Source: View original notebook on GitHub
Category: Machine Learning / Learn ML libraries
NumPy DOCS
- numpy is a fundamental package for scientific computation in Python.
- It provides a multidimensional array object for fast operation over it
- NumPy arrays are the main way we will be using NumPy module.
- Numpy arrays essentailly comes as vectors and matrices.
vectors are 1-d arrays
matrices are 2-d but can have single row or col.
An array object in numpy is of arbitrary homogeneous items(that is, all of the elements must be the same type)
Here are some of the things you’ll find in NumPy:
- ndarray, an efficient multidimensional array providing fast array-oriented arithmetic operations and flexible broadcasting capabilities.
- Mathematical functions for fast operations on entire arrays of data without having to write loops.
- Linear algebra, random number generation, and Fourier transform capabilities.
What advantages do NumPy arrays offer over (nested) Python lists?
- Two of NumPy’s features which are the basis of much of its power: vectorization and broadcasting not available in list.
- NumPy internally stores data in a contigiuous block of memory
- NumPy arrays also use much less memory than built-in Python sequences
- NumPy operations perform complex computations on entire arrays without the need for Python for loops(means less code). more...
list vs numpy
list (data of list is stored randomly in memory hence eacj index of list contains a reference)
import copy
li = [[1,2]]
li2 = li # creates a reference, changes both list if one is modified directly
- slicing
li2 = li[:] # creates shallow copy.
li2 = li.copy() # creates shallow copy.
li2 = copy.deepcopy(li) # creates deep copy, does not change anything
numpy(data is stored continuoslly in memory hence each index contains data)
- when we access a data in numpy array ,numpy boxes data in python object so even if u access same element two times u got two different boxes(
id(arr[0]) is not equal to id(arr[0])), yes it is not a typo.
- The data of Numpy arrays is internally stored as a contiguous C array. Each entry in the array is just a number. Python objects on the other hand require some housekeeping data, e.g. the reference count and a pointer to the type object. You can't simply have a raw pointer to a number in memory. For this reason, Numpy "boxes" a number in a Python object if you access an individual elemtent. This happens everytime you access an element, so even A[0] and A[0] are different objects:
A[0] is A[0]
False
This is at the heart of why Numpy can store arrays in a more memory-efficient way: It does not store a full Python object for each entry, and only creates these objects on the fly when needed. It is optimised for vectorised operations on the array, not for individual element access.
When you execute C = A[:] you are creating a new view for the same data. You are not making a copy. You will then have two different wrapper objects, pointed to by A and C respectively, but they are backed by the same buffer. The base attribute of an array refers to the array object it was originally created from:
A.base is None
True
>>> C.base is A
True
New views on the same data are particularly useful when combined with indexing, since you can get views that only include some slice of the original array, but are backed by the same memory.
To actually make a copy of an array, use the copy() method.
As a more general remark, you should not read too much into object identity in Python. In general, if x is y is true, you know that they are really the same object. However, if this returns false, they can still be two different proxies to the same object.
so
- arr = np.array([[1,2]])
- arr1 = arr # same as list, changes directly
- arr1 = arr[:] or arr.view() # changes directly
- arr1 = arr.copy() # creates a deep copy, no change
# To give you an idea of the performance difference, consider a NumPy array of one million integers, and the
# equivalent Python list:
import numpy as np
my_arr = np.arange(1000000)
my_list = list(range(1000000))
# Now let’s multiply each sequence by 2:
%time for _ in range(10): my_list2 = [x * 2 for x in my_list]
Output:
Wall time: 2.23 s
%time for _ in range(10): my_arr2 = my_arr * 2
Output:
Wall time: 29.6 ms
# see the difference above
The Basics
- NumPy’s main object is the homogeneous multidimensional array. It is a table of elements (usually numbers), all of the same type, indexed by a tuple of positive integers.
In NumPy dimensions are called
axes.- For example, the coordinates of a point in 3D space [1, 2, 1] has one axis. That axis has 3 elements in it, so we say it has a length of 3.
In the example below, the array has 2 axes(Dimensions). The first axis(#rows) has a length of 2, the second axis(#col) has a length of 3.[[ 1., 0., 0.],
[ 0., 1., 2.]]NumPy’s array class is called
ndarray. It is also known by the aliasarray.
Attributes of numpy Module/class
ndim- the number of axes (dimensions) of the array.
shape- the dimensions of the array. This is a tuple of integers indicating the size of the array in each dimension. For a matrix with n rows and m columns, shape will be (n,m). The length of the shape tuple is therefore the number of axes, ndim.
size- the total number of elements of the array. This is equal to the product of the elements of shape.
dtype- an object describing the type of the elements in the array. One can create or specify dtype’s using standard Python types. Additionally NumPy provides types of its own. numpy.int32, numpy.int16, and numpy.float64 are some examples.
itemsize- the size in bytes of each element of the array. For example, an array of elements of type float64 has itemsize 8 (=64/8), while one of type complex32 has itemsize 4 (=32/8). It is equivalent to ndarray.dtype.itemsize.
flags
import numpy as np
arr = np.array([[1,2,3],[4,5,6]])
print(arr.ndim)
print(arr.shape)
print(arr.size)
print(arr.dtype)
print(arr.itemsize) # sizeof(dtype)/8 = #_bytes
print('-------------------------')
print(arr.flags)
Output:
2
(2, 3)
6
int32
4
-------------------------
C_CONTIGUOUS : True
F_CONTIGUOUS : False
OWNDATA : True
WRITEABLE : True
ALIGNED : True
WRITEBACKIFCOPY : False
UPDATEIFCOPY : False
Creation of ndarray
- passing list/tuple to
array()
- passing list/tuple to
arr = np.array([1,2,3,4])
arr_2d = np.array([[1,2,3],[4,5,6]])
- using
arange(),zeros(),ones(),eye(),linspace()-> returns number of numbers(metioned as third parameter) evenly spaced numbers b/w start and stop.arange differs form range in fact that range returns list while arange return numpy array
reshape()-> reshape array from one dimension to other but numberof elements should not change or arr.size = srr_reshaped.sizenp.empty()=> returns array with garbage values.np.full()=> returns array with value specified as fill_value = value
# creating array using list
l = [1,2,3]
import numpy as np
arr = np.array(l)
type(arr)
Output:
numpy.ndarray
arr
Output:
array([1, 2, 3])
# creating 2-d arrayor matrix
mat = [[2,2],[3,4]]
mat
Output:
[[2, 2], [3, 4]]
type(mat)
Output:
list
np.array(mat)
Output:
array([[2, 2],
[3, 4]])
# changing datatype
c = np.array( [ [1,2], [3,4] ], dtype=complex )
c
Output:
array([[1.+0.j, 2.+0.j],
[3.+0.j, 4.+0.j]])
# using np.zeros(shape, type=float)
np.zeros(3)
Output:
array([0., 0., 0.])
np.zeros(3,int)
Output:
array([0, 0, 0])
np.zeros((4,4), bool)
Output:
array([[False, False, False, False],
[False, False, False, False],
[False, False, False, False],
[False, False, False, False]])
np.zeros((3,3,3), complex)
Output:
array([[[0.+0.j, 0.+0.j, 0.+0.j],
[0.+0.j, 0.+0.j, 0.+0.j],
[0.+0.j, 0.+0.j, 0.+0.j]],
[[0.+0.j, 0.+0.j, 0.+0.j],
[0.+0.j, 0.+0.j, 0.+0.j],
[0.+0.j, 0.+0.j, 0.+0.j]],
[[0.+0.j, 0.+0.j, 0.+0.j],
[0.+0.j, 0.+0.j, 0.+0.j],
[0.+0.j, 0.+0.j, 0.+0.j]]])
# using np.arange(start=0, stop, step=1, dtype=None)
np.arange(10)
Output:
array([0, 1, 2, 3, 4, 5, 6, 7, 8, 9])
np.arange(10, dtype=complex)
Output:
array([0.+0.j, 1.+0.j, 2.+0.j, 3.+0.j, 4.+0.j, 5.+0.j, 6.+0.j, 7.+0.j,
8.+0.j, 9.+0.j])
np.arange(3,15,3)
Output:
array([ 3, 6, 9, 12])
# np.ones()
np.ones(3)
Output:
array([1., 1., 1.])
np.ones((4,5))
Output:
array([[1., 1., 1., 1., 1.],
[1., 1., 1., 1., 1.],
[1., 1., 1., 1., 1.],
[1., 1., 1., 1., 1.]])
# np.linspace()
np.linspace(0,10,2)
Output:
array([ 0., 10.])
np.linspace(0,10,10)
Output:
array([ 0. , 1.11111111, 2.22222222, 3.33333333, 4.44444444,
5.55555556, 6.66666667, 7.77777778, 8.88888889, 10. ])
np.empty((2,3))
Output:
array([[2.67276450e+185, 1.69506143e+190, 1.75184137e+190],
[9.48819320e+077, 1.63730399e-306, 0.00000000e+000]])
np.full((2,3),fill_value=12)
Output:
array([[12, 12, 12],
[12, 12, 12]])
random class inside numpy class (generated random data following some distribution)
help(np.random)
Output:
Help on package numpy.random in numpy:
NAME
numpy.random
DESCRIPTION
========================
Random Number Generation
========================
==================== =========================================================
Utility functions
==============================================================================
random_sample Uniformly distributed floats over ``[0, 1)``.
random Alias for `random_sample`.
bytes Uniformly distributed random bytes.
random_integers Uniformly distributed integers in a given range.
permutation Randomly permute a sequence / generate a random sequence.
shuffle Randomly permute a sequence in place.
seed Seed the random number generator.
choice Random sample from 1-D array.
==================== =========================================================
==================== =========================================================
Compatibility functions
==============================================================================
rand Uniformly distributed values.
randn Normally distributed values.
ranf Uniformly distributed floating point numbers.
randint Uniformly distributed integers in a given range.
==================== =========================================================
==================== =========================================================
Univariate distributions
==============================================================================
beta Beta distribution over ``[0, 1]``.
binomial Binomial distribution.
chisquare :math:`\chi^2` distribution.
exponential Exponential distribution.
f F (Fisher-Snedecor) distribution.
gamma Gamma distribution.
geometric Geometric distribution.
gumbel Gumbel distribution.
hypergeometric Hypergeometric distribution.
laplace Laplace distribution.
logistic Logistic distribution.
lognormal Log-normal distribution.
logseries Logarithmic series distribution.
negative_binomial Negative binomial distribution.
noncentral_chisquare Non-central chi-square distribution.
noncentral_f Non-central F distribution.
normal Normal / Gaussian distribution.
pareto Pareto distribution.
poisson Poisson distribution.
power Power distribution.
rayleigh Rayleigh distribution.
triangular Triangular distribution.
uniform Uniform distribution.
vonmises Von Mises circular distribution.
wald Wald (inverse Gaussian) distribution.
... (output truncated)
np.random.randint(2,45,12)
Output:
array([17, 14, 18, 18, 11, 43, 24, 13, 9, 15, 26, 39])
np.random.rand(12)
Output:
array([0.99276065, 0.41500381, 0.25028878, 0.91046322, 0.04493408,
0.32149137, 0.14403006, 0.44153052, 0.32109934, 0.6895279 ,
0.72788751, 0.36467681])
np.random.randn(12)
Output:
array([-0.39758898, 1.09286962, 1.31434009, 1.03881812, 0.22472093,
-1.19395013, -0.59245433, 0.03993461, -0.05421944, 1.49273983,
-0.38237876, -0.37428445])
np.random.random((2,3)) # random() inside random()
Output:
array([[0.75163235, 0.46451207, 0.24398095],
[0.8308684 , 0.4203051 , 0.97528916]])
np.random.randn()
Output:
-0.16619521168481557
Note
>>> print(np.arange(10000))
[ 0 1 2 ..., 9997 9998 9999]
>>>
>>> print(np.arange(10000).reshape(100,100))
[[ 0 1 2 ..., 97 98 99]
[ 100 101 102 ..., 197 198 199]
[ 200 201 202 ..., 297 298 299]
...,
[9700 9701 9702 ..., 9797 9798 9799]
[9800 9801 9802 ..., 9897 9898 9899]
[9900 9901 9902 ..., 9997 9998 9999]]
To disable this behaviour and force NumPy to print the entire array, you can change the printing options using set_printoptions.
