GitHub Repo
https://github.com/Ajay2521/Face-Recognition-using-Siamese-Network
Deep learning algorithms fail to work well if we have only one training example. One-shot learning is a classification or object categorization task in which one or a few examples are used to classify many new examples. The principle behind one-shot learning is Humans learn new concepts with very little supervision. ### Problem in COnvNet - A small training set is really not enough to train a robust neural network for this task. The feature vectors trained do not contain important information that can be used for the recognition of future images. - Retraining the ConvNet every time the number of classes or dataset is increased is way too time and resource consuming. ### Solutions to one-shot learning #### 1. Siamese network for one-shot learning  #### Siamese networks are based on a similarity function, which does not require extensive training #### Takes input two images—one being the actual image and other the candidate image—and outputs the similarity between the two. The two input images, if very similar, output = lower value The two input images, if not similar, output = Higher value degree of difference between the two images is compared with the threshold value(𝜏) (which is a hyperparameter), if degree of difference between 2 image < threshold -> output = same person if degree of difference between 2 image > threshold -> output = different person #### Both the images to be compared are passed through the same networks, called sister networks, having the same parameters and shared weights. images are passed through a sequence of convolutional, pooling, and fully connected layers, and end up with a feature vector of a fixed size, say, 128 denoted by f(x1) as an encoding of the input image x1.
GitHub Repo
https://github.com/himanshub1007/Alzhimers-Disease-Prediction-Using-Deep-learning
# AD-Prediction Convolutional Neural Networks for Alzheimer's Disease Prediction Using Brain MRI Image ## Abstract Alzheimers disease (AD) is characterized by severe memory loss and cognitive impairment. It associates with significant brain structure changes, which can be measured by magnetic resonance imaging (MRI) scan. The observable preclinical structure changes provides an opportunity for AD early detection using image classification tools, like convolutional neural network (CNN). However, currently most AD related studies were limited by sample size. Finding an efficient way to train image classifier on limited data is critical. In our project, we explored different transfer-learning methods based on CNN for AD prediction brain structure MRI image. We find that both pretrained 2D AlexNet with 2D-representation method and simple neural network with pretrained 3D autoencoder improved the prediction performance comparing to a deep CNN trained from scratch. The pretrained 2D AlexNet performed even better (**86%**) than the 3D CNN with autoencoder (**77%**). ## Method #### 1. Data In this project, we used public brain MRI data from **Alzheimers Disease Neuroimaging Initiative (ADNI)** Study. ADNI is an ongoing, multicenter cohort study, started from 2004. It focuses on understanding the diagnostic and predictive value of Alzheimers disease specific biomarkers. The ADNI study has three phases: ADNI1, ADNI-GO, and ADNI2. Both ADNI1 and ADNI2 recruited new AD patients and normal control as research participants. Our data included a total of 686 structure MRI scans from both ADNI1 and ADNI2 phases, with 310 AD cases and 376 normal controls. We randomly derived the total sample into training dataset (n = 519), validation dataset (n = 100), and testing dataset (n = 67). #### 2. Image preprocessing Image preprocessing were conducted using Statistical Parametric Mapping (SPM) software, version 12. The original MRI scans were first skull-stripped and segmented using segmentation algorithm based on 6-tissue probability mapping and then normalized to the International Consortium for Brain Mapping template of European brains using affine registration. Other configuration includes: bias, noise, and global intensity normalization. The standard preprocessing process output 3D image files with an uniform size of 121x145x121. Skull-stripping and normalization ensured the comparability between images by transforming the original brain image into a standard image space, so that same brain substructures can be aligned at same image coordinates for different participants. Diluted or enhanced intensity was used to compensate the structure changes. the In our project, we used both whole brain (including both grey matter and white matter) and grey matter only. #### 3. AlexNet and Transfer Learning Convolutional Neural Networks (CNN) are very similar to ordinary Neural Networks. A CNN consists of an input and an output layer, as well as multiple hidden layers. The hidden layers are either convolutional, pooling or fully connected. ConvNet architectures make the explicit assumption that the inputs are images, which allows us to encode certain properties into the architecture. These then make the forward function more efficient to implement and vastly reduce the amount of parameters in the network. #### 3.1. AlexNet The net contains eight layers with weights; the first five are convolutional and the remaining three are fully connected. The overall architecture is shown in Figure 1. The output of the last fully-connected layer is fed to a 1000-way softmax which produces a distribution over the 1000 class labels. AlexNet maximizes the multinomial logistic regression objective, which is equivalent to maximizing the average across training cases of the log-probability of the correct label under the prediction distribution. The kernels of the second, fourth, and fifth convolutional layers are connected only to those kernel maps in the previous layer which reside on the same GPU (as shown in Figure1). The kernels of the third convolutional layer are connected to all kernel maps in the second layer. The neurons in the fully connected layers are connected to all neurons in the previous layer. Response-normalization layers follow the first and second convolutional layers. Max-pooling layers follow both response-normalization layers as well as the fifth convolutional layer. The ReLU non-linearity is applied to the output of every convolutional and fully-connected layer.  The first convolutional layer filters the 224x224x3 input image with 96 kernels of size 11x11x3 with a stride of 4 pixels (this is the distance between the receptive field centers of neighboring neurons in a kernel map). The second convolutional layer takes as input the (response-normalized and pooled) output of the first convolutional layer and filters it with 256 kernels of size 5x5x48. The third, fourth, and fifth convolutional layers are connected to one another without any intervening pooling or normalization layers. The third convolutional layer has 384 kernels of size 3x3x256 connected to the (normalized, pooled) outputs of the second convolutional layer. The fourth convolutional layer has 384 kernels of size 3x3x192 , and the fifth convolutional layer has 256 kernels of size 3x3x192. The fully-connected layers have 4096 neurons each. #### 3.2. Transfer Learning Training an entire Convolutional Network from scratch (with random initialization) is impractical[14] because it is relatively rare to have a dataset of sufficient size. An alternative is to pretrain a Conv-Net on a very large dataset (e.g. ImageNet), and then use the ConvNet either as an initialization or a fixed feature extractor for the task of interest. Typically, there are three major transfer learning scenarios: **ConvNet as fixed feature extractor:** We can take a ConvNet pretrained on ImageNet, and remove the last fully-connected layer, then treat the rest structure as a fixed feature extractor for the target dataset. In AlexNet, this would be a 4096-D vector. Usually, we call these features as CNN codes. Once we get these features, we can train a linear classifier (e.g. linear SVM or Softmax classifier) for our target dataset. **Fine-tuning the ConvNet:** Another idea is not only replace the last fully-connected layer in the classifier, but to also fine-tune the parameters of the pretrained network. Due to overfitting concerns, we can only fine-tune some higher-level part of the network. This suggestion is motivated by the observation that earlier features in a ConvNet contains more generic features (e.g. edge detectors or color blob detectors) that can be useful for many kind of tasks. But the later layer of the network becomes progressively more specific to the details of the classes contained in the original dataset. **Pretrained models:** The released pretrained model is usually the final ConvNet checkpoint. So it is common to see people use the network for fine-tuning. #### 4. 