Calculating area under a curve - trapezium rule

Question

Hi,
I'm trying to calculate the 'area under the curve' using the trapezium rule for a set of data points. i.e. I have a set of values, sampled at irregular intervals.
create table measurements (
  measurement_value integer,
  measurement_time timestamp
insert into measurements (measurement_value,measurement_time) values (50,TIMESTAMP'2011-01-01 00:00:00');
insert into measurements (measurement_value,measurement_time) values (40,TIMESTAMP'2011-01-01 01:00:00');
insert into measurements (measurement_value,measurement_time) values (40,TIMESTAMP'2011-01-01 03:00:00');
insert into measurements (measurement_value,measurement_time) values (50,TIMESTAMP'2011-01-01 03:30:00');
insert into measurements (measurement_value,measurement_time) values (40,TIMESTAMP'2011-01-01 05:00:00');
insert into measurements (measurement_value,measurement_time) values (40,TIMESTAMP'2011-01-01 06:00:00');
insert into measurements (measurement_value,measurement_time) values (60,TIMESTAMP'2011-01-01 07:00:00');I'd like to calculate the area of the 'curve' where the value was below a threshold of, e.g. 45.
I can use the trapezium rule and assume a straight line between subsequent measurement, i.e. for the first 2 data points, the value drops from 50 to 40 linearly over the hour, and so the area for those measurements is (30min * 5)/2. For the 2nd and 3rd data points, the area is (120min * 5), for the 3rd and 4th, the area is (15min * 5)/2. And so on. The values are not necessarily round numbers, almost any value is possible.
I've started by using 'lead' to obtain the subsequent measurement value and time, but I'm getting stuck dealing with the end points and non-trivial interpolation. Does anyone have any pointers to get me going?
Thanks,
Dan Scott

Answer 1

Maybe this will do it:
SELECT SUM(CASE
            WHEN curr_value > :threshold AND next_value < :threshold
            THEN -- Area of rectangle specified by threshold
                 (new_measurement_time_sec-curr_measurement_time_sec)*:threshold
               + -- Area of remaining trapezoid
                 ABS((next_measurement_time_sec - new_measurement_time_sec)*((next_value - :threshold)/2))
            WHEN next_value > :threshold AND curr_value < :threshold
            THEN -- Area of rectangle specified by threshold
                 (next_measurement_time_sec-new_measurement_time_sec)*:threshold
               + -- Area of remaining trapezoid
                 (new_measurement_time_sec - curr_measurement_time_sec)*((:threshold - curr_value)/2)
            WHEN (:threshold < curr_value AND :threshold < next_value)
              OR (:threshold > curr_value AND :threshold > next_value)
            THEN (next_measurement_time_sec-curr_measurement_time_sec)*:threshold
            ELSE -- Area of trapezoid
                 ABS((next_measurement_time_sec - curr_measurement_time_sec)*((next_value - curr_value)/2))
       END)  AS area
FROM   (
         SELECT measurement_time
              , curr_measurement_time_sec
              , next_measurement_time_sec
              , curr_value
              , next_value
              , (:threshold - LEAD(intercept) OVER (ORDER BY measurement_time))/LEAD(slope) OVER (ORDER BY measurement_time) AS new_measurement_time_sec
         FROM   ( SELECT measurement_time        
                       , TO_CHAR(measurement_time,'J')*24*60*60 + TO_CHAR(measurement_time,'SSSSS') AS curr_measurement_time_sec
                       , TO_CHAR(LEAD(measurement_time) OVER (ORDER BY measurement_time),'J')*24*60*60 + TO_CHAR(LEAD(measurement_time) OVER (ORDER BY measurement_time),'SSSSS') AS next_measurement_time_sec
                       , measurement_value                                        AS curr_value
                       , LEAD(measurement_value) OVER (ORDER BY measurement_time) AS next_value
                       , REGR_SLOPE(measurement_value, TO_CHAR(measurement_time,'J')*24*60*60 + TO_CHAR(measurement_time,'SSSSS')) OVER (ORDER BY measurement_time) AS slope
                       , REGR_INTERCEPT(measurement_value, TO_CHAR(measurement_time,'J')*24*60*60 + TO_CHAR(measurement_time,'SSSSS')) OVER (ORDER BY measurement_time) AS intercept
                  FROM   measurements
       )Results:
SQL> var threshold NUMBER;
SQL> exec :threshold := 45;
SQL> /
      AREA
1191437.2The units are the time values were converted to seconds since January 1, 4712 BC (roughly).
Edited by: Centinul on Jul 7, 2011 10:39 AM