>>>
>>> np.set_printoptions(threshold=np.nan)
print(np.arange(10000))
Output:
[ 0 1 2 ... 9997 9998 9999]
np.set_printoptions(threshold=np.nan)
print(np.arange(10000))
Output:
[ 0 1 2 3 4 5 6 7 8 9 10 11 12 13
14 15 16 17 18 19 20 21 22 23 24 25 26 27
28 29 30 31 32 33 34 35 36 37 38 39 40 41
42 43 44 45 46 47 48 49 50 51 52 53 54 55
56 57 58 59 60 61 62 63 64 65 66 67 68 69
70 71 72 73 74 75 76 77 78 79 80 81 82 83
84 85 86 87 88 89 90 91 92 93 94 95 96 97
98 99 100 101 102 103 104 105 106 107 108 109 110 111
112 113 114 115 116 117 118 119 120 121 122 123 124 125
126 127 128 129 130 131 132 133 134 135 136 137 138 139
140 141 142 143 144 145 146 147 148 149 150 151 152 153
154 155 156 157 158 159 160 161 162 163 164 165 166 167
168 169 170 171 172 173 174 175 176 177 178 179 180 181
182 183 184 185 186 187 188 189 190 191 192 193 194 195
196 197 198 199 200 201 202 203 204 205 206 207 208 209
210 211 212 213 214 215 216 217 218 219 220 221 222 223
224 225 226 227 228 229 230 231 232 233 234 235 236 237
238 239 240 241 242 243 244 245 246 247 248 249 250 251
252 253 254 255 256 257 258 259 260 261 262 263 264 265
266 267 268 269 270 271 272 273 274 275 276 277 278 279
280 281 282 283 284 285 286 287 288 289 290 291 292 293
294 295 296 297 298 299 300 301 302 303 304 305 306 307
308 309 310 311 312 313 314 315 316 317 318 319 320 321
322 323 324 325 326 327 328 329 330 331 332 333 334 335
336 337 338 339 340 341 342 343 344 345 346 347 348 349
350 351 352 353 354 355 356 357 358 359 360 361 362 363
364 365 366 367 368 369 370 371 372 373 374 375 376 377
378 379 380 381 382 383 384 385 386 387 388 389 390 391
392 393 394 395 396 397 398 399 400 401 402 403 404 405
406 407 408 409 410 411 412 413 414 415 416 417 418 419
420 421 422 423 424 425 426 427 428 429 430 431 432 433
434 435 436 437 438 439 440 441 442 443 444 445 446 447
448 449 450 451 452 453 454 455 456 457 458 459 460 461
462 463 464 465 466 467 468 469 470 471 472 473 474 475
476 477 478 479 480 481 482 483 484 485 486 487 488 489
490 491 492 493 494 495 496 497 498 499 500 501 502 503
504 505 506 507 508 509 510 511 512 513 514 515 516 517
518 519 520 521 522 523 524 525 526 527 528 529 530 531
532 533 534 535 536 537 538 539 540 541 542 543 544 545
546 547 548 549 550 551 552 553 554 555 556 557 558 559
560 561 562 563 564 565 566 567 568 569 570 571 572 573
574 575 576 577 578 579 580 581 582 583 584 585 586 587
588 589 590 5
... (output truncated)
Broadcasting(Basically applying some operation over various elements at once)
Numpy arrays differ from a normal Python list because of their ability to broadcast:
arr = np.arange(15)
arr
Output:
array([ 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14])
#Setting a value with index range (Broadcasting)
arr[0:5]=100
#Show
arr
Output:
array([100, 100, 100, 100, 100, 5, 6, 7, 8, 9, 10, 11, 12,
13, 14])
# Reset array, we'll see why I had to reset in a moment
arr = np.arange(0,11)
#Show
arr
Output:
array([ 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10])
#Important notes on Slices
slice_of_arr = arr[0:6]
#Show slice
slice_of_arr
Output:
array([0, 1, 2, 3, 4, 5])
#Change Slice
slice_of_arr[:]=99
#Show Slice again
slice_of_arr
Output:
array([99, 99, 99, 99, 99, 99])
Now note the changes also occur in our original array!
