Modeling Seating Choices & Assumed Repetative Placement in Confined, Populated Situations

Beginning as soon as children begin their academic careers in kindergarten, students are immediately assigned a seat at the start of each new school year. This special spot belongs to a student; their pencils, books, notes to friends, stickers, micro machines, etc. all reside within this personal space for the better part of a year. Day in and day out, kids enter a classroom and settle down to begin their day, not realizing that a pattern is being instilled in their young minds and they are - and will for many years to come - being trained.

Middle/high school rolls around and "periods" are introduced. Students move between several classrooms throughout the day to more assigned seats. Teachers almost always continue to utilize the practice seating charts. Although students may sometimes choose their own seats, the choice is usually established for the duration of semester/year with a seating chart. There are many beneficial reasons to continue the practice of assigned seats: often to assist the teacher learn the students names quickly, to make taking attendance efficient, or to separate certain individuals.


Cartesian Coordinates:
The location of a student in a classroom can be expressed by the function:


Where L = Location of a seated student in a classroom at any given time
d_ij = Cartesian coordinates of a desk in a classroom
Assuming an evenly distributed grid of i rows and j columns of desks:


As a student matriculates into a college setting, deeply ingrained norms follow and continue to show influence. Although students will now rarely be assigned a desk (although I had this Communications teacher once... oy.), these long standing tendencies are difficult to break. A few factors affect preference (front rows vs. back rows, friends, window seats, etc.) but overall it is quite common for students to return to the same seat each and every class period throughout the semester.

Spatial Contiguity:
Particular preferences demonstrate various types of contiguity. A preference for the front row is an example of limited rook contiguity. Whereas in chess, the the rook piece can move either vertically or horizontally an unlimited number of spaces across the board, a student with a front row preference chooses to move only in the ±i direction and not out of the first row (j=1) of seats. Further, a preference for a general area of a classroom, with variability of one is an example of king contiguity.


Back to preference, such a self-imposed location selection, may vary by more than one seat in any cardinal/intermediate direction. This is more of an example of limited queen contiguity - limited in terms of preference for the unique spot that is often retained by a student for the duration of the course. This limited change in distance is known as displacement.

Predicting Displacement:
The distance away from the originally chosen seat is known as displacement, and can be used as a limiting radius in predicting where a student will sit. This can be close to zero if the same seat is occupied every day, or can grow larger if a seat is chosen randomly throughout the semester.

Modeling displacement involves generally calculating the average Euclidean distance between two points, with a factor of arrival time used as a metric of normalization:

Simplified further:

Here's how it works:
Predicting a location's displacement (L_d) is largely derived by the Pythagorean theorem: c² = a² + b², where the Cartesian coordinates, i and j respectively, take the place of sides a and b, in an imaginary right triangle.

A set of coordinates are derived from producing an average overall change in each i and j direction.  Every day of attendance (k) has the opportunity to produce movement, so a given day's coordinate is subtracted from the previous location (k-1).  This is simplified by simply writing delta i, or the "change of i," etc.

The coordinates' squares are added together and the square root is divided by n number of classes to represent the average displacement (or side c in the imaginary triangle); but there's still a bit more going on in the denominator.

Arriving to class early significantly increases the probability of snagging your desired seat - especially in a front row / cute girl situation. Including the average amount of time that a student arrives before class as a friction factor (λ_t) in the equation reduces the n divisor, skewing the model's normalization with a negative (= lower displacement value = less variation = tendancy toward one seat) or positive (higher value = more variation = tendancy toward moving around the classroom) value. This is limited to arriving around 15 minutes before class to when ends when class begins, at zero. Further, an absolute value is applied to remove the negative value of "minutes early." Because this has the possibility of canceling out the n value (in that an average of arriving 5 minutes early on the 5th day of class results in a zero normalization value, and one cannot divide by zero), a finial simplification moves this up stairs into the numerator position.

Finally, to remove the negative square root value from the ± result, a concluding "absolute value" is applied to each of the variations of the formula.

This is simplified by taking the square root of the change of i and j squared, divided by n days of attendance minus the average early arrival time.

Direction:
Now that we have a predicted bounding radius, we can predict a direction of movement from the previous days of movement.  This change in direction is solved with an inverse trigonometric function: the arcsine of the average change of j movement, divided by the displacement model:

Combined:
The change of location can be modeled by a distance (displacement) and a direction of movement:

Formulas restated: