Numpy.nonzero

Примеры работы с NumPy

Подытожим все вышесказанное. Вот несколько примеров полезных инструментов NumPy, которые могут значительно облегчить процесс написания кода.


Математические формулы NumPy

Необходимость внедрения математических формул, которые будут работать с матрицами и векторами, является главной причиной использования NumPy. Именно поэтому NumPy пользуется большой популярностью среди представителей науки. В качестве примера рассмотрим формулу , которая является центральной для контролируемых моделей машинного обучения, что решают проблемы регрессии:

Реализовать данную формулу в NumPy довольно легко:

Главное достоинство NumPy в том, что его не заботит, если и содержат одно или тысячи значение (до тех пор, пока они оба одного размера). Рассмотрим пример, последовательно изучив четыре операции в следующей строке кода:

У обоих векторов и по три значения. Это значит, что в данном случае равно трем. После выполнения указанного выше вычитания мы получим значения, которые будут выглядеть следующим образом:

Затем мы можем возвести значения вектора в квадрат:

Теперь мы вычисляем эти значения:

Таким образом мы получаем значение ошибки некого прогноза и за качество модели.

Представление данных NumPy

Задумайтесь о всех тех типах данных, которыми вам понадобится оперировать, создавая различные модели работы (электронные таблицы, изображения, аудио и так далее). Очень многие типы могут быть представлены как n-мерные массивы:

The syntax of the NumPy zeros function

The NumPy zeros function enables you to create NumPy arrays that contain only zeros.

Importantly, this function enables you to specify the exact dimensions of the array. It also enables you to specify the exact data type.

The syntax works like this:

You basically call the function with the code .

Then inside the function, there is a set of arguments. The first positional argument is a tuple of values that specifies the dimensions of the new array.

Next, there’s an argument that enables you to specify the data type. If you don’t specify a data type, will use floats by default.

The syntax for using the zeros function is pretty straightforward, but it’s always easier to understand code when you have a few examples to work with. That being the case, let’s take a look at some examples.

Array Scalars¶

NumPy generally returns elements of arrays as array scalars (a scalar with an associated dtype). Array scalars differ from Python scalars, but for the most part they can be used interchangeably (the primary exception is for versions of Python older than v2.x, where integer array scalars cannot act as indices for lists and tuples). There are some exceptions, such as when code requires very specific attributes of a scalar or when it checks specifically whether a value is a Python scalar. Generally, problems are easily fixed by explicitly converting array scalars to Python scalars, using the corresponding Python type function (e.g., , , , , ).

A quick review of NumPy arrays

A NumPy array is basically like a container that holds numeric data that’s all of the same data type. I’m simplifying things a little, but that’s the essence of them.

We can create a very simple NumPy array as follows:

import numpy as np

np.array(,])

Which we can visually represent as follows:

Here, we’ve used the NumPy array function to create a 2-dimensional array with 2 rows and 6 columns. Notice as well that all of the data are integers. Again, in a NumPy array, all of the data must be of the same data type.

Keep in mind that NumPy arrays can be quite a bit more complicated as well. It’s possible to construct 3-dimensional arrays and N-dimensional arrays. For the sake of clarity though, we’ll work with 1 and 2-dimensional arrays in this post.

Creating empty arrays

There will be times when you’ll need to create an empty array. For example, you may be performing some computations and you need a structure to hold the results of those computations. In such a case, you might want to create an empty array …. that is, an array with all zeroes.

One way of doing this is with the NumPy function. You can create a an empty NumPy array by passing in a Python list with all zeros:

np.array(,])

The problem with this though is that it may not always be efficient. It’s a little cumbersome. And it would be very cumbersome if you needed to create a very large array or an array with high dimensions. For example, here’s some code to create an array with 3 rows and 30 columns that contains all zeros:

np.array(
,
,])

Honestly, this is a little ridiculous (and would be even more ridiculous if you made a larger array).

Hypothetically, you could also create a 1-dimensional array with the right number of zeros, and then reshape it to the correct shape using the NumPy reshape method.

np.zeros(90).reshape((3,30))

To be honest though, that method is also cumbersome and a little prone to errors.