arr # this is bacuse sequence slicing in numpy gives a view to array only not copy the elements
Output:
array([99, 99, 99, 99, 99, 99, 6, 7, 8, 9, 10])
Note that in all of cases where subsections of the array have been selected, the returned arrays are views.
arrays generated by basic slicing are always views of the original array.
Data is not copied, it's a view of the original array! This avoids memory problems!
#To get a copy, need to be explicit
arr_copy = arr.copy()
arr_copy
Output:
array([99, 99, 99, 99, 99, 99, 6, 7, 8, 9, 10])
Operation on numpy arrays(cann't be done on list )-
- Vectorization means applying operations element wise
- arrays are Broadcasted if arrays are of Different sizes if mismatched dimesion is 1.
arr = np.arange(10)
arr
Output:
array([0, 1, 2, 3, 4, 5, 6, 7, 8, 9])
arr+arr
Output:
array([ 0, 2, 4, 6, 8, 10, 12, 14, 16, 18])
arr-arr
Output:
array([0, 0, 0, 0, 0, 0, 0, 0, 0, 0])
arr/arr
Output:
c:\users\shaurya singhal\appdata\local\programs\python\python37-32\lib\site-packages\ipykernel_launcher.py:1: RuntimeWarning: invalid value encountered in true_divide
"""Entry point for launching an IPython kernel.
array([nan, 1., 1., 1., 1., 1., 1., 1., 1., 1.])
arr*arr
Output:
array([ 0, 1, 4, 9, 16, 25, 36, 49, 64, 81])
1/arr
Output:
c:\users\shaurya singhal\appdata\local\programs\python\python37-32\lib\site-packages\ipykernel_launcher.py:1: RuntimeWarning: divide by zero encountered in true_divide
"""Entry point for launching an IPython kernel.
array([ inf, 1. , 0.5 , 0.33333333, 0.25 ,
0.2 , 0.16666667, 0.14285714, 0.125 , 0.11111111])
arr**arr
Output:
array([ 1, 1, 4, 27, 256, 3125,
46656, 823543, 16777216, 387420489], dtype=int32)
arr
Output:
array([0, 1, 2, 3, 4, 5, 6, 7, 8, 9])
arr<5
Output:
array([ True, True, True, True, True, False, False, False, False,
False])
arr[arr<4]
Output:
array([0, 1, 2, 3])
The Python keywords and and or do not work with boolean arrays. Use & (and) and | (or) instead.
arr[arr>4]
Output:
array([5, 6, 7, 8, 9])
# Product of matrices
a=np.array([[1,2],[4,5]])
a
Output:
array([[1, 2],
[4, 5]])
a*a # element product
Output:
array([[ 1, 4],
[16, 25]])
a @ a # matrix product (a and b must be compatible matrix i.e a-cols == b-rows)
Output:
array([[ 9, 12],
[24, 33]])
a.dot(a) # matrix product
Output:
array([[ 9, 12],
[24, 33]])
# max ,min
arr = np.random.random((2,3))
arr
Output:
array([[0.82548126, 0.33824021, 0.1205504 ],
[0.6991525 , 0.88811133, 0.64167258]])
arr.max()
Output:
0.8881113349325174
arr.min()
Output:
0.12055039839690174
print(arr.argmin(axis=0))
Output:
[1 0 0]
np.min(arr)
Output:
0.12055039839690174
np.sum(arr)
Output:
3.5132082811098466
arr
Output:
array([[0.82548126, 0.33824021, 0.1205504 ],
[0.6991525 , 0.88811133, 0.64167258]])
# find max in particular axis
arr.max(axis = 0) # find max in cols
Output:
array([0.82548126, 0.88811133, 0.64167258])
arr.max(axis=1) # max in rows
Output:
array([0.82548126, 0.88811133])
Universal Functions of NumPy(or mathematical functions)
A universal function (or ufunc for short) is a function that operates on ndarrays in an element-by-element fashion, supporting array broadcasting, type casting, and several other standard features. That is, a ufunc is a “vectorized” wrapper for a function that takes a fixed number of specific inputs and produces a fixed number of specific outputs.
