"A fact is a simple statement that everyone believes. It is innocent, unless found guilty. A hypothesis is a novel suggestion that no one wants to believe. It is guilty, until found effective."

Edward Teller

The mathematics used to calculate the total quantitative-qualitative weight, e.g. T-Value, for an arrangement of ridge features is based on the product rule.  Each ridge formation "shape in position" or "shaposition" is assumed to be relatively statistically independent events.  As a result, the weights of the individual ridge formations (derived by the inverse of their frequencies) are multiplied together to calculate the aggregate or total weight for an aggregate arrangement of ridge features. 

The scientific validity of the multiplication rule depends on whether fingerprint ridge formation types in position are statistically independent.  Although the orientation and type of ridge features can under specific circumstances be reliably predicted, e.g., the orientation of directional ridge features in a funnel, the precise positioning of ridge formation types cannot be reliably predicted (see Pattern Force 1/2 and Pattern Force 2/2). 

The assumption of statistical independence is applied to ridge feature shape in position whereby the frequency of a ridge formation’s type (i.e. ending ridge unit, bifurcating ridge unit, single ridge unit (dot), and so on) combines with (e.g., multiplies to) the frequency of its position (i.e. in terms of intervening ridge count to its nearest neighbor), and as a result the product of the probability for each defines its "shaposition" rarity and subsequent value. 

Ridge Formation Types in Position Do Not Repeat

The presence of any specific ridge formation type located in a specific position in a fingerprint does not cause the presence of another specific ridge formation type located in a specific position in that fingerprint.  As a result, fingerprint ridge formation types in position cannot be predicted. 

The following experiment illustrates the unpredictability of ridge formations or ridge unit types in position:  Cut out a .5mm x.5mm area in the middle of a piece of paper (this is a very tiny hole).  Place the paper over a random fingerprint and locate a ridge unit type through the hole.  Identify the ridge unit type.  Without moving the paper, predict the adjacent or nearest neighbor ridge unit type in position.  It cannot be predicted.  Although it cannot be predicted, the probability that one ridge unit type will occur can be calculated.  For example given the above scenario, and based on the frequency of occurrence for ridge unit types, the probability is approximately 86.9% that the adjacent or nearest neighbor ridge unit will be a continuous ridge unit.  Thus it may be stated that individual ridge unit types in position cannot be predicted, but only their probabilities.

The term “independent variable” refers to an event that is unpredictable and random.  In general ridge unit types in position are unpredictable events based on the absence of repeatable arrangements, however, at the same time some ridge formations exhibit non-randomness in orientation due to, for example, pattern force in a diminishing area where some groups of ridges are all forced to terminate in the same direction (see Pattern Force).  In essence, ridge formations exhibit properties of either increased randomness or decreased randomness due to the absence or presence of pattern force.  Although some minutiae exhibit this “less random” variable, they nevertheless retain the property of unpredictability with regards to specific ridge unit type in specific position. 

The theory of a less random event contained within in a random event framework may be illustrated by the following example:

A flock of geese in flight may be considered a collection of independent events in which each goose bears a non-randomness or predictability factor with regards to orientation during flight.  Each individual goose by itself may be considered a random event that displays relatively unpredictable behavior with regards to flight orientation, however, when joined with other geese in flight, the variable of flight orientation is impacted by the natural forces that create the propensity for the geese to fly together, in formation, in the same direction.  Each goose tends to fly in the same direction as the flock, which makes each goose predictable with regards to flight orientation.  As a result, each independent goose observed flying in a flock may be considered more predicable and less random than the independent goose by itself.  Each is more or less an independent event, however one may be considered more predictable than the other.

Similarly, ridge formations located in a diminishing area ridge funnel are less random with regards to orientation but otherwise retain the quality of unpredictability with respect to specific ridge unit type in specific position.  Ridge unit types found in areas impacted by pattern force are considered less random.  However, they retain their independent nature and are still subject to the product rule.

It is significant to add that although the exact random process governing minutiae formation is still unknown, during development ridge formation initiates at several points and spreads out such that developing ridge “fields” ultimately meet .  If ridge formations develop at several points on a finger or in separate ridge fields, then specific ridge unit types, i.e. ending ridges, bifurcations, and so on, develop independent from those in neighboring fields.  In addition, ridge units are not only subjected to differential growth factors while developing into rows, they are also subjected to a random, natural growth factor in relation to their shapes.  The combinative factors of differential, random growth and minutiae formation in separate ridge fields supports the theory of independence for ridge unit types in position.  Although clusters of “look-alike” ridge formations have been recorded (see Error Rate in Terms of Look-alikes), the consistent and reliable non-repeatability of ridge unit types in arrangement further supports their random, unpredictable and therefore independent nature.  Therefore, ridge unit types in position are considered fundamentally unpredictable events, with both random and less random properties, but nevertheless subject to the probability rule for independent variables where ridge unit types with more random properties are given expanded weight and those with less random properties are given reduced weight based on the presence [or absence] of pattern force.

The work of Newell and Kuchen, see "A model for fingerprint formation", provides additional foundation for the fact that individual ridge formation types do not depend on each other specifically and are fundamentally dependent on the stresses and forces that occur at the basal layer during fetal development.

The observations by Newell and Kuchen provides further support of the fallacy to consider "compound" ridge formation types, i.e. enclosures, short ridges, spurs, etc., as independent events and use them in frequency studies and apply any weights accorded to them to the product rule in order to attempt to define quantitative weights or match probabilities for aggregate ridge formation types.

Different minutiae densities in different fingerprint regions have been reported [5] , however a correlation exists between minutiae density and ridge count.  Fingerprint regions with higher minutiae density display lower intervening ridge counts between nearest neighbors and regions with lower minutiae density display higher intervening ridge counts.  Since ridge formation variables must maintain their independence from each other in order to satisfy the product rule, for purposes of simplicity, the ridge count variable was favored over minutiae density and applied to the model (see Intervening Ridge Count).

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Ridge formations exhibit properties of either increased randomness or decreased randomness in terms of orientation due to the absence or presence of pattern force.  Although some minutiae exhibit this “less random” variable, they nevertheless retain the property of unpredictability with regards to specific ridge unit type in specific position, and therefore remain subject to the product rule.

The probability rule for independent variables, or product rule, states the probability of the simultaneous occurrences of two independent events equals the product of the probabilities of each event [1].  With regards to fingerprints, if probabilities can be established for the presence of any independent event that occurs in a fingerprint, then the probabilities of all parts of a fingerprint that are deemed independent, when multiplied equal the probability for the whole fingerprint.