Mapping the Machine Learning Time Feature to a Circle to Compute Distance

One of the standard questions in machine learning is to compute the distance between two time intervals. If did a standard time interval difference between 1:00 am and 12:59 PM, there would be a lot of minutes between the two in terms of elapsed time. However, from a human activity perspective, there isn't much of a difference between the two. They are done really early in the morning.

One approach to solving the problem is to map the times 00:00 to 23:59 to a circle and then compute the chord length between the two mappings. For a discussion on how to compute chord length, please refer to the Wikipedia article "Circular Segment."

Below is a picture of a circle with the cord length denoted by c

 

The general equation to compute the length of the chord is given by the equation below

For simplicity will pick R to be 1. The above equation now simplifies to the following

The above concepts and math was implemented in the following code using Python 3.6.

import math

def create_hours_minutes_in_a_day():
    number_of_hours_in_day = 24
    number_of_minutes_in_an_hour = 60
    number_of_minutes_in_a_day = 0
    hours_minutes_in_day = []
    for hour in range(0, number_of_hours_in_day):
        if hour < 10:
            hour_to_print = '0' + str(hour)
        else:
            hour_to_print = str(hour)
        for minute in range(0, number_of_minutes_in_an_hour):
            if minute < 10:
                minute_to_print = '0' + str(minute)
            else:
                minute_to_print = str(minute)
            hours_minutes_in_day.append(hour_to_print+':'+minute_to_print)
            number_of_minutes_in_a_day += 1
    return hours_minutes_in_day, number_of_minutes_in_a_day

def map_hour_minute_in_day_to_circle(hour_colon_minute_in_day, number_of_minutes):
    number_of_degrees_in_a_circle = 360
    increment_size_in_degrees = number_of_degrees_in_a_circle / number_of_minutes
    current_angle_in_degrees = 0
    map_hour_minute_to_circle = dict()
    for x in hour_colon_minute_in_day:
        map_hour_minute_to_circle[x] = current_angle_in_degrees
        current_angle_in_degrees += increment_size_in_degrees
    return map_hour_minute_to_circle

def compute_arc_length_between_time_interval(mapping_of_hour_minute_to_angle, time_a, time_b):
    # Since we want the relative distance betwee two time intervals, 
    # can set the circle to a radius = 1
    angle_a = mapping_of_hour_minute_to_angle[time_a]
    angle_b = mapping_of_hour_minute_to_angle[time_b]
    chord_length = 2 * math.sin ( math.radians(angle_b - angle_a) / 2 ) 
    print("Time A: ", time_a, "Angle A: ", angle_a)
    print("Time b: ", time_b, "Angle A: ", angle_b)
    print("Chord: ", chord_length)
    print("---")

hour_colon_minute_in_day, number_of_minutes = create_hours_minutes_in_a_day()

map_hour_minute_to_angle = map_hour_minute_in_day_to_circle(hour_colon_minute_in_day, number_of_minutes)

compute_arc_length_between_time_interval(map_hour_minute_to_angle, '00:00', '06:00')

compute_arc_length_between_time_interval(map_hour_minute_to_angle, '00:00', '12:00')

compute_arc_length_between_time_interval(map_hour_minute_to_angle, '00:00', '18:00')

compute_arc_length_between_time_interval(map_hour_minute_to_angle, '00:00', '23:59')

The output from the above code is below

Time A:  00:00 Angle A:  0
Time b:  06:00 Angle A:  90.0
Chord:  1.4142135623730951
---
Time A:  00:00 Angle A:  0
Time b:  12:00 Angle A:  180.0
Chord:  2.0
---
Time A:  00:00 Angle A:  0
Time b:  18:00 Angle A:  270.0
Chord:  1.4142135623730951
---
Time A:  00:00 Angle A:  0
Time b:  23:59 Angle A:  359.75
Chord:  0.0043633196686735124
---