import math
print(math.sqrt(4))
print(math.sin(math.pi/6))
Output:
2.0
0.49999999999999994
# using numpy
np.sqrt(4)
Output:
2.0
np.sin(np.pi/6)
Output:
0.49999999999999994
np.exp(3)
Output:
20.085536923187668
np.exp(np.array([0,1,2]))
Output:
array([1. , 2.71828183, 7.3890561 ])
Indexing a 2D array (matrices)
The general format is arr_2d[row][col] or arr_2d[row,col].
- when evaluating first form arr_2d[row][col] , first bracket([row]) is evaluated and then second one; make it hard to select submatrix from original matrix arr_2d becoz after first bracket execution also it will remain same nd matrix as it is original thus making impossible to select submatrix.
- use
arr_2d[row,col]for selecting sub matrix in python becoz it is evaluated as it is and selecting a subpart
arr_2d = np.array(([5,10,15],[20,25,30],[35,40,45]))
#Show
arr_2d
Output:
array([[ 5, 10, 15],
[20, 25, 30],
[35, 40, 45]])
#Indexing row
arr_2d[1]
Output:
array([20, 25, 30])
# Format is arr_2d[row][col] or arr_2d[row,col]
# Getting individual element value
arr_2d[1][0]
Output:
20
# Getting individual element value
arr_2d[1,0]
Output:
20
arr_2d
Output:
array([[ 5, 10, 15],
[20, 25, 30],
[35, 40, 45]])
# 2D array slicing
#Shape (2,2) from top right corner
arr_2d[:2,1:]
Output:
array([[10, 15],
[25, 30]])
arr_2d[:2][1:] # see result in different answer thats what we dont wanted
Output:
array([[20, 25, 30]])
#Shape bottom row
arr_2d[2]
Output:
array([35, 40, 45])
#Shape bottom row
arr_2d[2,:]
Output:
array([35, 40, 45])
Fancy Indexing
Fancy indexing allows you to select entire rows or columns out of order,to show this, let's quickly build out a numpy array:
#Set up matrix
arr2d = np.zeros((10,10))
arr2d.shape
Output:
(10, 10)
# no of columns of array
arr_length = arr2d.shape[1]
print(arr_length)
Output:
10
#Set up array
for i in range(arr_length):
arr2d[i] = i+1
arr2d
Output:
array([[ 1., 1., 1., 1., 1., 1., 1., 1., 1., 1.],
[ 2., 2., 2., 2., 2., 2., 2., 2., 2., 2.],
[ 3., 3., 3., 3., 3., 3., 3., 3., 3., 3.],
[ 4., 4., 4., 4., 4., 4., 4., 4., 4., 4.],
[ 5., 5., 5., 5., 5., 5., 5., 5., 5., 5.],
[ 6., 6., 6., 6., 6., 6., 6., 6., 6., 6.],
[ 7., 7., 7., 7., 7., 7., 7., 7., 7., 7.],
[ 8., 8., 8., 8., 8., 8., 8., 8., 8., 8.],
[ 9., 9., 9., 9., 9., 9., 9., 9., 9., 9.],
[10., 10., 10., 10., 10., 10., 10., 10., 10., 10.]])
Fancy indexing allows the following
arr2d[[2,4,6,8]]
Output:
array([[3., 3., 3., 3., 3., 3., 3., 3., 3., 3.],
[5., 5., 5., 5., 5., 5., 5., 5., 5., 5.],
[7., 7., 7., 7., 7., 7., 7., 7., 7., 7.],
[9., 9., 9., 9., 9., 9., 9., 9., 9., 9.]])