3D Autoencoder and Convolutional Neural Network We take a two-stage approach where we first train a 3D sparse autoencoder to learn filters for convolution operations, and then build a convolutional neural network whose first layer uses the filters learned with the autoencoder.  #### 4.1. Sparse Autoencoder An autoencoder is a 3-layer neural network that is used to extract features from an input such as an image. Sparse representations can provide a simple interpretation of the input data in terms of a small number of \parts by extracting the structure hidden in the data. The autoencoder has an input layer, a hidden layer and an output layer, and the input and output layers have same number of units, while the hidden layer contains more units for a sparse and overcomplete representation. The encoder function maps input x to representation h, and the decoder function maps the representation h to the output x. In our problem, we extract 3D patches from scans as the input to the network. The decoder function aims to reconstruct the input form the hidden representation h. #### 4.2. 3D Convolutional Neural Network Training the 3D convolutional neural network(CNN) is the second stage. The CNN we use in this project has one convolutional layer, one pooling layer, two linear layers, and finally a log softmax layer. After training the sparse autoencoder, we take the weights and biases of the encoder from trained model, and use them a 3D filter of a 3D convolutional layer of the 1-layer convolutional neural network. Figure 2 shows the architecture of the network. #### 5. Tools In this project, we used Nibabel for MRI image processing and PyTorch Neural Networks implementation.
GitHub Repo
https://github.com/4fox123/LATEST-TRENDS-in-UI-UX---4Fox-Solutions
COVID has certainly changed the dynamics of the IT industry. As a businessman, one needs to cohere to the latest UI UX trends. UI/UX trends are altered based on the varying taste of the user. A user visits thousands of websites every day and as a businessman, everyone is trying to grab the attention of the visitor. Standing out from the competitors is hard to crack nowadays. The online platform that grabs the attention is the one that scores the game. A good design surely adds value and makes your customers visit you again. Though, the design is changing all the time. Each year, UI trends change, new features are introduced, and out-dated versions lose their way. So, you need to stay updated and keep your business up to date as well. In this article, we study the latest UI/UX trends that help you in your business growth. So let's dive into What is UI/UX? UI/UX is becoming a popular term and it’s also becoming better known in the industry. UX or User Experience essentially understands the customer their needs and designing an experience for the customer according to that which covers various aspects such as tech, such as understanding needs, what they see, what they experience with all their senses is what user experience is all about. A user experience with your site is known as User Experience. A UX/UI Designer is responsible for creating user-friendly interfaces and helps the clients to better understand the use of technological products UI or user interface is one part of the user experience, it’s one of the touch points the customer interfaces with, that is what they see and that’s what they touch now there is on touch devices or if it’s a computer then it’s something that you use online but that interface that communicated the brand or the service to the customer is the user interface and because that is a very important way or the very useful way to communicate with the customer, it plays a huge role in the communication of the service or the product to the customer. User Experience/ User Interface Design Trends Minimalism If we talk in terms of Design, then less is always more, a golden rule of website design. Limiting the number of colors, trying different proportions and compositions are good. It's because an average user sees many discount ads and gets constant notifications and the designer always try to simplify the way they present information. Simplified UX Avoid inserting extra steps which users found non-mandatory. Try to minimize the elements. Now the latest UX is Simplified registration and sign-in. Users don't want to remember extra passwords, so making use of their phone number as a password is a bit simplified. If you can omit unnecessary steps, do it. If you can omit some fields in the form, do it. Blurred, colorful background User Interface design helps to make the website more stylish, elegant, and eye-soothing. In earlier use, 2-3 colors in linear gradients were in trend but now it goes up to 10. In short, an overlay is trending now. However, you need to be careful while playing with the gradients. Neomorphism Neomorphism is gaining popularity because of its subtle yet original appearance that merges skeuomorphism with flat design. At present, the main purpose of products is to produce different user interface items utilizing geomorphic. Unique and absurd 2D illustrations 2D illustrations contribute to several aspects of design in production, including Environment design, Prop design, PreVis design, drawers, clean-up, rotations, and color design. Illustrations stay on top of user interface trends. It is getting fancier with time. The picture quality is in SVG format because like other formats such as PNG, GIF, and JPEG the picture quality is damaged when we increase the screen resolution and this is not the issue in SVG format. Because the vector format can be increased and decreased with no loss in quality. (VUI )Voice User Interface Users may use voice commands to engage with a UI or to talk. A voice-activated interface prevents consumers from using the interface. UX design teams compete with the latest upgrades and advances in this industry more than ever before. VUI is widely used in apps for translation. It hears in your language and translates into the language you want. ALEXA, SIRI is the best example of a voice user interface. With VUI you can easily communicate with people who do not know your language. Mobile-first approach Never ignore mobile accessibility as almost a 5.27billion people use a mobile phone and the chances of using social media platforms are 90%. So what if you ignore mobile visibility and more focus on the desktop appearance of your website? In this case, you lose a large amount of audience from your end. So always use the mobile approach in mind and create and design your website accordingly. Icons Icons are generally visually expressing objects, actions, and ideas. To communicate with the user we need different UI/UX. There are different picture representations of icons that are visually appealing. Conveying meaning in less space and creative ways is more powerful. Choose icons from the same family as they are the same size. It is also vital to understand that not all UX design trends are required for a product or website. The usefulness could only be complicated. We would rather keep up with the trends and only use them if they match the needs of your users and are likely to be working for your company. Synopsis: In 2021, design is something less complicated, more diverse, delightful, and satisfactory. Make sure you adhere to all these trends for reaching out to the million customers out there. Hope these trends are helpful for you to know where we stand in UI/UX nowadays. UI/UX design is to help users achieve their goals. Always ensure the design is relevant and valuable for consumers.
GitHub Repo
https://github.com/Aryia-Behroziuan/numpy
Quickstart tutorial 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 (n>=2) 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: Understand the difference between one-, two- and n-dimensional arrays in NumPy; Understand how to apply some linear algebra operations to n-dimensional arrays without using for-loops; Understand axis and shape properties for n-dimensional arrays. 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 non-negative 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 pictured below, the array has 2 axes. The first axis has a length of 2, the second axis has a length of 3. [[ 1., 0., 0.], [ 0., 1., 2.]] NumPy’s array class is called ndarray. It is also known by the alias array. Note that numpy.array is not the same as the Standard Python Library class array.array, which only handles one-dimensional arrays and offers less functionality. The more important attributes of an ndarray object are: ndarray.ndim the number of axes (dimensions) of the array. ndarray.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. ndarray.size the total number of elements of the array. This is equal to the product of the elements of shape. ndarray.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. ndarray.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. ndarray.data the buffer containing the actual elements of the array. Normally, we won’t need to use this attribute because we will access the elements in an array using indexing facilities. An example >>> import numpy as np a = np.arange(15).reshape(3, 5) a array([[ 0, 1, 2, 3, 4], [ 5, 6, 7, 8, 9], [10, 11, 12, 13, 14]]) a.shape (3, 5) a.ndim 2 a.dtype.name 'int64' a.itemsize 8 a.size 15 type(a) <class 'numpy.ndarray'> b = np.array([6, 7, 8]) b array([6, 7, 8]) type(b) <class 'numpy.ndarray'> Array Creation There are several ways to create arrays. For example, you can create an array from a regular Python list or tuple using the array function. The type of the resulting array is deduced from the type of the elements in the sequences. >>> >>> import numpy as np >>> a = np.array([2,3,4]) >>> a array([2, 3, 4]) >>> a.dtype dtype('int64') >>> b = np.array([1.2, 3.5, 5.1]) >>> b.dtype dtype('float64') A frequent error consists in calling array with multiple arguments, rather than providing a single sequence as an argument. >>> >>> a = np.array(1,2,3,4) # WRONG Traceback (most recent call last): ... TypeError: array() takes from 1 to 2 positional arguments but 4 were given >>> a = np.array([1,2,3,4]) # RIGHT array transforms sequences of sequences into two-dimensional arrays, sequences of sequences of sequences into three-dimensional arrays, and so on. >>> >>> b = np.array([(1.5,2,3), (4,5,6)]) >>> b array([[1.5, 2. , 3. ], [4. , 5. , 6. ]]) The type of the array can also be explicitly specified at creation time: >>> >>> c = np.array( [ [1,2], [3,4] ], dtype=complex ) >>> c array([[1.+0.j, 2.+0.j], [3.+0.j, 4.+0.j]]) Often, the elements of an array are originally unknown, but its size is known. Hence, NumPy offers several functions to create arrays with initial placeholder content. These minimize the necessity of growing arrays, an expensive operation. The function zeros creates an array full of zeros, the function ones creates an array full of ones, and the function empty creates an array whose initial content is random and depends on the state of the memory. By default, the dtype of the created array is float64. >>> >>> np.zeros((3, 4)) array([[0., 0., 0., 0.], [0., 0., 0., 0.], [0., 0., 0., 0.]]) >>> np.ones( (2,3,4), dtype=np.int16 ) # dtype can also be specified array([[[1, 1, 1, 1], [1, 1, 1, 1], [1, 1, 1, 1]], [[1, 1, 1, 1], [1, 1, 1, 1], [1, 1, 1, 1]]], dtype=int16) >>> np.empty( (2,3) ) # uninitialized array([[ 3.73603959e-262, 6.02658058e-154, 6.55490914e-260], # may vary [ 5.30498948e-313, 3.14673309e-307, 1.00000000e+000]]) To create sequences of numbers, NumPy provides the arange function which is analogous to the Python built-in range, but returns an array. >>> >>> np.arange( 10, 30, 5 ) array([10, 15, 20, 25]) >>> np.arange( 0, 2, 0.3 ) # it accepts float arguments array([0. , 0.3, 0.6, 0.9, 1.2, 1.5, 1.8]) When arange is used with floating point arguments, it is generally not possible to predict the number of elements obtained, due to the finite floating point precision. For this reason, it is usually better to use the function linspace that receives as an argument the number of elements that we want, instead of the step: >>> >>> from numpy import pi >>> np.linspace( 0, 2, 9 ) # 9 numbers from 0 to 2 array([0. , 0.25, 0.5 , 0.75, 1. , 1.25, 1.5 , 1.75, 2. ]) >>> x = np.linspace( 0, 2*pi, 100 ) # useful to evaluate function at lots of points >>> f = np.sin(x) See also array, zeros, zeros_like, ones, ones_like, empty, empty_like, arange, linspace, numpy.random.Generator.rand, numpy.random.Generator.randn, fromfunction, fromfile Printing Arrays When you print an array, NumPy displays it in a similar way to nested lists, but with the following layout: the last axis is printed from left to right, the second-to-last is printed from top to bottom, the rest are also printed from top to bottom, with each slice separated from the next by an empty line. One-dimensional arrays are then printed as rows, bidimensionals as matrices and tridimensionals as lists of matrices. >>> >>> a = np.arange(6) # 1d array >>> print(a) [0 1 2 3 4 5] >>> >>> b = np.arange(12).reshape(4,3) # 2d array >>> print(b) [[ 0 1 2] [ 3 4 5] [ 6 7 8] [ 9 10 11]] >>> >>> c = np.arange(24).reshape(2,3,4) # 3d array >>> print(c) [[[ 0 1 2 3] [ 4 5 6 7] [ 8 9 10 11]] [[12 13 14 15] [16 17 18 19] [20 21 22 23]]] See below to get more details on reshape. If an array is too large to be printed, NumPy automatically skips the central part of the array and only prints the corners: >>> >>> 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=sys.maxsize) # sys module should be imported Basic Operations Arithmetic operators on arrays apply elementwise. A new array is created and filled with the result. >>> >>> a = np.array( [20,30,40,50] ) >>> b = np.arange( 4 ) >>> b array([0, 1, 2, 3]) >>> c = a-b >>> c array([20, 29, 38, 47]) >>> b**2 array([0, 1, 4, 9]) >>> 10*np.sin(a) array([ 9.12945251, -9.88031624, 7.4511316 , -2.62374854]) >>> a<35 array([ True, True, False, False]) Unlike in many matrix languages, the product operator * operates elementwise in NumPy arrays. The matrix product can be performed using the @ operator (in python >=3.5) or the dot function or method: >>> >>> A = np.array( [[1,1], ... [0,1]] ) >>> B = np.array( [[2,0], ... [3,4]] ) >>> A * B # elementwise product array([[2, 0], [0, 4]]) >>> A @ B # matrix product array([[5, 4], [3, 4]]) >>> A.dot(B) # another matrix product array([[5, 4], [3, 4]]) Some operations, such as += and *=, act in place to modify an existing array rather than create a new one. >>> >>> rg = np.random.default_rng(1) # create instance of default random number generator >>> a = np.ones((2,3), dtype=int) >>> b = rg.random((2,3)) >>> a *= 3 >>> a array([[3, 3, 3], [3, 3, 3]]) >>> b += a >>> b array([[3.51182162, 3.9504637 , 3.14415961], [3.94864945, 3.31183145, 3.42332645]]) >>> a += b # b is not automatically converted to integer type Traceback (most recent call last): ... numpy.core._exceptions.UFuncTypeError: Cannot cast ufunc 'add' output from dtype('float64') to dtype('int64') with casting rule 'same_kind' When operating with arrays of different types, the type of the resulting array corresponds to the more general or precise one (a behavior known as upcasting). >>> >>> a = np.ones(3, dtype=np.int32) >>> b = np.linspace(0,pi,3) >>> b.dtype.name 'float64' >>> c = a+b >>> c array([1. , 2.57079633, 4.14159265]) >>> c.dtype.name 'float64' >>> d = np.exp(c*1j) >>> d array([ 0.54030231+0.84147098j, -0.84147098+0.54030231j, -0.54030231-0.84147098j]) >>> d.dtype.name 'complex128' Many unary operations, such as computing the sum of all the elements in the array, are implemented as methods of the ndarray class. >>> >>> a = rg.random((2,3)) >>> a array([[0.82770259, 0.40919914, 0.54959369], [0.02755911, 0.75351311, 0.53814331]]) >>> a.sum() 3.1057109529998157 >>> a.min() 0.027559113243068367 >>> a.max() 0.8277025938204418 By default, these operations apply to the array as though it were a list of numbers, regardless of its shape. However, by specifying the axis parameter you can apply an operation along the specified axis of an array: >>> >>> b = np.arange(12).reshape(3,4) >>> b array([[ 0, 1, 2, 3], [ 4, 5, 6, 7], [ 8, 9, 10, 11]]) >>> >>> b.sum(axis=0) # sum of each column array([12, 15, 18, 21]) >>> >>> b.min(axis=1) # min of each row array([0, 4, 8]) >>> >>> b.cumsum(axis=1) # cumulative sum along each row array([[ 0, 1, 3, 6], [ 4, 9, 15, 22], [ 8, 17, 27, 38]]) Universal Functions NumPy provides familiar mathematical functions such as sin, cos, and exp. In NumPy, these are called “universal functions”(ufunc). Within NumPy, these functions operate elementwise on an array, producing an array as output. >>> >>> B = np.arange(3) >>> B array([0, 1, 2]) >>> np.exp(B) array([1. , 2.71828183, 7.3890561 ]) >>> np.sqrt(B) array([0. , 1. , 1.41421356]) >>> C = np.array([2., -1., 4.]) >>> np.add(B, C) array([2., 0., 6.]) See also all, any, apply_along_axis, argmax, argmin, argsort, average, bincount, ceil, clip, conj, corrcoef, cov, cross, cumprod, cumsum, diff, dot, floor, inner, invert, lexsort, max, maximum, mean, median, min, minimum, nonzero, outer, prod, re, round, sort, std, sum, trace, transpose, var, vdot, vectorize, where Indexing, Slicing and Iterating One-dimensional arrays can be indexed, sliced and iterated over, much like lists and other Python sequences. >>> >>> a = np.arange(10)**3 >>> a array([ 0, 1, 8, 27, 64, 125, 216, 343, 512, 729]) >>> a[2] 8 >>> a[2:5] array([ 8, 27, 64]) # equivalent to a[0:6:2] = 1000; # from start to position 6, exclusive, set every 2nd element to 1000 >>> a[:6:2] = 1000 >>> a array([1000, 1, 1000, 27, 1000, 125, 216, 343, 512, 729]) >>> a[ : :-1] # reversed a array([ 729, 512, 343, 216, 125, 1000, 27, 1000, 1, 1000]) >>> for i in a: ... print(i**(1/3.)) ... 