There must be a better way, right?

Yes, there is.

Enter, the NumPy zeros function.

3.1. Прежде чем читать

Нужно немного знать Python. Причем «немного» означает действительно немного и вовсе не означает, что перед чтением данного руководства вам нужно досконально изучить этот язык. Открытой вкладки с официальным руководством окажется вполне достаточно.

Все примеры выполнены в консоли IDE Spyder дистрибутива Anaconda на Python версии 3.5. и NumPy версии 1.14.0. Приводимые примеры так же будут работать в любом другом дистрибутиве Python 3.х версии и последней версией пакета NumPy. Но если некоторые примеры все же не работают, то ознакомьтесь с официальной документацией вашего дистрибутива, возможно причина связана с его особенностями.

Например, если в своем дистрибутиве вы обнаружили последнюю версию IDE Spyder, то в ней нет Python консоли, к которой привыкают многие новички, учившиеся экспериментировать с кодом в IDLE. При этом новичкам может так же показаться, что и все примеры, представленные здесь, тоже лучше выполнять в Python консоли. Но нет, Python консоль использовалась автором лишь по техническим причинам, которые связаны с редактурой, версткой и дизайном кода. Консоль IPython имеет гораздо больше преимуществ.

Prerequisites¶

Before reading this tutorial you should know a bit of Python. If you would like to refresh your memory, take a look at the Python tutorial.

If you wish to work the examples in this tutorial, you must also have some software installed on your computer. Please see https://scipy.org/install.html for instructions.

Learner profile

This tutorial is intended as a quick overview of algebra and arrays in NumPy and want to understand how n-dimensional () arrays are represented and can be manipulated. In particular, if you don’t know how to apply common functions to n-dimensional arrays (without using for-loops), or if you want to understand axis and shape properties for n-dimensional arrays, this tutorial might be of help.

Learning Objectives

After this tutorial, you should be able to:

1.4.1.6. Copies and views¶

A slicing operation creates a view on the original array, which is just a way of accessing array data. Thus the original array is not copied in memory. You can use to check if two arrays share the same memory block. Note however, that this uses heuristics and may give you false positives.

When modifying the view, the original array is modified as well:

>>> a = np.arange(10)
>>> a
array()
>>> b = a)
>>> np.may_share_memory(a, b)
True
>>> b = 12
>>> b
array()
>>> a   # (!)
array()

>>> a = np.arange(10)
>>> c = a)

>>> np.may_share_memory(a, c)
False

This behavior can be surprising at first sight… but it allows to save both memory and time.

Access 3-D Arrays

To access elements from 3-D arrays we can use comma separated integers representing the dimensions and the index of the element.

Example

Access the third element of the second array of the first array:

import numpy as nparr = np.array(, ], , ]]) print(arr)

Example Explained

prints the value .

And this is why:

The first number represents the first dimension, which contains two arrays: , ] and: , ] Since we selected , we are left with the first array: , ]

The second number represents the second dimension, which also contains two arrays:

and:

Since we selected , we are left with the second array:

The third number represents the third dimension, which contains three values: 4 5 6 Since we selected , we end up with the third value: 6

1.4.1.7. Fancy indexing¶

Tip

NumPy arrays can be indexed with slices, but also with boolean or integer arrays (masks). This method is called fancy indexing. It creates copies not views.

Using boolean masks

>>> np.random.seed(3)
>>> a = np.random.randint(, 21, 15)
>>> a
array()
>>> (a % 3 == )
array()
>>> mask = (a % 3 == )
>>> extract_from_a = amask # or,  a
>>> extract_from_a           # extract a sub-array with the mask
array()

Indexing with a mask can be very useful to assign a new value to a sub-array:

>>> aa % 3 ==  = -1
>>> a
array()

Indexing with an array of integers

>>> a = np.arange(, 100, 10)
>>> a
array()

Indexing can be done with an array of integers, where the same index is repeated several time:

>>> a]  # note:  is a Python list
array()

New values can be assigned with this kind of indexing:

>>> a] = -100
>>> a
array()

Tip

When a new array is created by indexing with an array of integers, the new array has the same shape as the array of integers:

>>> a = np.arange(10)
>>> idx = np.array(, 9, 7]])
>>> idx.shape
(2, 2)
>>> aidx
array(,
       ])

The image below illustrates various fancy indexing applications

Exercise: Fancy indexing

  • Again, reproduce the fancy indexing shown in the diagram above.
  • Use fancy indexing on the left and array creation on the right to assign values into an array, for instance by setting parts of the array in the diagram above to zero.