#Allows in any order
arr2d[[6,4,2,7]]
Output:
array([[7., 7., 7., 7., 7., 7., 7., 7., 7., 7.],
[5., 5., 5., 5., 5., 5., 5., 5., 5., 5.],
[3., 3., 3., 3., 3., 3., 3., 3., 3., 3.],
[8., 8., 8., 8., 8., 8., 8., 8., 8., 8.]])
numpy for statistics
- min , max
- mean
- average => weighted mean
- median
- mode
- std
- variance
import numpy as np
np.min([1,2,3])
Output:
1
np.max([3,34,5])
Output:
34
np.mean([2,3,4])
Output:
3.0
np.average([1,5,4,2,0], weights=[1,2,3,4,5]) # sum(wi * xi) /sum(wi)
Output:
2.066666666666667
(1*1 + 5*2 + 4*3 + 2*4 + 0*5 ) / 15
Output:
2.066666666666667
np.median([2,3,4,65,75,8,67,8,9,654,1,83])
Output:
8.5
# using formula
a = np.array([2,3,4])
u = np.mean(a)
std = np.sqrt(np.mean((a-u)**2))
print(std)
Output:
0.816496580927726
# using inbuilt std function
np.std([2,3,4])
Output:
0.816496580927726
np.var([2,3,4])
Output:
0.6666666666666666
Stacking together different arrays
np.hstack()- To stack arrays along horizontal axis.np.vstack()- To stack arrays along vertical axis.np.row_stack()- To stack 1-D arrays as rows into 2-D arraysnp.column_stack()- To stack 1-D arrays as column into 2-D arraysnp.concatenate(): To stack arrays along specified axis (axis is passed as argument).- can stack more than two arrays
Splitting one array into several smaller ones
hsplit()vsplit()
a = np.floor(10*np.random.random((2,12)))
a
array([[ 9., 5., 6., 3., 6., 8., 0., 7., 9., 7., 2., 7.],
[ 1., 4., 9., 2., 2., 1., 0., 6., 2., 2., 4., 0.]])
np.hsplit(a,3) # Split a into 3
[array([[ 9., 5., 6., 3.],
[ 1., 4., 9., 2.]]), array([[ 6., 8., 0., 7.],
[ 2., 1., 0., 6.]]), array([[ 9., 7., 2., 7.],
[ 2., 2., 4., 0.]])]
np.hsplit(a,(3,4)) # Split a after the third and the fourth column
[array([[ 9., 5., 6.],
[ 1., 4., 9.]]), array([[ 3.],
[ 2.]]), array([[ 6., 8., 0., 7., 9., 7., 2., 7.],
[ 2., 1., 0., 6., 2., 2., 4., 0.]])]
Copies and Views
When operating and manipulating arrays, their data is sometimes copied into a new array and sometimes not. This is often a source of confusion for beginners. There are three cases:
No Copy at All
Simple assignments make no copy of array objects or of their data.
a = np.arange(12)
b = a # no new object is created
b is a # a and b are two names for the same ndarray object
True
b.shape = 3,4 # changes the shape of a
a.shape
(3, 4)
View or Shallow Copy(view())
Different array objects can share the same data. The view method creates a new array object that looks at the same data.
c = a.view()
c is a
False
c.base is a # c is a view of the data owned by a
True
c.flags.owndata
False
c.shape = 2,6 # a's shape doesn't change
a.shape
(3, 4)
c[0,4] = 1234 # a's data changes
a
array([[ 0, 1, 2, 3],
[1234, 5, 6, 7],
[ 8, 9, 10, 11]])
Slicing an array returns a view of it:
>>> s = a[ : , 1:3]
>>> s[:] = 10 # s[:] is a view of s. Note the difference between s=10 and s[:]=10
>>> a
array([[ 0, 10, 10, 3],
[1234, 10, 10, 7],
[ 8, 10, 10, 11]])
Deep Copy
The copy method makes a complete copy of the array and its data.
d = a.copy() # a new array object with new data is created
d is a
False
d.base is a # d doesn't share anything with a
False
d[0,0] = 9999
a
array([[ 0, 10, 10, 3],
[1234, 10, 10, 7],
[ 8, 10, 10, 11]])
Broadcasting rules
The first rule of broadcasting is that if all input arrays do not have the same number of dimensions, a “1” will be repeatedly prepended to the shapes of the smaller arrays until all the arrays have the same number of dimensions.
