A random process is a repeating process in which the outcome follows no deterministic pattern, but follows a probability distribution.  The probability for fingerprint ridge formations in position as unpredictable events may be described in terms of a simple game die.  A game die has 6 sides with 1, 2, 3, 4, 5 and 6 numbers of dots on each side.  Following a large number of tosses the numbers of dots have the propensity to average.  Based on the Law of Large Numbers, the average number of dots likely to occur equals precisely 3.5.  The law of large numbers basically states that as the number of trials of a random process increases, the percentage difference between the expected and actual values goes to zero [6].  As a result, it describes the long-term stability for the probability of occurrence of a random variable.  Given a random sample of fingerprints, each with a random number of independent ridge formation observations, the average of these observations will eventually approach and stay close to the population mean.  Therefore similar to the die scenario, the average number of ridge unit types that occur in fingerprints can be determined. 

The Perfect "Fingerprint Die"

The perfect “fingerprint die” would contain all possible fingerprints and therefore have about 66 billion sides representative of the world fingerprint population, e.g., 6.6 billion people multiplied by 10 for numbers of fingers for each person.  If such a die was tossed enough times, the number of fingerprint patterns and ridge unit types contained within each pattern, for each toss, wound essentially average.  The fact that a certain number of fingerprint pattern forces and a certain number of ridge unit types may appear following a single roll in no way effects or impacts the results of the next roll.  Each roll is completely independent of the next in which each fingerprint that comes up on the roll remains on the die for the next roll. 

Although no such fingerprint die can be actually employed, the random selection of fingerprints from either ten-print files or individuals suffices since the likelihood of the same fingerprint repeating on any such die throw is so small that it should be considered a practical impossibility.  After a large enough number of “rolls” takes place and ridge formations found in each are counted, the average frequency of occurrence for each ridge formation is established. The frequency of occurrence is then put in terms of percentage distribution with regards to total population.  The inverse of the frequency of occurrence is then used to represent its quantitative weight.

Names for Ridge Feature Shapes

Historically compound ridge formations that resemble particular shapes have been named and these names have been passed down in literature and subsequently used in probability studies.  For example, spurs, enclosures, short ridges and so on, have been treated as independent events and probabilities based on their frequencies have been published [5] [7].  The names attributed to these ridge formations are merely descriptions for particular shapes they collectively form and/or resemble.  However the notion that any compound minutiae type is a single independent event is a fallacy and as a result necessarily distorts any probability calculation based on it. 

Example

Osterburg found that the frequency of occurrence for a bifurcation was .0382 and a “lake” was .0064 with quantitative weights (defined by the inverse of the frequency of occurrence) of 26 and 156 respectively.  However, a “lake” is actually comprised of two bifurcating ridge units and a small number of continuous ridge units in which each continuous ridge unit bears a quantitative weight of 1.3055 (see Osterburg Frequency Table). 

Based on the product rule, the aggregate quantitative weight for two clear, reliable bifurcating ridge units and a number of clear, reliable continuous ridge units can be precisely defined.  The aggregate weight for just the two bifurcations is 26 x 26, or 676.  The quantitative weight, however, attributed for a single “lake” is only 156.  The idea that a lake is a single independent event results in a 4-fold distortion factor.  If the quantitative weight for merely 4 continuous ridge units, which represents the sides of the “lake”, is included in the aggregate, the distortion factor is 12-fold.  

A lake or enclosure is two bifurcations

A compound ridge formation historically known as a lake or enclosure is fundamentally comprised of two bifurcating ridge units. The above figure has a total of 2 bifurcating ridge units, 7 continuous ridge units, and 9 pores.

Similarly, the following compound ridge formation types can be simplified as follows:

A crossover is two bifurcations 

The above "crossover" is fundamentally 2 bifurcations.  The above figure has a total of 2 bifurcating ridge units, 9 continuous ridge units, and 11 pores.

A spur is 1 ending ridge and 1 bifurcation 

The above "spur" is fundamentally 1 ending ridge and 1 bifurcation.  The above figure has a total of 1 ending ridge unit, 1 bifurcating ridge unit, 7 continuous ridge units, and 9 pores. 

A short ridge is two ending ridges 

The above "short ridge" is fundamentally 2 ending ridges.  The above figure has a total of 2 ending ridge units, 1 continuous ridge units, and 3 pores. 

The Average Ridge Unit

In order to develop a model based on a ridge unit approach, the general dimensions for the average ridge unit needed to be defined.  Cummins defined the average width of an adult ridge unit to be approximately .018 inch, or .45mm [8].  No information was found in literature that defined the average length of an adult ridge unit and it was hypothesized that the width and length of a single ridge unit was relatively equal.  

A simple study was performed to verify the length of the average ridge unit as follows:  Based on empirical data, a pore is inherent to all ridge units.  The pore defines ridge unit location, which means ridge unit length can be determined by distal measurements between pores.  Based on a random selection of twenty flat fingerprint impressions the number of pores found in randomly selected 5mm lengths of continuous ridges in each impression, were counted.  Only clear, distinct fingerprint impressions displaying clear pore detail were used.  The number of pores per 5mm continuous ridge consistently ranged from 9 to 12 with an average of 11.1.  This corresponded to 1 pore per .45mm length of ridge, thus verifying the hypothesis.

 

The Osterburg Study

For purposes of objectivity, the results from the Osterburg study were used as a foundation to define relative frequency and therefore estimate, in part, quantitative weights for individual ridge formation types (see Osterburg Frequency Table).  The study was critically reviewed by Dr. David A. Stoney who wrote the following:

"The Osterburg model is appealing because it is simple to apply and is statistically sophisticated.  It is particularly useful in the comparison of individuality among different fingerprints.  If some statistical standard of configuration of minutia is defined, the model provides a means to compare other minutia configurations to the standard [9].” 

Upon scientific review of the Osterburg model, the 39 fingerprint sample size was never criticized as insufficient [9] [10].  As a result, the T-Model’s application of the same fingerprint sample and grid structure conversion from 8,591 to 23,136 independent cells, was deemed equally valid, if not more so. 

Survey Sample Size and Margin of Error

In general, the margin of error in a sample “survey” equals 1 divided by the square root of the number of independent events surveyed within that sample. The formula is derived from the standard deviation of the proportion of times that a researcher gets a sample "right," given a whole bunch of samples [65].

Upon initial inspection, a sample size of only 39 fingerprints does not seem like a sufficient sample size needed to accurately reflect what is happening in fingerprints.  In fact a sampling of only 39 equates to a margin or error of 16.01% (not very good).  All expert fingerprint examiners know from experience that 39 fingerprints is not enough to accurately calculate the average number of times every single ridge feature occurs in fingerprints.  For example, burn marks are rarely found in fingerprints and in order to define the number of times a burn mark occurs in fingerprints would require a much larger fingerprint sample.  However, a burn mark is an anomaly, which does not reflect the ridge formation marks typically found in fingerprints.

The ridge formation marks typically found in fingerprints are ridge units.  Based on a ridge unit approach to illustrate ridge formation marks that occur in fingerprints, the most frequently occurring ridge unit type found is the continuous ridge unit.  The second most frequently occurring ridge unit type is the ending ridge unit.  The third is the bifurcating ridge unit and the fourth is the single ridge unit or “dot”.  These are the most frequently occurring ridge unit types found in fingerprints.  Not surprisingly, they are also the most common ridge unit types used by fingerprint examiners to make identifications.  It is significant to note that the frequency for anomalous fingerprint features such as burn marks can be easily and quickly determined by taking a larger fingerprint sample (see Ridge Unit Weights 2/3).

A fingerprint sample of 39 represents a collection of approximately 23,136 ridge units.  Based on the theory that ridge units are [more or less] random events in terms of “ridge unit type in position”, the sampling of 23,136 independent ridge unit events represents a sample size with a margin or error of only 0.65% (extremely good).  This margin of error reflects the chance that the ridge units found in a latent fingerprint do not accurately represent the fingerprint population as a whole.  It is significant to note that fingerprint population size is not a factor or in any way determines how big the sample size needs to be.   For example, the relevant fingerprint population of persons who could have committed a crime, for example, 100 fingerprints based on a boat population of 10 people, or a world fingerprint population of 66+ billion, will not affect how big the sample needs to be in order to come within the desired margin of error.  The Math Gods just don't care [65].

Furthermore, it is significant to mention here that survey data is a measure and all measure is imprecise.  Margin of error reveals the imprecision that is inherent in survey data.  In other words, survey data provides a range, not a specific number.  A 0.65% error rate equates to a 99.35% level of confidence which means that 0.65 percent of ridge unit surveys should be off the wall with numbers that don't make much sense.  It means if 1000 ridge unit surveys were conducted, a range of about 65 of them should provide results that are a bit “wacky”.   This means that fingerprint examiners, scientists or other researchers who conduct a ridge unit survey should understand that unexpected numbers that seem way out of line may come up even when proper guidelines are followed.  The chance that results from the Osterburg survey are “way out of line” is unlikely--nevertheless, additional independent ridge unit surveys should be performed to confirm that the Osterburg survey was not one of the 65 wacky ones (see Solicitation).

