PostgreSQL: interpolate missing value

Question

I have a table in PostgreSQL with a timestamp and a value.
I would like to interpolate the missing values under "lat".

The value under "lat" are tidalheights above a datum. For the purpose it is ok to interpolate the missing value linear between the two known values.

What is the best method to do so in PostgreSQL?

Edit 20200825

I have solved this problem in a different way using the QGIS fieldcalculator. Problem with this method: it takes a long time and it the process runs client-side and I would like to run it directly on the server.

In steps, my workflow was:

1. The interval between the recorded "lat" values is 10 minutes. I calculated the increment per minute between two recorded values and stored this in an extra column called "tidal_step" at the recorded "lat"-value. (i stored the timestamp also as an "epoch" in a column)

In QGIS:

tidal_step =
-- the lat value @ the epoch, 10 minutes or 600000 miliseconds from the current epoch: 
(attribute(get_feature('werkset','epoch',("epoch"+'600000')),'lat') -
-- the lat value @ the current
attribute(get_feature('werkset','epoch',"epoch"),'lat'))
/10

for the first two values from the example image that results in: (4.95 - 5.07) /10 = -0.012

2. I determined the amount of minutes of the "lat"-value to be interpolated, past the last recorded instance where a "lat" value was recorded and stored this in a column: "min_past_rec"

In QGIS:

left(
right("timestamp",8) --this takes the timestamp and goes 8 charakters from the right
,1) -- this takes the string from the previous right( and goes 1 character left

for the first value in the example: 2019-01-01 00:15:15 returns: '5' This is 5 minutes past the last recorded value.

4. I interpolated the missing values by adding the ("min_past_rec" * "tidal_step") to the last recorded "lat" value and stored this in in column called "lat_interpolated"

In QGIS

CASE
WHEN "lat" = NULL 
THEN
-- minutes pas the last recorded instance:
("min_past_rec" *
-- the "tidal_step" at the last recorded "lat"-value:
(attribute(get_feature('werkset','epoch',
("epoch" - --the epoch of the "lat" value to be interpolated minus:

left(right("timestamp",8),1) * 600000 -- = the amount of minutes after the last recorded instance.
+ left(right("timestamp",6),2) * 1000) --  and the amount of seconds after the last recorded instance.
),'tidal_step')) +

-- the last recorded "lat"-value

(attribute(get_feature('werkset','epoch',("epoch" - left(right("timestamp",8),1) * 600000 + left(right("timestamp",6),2) * 1000)),'lat'))

With data from the example:

2019-01-01 00:17:33:

"lat_interpolated" = "min_past_rec" * "tidal_step" + "lat" = 
7*-0.012 + 4.95 = 4.866

5. delete obsolete columns from database

Which statements/script should I use in PostgreSQL to perform the same task?

Answer 1

I have a (partial) solution - what I did was the following (see the fiddle available here):

The algorithm I used for interpolating was

• if there's a sequence of 1 NULL, take the average of the value above and the value below.

• A sequence of 2 NULLs, the top assigned value is the average of the two records above it and the bottom assigned one is the average of the two records below.

In order to do this, I did the following:

Create a table:

CREATE TABLE data 
(
  s SERIAL PRIMARY KEY, 
  t TIMESTAMP, 
  lat NUMERIC
);

Populate it with some sample data:

INSERT INTO data (t, lat)
VALUES
('2019-01-01 00:00:00', 5.07),
('2019-01-01 01:00:00', 4.60),
('2019-01-01 02:00:00', NULL),
('2019-01-01 03:00:00', NULL),
('2019-01-01 04:00:00', 4.7),
('2019-01-01 05:00:00', 4.20),
('2019-01-01 06:00:00', NULL),
('2019-01-01 07:00:00', 4.98),
('2019-01-01 08:00:00', 4.50);

Note that records 3 & 4 and 7 are NULL.

And then I ran my first query:

WITH cte1 AS
(
  SELECT d1.s,
    d1.t AS t1, d1.lat AS l1,
    LAG(d1.lat, 2)  OVER (ORDER BY t ASC) AS lag_t1_2,
    LAG(d1.lat, 1)  OVER (ORDER BY t ASC) AS lag_t1,
    LEAD(d1.lat, 1) OVER (ORDER BY t ASC) AS lead_t1,
    LEAD(d1.lat, 2) OVER (ORDER BY t ASC) AS lead_t1_2
  FROM data d1
),
cte2 AS
(
  SELECT 
    d2.t AS t2, d2.lat AS l2, 
    LAG(d2.lat, 1) OVER(ORDER BY t DESC) AS lag_t2,
    LEAD(d2.lat, 1) OVER(ORDER BY t DESC) AS lead_t2
  FROM data d2
),
cte3 AS
(
  SELECT t1.s,
    t1.t1,  t1.lag_t1_2, t1.lag_t1, t2.lag_t2, t1.l1, t2.l2, 
    t1.lead_t1, t2.lead_t2, t1.lead_t1_2
  FROM cte1 t1
  JOIN cte2 t2
  ON t1.t1 = t2.t2
)
SELECT * FROM cte3;

Result (spaces mean NULL - it's much clearer on the fiddle):

s   t1  lag_t1_2    lag_t1  lag_t2  l1  l2  lead_t1     lead_t2     lead_t1_2
1   2019-01-01 00:00:00             4.60    5.07    5.07    4.60        
2   2019-01-01 01:00:00         5.07        4.60    4.60        5.07    
3   2019-01-01 02:00:00     5.07    4.60                    4.60    4.7
4   2019-01-01 03:00:00     4.60        4.7             4.7         4.20
5   2019-01-01 04:00:00             4.20    4.7     4.7     4.20        
6   2019-01-01 05:00:00         4.7         4.20    4.20        4.7     4.98
7   2019-01-01 06:00:00     4.7     4.20    4.98            4.98    4.20    4.50
8   2019-01-01 07:00:00     4.20        4.50    4.98    4.98    4.50        
9   2019-01-01 08:00:00         4.98        4.50    4.50        4.98 

Note the use of the LAG() and LEAD() Window functions (documentation). I've used them on the same table, but sorted differently.