The second rule of broadcasting ensures that arrays with a size of 1 along a particular dimension act as if they had the size of the array with the largest shape along that dimension. The value of the array element is assumed to be the same along that dimension for the “broadcast” array.
In order to broadcast, the size of the trailing axes for both arrays in an operation must either be the same size or one of them must be one.
Let us see some examples:
A(2-D array): 4 x 3 B(1-D array): 3 Result : 4 x 3 A(4-D array): 7 x 1 x 6 x 1 B(3-D array): 3 x 1 x 5 Result : 7 x 3 x 6 x 5 But this would be a mismatch:
A: 4 x 3 B: 4 Now, let us see
Array Sorting
There is a simple np.sort method for sorting NumPy arrays.
import numpy as np
arr = np.array([[1, 4, 2],
[3, 4, 6],
[0, -1, 5]])
np.sort(arr, axis=1)
Output:
array([[ 1, 2, 4],
[ 3, 4, 6],
[-1, 0, 5]])
np.sort(arr, axis=0)
Output:
array([[ 0, -1, 2],
[ 1, 4, 5],
[ 3, 4, 6]])
np.sort(arr, axis=None)
Output:
array([-1, 0, 1, 2, 3, 4, 4, 5, 6])
Linear Algebra
The Linear Algebra module of NumPy offers various methods to apply linear algebra on any numpy array.
You can find:
- rank, determinant, trace, etc. of an array.
- eigen values of matrices
- matrix and vector products (dot, inner, outer,etc. product), matrix exponentiation
- solve linear or tensor equations and much more! Now, let us assume that we want to solve this linear equation set:
x + 2*y = 8
3*x + 4*y = 18
This problem can be solved using linalg.solve method as shown in example below:
import numpy as np
# coefficients
a = np.array([[1, 2], [3, 4]])
# constants
b = np.array([8, 18])
np.linalg.solve(a, b)
Output:
array([2., 3.])
A = np.array([[6, 1, 1],
[4, -2, 5],
[2, 8, 7]])
# rank of matrix
np.linalg.matrix_rank(A)
Output:
3
# trace of matrix- diagonal sum
np.trace(A)
Output:
11
# determinant of matrix
np.linalg.det(A)
Output:
-306.0
# inverse of matrix
np.linalg.inv(A)
Output:
array([[ 0.17647059, -0.00326797, -0.02287582],
[ 0.05882353, -0.13071895, 0.08496732],
[-0.11764706, 0.1503268 , 0.05228758]])
# matrix exponentiation
np.linalg.matrix_power(A,3)
Output:
array([[336, 162, 228],
[406, 162, 469],
[698, 702, 905]])
Saving and loading numpy arrays
The .npy format is the standard binary file format in NumPy for persisting a single arbitrary NumPy array on disk. The format stores all of the shape and dtype information necessary to reconstruct the array correctly even on another machine with a different architecture. The format is designed to be as simple as possible while achieving its limited goals.
The .npz format is the standard format for persisting multiple NumPy arrays on disk. A .npz file is a zip file containing multiple .npy files, one for each array.
np.save(filename.npy, array) : saves a single array in npy format.
np.savez(filename.npz, array_1[, array_2]) : saves multiple numpy arrays in npz format.
np.load(filename) : load a npy or npz format file.
import numpy as np
a = np.array([[1,2,3],
[4,5,6]])
b = np.array([[6,5,4],
[3,2,1]])
np.save("a.npy", a)
a = np.load("a.npy")
a
Output:
array([[1, 2, 3],
[4, 5, 6]])
np.savez("AB.npz", x=a, y=b)
arr = np.load("AB.npz")
arr['x']
Output:
array([[1, 2, 3],
[4, 5, 6]])
arr['y']
Output:
array([[6, 5, 4],
[3, 2, 1]])