"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 Formulae

T-Model formulae are used to calculate the following values:

1.  Number of Look-alikes (L) in the Relevant Population (RP)

2.  Total quantitative-qualitative value or T-Value (T) for an arrangement of ridge features.

Background

The total value for the aggregate quantitative weight for the average total volume of ridge formations present in the average flat fingerprint was needed in order to develop formulae to calculate numbers of look-alikes in a given fingerprint population.  In order to design formulae that could be applied to all fingerprints, the approximate total quantitative weight e.g., T-Value, for the average flat fingerprint needed to be estimated (the average flat fingerprint was used because it is the best representation of a latent or evidence fingerprint).  This 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 quantitative weight for each was raised to a power equivalent to the number of occurrences for that ridge formation type.  The quantitative weights for each group of ridge formation types were multiplied, which established the aggregate quantitative weight, e.g., T-Value, for the average flat fingerprint (see Table 7 below).   The T-Value was then applied to formulae in order to define duplication likelihoods based on relevant fingerprint population. 

T-Value for the Average Flat Fingerprint

Ridge Feature Shapes (Types)

Based on frequency of occurrence, the average flat fingerprint contains the following number of ridge features bearing the following weight:

44.46 ending ridges each weighted 13.34 for an aggregate weight of 13.34^44.46.

23.71 bifurcations each weighted 25.01 for an aggregate weight of 25.01^23.71.

6.05 dots each weighted 98.03 for an aggregate weight of 98.03^6.05.*

.0256 trifurcations each weighted 23136 for an aggregate weight of 23136^.0256.

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

.4358 delta areas each weighted 1360 for an aggregate weight of 1360^.4358.

2.9 creases each weighted 204.4 for an aggregate weight of 204.4^2.9.

.41 scars each weighted 1446 for an aggregate weight of 1446^.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:  1.71 x 10^117 

*Note:  Based on validation study for dots completed June 2010, the value for a single dot is refined to 45, e.g., a probability of 1/45, and the value for a cluster dot ranges from 3 to 9, e.g., a probability of 1/3 to 1/9 (see Validation Study). At this time, for purposes of conservatism, the upper-bound value for a dot derived from the Osterburg study, e.g., 98, a probability of 1/98, is used by the T-Model only to estimate the conservative, theoretical value for the average flat fingerprint to calculate numbers of "fingerprint parts" in the average flat fingerprint for a given arrangement of ridge features.  The higher the value is for the number of fingerprint parts, the more it benefits the defendant, since an increase in numbers of parts means that the T-Value needed to infer identification must be higher.

Ridge Feature Positions 

Aggregate weight for ridge feature positions is included as follows:

Expansion factors for average intervening ridge counts between nearest neighbors were  factored into the equation to define the T-Value for the average flat fingerprint as follows: 

Based on 3,738 minutiae found in 39 flat fingerprints (not including continuous ridges, pores, scars or creases) there is an average of 96 minutiae per flat fingerprint.  As a result there are 48 pairs of nearest neighbor ridge formations bearing intervening ridge counts.  Based on the ridge count frequency of occurrence study, 64.52% minutiae pairs had a ridge count equal to 0 or 1 and subsequently possessed no expansion factor.  The remaining minutiae pairs had ridge counts greater than 1 and subsequent expansion factors were defined as follows:

11.8128 ridge formation types had an average of 2 ridge counts with an expansion factor of 2.5 each (24.61% of 48 = 11.8128).  As a result the expansion factor equals 2.5 ^ 11.8128, or 50,209.48.

3.1968 ridge formation types have an average of 3 ridge counts with an expansion factor of 9.5 each (6.66% of 48 = 3.1968).  As a result the expansion factor equals 9.5 ^ 3.1968, or 1,335.32.

1.1472 ridge formation types have an average of 4 ridge counts with an expansion factor of 26.5 each (2.39% of 48 = 1.1472).  As a result the expansion factor equals 26.5 ^ 1.1472, or 46.43.

.6144 ridge formation types have an average of 5 ridge counts with an expansion factor of 50.0 each (1.28% of 48 = .6144).  As a result the expansion factor equals 50 ^ .6144, or 11.06.

.1632 ridge formation types have an average of 6 ridge counts with an expansion factor of 188.5 each (.34% of 48 = .1632).  As a result the expansion factor equals 188.5 ^ .1632, or 2.35.

.0816 ridge formation types have an average of 7 ridge counts with an expansion factor of 377.5 each (.17% of 48 = .0816).  As a result the ridge count expansion factor equals 377.5 ^ .0816, or 1.62.

The aggregate expansion factor for average ridge counts was subsequently defined as follows:  

(50,209.48) x (1,335.32) x (46.43) x (11.06) x (2.35) x (1.62) = 131,071,347,656

Note:  The combined aggregate weight for Level III ridge features in the average flat fingerprint is defined 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 combined aggregate weight for the above two (2) level III ridge feature types is as follows:  2.37 x 10^120.

The total quantitative weight for the above average number of Level II and Level III ridge feature types and expansion factors for ridge counts were multiplied.  As a result, the T-Value for the average flat fingerprint was defined as 2.45 x 10^239.  For purposes of conservativeness this figure may be rounded up to the nearest factor of 10 and defined as 10^240.

The value 10^240 equates to the total 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 ridge 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.  The removal of Level III ridge features from the value 10^240 establishes the value for the average flat fingerprint that should be applied to T-Model formula when casework involves primarily Level I and level II ridge features.  That value is 1.71 x 10^117.  For purposes of simplicity and conservatism, this value is rounded up as follows:

10^120

The value 10^120 represents the reciprocal of the fingerprint match probability, e.g., 1/10^120, for the above number of Level I and Level II ridge features types in position present in the average flat fingerprint, e.g., not Level III.  This value is used to refine the relevant population group that a particular arrangement of Level I and Level II ridge feature types in position can exist as a close match or look-alike in another part of another finger in another person.

Underlying Assumption

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 formulae.  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, vlaue 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).

The Formulae

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 previoulsy 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

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 as follows:

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