used or needed by latent print examiners to establish inference for identification.  As a result the value 10^240 is applied to the T-Model only when Level III ridge features are needed to establish inference for identification. 

Otherwise, only the aggregate value for Level I and Level II ridge features in the average flat fingerprint should be used. 

Underlying Assumptions

The discriminating value of 10^120 for the average flat fingerprint is not directly testable and therefore remains an underlying assumption in the T-Model formula.  Although this value is not directly testable, it can be indirectly tested by "close match" experiments in order to determine it's influence on the accuracy for the model to predict numbers of close matches likely to be present in a given population group.  Based on results from these experiments, the value 10^120 has, at this time, shown to be a relatively accurate, yet conservative, value needed to make the most accurate, reliable predictions for the number of close matches likely to be present in a given population group (see Validation Study).

Formula Foundation

The formula "L = 120(R)/T(Log T)" is the result of developing and combining the below two formulas and simply eliminating the variable "P" using simple mathematics:

T ^ P = 10 ^ 120

L = (R) (P) / T

where "P" equals the number of "parts" in a fingerprint.

The above tw formulas were developed as follows: 

The Number of "Parts (P)" in the Average Flat Fingerprint

The value for the estimated number of "parts" that an arrangement of ridge features bearing a given T-Value can exist in the average flat fingerprints estimated as follows:

The log of 10^120 divided by the log of the T-Value (T) for the aggregate quantitative - qualitative volume of ridge formations found in agreement between a latent fingerprint impression and an exemplar establishes the numbers of parts (P) where the part represents the amount of ridge detail, for example, in a latent.  P is the exponential power that T must be raised to equal the value for the average flat fingerprint (F). 

Since the value for P (fingerprint parts) is needed to calculate the estimated number of close matches or look-alikes for an arrangement of ridge features, the following formula may be used:

(T) ^ (P) = F

Where,

T = T-Value for the arrangement of ridge features.
P = Number of Parts in the average flat fingerprint (F)
F = 10^120 (the T-Value for the average flat fingerprint)

This equation is based on the concept that the product of the parts equals the whole.  This concept is demonstrated by the following math example:

2 x 2 x 2 x 2 x 2 (product of parts) = 32 (whole)

Based on this example, the number of parts equals 5, i.e. there are five parts with a value of 2 each.  If the T-Value for each part equals 2, then based on the above formula, the total value for the whole equals 32, or (2)^5.

Number of Look-alikes (L) in the Relevant Fingerprint Population  

The number of look-alikes (L) present in a given fingerprint population is calculated by multiplying relevant fingerprint population (R) to the number of parts (P) divided by the T-Value (T) for the latent or pair of corresponding impressions.  The following formula is used to define the number of look-alikes (L) likely to exist:

Where,

L = Number of Look-alikes
R = Relevant Fingerprint Population
P = Number of parts in the whole average flat fingerprint
T = T-Value for the particular arrangement of ridge features in a single impression or in agreement in two impressions

The formula L = (R)(P)/T is based on simple math where the number of look-alikes (L) is determined by multiplying the number of parts (P) for the designated latent found in the average whole flat fingerprint (previously defined by the equation T^P=10^120) and divided by the T-Value for the same latent.  When relevant fingerprint population (R) is multiplied by the number of parts (P), which is previously established by the equation T^P=10^120, the total number of parts equivalent to the latent in the given fingerprint population is defined.  When that value is then divided by the total value for the latent, the number of parts bearing that same value, i.e. look-alikes, that are likely to occur within that population, is estimated.

How to Calculate the T-Value for Fingerprint Ridge Features

It is significant to mention that the T-Value for an arrangement of ridge features in a fingerprint impression or shared features in two fingerprint impressions is calculated by multiplying the values for the following variables for each individual ridge feature, and then multiplying the product of each against each other:

T-Value =

Quantitative weight for ridge formation type multiplied by

Quantitative weight for ridge formation position, multiplied by

Qualitative reduction factor for reduced clarity and reliability, multiplied by

Qualitative reduction factor for reduced quality of agreement.

The final product defines the T-Value for the aggregate quantitative-qualitative amount of ridge formations present in either a single fingerprint impression or shared fingerprint impressions.

Sufficiency to infer positive identification depends on relevant population for the case at hand because as the relevant population increase so does the number of look-alikes likely to occur.

"It is the frequency of events within a given class relative to the population which so determines its probability."

Roy Huber