Converting Data Type on Existing Arrays

The best way to change the data type of an existing array, is to make a copy of the array with the method.

The function creates a copy of the array, and allows you to specify the data type as a parameter.

The data type can be specified using a string, like for float, for integer etc. or you can use the data type directly like for float and for integer.

Example

Change data type from float to integer by using as parameter value:

import numpy as nparr = np.array()newarr = arr.astype(‘i’)print(newarr)print(newarr.dtype)

Example

Change data type from float to integer by using as parameter value:

import numpy as nparr = np.array()newarr = arr.astype(int)print(newarr)print(newarr.dtype)

Example

Change data type from integer to boolean:


import numpy as nparr = np.array()newarr = arr.astype(bool)print(newarr)print(newarr.dtype)

Table of Rough MATLAB-NumPy Equivalents¶

The table below gives rough equivalents for some common MATLAB expressions. These are not exact equivalents, but rather should be taken as hints to get you going in the right direction. For more detail read the built-in documentation on the NumPy functions.

In the table below, it is assumed that you have executed the following commands in Python:

from numpy import *
import scipy.linalg

Also assume below that if the Notes talk about “matrix” that the arguments are two-dimensional entities.

General Purpose Equivalents

MATLAB

numpy

Notes

or or (in Ipython)

get help on the function func

find out where func is defined

or (in Ipython)

print source for func (if not a native function)

short-circuiting logical AND operator (Python native operator); scalar arguments only

short-circuiting logical OR operator (Python native operator); scalar arguments only

, , ,

complex numbers

Distance between 1 and the nearest floating point number.

integrate an ODE with Runge-Kutta 4,5

integrate an ODE with BDF method

Обработка изображений в NumPy

Изображение является матрицей пикселей по размеру (высота х ширина).

Если изображение черно-белое, то есть представленное в полутонах, каждый пиксель может быть представлен как единственное число. Обычно между 0 (черный) и 255 (белый). Хотите обрезать квадрат размером пикселей в верхнем левом углу картинки? Просто попросите в NumPy .

Вот как выглядит фрагмент изображения:

Если изображение цветное, каждый пиксель представлен тремя числами. Здесь за основу берется цветовая модель RGB — красный (R), зеленый (G) и синий (B).

В данном случае нам понадобится третья размерность, так как каждая клетка вмещает только одно число. Таким образом, цветная картинка будет представлена массивом с размерностями: (высота х ширина х 3).

Field Access¶

See also

,

If the object is a structured array the of the array can be accessed by indexing the array with strings, dictionary-like.

Indexing returns a new to the array, which is of the same shape as x (except when the field is a sub-array) but of data type and contains only the part of the data in the specified field. Also scalars can be “indexed” this way.

Indexing into a structured array can also be done with a list of field names, e.g. . As of NumPy 1.16 this returns a view containing only those fields. In older versions of numpy it returned a copy. See the user guide section on for more information on multifield indexing.

If the accessed field is a sub-array, the dimensions of the sub-array are appended to the shape of the result.

1.4.1.5. Indexing and slicing¶

The items of an array can be accessed and assigned to the same way as other Python sequences (e.g. lists):

>>> a = np.arange(10)
>>> a
array()
>>> a], a2], a-1
(0, 2, 9)

Warning

Indices begin at 0, like other Python sequences (and C/C++). In contrast, in Fortran or Matlab, indices begin at 1.

