NumPy Arrays and Graphs: Quick Revision

A brief note on NumPy Arrays and Graphs. Read it before a test. Try each question at the end before you open its answer.

Arrays: many numbers of one type

• A NumPy array holds many numbers of one type and does maths on all of them at once. Start with import numpy as np.
• np.array([1, 2, 3]) makes an array from a list. np.zeros(5), np.ones(5) and np.full(5, 7) make filled arrays.
• np.arange(0, 10, 2) stops before 10. np.linspace(0, 1, 11) gives 11 numbers and includes both ends.
• shape, ndim, dtype and len describe an array. One decimal makes the whole array decimals.
• astype(int) cuts off the decimal part. It does not round.

Maths on whole arrays

• A formula written for one number works on a whole array, with no loop: v = 4.9 * t**2.
• Two arrays work place by place, so they must have the same length.
• Use the NumPy functions np.sqrt, np.exp and np.sin on arrays. Angles go in radians: np.sin(np.deg2rad(30)).
• A comparison gives an array of True and False. (a > 5).sum() counts the True values.

Picking numbers

• Indexes and slices work as for lists: a[0], a[-1], a[2:5].
• A slice is a view of the same numbers, not a copy. Change it, and the original changes. Use .copy() for a real copy.
• A mask picks: a[a > 5]. Join conditions with &, | and ~, each in brackets: a[(a > 2) & (a < 9)].
• np.where(a > 0, a, 0) chooses at every place. np.clip(a, 0, 100) keeps numbers between two limits.

Summaries and tables

• sum, mean, min, max, argmax, np.median, cumsum and np.diff turn many numbers into a few.
• std() divides by $n$. For a sample, write std(ddof=1).
• A table is a two-dimensional array. a[1, 2] is row 1, column 2; a[:, 0] is the first column.
• axis=0 works down the columns and gives one value per column. axis=1 gives one value per row.
• reshape changes the shape, .T swaps rows and columns, and broadcasting joins a column and a row into a table.

Graphs with matplotlib

• plt.plot(x, y) joins the points; x comes first. Name the axes with plt.xlabel and plt.ylabel, with the unit.
• A format such as "o-" sets markers and a line. Give each line a label, then call plt.legend().
• For a smooth curve of a function, use np.linspace with at least 100 points.
• fig, ax = plt.subplots(2, 1) gives several graphs in one picture. Call ax.plot, ax.set_xlabel and so on.
• ax.errorbar shows uncertainties, and a log scale turns an exponential into a straight line.

• From positions at known times, np.diff(x) / np.diff(t) gives the speeds. There is one fewer, and each belongs to the middle of its gap.

A worked example

Three weather stations record the noon temperature, in °C, on four days. Each row is a station and each column is a day.

1. temps.mean(axis=1) works along each row, so there is one mean per station: 33.25 °C, 22.5 °C and 28.5 °C.
2. temps.max(axis=0) works down each column, so there is one value per day: 31, 33, 35 and 34 °C.
3. temps > 30 is a table of True and False. .sum() counts the True values: 4 readings, all at the first station. A reading of 30 °C is not above 30 °C.
4. temps[0].argmax() gives the place of the biggest number in row 0, which is 2. Counting from 0, that is the third day.

Practice with new numbers

A program to try

A cup of tea cools in a room at 25 °C. The program works out its temperature every half minute, finds when it falls below 50 °C, and draws the graph.

Mistakes to avoid

• A list instead of an array. [1, 2] * 2 repeats the list. Make an array first for maths.
• Arrays of different lengths. np.array([1, 2, 3]) + np.array([1, 2]) stops with ValueError: operands could not be broadcast together with shapes (3,) (2,).
• and between arrays, or a missing bracket. a[a > 2 and a < 5] stops with ValueError: The truth value of an array with more than one element is ambiguous. Use a.any() or a.all(). Write a[(a > 2) & (a < 5)].
• math.sin on an array. It stops with TypeError: only 0-dimensional arrays can be converted to Python scalars. Use np.sin.
• Degrees in np.sin. np.sin(30) treats 30 as radians. Use np.deg2rad first.
• x and y of different lengths in a graph. plt.plot stops with ValueError: x and y must have same first dimension.
• A graph without units. A reader cannot guess them.

Check yourself

1. How many numbers does np.arange(2, 20, 3) hold, and which is the last one?

2. A table scores has 5 rows, one for each student, and 3 columns, one for each test. What shape does scores.mean(axis=0) have, and what does each number mean?

3. Rina writes b = a[1:4] and then b[0] = 0. Does the array a change?

4. Which graph suits each set of data: the heights of 200 students, and the length of a spring for six loads?

5. A trolley's position is measured at 0, 0.5, 1.0 and 1.5 s. How many speeds does np.diff(x) / np.diff(t) give, and at which times do they belong?