COMP1730/COMP6730 Programming for Scientists
Sequence types
Announcements
* Mid-semester exam: Monday, week 7, 16th Sep (right after semester break).
– comp1730: 6:30pm – comp6730: 8:30pm
* Questionnaire of schedule conflicts in weeks 6 and 7:
– If you have other exams at the same time with your labs, please fill out this survey until end of week 5 (Fri 23rd, 6pm)!
Lecture outline
* Sequence data types
* NumPy arrays
* Indexing, length & slicing
Sequences
* A sequence contains zero or more values. * Each value in a sequence has a position, or
index, ranging from 0 to n − 1.
* The indexing operator can be applied to all sequence types, and returns the value at a specified position in the sequence.
– Indexing is done by writing the index in square brackets after the sequence value, like so:
sequence[pos]
Sequence data types
* python has three built-in sequence types: – strings (str) contain only text;
– lists (list) can contain a mix of value types; – tuples (tuple) are like lists, but immutable.
* Sequence types provided by other modules: – NumPy arrays (numpy.ndarray).
NumPy arrays
NumPy and SciPy
* The NumPy and SciPy libraries are not part of the python standard library, but often considered essential for scientific / engineering applications.
* The NumPy and SciPy libraries provide
– an n-dimensional array data type (ndarray);
– fast math operations on arrays/matrices;
– linear algebra, Fourier transform, random
number generation, signal processing,
optimisation, and statistics functions;
– plotting (via matplotlib).
* Documentation: numpy.org and scipy.org.
Problem: Sensor modelling
* Time series of two measurements:
* IR sensor
(% of range)
* Tachometer (1/360th rev.)
* Is there a linear relation between x and y?
* Fitastraightline(y=ax+b)asclosetoallof the points as possible.
– This can be done by solving a least-squares optimisation problem.
– Simpler idea: Calculate the average slope between pairs of (adjacent) points.
* Need to remove or ignore “outliers”.
* Calculate residuals (ri = yi − (axi + b)) and check if they are normally distributed.
The NumPy ndarray type
* ndarray is a sequence type.
* All values in an array must be of the same type.
* Typically numbers (integers, floating point or
complex) or Booleans, but can be any type.
>>> import numpy as np
>>> x = np.array([1.0, 2, 3])
>>> x
array([ 1., 2., 3.])
>>> type(x)
>>> x.dtype
dtype(’float64’)
Indexing & length
array:
index: 0 1 2 3 4 -5 -4 -3 -2 -1
* In python, all sequences are indexed from 0.
* The index must be an integer.
* python also allows indexing from the sequence end using negative indices, starting with -1.
* The length of a sequence is the number of elements, not the index of the last element.
3.0
1.5
0.0
-1.5
-3.0
* len(sequence) returns sequence length.
* Sequence elements are accessed by writing the
index in square brackets, [].
>>> x = np.array([3,1.5,0,-1.5,-3]) >>> x[1]
1.5
>> x[-1]
-3.0
>>> len(x)
5
>>> x[5]
IndexError: index 5 is out of bounds
for axis 0 with size 5
More operations on NumPy arrays
Creating 1-dimensional arrays
>>> np.zeros(5)
array([ 0., 0., 0., 0., 0.])
>>> np.ones(3) * 5
array([ 5., 5., 5.])
>>> np.linspace(3, -3, 5)
array([3. , 1.5, 0. , -1.5, -3. ])
>>> import numpy.random as rnd
>>> rnd.normal(0, 2, 10) array([0.11224282, -1.84772958, …
Element-wise operators
* Arithmetic (+,-,*,/,**,//,%), comparison (==,!=,<,>,<=,>=) and logical (&,|) operators can be applied to
– an ndarray and a single value: the operation is done between each element of the array and the value; or
– two ndarrays of the same size: the operation is done between pairs of elements in equal positions.
>>> x = np.array([-2.,-1.,0.,1.,2.]) >>> -(x ** 2) + 2
array([-2., 1., 2., 1., -2.])
>>> y = 2 ** x
>>> y
array([ 0.25, 0.5, 1., 2., 4.])
>>> x + y
array([ -1.75, -0.5, 1., 3., 6.])
>>> x + array([1., -1.])
ValueError: operands could not be
broadcast with shapes (5,) (2,)
* NumPy provides most math functions (e.g., cos, exp, log, sqrt, etc) that also work element-wise on arrays.
>>> x = np.linspace(-np.pi, np.pi, 9)
>>> np.cos(x)
array([-1.00e+00, -7.07e-01,6.12e-17,
7.07e-01,1.00e+00,7.07e-01,
6.12e-17, -7.07e-01, -1.00e+00])
>>> np.sqrt(x)
RuntimeWarning:invalid value …
array([ nan, nan, 0., 1., 1.41421356])
Functions of arrays
>>> x = np.linspace(-1, 3, 5)
>>> np.min(x ** 2)
0.0
>>> np.max(x)
3.0
>>> np.sum(x)
5.0
>>> np.mean(x)
1.0
>>> np.std(x)
1.4142135623730951
Generalised indexing
* Most python sequence types support slicing – accessing a subsequence by indexing a range of positions:
sequence[start:end] * Unique to NumPy array:
– Indexing with an array of integers selects elements from the positions in the index array.
– Indexing with an array of Booleans selects elements from the positions where the index array contains True.
Slicing
* The slice range is “half-open”: start index is included, end index is one after last included element.
>>> x = np.array([3,1.5,0,-1.5,-3]) >>> x[1:4]
array([ 1.5, 0, -1.5])
start end
array:
index: 0 1 2 3 4
3.0
1.5
0.0
-1.5
-3.0
Indexing with arrays
>>> x = np.array([3,1.5,0,-1.5,-3]) >>> i = np.array([0,1,4])
>>> x[i]
array([ 3., 1.5., -3.])
>>> j = (x == np.floor(x))
>>> j
array([True, False, True, False, True])
>>> x[j]
array([ 3., 0., -3.])
# select “good” sample points:
ok = (y > np.min(y)) & (y < np.max(y))# compute y and x difference:dy = y[ok][1:] – y[ok][0:-1]dx = x[ok][1:] – x[ok][0:-1]# average slope:a = np.mean(dy / dx)# find average intercept:b = np.mean(y[ok] – a * x[ok])# compute residuals:r = y[ok] – (a * x[ok] + b)…or…import scipyok = (y > np.min(y)) & (y < np.max(y))# fit a 1st degree polynomial:p = scipy.polyfit(x[ok], y[ok], 1)# calculate r = y – p(x)r = y[ok] – scipy.polyval(p, x[ok])
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