9.999999999999998 1.0 9.999999999999998 3.0 9.999999999999998 4.999999999999999 5.999999999999999 6.999999999999999 7.999999999999999 8.999999999999998 Multidimensional arrays can have one index per axis. These indices are given in a tuple separated by commas: >>> >>> def f(x,y): ... return 10*x+y ... >>> b = np.fromfunction(f,(5,4),dtype=int) >>> b array([[ 0, 1, 2, 3], [10, 11, 12, 13], [20, 21, 22, 23], [30, 31, 32, 33], [40, 41, 42, 43]]) >>> b[2,3] 23 >>> b[0:5, 1] # each row in the second column of b array([ 1, 11, 21, 31, 41]) >>> b[ : ,1] # equivalent to the previous example array([ 1, 11, 21, 31, 41]) >>> b[1:3, : ] # each column in the second and third row of b array([[10, 11, 12, 13], [20, 21, 22, 23]]) When fewer indices are provided than the number of axes, the missing indices are considered complete slices: >>> >>> b[-1] # the last row. Equivalent to b[-1,:] array([40, 41, 42, 43]) The expression within brackets in b[i] is treated as an i followed by as many instances of : as needed to represent the remaining axes. NumPy also allows you to write this using dots as b[i,...]. The dots (...) represent as many colons as needed to produce a complete indexing tuple. For example, if x is an array with 5 axes, then x[1,2,...] is equivalent to x[1,2,:,:,:], x[...,3] to x[:,:,:,:,3] and x[4,...,5,:] to x[4,:,:,5,:]. >>> >>> c = np.array( [[[ 0, 1, 2], # a 3D array (two stacked 2D arrays) ... [ 10, 12, 13]], ... [[100,101,102], ... [110,112,113]]]) >>> c.shape (2, 2, 3) >>> c[1,...] # same as c[1,:,:] or c[1] array([[100, 101, 102], [110, 112, 113]]) >>> c[...,2] # same as c[:,:,2] array([[ 2, 13], [102, 113]]) Iterating over multidimensional arrays is done with respect to the first axis: >>> >>> for row in b: ... print(row) ... [0 1 2 3] [10 11 12 13] [20 21 22 23] [30 31 32 33] [40 41 42 43] However, if one wants to perform an operation on each element in the array, one can use the flat attribute which is an iterator over all the elements of the array: >>> >>> for element in b.flat: ... print(element) ... 0 1 2 3 10 11 12 13 20 21 22 23 30 31 32 33 40 41 42 43 See also Indexing, Indexing (reference), newaxis, ndenumerate, indices Shape Manipulation Changing the shape of an array An array has a shape given by the number of elements along each axis: >>> >>> a = np.floor(10*rg.random((3,4))) >>> a array([[3., 7., 3., 4.], [1., 4., 2., 2.], [7., 2., 4., 9.]]) >>> a.shape (3, 4) The shape of an array can be changed with various commands. Note that the following three commands all return a modified array, but do not change the original array: >>> >>> a.ravel() # returns the array, flattened array([3., 7., 3., 4., 1., 4., 2., 2., 7., 2., 4., 9.]) >>> a.reshape(6,2) # returns the array with a modified shape array([[3., 7.], [3., 4.], [1., 4.], [2., 2.], [7., 2.], [4., 9.]]) >>> a.T # returns the array, transposed array([[3., 1., 7.], [7., 4., 2.], [3., 2., 4.], [4., 2., 9.]]) >>> a.T.shape (4, 3) >>> a.shape (3, 4) The order of the elements in the array resulting from ravel() is normally “C-style”, that is, the rightmost index “changes the fastest”, so the element after a[0,0] is a[0,1]. If the array is reshaped to some other shape, again the array is treated as “C-style”. NumPy normally creates arrays stored in this order, so ravel() will usually not need to copy its argument, but if the array was made by taking slices of another array or created with unusual options, it may need to be copied. The functions ravel() and reshape() can also be instructed, using an optional argument, to use FORTRAN-style arrays, in which the leftmost index changes the fastest. The reshape function returns its argument with a modified shape, whereas the ndarray.resize method modifies the array itself: >>> >>> a array([[3., 7., 3., 4.], [1., 4., 2., 2.], [7., 2., 4., 9.]]) >>> a.resize((2,6)) >>> a array([[3., 7., 3., 4., 1., 4.], [2., 2., 7., 2., 4., 9.]]) If a dimension is given as -1 in a reshaping operation, the other dimensions are automatically calculated: >>> >>> a.reshape(3,-1) array([[3., 7., 3., 4.], [1., 4., 2., 2.], [7., 2., 4., 9.]]) See also ndarray.shape, reshape, resize, ravel Stacking together different arrays Several arrays can be stacked together along different axes: >>> >>> a = np.floor(10*rg.random((2,2))) >>> a array([[9., 7.], [5., 2.]]) >>> b = np.floor(10*rg.random((2,2))) >>> b array([[1., 9.], [5., 1.]]) >>> np.vstack((a,b)) array([[9., 7.], [5., 2.], [1., 9.], [5., 1.]]) >>> np.hstack((a,b)) array([[9., 7., 1., 9.], [5., 2., 5., 1.]]) The function column_stack stacks 1D arrays as columns into a 2D array. It is equivalent to hstack only for 2D arrays: >>> >>> from numpy import newaxis >>> np.column_stack((a,b)) # with 2D arrays array([[9., 7., 1., 9.], [5., 2., 5., 1.]]) >>> a = np.array([4.,2.]) >>> b = np.array([3.,8.]) >>> np.column_stack((a,b)) # returns a 2D array array([[4., 3.], [2., 8.]]) >>> np.hstack((a,b)) # the result is different array([4., 2., 3., 8.]) >>> a[:,newaxis] # view `a` as a 2D column vector array([[4.], [2.]]) >>> np.column_stack((a[:,newaxis],b[:,newaxis])) array([[4., 3.], [2., 8.]]) >>> np.hstack((a[:,newaxis],b[:,newaxis])) # the