The usual python idiom for reversing a sequence is supported:

>>> a)

For multidimensional arrays, indexes are tuples of integers:

>>> a = np.diag(np.arange(3))
>>> a
array(,
       ,
       ])
>>> a1, 1
1
>>> a2, 1 = 10 # third line, second column
>>> a
array(,
       ,
       ])
>>> a1
array()

Note

  • In 2D, the first dimension corresponds to rows, the second to columns.
  • for multidimensional , is interpreted by taking all elements in the unspecified dimensions.

Slicing: Arrays, like other Python sequences can also be sliced:

>>> a = np.arange(10)
>>> a
array()
>>> a293 # 
array()

Note that the last index is not included! :

>>> a)

All three slice components are not required: by default, start is 0, end is the last and step is 1:

>>> a13
array()
>>> a)
>>> a3:]
array()

A small illustrated summary of NumPy indexing and slicing…


You can also combine assignment and slicing:

>>> a = np.arange(10)
>>> a5:] = 10
>>> a
array()
>>> b = np.arange(5)
>>> a5:] = b)

Exercise: Indexing and slicing

  • Try the different flavours of slicing, using , and : starting from a linspace, try to obtain odd numbers counting backwards, and even numbers counting forwards.

  • Reproduce the slices in the diagram above. You may use the following expression to create the array:

    >>> np.arange(6) + np.arange(, 51, 10),
           ,
           ,
           ,
           ,
           ])
    

Exercise: Array creation

Create the following arrays (with correct data types):

,
 1, 1, 1, 1],
 1, 1, 1, 2],
 1, 6, 1, 1]]

,
 2., , , , ],
 , 3., , , ],
 , , 4., , ],
 , , , 5., ],
 , , , , 6.]]

Par on course: 3 statements for each

Hint: Individual array elements can be accessed similarly to a list, e.g. or .

Hint: Examine the docstring for .

Recommendations

We’ll start with recommendations based on the user’s experience level and operating system of interest. If you’re in between “beginning” and “advanced”, please go with “beginning” if you want to keep things simple, and with “advanced” if you want to work according to best practices that go a longer way in the future.

Beginning users

On all of Windows, macOS, and Linux:

  • Install Anaconda (it installs all packages you need and all other tools mentioned below).
  • For writing and executing code, use notebooks in JupyterLab for exploratory and interactive computing, and Spyder or Visual Studio Code for writing scripts and packages.
  • Use Anaconda Navigator to manage your packages and start JupyterLab, Spyder, or Visual Studio Code.

Windows or macOS

  • Install Miniconda.
  • Keep the conda environment minimal, and use one or more

    to install the package you need for the task or project you’re working on.

  • Unless you’re fine with only the packages in the channel, make your default channel via .

Linux

If you’re fine with slightly outdated packages and prefer stability over being able to use the latest versions of libraries:

  • Use your OS package manager for as much as possible (Python itself, NumPy, and other libraries).
  • Install packages not provided by your package manager with .

If you use a GPU:

  • Install Miniconda.
  • Keep the conda environment minimal, and use one or more

    to install the package you need for the task or project you’re working on.

  • Use the conda channel ( doesn’t have good support for GPU packages yet).

Otherwise:

  • Install Miniforge.
  • Keep the conda environment minimal, and use one or more

    to install the package you need for the task or project you’re working on.

Alternative if you prefer pip/PyPI

For users who know, from personal preference or reading about the main differences between conda and pip below, they prefer a pip/PyPI-based solution, we recommend:

  • Install Python from, for example, python.org, Homebrew, or your Linux package manager.
  • Use Poetry as the most well-maintained tool that provides a dependency resolver and environment management capabilities in a similar fashion as conda does.