The Validity of Any Idea is Experiment 

In order to further verify the validity of only 39 fingerprints as sufficient to define relative accurate minutiae frequencies found in the average fingerprint, the following experiment was performed: 

The ending ridge is the most frequently occurring significantly weighted minutiae found in fingerprints and therefore was selected for this study.  The average number of ending ridges found in a random selection of 39 fingerprints based on extrapolation of the Osterburg study was 44.46.  The number 44.46 is approximately equal to 45.50, which represents the average number of dots counted as a result of 13 dice rolled a large enough number of times so that the average frequency of occurrence remains stable.  Dice are independent events and were used to emulate frequency of occurrence for ending ridge events absent pattern force.  The 13 dice were rolled precisely 39 times and the results were then averaged.  This was done 10 separate times using a random die roller [11].   The average number of dots, i.e. pseudo-ending ridge events, was then compared to the known average of 45.50.  The average numbers of dots counted for each set were 45.33, 44.51, 46.10, 46.07, 45.28, 46.20, 45.51, 46.25, 46.30 and 44.89 which equates to a percentage difference of 0.5%, 2.2%, 1.3%, 1.3%, 0.5%, 1.6%, 0.1%, 1.8%, 1.7% and 1.4% respectively.  The average percentage difference was 1.1%. 

These percentage differences were deemed consistently small enough to validate 39 as a sufficient number of fingerprints to be used as a basis to define relative frequency of occurrences for ridge formation types.  Although a larger fingerprint sample will cause frequency averages to remain steady, any change should be more or less fractional and if for purposes of conservativeness subsequent quantitative weights are simplified and rounded down to the nearest whole numbers, half or quarter fractions, then any number of additional fingerprints should not change these rounded lower bound conservative values.

It is significant to note the Osterburg model is convenient because the aggregate grid of 8,591 1mm^2 cells in the 39 randomly selected fingerprints corresponds more to a flat fingerprint impression, as opposed to a rolled fingerprint impression.  Based on a simple study, friction ridge skin most often contacting a given substrate is generally the central portion of a finger and not its periphery.  Thus it may be inferred that the flat fingerprint impression represents ridge formations that are typically found in the chance, accidental or latent fingerprint impression.  See the below study.

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In terms of compound minutiae, in order to insure independence and avoid distortion when defining quantitative weights for ridge formation types, they should not be combined or coupled-up in any way.   With exception to core and delta formations, no combinative ridge formation types are used, which means the notion of a “spur”, “lake”, “short ridge” or any other type of compound minutiae is completely discarded.  As a result, quantitative weights for ridge formations are based on the frequency of occurrence for the most singular, and therefore, independent ridge formation type fundamental to all ridge formations:  the ridge unit.

 

"The ridge unit approach to illustrate a ridge is fine."           

                                David Ashbaugh

Osterburg took the center portion of a fingerprint which most reflects information found in a flat fingerprint on an exemplar ten-print record compared to its counterpart rolled impression.  It was hypothesized that a flat fingerprint best represented ridge detail found in the average "useable" latent fingerprint, e.g., latent print impression bearing value for a conclusion of identification or exclusion. In order to corroborate this idea, the following study was performed:

Study

A frequency of occurrence study was performed on a random sample of 50 identified latent fingerprints at the San Jose Police Department Central Identification Unit as follows:  The numbers of identified latent fingerprints with delta, out-of-delta, core and periphery regions were counted with the following results:

Region of Interest        Percentage Distribution

Core                                 81%   
Out-of-Delta                      75%   
Delta                                35%   
Periphery                           6%   

As a result of the above study, it was found that approximately 94% of latent print identifications examined displayed no periphery, and 81% displayed a core, which are both consistent with information found in the flat fingerprint impression as opposed to information found in a rolled fingerprint impression which includes periphery ridge feature information. 

Based on these findings, it may be said with relative confidence that ridge formation types present in the average flat fingerprint (compared to the average rolled fingerprint) better represent ridge features found in the average latent fingerprint. Therefore, the Osterburg study is appealing because the results are more consistent with the flat fingerprint as opposed to the rolled fingerprint, and subsequently better reflect ridge feature information  found in the average latent fingerprint.