result is the same array([[4., 3.], [2., 8.]]) On the other hand, the function row_stack is equivalent to vstack for any input arrays. In fact, row_stack is an alias for vstack: >>> >>> np.column_stack is np.hstack False >>> np.row_stack is np.vstack True In general, for arrays with more than two dimensions, hstack stacks along their second axes, vstack stacks along their first axes, and concatenate allows for an optional arguments giving the number of the axis along which the concatenation should happen. Note In complex cases, r_ and c_ are useful for creating arrays by stacking numbers along one axis. They allow the use of range literals (“:”) >>> >>> np.r_[1:4,0,4] array([1, 2, 3, 0, 4]) When used with arrays as arguments, r_ and c_ are similar to vstack and hstack in their default behavior, but allow for an optional argument giving the number of the axis along which to concatenate. See also hstack, vstack, column_stack, concatenate, c_, r_ Splitting one array into several smaller ones Using hsplit, you can split an array along its horizontal axis, either by specifying the number of equally shaped arrays to return, or by specifying the columns after which the division should occur: >>> >>> a = np.floor(10*rg.random((2,12))) >>> a array([[6., 7., 6., 9., 0., 5., 4., 0., 6., 8., 5., 2.], [8., 5., 5., 7., 1., 8., 6., 7., 1., 8., 1., 0.]]) # Split a into 3 >>> np.hsplit(a,3) [array([[6., 7., 6., 9.], [8., 5., 5., 7.]]), array([[0., 5., 4., 0.], [1., 8., 6., 7.]]), array([[6., 8., 5., 2.], [1., 8., 1., 0.]])] # Split a after the third and the fourth column >>> np.hsplit(a,(3,4)) [array([[6., 7., 6.], [8., 5., 5.]]), array([[9.], [7.]]), array([[0., 5., 4., 0., 6., 8., 5., 2.], [1., 8., 6., 7., 1., 8., 1., 0.]])] vsplit splits along the vertical axis, and array_split allows one to specify along which axis to split. 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 objects or their data. >>> >>> a = np.array([[ 0, 1, 2, 3], ... [ 4, 5, 6, 7], ... [ 8, 9, 10, 11]]) >>> b = a # no new object is created >>> b is a # a and b are two names for the same ndarray object True Python passes mutable objects as references, so function calls make no copy. >>> >>> def f(x): ... print(id(x)) ... >>> id(a) # id is a unique identifier of an object 148293216 # may vary >>> f(a) 148293216 # may vary View or Shallow Copy 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 = c.reshape((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] # spaces added for clarity; could also be written "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]]) Sometimes copy should be called after slicing if the original array is not required anymore. For example, suppose a is a huge intermediate result and the final result b only contains a small fraction of a, a deep copy should be made when constructing b with slicing: >>> >>> a = np.arange(int(1e8)) >>> b = a[:100].copy() >>> del a # the memory of ``a`` can be released. If b = a[:100] is used instead, a is referenced by b and will persist in memory even if del a is executed. Functions and Methods Overview Here is a list of some useful NumPy functions and methods names ordered in categories. See Routines for the full list. Array Creation arange, array, copy, empty, empty_like, eye, fromfile, fromfunction, identity, linspace, logspace, mgrid, ogrid, ones, ones_like, r_, zeros, zeros_like Conversions ndarray.astype, atleast_1d, atleast_2d, atleast_3d, mat Manipulations array_split, column_stack, concatenate, diagonal, dsplit, dstack, hsplit, hstack, ndarray.item, newaxis, ravel, repeat, reshape, resize, squeeze, swapaxes, take, transpose, vsplit, vstack Questions all, any, nonzero, where Ordering argmax, argmin, argsort, max, min, ptp, searchsorted, sort Operations choose, compress, cumprod, cumsum, inner, ndarray.fill, imag, prod, put, putmask, real, sum Basic Statistics cov, mean, std, var Basic Linear Algebra cross, dot, outer, linalg.svd, vdot Less Basic Broadcasting rules Broadcasting allows universal functions to deal in a meaningful way with inputs that do not have exactly the same shape. 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. After application of the broadcasting rules, the sizes of all arrays must match. More details can be found in Broadcasting. Advanced indexing and index tricks NumPy offers more indexing facilities than regular Python sequences. In addition to indexing by integers and slices, as we saw before, arrays can be indexed by arrays of integers and arrays of booleans. Indexing with Arrays of Indices >>> >>> a = np.arange(12)**2 # the first 12 square numbers >>> i = np.array([1, 1, 3, 8, 5]) # an array of indices >>> a[i] # the elements of a at the positions i array([ 1, 1, 9, 64, 25]) >>> >>> j = np.array([[3, 4], [9, 7]]) # a bidimensional array of indices >>> a[j] # the same shape as j array([[ 9, 16], [81, 49]]) When the indexed array a is multidimensional, a single array of indices refers to the first dimension of a. The following example shows this behavior by converting an image of labels into a color image using a palette. >>> >>> palette = np.array([[0, 0, 