Percentile rank that takes sorted argument (or same functionality) in PostgreSQL 9.3.5

Question

How do I calculate the percentile of the nth element in a field relative to the ones previous to it in a time series for each subset (partition) of my data?

For example, this is exactly what I want to do, except it is a cumulative sum, not a percentile:

select * 
from 
(SELECT trade_date, daily_val, id, company_id, sum(daily_val)
    over (partition by company_id order by trade_date rows between unbounded preceding and current row) AS sum
FROM daily_data
where (company_id = 858 or company_id = 726) and wh_calc_id = 156) as x
order by company_id,trade_date

This is what I want to do:

select * 
from 
(SELECT trade_date, daily_val, id, company_id, percent_rank(daily_val)
    over (partition by company_id order by trade_date rows between unbounded preceding and current row) AS pctl
FROM daily_data
where (company_id = 858 or company_id = 726) and wh_calc_id = 156) as x
order by company_id,trade_date

(Note: the only difference between the two is sum() vs percent_rank())

It seems I need a percentile rank function available in PostgreSQL 9.5 that takes a sorted expression argument. I know it exists for more recent versions:
https://www.postgresql.org/docs/9.5/static/functions-aggregate.html

And the 9.3 version doesn't seem to take an argument:
https://www.postgresql.org/docs/9.3/static/functions-window.html

I really don't want to upgrade PostgreSQL as I already have a lot going on right now.

Here is some simplified example output of what I want (created in Excel):

trade_date  daily_val   pctl    company_id
1/1/1900    1484.881621 1.000000    1
1/2/1900    1465.465315 0.000000    1
1/3/1900    1471.244094 0.500000    1
1/4/1900    1475.109578 0.666000    1
1/5/1900    1484.881621 0.750000    1
1/6/1900    1508.863507 1.000000    1
1/7/1900    1519.089431 1.000000    1
1/8/1900    1504.819339 0.714000    1
1/9/1900    1523.210769 1.000000    1
1/10/1900   1514.990335 0.777000    1
1/11/1900   1550.595515 1.000000    1
1/12/1900   1529.454907 0.909000    1
1/13/1900   1546.324863 0.916000    1
1/14/1900   1542.077672 0.846000    1
1/15/1900   1561.396941 1.000000    1
1/16/1900   1519.089431 0.533000    1
1/17/1900   1498.774337 0.312000    1
1/18/1900   1486.847801 0.294000    1
1/19/1900   1488.819195 0.333000    1
1/20/1900   1506.83871  0.473000    1
1/21/1900   1500.777505 0.400000    1
1/1/1900    149.113683  1.000000    5
1/2/1900    155.505015  1.000000    5
1/3/1900    154.15655   0.500000    5
1/4/1900    160.425942  1.000000    5
1/5/1900    156.29799   0.750000    5
1/6/1900    164.058078  1.000000    5
1/7/1900    159.935225  0.666000    5
1/8/1900    158.537455  0.571000    5
1/9/1900    154.147639  0.125000    5
1/10/1900   155.722962  0.444000    5
1/11/1900   155.505015  0.300000    5
1/12/1900   162.766282  0.909000    5
1/13/1900   169.264949  1.000000    5
1/14/1900   167.73179   0.923000    5
1/15/1900   162.280915  0.714000    5
1/16/1900   159.906452  0.533000    5
1/17/1900   161.04602   0.687000    5
1/18/1900   157.656379  0.411000    5
1/19/1900   154.478044  0.166000    5
1/20/1900   154.576546  0.210000    5
1/21/1900   154.147639  0.050000    5

ntile doesn't work either.

Answer 1

In Postgres 9.4+, you can use the hypothetical-set aggregate function percent_rank() in a LATERAL join like this:

SELECT *
FROM   daily_data d, LATERAL (
   SELECT round(percent_rank(d.daily_val) WITHIN GROUP (ORDER BY daily_val)::numeric
              , 6) AS pctl_calc
   FROM   daily_data
   WHERE  company_id = d.company_id
   AND    trade_date < d.trade_date
   ) x
ORDER  BY company_id, trade_date;

The CROSS JOIN LATERAL (, LATERAL for short) is safe, because a subquery with an aggregate function always returns a row. (Else you might want LEFT JOIN LATERAL (...) x ON true.)

Minor difference: The first row in each group returns 0, not 1 like in your example. Easily fixed, if you need it.

This would do the same job in Postgres 9.3:

SELECT d.*
     , round(CASE WHEN x.ct = 0 THEN numeric '1'  -- or 0 ?
                  ELSE x.ct_lt / x.ct END, 6) AS pctl_calc
FROM   daily_data d, LATERAL (
   SELECT count(daily_val) AS ct
        , count(daily_val < d.daily_val OR NULL)::numeric As ct_lt
   FROM   daily_data
   WHERE  company_id = d.company_id
   AND    trade_date < d.trade_date
   ) x
ORDER  BY company_id, trade_date;

I use count(daily_val) for the full count, not count(*) to ignore NULL values - necessary if daily_val can be NULL, .

SQL Fiddle.

Or, without LATERAL join for even older versions. May be faster, too:

SELECT company_id, trade_date, daily_val, pctl
     , round(CASE WHEN ct = 0 THEN numeric '1'  -- 0 ?
                  ELSE ct_lt / ct END, 6) AS pctl_calc
FROM  (
   SELECT d.*
        , count(x.daily_val) AS ct
        , count(x.daily_val < d.daily_val OR NULL)::numeric AS ct_lt
   FROM   daily_data d
   LEFT   JOIN daily_data x ON x.company_id = d.company_id
                           AND x.trade_date < d.trade_date
   GROUP  BY d.id  -- pg 9.1+
   ) sub
ORDER  BY company_id, trade_date;

Assuming id is the PK for a simplified GROUP BY, requires pg 9.1 though.