Statistical aggregates (Corr, RegrSlope, RegrIntercept, RegrR2, ...)

# ======================================================================
# PostgreSQL's linear-regression aggregates. `RegrSlope`/`RegrIntercept` fit y =
# slope·x + intercept, `Corr` is the correlation (−1..1), and `RegrR2` is the
# goodness of fit. Note the keyword form `y=`, `x=` (dependent variable first,
# as in SQL).
# ======================================================================

# --- Django ORM ---
Measurement.objects.values("series").annotate(
    n=RegrCount(y="y", x="x"),
    corr=Corr(y="y", x="x"),
    slope=RegrSlope(y="y", x="x"),
    intercept=RegrIntercept(y="y", x="x"),
    r2=RegrR2(y="y", x="x"),
    cov=CovarPop(y="y", x="x"),
    avg_x=RegrAvgX(y="y", x="x"), avg_y=RegrAvgY(y="y", x="x"),
    sxx=RegrSXX(y="y", x="x"), syy=RegrSYY(y="y", x="x"), sxy=RegrSXY(y="y", x="x"),
).order_by("series")

# --- generated SQL ---
#   SELECT "examples_measurement"."series" AS "series", REGR_COUNT("examples_measurement"."y", "examples_measurement"."x") AS "n", CORR("examples_measurement"."y", "examples_measurement"."x") AS "corr", REGR_SLOPE("examples_measurement"."y", "examples_measurement"."x") AS "slope", REGR_INTERCEPT("examples_measurement"."y", "examples_measurement"."x") AS "intercept", REGR_R2("examples_measurement"."y", "examples_measurement"."x") AS "r2", COVAR_POP("examples_measurement"."y", "examples_measurement"."x") AS "cov", REGR_AVGX("examples_measurement"."y", "examples_measurement"."x") AS "avg_x", REGR_AVGY("examples_measurement"."y", "examples_measurement"."x") AS "avg_y", REGR_SXX("examples_measurement"."y", "examples_measurement"."x") AS "sxx", REGR_SYY("examples_measurement"."y", "examples_measurement"."x") AS "syy", REGR_SXY("examples_measurement"."y", "examples_measurement"."x") AS "sxy"
#   FROM "examples_measurement"
#   GROUP BY 1
#   ORDER BY 1 ASC

# ----------------------------------------------------------------------
# series 'linear' (n=10)
# ----------------------------------------------------------------------

# Result:
#   fit:  y = 1.994 * x + 1.033
#   corr=1.000  r2=1.000  covar_pop=16.450
#   avg_x=5.50  avg_y=12.00
#   sxx=82.500  syy=328.100  sxy=164.500  (slope = sxy/sxx = 1.994)

# ----------------------------------------------------------------------
# series 'noisy' (n=10)
# ----------------------------------------------------------------------

# Result:
#   fit:  y = -0.018 * x + 4.800
#   corr=-0.020  r2=0.000  covar_pop=-0.150
#   avg_x=5.50  avg_y=4.70
#   sxx=82.500  syy=68.100  sxy=-1.500  (slope = sxy/sxx = -0.018)