Vectorized operations and slices

Vectorized operations, axes, and masks

Vectorization (vectorization) describes an operation on an entire array at once. values + 2 adds two to each element, and values * values multiplies elementwise. The @ operator performs matrix multiplication, so do not confuse it with *. Python lists behave differently: [1, 2] * 2 repeats the list.

import numpy as np

celsius = np.array([0.0, 10.0, 20.0])
fahrenheit = celsius * 1.8 + 32
print(fahrenheit)
mask = (celsius >= 5) & (celsius <= 20)
print(mask)
print(celsius[mask])

Output: [32. 50. 68.], [False True True], [10. 20.]. A comparison returns an array of Boolean values, a mask (mask). Elementwise combinations require &, |, and ~, rather than Python's and, or, and not; put each condition in parentheses because of operator precedence. Writing if array: for an array of several elements does not specify whether you mean “at least one” or “all.” Use mask.any() or mask.all() according to the intended meaning.

What axis removes

mean, sum, min, and max can calculate a result over the entire array or along one axis. The axis argument names the axis to reduce. For a measurement×sensor matrix, axis=0 removes measurements and gives a result for each sensor; axis=1 removes sensors and gives a result for each measurement (Fig. 13.1).

Figure 13.1. Matrix axes and the shape of an aggregation result

import numpy as np

data = np.array([[10, 20, 30], [12, 22, 32]], dtype=float)
print(data.mean(axis=0))
print(data.mean(axis=1))
print(data.sum())
print(data.mean(axis=1, keepdims=True).shape)

Output: [11. 21. 31.], [20. 22.], 126.0, (2, 1). keepdims=True preserves the reduced axis with length one, making it easier to subtract the mean from each row afterward. Handle empty data before min or mean: there is no universally “correct average” for an empty sample.

Slices, copies, and broadcasting

A basic NumPy slice usually creates a view (view) that references the same data. Changing such a view may change the original array. In contrast, indexing with a list of indices or a Boolean mask creates a separate array (Fig. 13.2). Do not automatically apply NumPy's behavior to Python lists or pandas tables.

Figure 13.2. Shared slice storage and independent selection data

import numpy as np

a = np.array([10, 20, 30])
view = a[1:]
selected = a[a > 10]
view[0] = 99
selected[0] = -1
print(a.tolist())
print(selected.tolist())
print(np.shares_memory(a, view))

Output: [10, 99, 30], [-1, 30], True. When a function must return an independent result, write a[1:].copy(). This is not always the cheapest operation in memory terms, but it makes the contract unambiguous. Do not copy large arrays unnecessarily.

Broadcasting

Broadcasting (broadcasting) lets you combine compatible shapes without a manual loop. Dimensions are compared from right to left: they must match, or one must equal one. A missing leading axis is treated as length one. The rules are described at https://numpy.org/doc/stable/user/basics.broadcasting.html.

import numpy as np

measurements = np.array([[10., 20., 30.], [12., 22., 32.]])
offsets = np.array([1., -1., 2.])
corrected = measurements + offsets
centered = corrected - corrected.mean(axis=1, keepdims=True)
print(corrected)
print(np.allclose(centered.mean(axis=1), 0.0, atol=1e-10))

Output:

[[11. 19. 32.]
 [13. 21. 34.]]
True

Shape (2, 3) is compatible with (3,), but not (2,). A separate offset for each row requires a column of shape (2, 1). Check the combination of (n, 1) and (n,) especially carefully: it produces an (n, n) matrix even though the author may have expected n values. State the expected shape after each important step.

Random numbers and approximate equality

For a learning generator, use np.random.default_rng with a recorded initial seed. The same seed in the same environment lets you repeat an experiment. For long-term archiving, also save the data itself and the versions: a seed is not a universal interchange format across all future generator implementations. This generator is not intended for passwords or tokens.

For real values, use np.isclose and np.allclose with explicitly chosen atol and rtol. They specify the permitted absolute and relative errors and should be chosen according to the problem's units. Rounding solely for printing should not affect subsequent calculations. A measurement's deviation from the mean is not proof of sensor failure without an additional model and validation.