0], # black ... [255, 0, 0], # red ... [0, 255, 0], # green ... [0, 0, 255], # blue ... [255, 255, 255]]) # white >>> image = np.array([[0, 1, 2, 0], # each value corresponds to a color in the palette ... [0, 3, 4, 0]]) >>> palette[image] # the (2, 4, 3) color image array([[[ 0, 0, 0], [255, 0, 0], [ 0, 255, 0], [ 0, 0, 0]], [[ 0, 0, 0], [ 0, 0, 255], [255, 255, 255], [ 0, 0, 0]]]) We can also give indexes for more than one dimension. The arrays of indices for each dimension must have the same shape. >>> >>> a = np.arange(12).reshape(3,4) >>> a array([[ 0, 1, 2, 3], [ 4, 5, 6, 7], [ 8, 9, 10, 11]]) >>> i = np.array([[0, 1], # indices for the first dim of a ... [1, 2]]) >>> j = np.array([[2, 1], # indices for the second dim ... [3, 3]]) >>> >>> a[i, j] # i and j must have equal shape array([[ 2, 5], [ 7, 11]]) >>> >>> a[i, 2] array([[ 2, 6], [ 6, 10]]) >>> >>> a[:, j] # i.e., a[ : , j] array([[[ 2, 1], [ 3, 3]], [[ 6, 5], [ 7, 7]], [[10, 9], [11, 11]]]) In Python, arr[i, j] is exactly the same as arr[(i, j)]—so we can put i and j in a tuple and then do the indexing with that. >>> >>> l = (i, j) # equivalent to a[i, j] >>> a[l] array([[ 2, 5], [ 7, 11]]) However, we can not do this by putting i and j into an array, because this array will be interpreted as indexing the first dimension of a. >>> >>> s = np.array([i, j]) # not what we want >>> a[s] Traceback (most recent call last): File "<stdin>", line 1, in <module> IndexError: index 3 is out of bounds for axis 0 with size 3 # same as a[i, j] >>> a[tuple(s)] array([[ 2, 5], [ 7, 11]]) Another common use of indexing with arrays is the search of the maximum value of time-dependent series: >>> >>> time = np.linspace(20, 145, 5) # time scale >>> data = np.sin(np.arange(20)).reshape(5,4) # 4 time-dependent series >>> time array([ 20. , 51.25, 82.5 , 113.75, 145. ]) >>> data array([[ 0. , 0.84147098, 0.90929743, 0.14112001], [-0.7568025 , -0.95892427, -0.2794155 , 0.6569866 ], [ 0.98935825, 0.41211849, -0.54402111, -0.99999021], [-0.53657292, 0.42016704, 0.99060736, 0.65028784], [-0.28790332, -0.96139749, -0.75098725, 0.14987721]]) # index of the maxima for each series >>> ind = data.argmax(axis=0) >>> ind array([2, 0, 3, 1]) # times corresponding to the maxima >>> time_max = time[ind] >>> >>> data_max = data[ind, range(data.shape[1])] # => data[ind[0],0], data[ind[1],1]... >>> time_max array([ 82.5 , 20. , 113.75, 51.25]) >>> data_max array([0.98935825, 0.84147098, 0.99060736, 0.6569866 ]) >>> np.all(data_max == data.max(axis=0)) True You can also use indexing with arrays as a target to assign to: >>> >>> a = np.arange(5) >>> a array([0, 1, 2, 3, 4]) >>> a[[1,3,4]] = 0 >>> a array([0, 0, 2, 0, 0]) However, when the list of indices contains repetitions, the assignment is done several times, leaving behind the last value: >>> >>> a = np.arange(5) >>> a[[0,0,2]]=[1,2,3] >>> a array([2, 1, 3, 3, 4]) This is reasonable enough, but watch out if you want to use Python’s += construct, as it may not do what you expect: >>> >>> a = np.arange(5) >>> a[[0,0,2]]+=1 >>> a array([1, 1, 3, 3, 4]) Even though 0 occurs twice in the list of indices, the 0th element is only incremented once. This is because Python requires “a+=1” to be equivalent to “a = a + 1”. Indexing with Boolean Arrays When we index arrays with arrays of (integer) indices we are providing the list of indices to pick. With boolean indices the approach is different; we explicitly choose which items in the array we want and which ones we don’t. The most natural way one can think of for boolean indexing is to use boolean arrays that have the same shape as the original array: >>> >>> a = np.arange(12).reshape(3,4) >>> b = a > 4 >>> b # b is a boolean with a's shape array([[False, False, False, False], [False, True, True, True], [ True, True, True, True]]) >>> a[b] # 1d array with the selected elements array([ 5, 6, 7, 8, 9, 10, 11]) This property can be very useful in assignments: >>> >>> a[b] = 0 # All elements of 'a' higher than 4 become 0 >>> a array([[0, 1, 2, 3], [4, 0, 0, 0], [0, 0, 0, 0]]) You can look at the following example to see how to use boolean indexing to generate an image of the Mandelbrot set: >>> import numpy as np import matplotlib.pyplot as plt def mandelbrot( h,w, maxit=20 ): """Returns an image of the Mandelbrot fractal of size (h,w).""" y,x = np.ogrid[ -1.4:1.4:h*1j, -2:0.8:w*1j ] c = x+y*1j z = c divtime = maxit + np.zeros(z.shape, dtype=int) for i in range(maxit): z = z**2 + c diverge = z*np.conj(z) > 2**2 # who is diverging div_now = diverge & (divtime==maxit) # who is diverging now divtime[div_now] = i # note when z[diverge] = 2 # avoid diverging too much return divtime plt.imshow(mandelbrot(400,400)) ../_images/quickstart-1.png The second way of indexing with booleans is more similar to integer indexing; for each dimension of the array we give a 1D boolean array selecting the slices we want: >>> >>> a = np.arange(12).reshape(3,4) >>> b1 = np.array([False,True,True]) # first dim selection >>> b2 = np.array([True,False,True,False]) # second dim selection >>> >>> a[b1,:] # selecting rows array([[ 4, 5, 6, 7], [ 8, 