Python NumPy

NumPy IntroNumPy Getting StartedNumPy Creating ArraysNumPy Array IndexingNumPy Array SlicingNumPy Data TypesNumPy Copy vs ViewNumPy Array ShapeNumPy Array ReshapeNumPy Array IteratingNumPy Array JoinNumPy Array SplitNumPy Array SearchNumPy Array SortNumPy Array FilterNumPy Random Random Intro Data Distribution Random Permutation Seaborn Module Normal Distribution Binomial Distribution Poisson Distribution Uniform Distribution Logistic Distribution Multinomial Distribution Exponential Distribution Chi Square Distribution Rayleigh Distribution Pareto Distribution Zipf Distribution

NumPy ufunc ufunc Intro ufunc Create Function ufunc Simple Arithmetic ufunc Rounding Decimals ufunc Logs ufunc Summations ufunc Products ufunc Differences ufunc Finding LCM ufunc Finding GCD ufunc Trigonometric ufunc Hyperbolic ufunc Set Operations

Overflow Errors¶

The fixed size of NumPy numeric types may cause overflow errors when a value requires more memory than available in the data type. For example, evaluates correctly for 64-bit integers, but gives 1874919424 (incorrect) for a 32-bit integer.

>>> np.power(100, 8, dtype=np.int64)
10000000000000000
>>> np.power(100, 8, dtype=np.int32)
1874919424

The behaviour of NumPy and Python integer types differs significantly for integer overflows and may confuse users expecting NumPy integers to behave similar to Python’s . Unlike NumPy, the size of Python’s is flexible. This means Python integers may expand to accommodate any integer and will not overflow.

NumPy provides and to verify the minimum or maximum values of NumPy integer and floating point values respectively

>>> np.iinfo(int) # Bounds of the default integer on this system.
iinfo(min=-9223372036854775808, max=9223372036854775807, dtype=int64)
>>> np.iinfo(np.int32) # Bounds of a 32-bit integer
iinfo(min=-2147483648, max=2147483647, dtype=int32)
>>> np.iinfo(np.int64) # Bounds of a 64-bit integer
iinfo(min=-9223372036854775808, max=9223372036854775807, dtype=int64)

If 64-bit integers are still too small the result may be cast to a floating point number. Floating point numbers offer a larger, but inexact, range of possible values.

Filtering Arrays

Getting some elements out of an existing array and creating a new array out of them is called filtering.

In NumPy, you filter an array using a boolean index list.

A boolean index list is a list of booleans corresponding to indexes in the array.

If the value at an index is that element is contained in the filtered array, if the value at that index is that element is excluded from the filtered array.

Example

Create an array from the elements on index 0 and 2:

import numpy as nparr = np.array()x = newarr = arrprint(newarr)

The example above will return , why?

Because the new filter contains only the values where the filter array had the value , in this case, index 0 and 2.

Intrinsic NumPy Array Creation¶

NumPy has built-in functions for creating arrays from scratch:

zeros(shape) will create an array filled with 0 values with the specified shape. The default dtype is float64.

>>> np.zeros((2, 3))
array(, ])

ones(shape) will create an array filled with 1 values. It is identical to zeros in all other respects.

arange() will create arrays with regularly incrementing values. Check the docstring for complete information on the various ways it can be used. A few examples will be given here:

>>> np.arange(10)
array()
>>> np.arange(2, 10, dtype=float)
array()
>>> np.arange(2, 3, 0.1)
array()

Note that there are some subtleties regarding the last usage that the user should be aware of that are described in the arange docstring.

linspace() will create arrays with a specified number of elements, and spaced equally between the specified beginning and end values. For example:

>>> np.linspace(1., 4., 6)
array()

The advantage of this creation function is that one can guarantee the number of elements and the starting and end point, which arange() generally will not do for arbitrary start, stop, and step values.

indices() will create a set of arrays (stacked as a one-higher dimensioned array), one per dimension with each representing variation in that dimension. An example illustrates much better than a verbal description:

>>> np.indices((3,3))
array(, , ], , , ]])

1.4.1.1. What are NumPy and NumPy arrays?¶

NumPy arrays

Python objects:
  • high-level number objects: integers, floating point
  • containers: lists (costless insertion and append), dictionaries (fast lookup)
NumPy provides:
  • extension package to Python for multi-dimensional arrays
  • closer to hardware (efficiency)
  • designed for scientific computation (convenience)
  • Also known as array oriented computing
>>> import numpy as np
>>> a = np.array()
>>> a
array()

Tip

For example, An array containing:

  • values of an experiment/simulation at discrete time steps
  • signal recorded by a measurement device, e.g. sound wave
  • pixels of an image, grey-level or colour
  • 3-D data measured at different X-Y-Z positions, e.g. MRI scan

Why it is useful: Memory-efficient container that provides fast numerical operations.

