[SOLVED] math python statistic COMP1730/COMP6730 Programming for Scientists

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])