This and using the OFFSET option means that from my original single lat column, I now have 6 extra columns of "generated" data which are very useful for assigning values to the missing NULL values. The last (partial) piece of the puzzle is shown below (full SQL query is at the bottom of this post and also in the fiddle).

cte4 AS
(
  SELECT t1.s,
  t1.l1 AS lat,
    CASE 
      WHEN (t1.l1 IS NOT NULL) THEN t1.l1
      WHEN (t1.l1 IS NULL) AND (t1.l2) IS NULL AND (t1.lag_t1 IS NOT NULL)
        AND (t1.lag_t2 IS NOT NULL) THEN ROUND((t1.lag_t1 + t1.lag_t2)/2, 2) 
      WHEN (t1.lag_t2 IS NULL) AND (t1.l1 IS NULL) AND (t1.l2 IS NULL) 
        AND (t1.lead_t1 IS NULL) THEN ROUND((t1.lag_t1 + t1.lag_t1_2)/2, 2)
      WHEN (t1.l1 IS NULL) AND (t1.l2 IS NULL)  AND (t1.lag_t1 IS NULL)
        AND (t1.lead_t2 IS NULL) THEN ROUND((t1.lead_t1 + t1.lead_t1_2)/2, 2)
      ELSE 0
    END AS final_val
  FROM cte3 t1
)
SELECT s, lat, final_val FROM cte4;

Final result:

s    lat    final_val
1   5.07         5.07
2   4.60         4.60
3   NULL         4.84
4   NULL         4.45
5   4.7           4.7
6   4.20         4.20
7   NULL         4.59
8   4.98         4.98
9   4.50         4.50

So, you can see that the calculated value for record 7 is the average of records 6 & 8 and record 3 is the average of records 1 & 2 and the assigned value for record 4 is the average of 5 & 6. This was enabled by the use of the OFFSET option for the LAG() and LEAD() functions. If you get sequences of 3 NULLs, then you'll have to use an OFFSET of 3 and so on.

I'm not really happy with this solution - it involves hard-coding for the number of NULLs and those CASE statements will become even more complex and horrible. Ideally some sort of RECURSIVE CTE solution is required, but I HTH!

=============================== Full Query ========================

WITH cte1 AS
(
  SELECT d1.s,
    d1.t AS t1, d1.lat AS l1,
    LAG(d1.lat, 2)  OVER (ORDER BY t ASC) AS lag_t1_2,
    LAG(d1.lat, 1)  OVER (ORDER BY t ASC) AS lag_t1,
    LEAD(d1.lat, 1) OVER (ORDER BY t ASC) AS lead_t1,
    LEAD(d1.lat, 2) OVER (ORDER BY t ASC) AS lead_t1_2
  FROM data d1
),
cte2 AS
(
  SELECT 
    d2.t AS t2, d2.lat AS l2, 
    LAG(d2.lat, 1) OVER(ORDER BY t DESC) AS lag_t2,
    LEAD(d2.lat, 1) OVER(ORDER BY t DESC) AS lead_t2
  FROM data d2
),
cte3 AS
(
  SELECT t1.s,
    t1.t1,  t1.lag_t1_2, t1.lag_t1, t2.lag_t2, t1.l1, t2.l2, 
    t1.lead_t1, t2.lead_t2, t1.lead_t1_2
  FROM cte1 t1
  JOIN cte2 t2
  ON t1.t1 = t2.t2
),
cte4 AS
(
  SELECT t1.s,
  t1.l1 AS lat,
    CASE 
      WHEN (t1.l1 IS NOT NULL) THEN t1.l1
      WHEN (t1.l1 IS NULL) AND (t1.l2) IS NULL AND (t1.lag_t1 IS NOT NULL)
        AND (t1.lag_t2 IS NOT NULL) THEN ROUND((t1.lag_t1 + t1.lag_t2)/2, 2) 
      WHEN (t1.lag_t2 IS NULL) AND (t1.l1 IS NULL) AND (t1.l2 IS NULL) 
        AND (t1.lead_t1 IS NULL) THEN ROUND((t1.lag_t1 + t1.lag_t1_2)/2, 2)
      WHEN (t1.l1 IS NULL) AND (t1.l2 IS NULL)  AND (t1.lag_t1 IS NULL)
        AND (t1.lead_t2 IS NULL) THEN ROUND((t1.lead_t1 + t1.lead_t1_2)/2, 2)
      ELSE 0
    END AS final_val,
    t1.lead_t1_2
  FROM cte3 t1
)
SELECT s, lat, final_val, lead_t1_2 FROM cte4;