In : L = range(1000)

In : %timeit i**2 for i in L
1000 loops, best of 3: 403 us per loop

In : a = np.arange(1000)

In : %timeit a**2
100000 loops, best of 3: 12.7 us per loop

NumPy Reference documentation

  • On the web: http://docs.scipy.org/

  • Interactive help:

    In : np.array?
    String Form:<built-in function array>
    Docstring:
    array(object, dtype=None, copy=True, order=None, subok=False, ndmin=0, ...
    
  • Looking for something:

    >>> np.lookfor('create array') 
    Search results for 'create array'
    ---------------------------------
    numpy.array
        Create an array.
    numpy.memmap
        Create a memory-map to an array stored in a *binary* file on disk.
    
    In : np.con*?
    np.concatenate
    np.conj
    np.conjugate
    np.convolve
    

Notes¶

Submatrix: Assignment to a submatrix can be done with lists of indexes using the command. E.g., for 2d array , one might do: .

HELP: There is no direct equivalent of MATLAB’s command, but the commands and will usually list the filename where the function is located. Python also has an module (do ) which provides a that often works.

INDEXING: MATLAB uses one based indexing, so the initial element of a sequence has index 1. Python uses zero based indexing, so the initial element of a sequence has index 0. Confusion and flamewars arise because each has advantages and disadvantages. One based indexing is consistent with common human language usage, where the “first” element of a sequence has index 1. Zero based indexing simplifies indexing. See also a text by prof.dr. Edsger W. Dijkstra.

RANGES: In MATLAB, can be used as both a range literal and a ‘slice’ index (inside parentheses); however, in Python, constructs like can only be used as a slice index (inside square brackets). Thus the somewhat quirky object was created to allow numpy to have a similarly terse range construction mechanism. Note that is not called like a function or a constructor, but rather indexed using square brackets, which allows the use of Python’s slice syntax in the arguments.

LOGICOPS: & or | in NumPy is bitwise AND/OR, while in Matlab & and | are logical AND/OR. The difference should be clear to anyone with significant programming experience. The two can appear to work the same, but there are important differences. If you would have used Matlab’s & or | operators, you should use the NumPy ufuncs logical_and/logical_or. The notable differences between Matlab’s and NumPy’s & and | operators are:

  • Non-logical {0,1} inputs: NumPy’s output is the bitwise AND of the inputs. Matlab treats any non-zero value as 1 and returns the logical AND. For example (3 & 4) in NumPy is 0, while in Matlab both 3 and 4 are considered logical true and (3 & 4) returns 1.

  • Precedence: NumPy’s & operator is higher precedence than logical operators like < and >; Matlab’s is the reverse.

If you know you have boolean arguments, you can get away with using NumPy’s bitwise operators, but be careful with parentheses, like this: z = (x > 1) & (x < 2). The absence of NumPy operator forms of logical_and and logical_or is an unfortunate consequence of Python’s design.

Creating the Filter Array

In the example above we hard-coded the and values, but the common use is to create a filter array based on conditions.

Example

Create a filter array that will return only values higher than 42:

import numpy as nparr = np.array()# Create an empty listfilter_arr = []# go through each element in arrfor element in arr:  # if the element is higher than 42, set the value to True, otherwise False:  if element > 42:    filter_arr.append(True)  else:    filter_arr.append(False)newarr = arrprint(filter_arr)print(newarr)

Example

Create a filter array that will return only even elements from the original array:

import numpy as nparr = np.array()# Create an empty listfilter_arr = []# go through each element in arrfor element in arr:  # if the element is completely divisble by 2, set the value to True, otherwise False  if element % 2 == 0:    filter_arr.append(True)  else:    filter_arr.append(False)newarr = arrprint(filter_arr)print(newarr)


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