Running totals and differences

sum collapses an axis to one number and throws away the journey. Cumulative operations keep the journey: same length out, showing the value at every point along the way.

x = np.array([3, 5, 2, 8, 4])
x.sum()        # 22
x.cumsum()     # [ 3  8 10 18 22]   <- same length, last value is the sum

Anywhere you'd otherwise write a loop with a running variable — a bank balance, a cumulative share of the total, a race position over time — this is the one-liner.

diff goes the other way

np.diff(x)         # [ 2 -3  6 -4]   <- gaps between neighbours

Note the length: one shorter. Five values have four gaps between them. That catches people out when they try to put the result back next to the original, and there's a fix built in — np.diff(x, prepend=0) keeps the length by assuming a zero in front.

diff and cumsum undo each other, which is a genuinely useful relationship: it's how you convert between "total so far" and "change this period", the two ways every dataset of running numbers gets stored.

The rest of the family

x.cumprod()                  # running product — compound growth
np.maximum.accumulate(x)     # running maximum — the peak so far
np.minimum.accumulate(x)     # running minimum
np.diff(x, n=2)              # differences of the differences
np.gradient(x)               # central differences — same length out

np.maximum.accumulate is worth knowing by name. Combined with cumsum, it's how you compute a drawdown — the gap between the running peak and the current value — in two lines and no loops:

balance  = x.cumsum()
peak     = np.maximum.accumulate(balance)
drawdown = peak - balance

That .accumulate suffix works on any ufunc that takes two arguments, so np.add.accumulate is just cumsum spelled the long way.

Along an axis

On 2-D data these take axis like any reduction — but since nothing collapses, the output shape matches the input:

cups.cumsum(axis=0)    # running total down the days, per stall
cups.cumsum(axis=1)    # running total across the stalls, per day
cups.cumsum()          # flattens first, then accumulates — rarely what you want

That last one is the trap: with no axis, the array is flattened before accumulating, so you get one long run across all the rows joined end to end. Always name the axis on 2-D data.