Numpy and Scipy are the most common Python package for mathematical and numerical routines in precompiled, fast functions. The NumPy package provides basic routines for manipulating large arrays and matrices of numeric data.
The SciPy package extends the functionality of NumPy with a substantial collection of useful algorithms, like minimization, Fourier transformation, regression, and other applied mathematical techniques.
References
First we use the following convention when importing numpy and scipy
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import numpy as np
import scipy as cp
We can create a ndarray from a Python list like this
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a = np.array([1,2,3,4])
b = np.array([5,6])
print(a)
print(b)
Each array has its own type, data type and shape:
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print(type(a))
print(a.dtype)
a.shape
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In this case, our array is a 1-dimensional array, thus the shape (4,). Surely we can have multi-dimensional array like this:
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a2 = np.array ([[1,2,3], [4,5,6]])
print(a2.shape)
print(len(a2))
print(a2.ndim)
Note that len function returns the first dimension. ndim property returns the number of dimensions.
If you want to get the number of elements in an array, you can use the size property.
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a2.size
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You can specify the type of the array elements in the constructor like this:
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a3 = np.array([4.5, 7.1, 6.2], float)
a3.dtype
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You can convert array type from one type to another:
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print(a3.astype(str))
You can reshape array using tuples that define new dimension
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a = np.array(range(10), float)
a = a.reshape((5, 2))
a
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Convert to list:
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a.tolist()
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Convert to binary string
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binary_string = a.tostring()
print(binary_string)
a = np.fromstring(binary_string)
The arange function is similar to the range function but returns an array:
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np.arange(5, dtype=float)
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Array can be copied from existing ones
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b = a.copy ()
b
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print(np.zeros(10))
print(np.ones(20))
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print(np.zeros((3,5)))
print(np.ones((5,7)))
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np.empty((2,3,2))
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The zeros_like and ones_like functions create a new array with the same dimensions and type of an existing one:
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a = np.array([[1, 2, 3], [4, 5, 6]], float)
np.zeros_like(a)
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To create an identity matrix of a given size:
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np.identity(4, dtype=float)
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The eye function returns matrices with ones along the kth diagonal:
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np.eye(4, k=1, dtype=float)
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Fill array with single value
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a.fill(0)
a
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Transpose array:
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a = np.array([[1, 2, 3], [4, 5, 6]], float)
print(a.transpose())
print(a.T) # shorthand
print(a.T.shape)
Flatten a multidimensional array to a 1 dimensional
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a.flatten()
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Concatenate 1-dimensional arrays
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a = np.array([1,2], float)
b = np.array([3,4,5,6], float)
c = np.array([7,8,9], float)
np.concatenate((a, b, c))
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If an array has more than one dimension, it is possible to specify the axis along which multiple arrays are concatenated. By default (without specifying the axis), NumPy concatenates along the first dimension:
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a = np.array([[1, 2], [3, 4]], float)
b = np.array([[5, 6], [7,8]], float)
print(np.concatenate((a,b)))
print(np.concatenate((a,b), axis=0))
print(np.concatenate((a,b), axis=1))
Finally, the dimensionality of an array can be increased using the newaxis constant in bracket notation
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a = np.array([1, 2, 3], float)
print(a)
print (a[:, np.newaxis])
print (a[:,np.newaxis].shape)
print(a[np.newaxis,:])
print(a[np.newaxis,:].shape)
The in statement can be used to test if values are present in an array:
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print(7.1 in a3)
print(2 in a3)
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a = np.array([1,2,3], float)
b = np.array([5,2,6], float)
print (a + b)
print (a * b)
print (b / a)
print (a % b)
print (b**a)
For two-dimensional arrays, multiplication remains elementwise and does not correspond to matrix multiplication.
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a = np.array([[1,2], [3,4]], float)
b = np.array([[2,0], [1,3]], float)
a * b
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Errors are thrown if arrays do not match in size
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a = np.array([1,2,3], float)
b = np.array([4,5], float)
# a + b should throw error
NumPy offers a large library of common mathematical functions that can be applied elementwise to arrays. abs, sign, sqrt, log, log10, exp, sin, cos, tan, arcsin, arccos, arctan, sinh, cosh, tanh, arcsinh, arccosh, arctanh, floor, ceil, rint
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a = np.array( [0, 1.5 ,2.4,3.7] )
print (np.sqrt(a))
print (np.floor(a))
print (np.sin(a))
print (np.cos(a))
Also included in the NumPy module are two important mathematical constants:
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print (np.pi)
print (np.e)
Other array manipulation functions:
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# iterate array like a list
a = np.array([1, 4, 5], int)
for x in a:
print(x)
# loop through multi dimensional array
a = np.array([[1, 2], [3, 4], [5, 6]], float)
for x in a:
print(x)
for x,y in a:
print (x * y)
# basic array operation
print (a.sum())
print (np.sum(a))
print (a.prod())
print (np.prod(a))
print (a.min())
print (a.max())
# print the indice of the min and max value
print (a.argmin())
print (a.argmax())
# multidimensional array can specify the axis
a = np.array([[0, 2], [3, -1], [3, 5]], float)
print (a.min(axis=1))
print (a.max(axis=0))
# sort, clip
a = np.array([6, 2, 5, -1, 0, 3, 7], float)
print (sorted(a))
print (a.clip(0, 5))
# uniq
print (np.unique(a))
# diagonal
a = np.array([[0, 2], [3, -1], [3, 5]], float)
print (a.diagonal())
Comparison operator and value testing
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a = np.array([1, 3, 0], float)
b = np.array([0, 3, 2], float)
print (a > b)
print (a < b)
print (a == b)
print (np.any(a > b))
print (np.all(a > b))
a = np.array([1, 3, 0], float)
print (np.logical_and(a > 0, a < 3))
print (np.logical_not(b))
c = np.array([False, True, False], bool)
print (np.logical_or(b, c))
Some testing functions
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a = np.array([1, 3, 0], float)
print (np.where(a != 0, 1 / a, a))
a = np.array([[0, 1], [3, 0]], float)
print (a.nonzero())
a = np.array([1, np.NaN, np.Inf], float)
print (np.isnan(a))
print (np.isfinite(a))
Array item selection:
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a = np.array([[6, 4], [5, 9]], float)
print (a[a >= 6])
Array selection using integer array
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a = np.array([2, 4, 6, 8], float)
b = np.array([0, 0, 1, 3, 2, 1], int)
print (a[b])
Take and put can be used as well
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a = np.array([2, 4, 6, 8], float)
b = np.array([0, 0, 1, 3, 2, 1], int)
print (a.take(b))
a = np.array([0, 1, 2, 3, 4, 5], float)
b = np.array([9, 8, 7], float)
a.put([0, 3], b)
print (a)
scipy can create polynomial
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p = cp.poly1d([3,4,5])
print(p)
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print (p(1))
print (p([2,3]))
Do integral and derivative
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print(p.integ())
print(p.deriv())
numpy provide basic statistics function: mean, var, std, cov
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a = np.random.randn(100)
print(np.mean(a))
print(np.var(a))
print(np.std(a))
print(np.cov(a))
The median can be found:
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np.median(a)
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The covariance for data can be found for multiple variables:
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a = np.array([[1,2,3,4,5], [6,5,2,2,2]], float)
print(np.cov(a))
scipy provide more advanced functions
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# Create a normal distribution with mean 0.5, variance 1
import scipy.stats
rv = cp.stats.norm()
print (rv.stats())
print (rv.mean(), rv.std(), rv.var())
# get cdf, pdf of a particular point
print(rv.pdf(0))
print(rv.cdf(0))
Set random seed
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np.random.seed(213412)
An array of random numbers in the half-open interval [0.0, 1.0) can be generated:
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np.random.randn ()
print (np.random.random())
print( np.random.randint(5, 10))
print ( np.random.rand(5))
print (np.random.rand (2,4))
NumPy also includes generators for many other distributions: Beta, binomial, chi-square, Dirichlet, exponential, F, Gamma, geometric, Gumbel, hypergeometric, Laplace, logistic, lognormal, logarithmic, multinomial, multivariate, negative binomial, noncentral chi-square, noncentral F, normal, Pareto, Poisson, power, Rayleigh, Cauchy, student's t, triangular, von Mises, Wald, Weibull, and Zipf distributions
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print(np.random.poisson(6.0))
print (np.random.normal())
print ( np.random.normal(1.5, 4.0))
print ( np.random.normal(size=5))
The random module can also be used to randomly shuffle the order of items in a list
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a = list(range(10))
np.random.shuffle(a)
a
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