9, 10, 11]]) >>> >>> a[b1] # same thing array([[ 4, 5, 6, 7], [ 8, 9, 10, 11]]) >>> >>> a[:,b2] # selecting columns array([[ 0, 2], [ 4, 6], [ 8, 10]]) >>> >>> a[b1,b2] # a weird thing to do array([ 4, 10]) Note that the length of the 1D boolean array must coincide with the length of the dimension (or axis) you want to slice. In the previous example, b1 has length 3 (the number of rows in a), and b2 (of length 4) is suitable to index the 2nd axis (columns) of a. The ix_() function The ix_ function can be used to combine different vectors so as to obtain the result for each n-uplet. For example, if you want to compute all the a+b*c for all the triplets taken from each of the vectors a, b and c: >>> >>> a = np.array([2,3,4,5]) >>> b = np.array([8,5,4]) >>> c = np.array([5,4,6,8,3]) >>> ax,bx,cx = np.ix_(a,b,c) >>> ax array([[[2]], [[3]], [[4]], [[5]]]) >>> bx array([[[8], [5], [4]]]) >>> cx array([[[5, 4, 6, 8, 3]]]) >>> ax.shape, bx.shape, cx.shape ((4, 1, 1), (1, 3, 1), (1, 1, 5)) >>> result = ax+bx*cx >>> result array([[[42, 34, 50, 66, 26], [27, 22, 32, 42, 17], [22, 18, 26, 34, 14]], [[43, 35, 51, 67, 27], [28, 23, 33, 43, 18], [23, 19, 27, 35, 15]], [[44, 36, 52, 68, 28], [29, 24, 34, 44, 19], [24, 20, 28, 36, 16]], [[45, 37, 53, 69, 29], [30, 25, 35, 45, 20], [25, 21, 29, 37, 17]]]) >>> result[3,2,4] 17 >>> a[3]+b[2]*c[4] 17 You could also implement the reduce as follows: >>> >>> def ufunc_reduce(ufct, *vectors): ... vs = np.ix_(*vectors) ... r = ufct.identity ... for v in vs: ... r = ufct(r,v) ... return r and then use it as: >>> >>> ufunc_reduce(np.add,a,b,c) array([[[15, 14, 16, 18, 13], [12, 11, 13, 15, 10], [11, 10, 12, 14, 9]], [[16, 15, 17, 19, 14], [13, 12, 14, 16, 11], [12, 11, 13, 15, 10]], [[17, 16, 18, 20, 15], [14, 13, 15, 17, 12], [13, 12, 14, 16, 11]], [[18, 17, 19, 21, 16], [15, 14, 16, 18, 13], [14, 13, 15, 17, 12]]]) The advantage of this version of reduce compared to the normal ufunc.reduce is that it makes use of the Broadcasting Rules in order to avoid creating an argument array the size of the output times the number of vectors. Indexing with strings See Structured arrays. Linear Algebra Work in progress. Basic linear algebra to be included here. Simple Array Operations See linalg.py in numpy folder for more. >>> >>> import numpy as np >>> a = np.array([[1.0, 2.0], [3.0, 4.0]]) >>> print(a) [[1. 2.] [3. 4.]] >>> a.transpose() array([[1., 3.], [2., 4.]]) >>> np.linalg.inv(a) array([[-2. , 1. ], [ 1.5, -0.5]]) >>> u = np.eye(2) # unit 2x2 matrix; "eye" represents "I" >>> u array([[1., 0.], [0., 1.]]) >>> j = np.array([[0.0, -1.0], [1.0, 0.0]]) >>> j @ j # matrix product array([[-1., 0.], [ 0., -1.]]) >>> np.trace(u) # trace 2.0 >>> y = np.array([[5.], [7.]]) >>> np.linalg.solve(a, y) array([[-3.], [ 4.]]) >>> np.linalg.eig(j) (array([0.+1.j, 0.-1.j]), array([[0.70710678+0.j , 0.70710678-0.j ], [0. -0.70710678j, 0. +0.70710678j]])) Parameters: square matrix Returns The eigenvalues, each repeated according to its multiplicity. The normalized (unit "length") eigenvectors, such that the column ``v[:,i]`` is the eigenvector corresponding to the eigenvalue ``w[i]`` . Tricks and Tips Here we give a list of short and useful tips. “Automatic” Reshaping To change the dimensions of an array, you can omit one of the sizes which will then be deduced automatically: >>> >>> a = np.arange(30) >>> b = a.reshape((2, -1, 3)) # -1 means "whatever is needed" >>> b.shape (2, 5, 3) >>> b array([[[ 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]]]) Vector Stacking How do we construct a 2D array from a list of equally-sized row vectors? In MATLAB this is quite easy: if x and y are two vectors of the same length you only need do m=[x;y]. In NumPy this works via the functions column_stack, dstack, hstack and vstack, depending on the dimension in which the stacking is to be done. For example: >>> >>> x = np.arange(0,10,2) >>> y = np.arange(5) >>> m = np.vstack([x,y]) >>> m array([[0, 2, 4, 6, 8], [0, 1, 2, 3, 4]]) >>> xy = np.hstack([x,y]) >>> xy array([0, 2, 4, 6, 8, 0, 1, 2, 3, 4]) The logic behind those functions in more than two dimensions can be strange. See also NumPy for Matlab users Histograms The NumPy histogram function applied to an array returns a pair of vectors: the histogram of the array and a vector of the bin edges. Beware: matplotlib also has a function to build histograms (called hist, as in Matlab) that differs from the one in NumPy. The main difference is that pylab.hist plots the histogram automatically, while numpy.histogram only generates the data. >>> import numpy as np rg = np.random.default_rng(1) import matplotlib.pyplot as plt # Build a vector of 10000 normal deviates with variance 0.5^2 and mean 2 mu, sigma = 2, 0.5 v = rg.normal(mu,sigma,10000) # Plot a normalized histogram with 50 bins plt.hist(v, bins=50, density=1) # matplotlib version (plot) # Compute the histogram with numpy and then plot it (n, bins) = np.histogram(v, bins=50, density=True) # NumPy version (no plot) plt.plot(.5*(bins[1:]+bins[:-1]), n) ../_images/quickstart-2.png Further reading The Python tutorial NumPy Reference SciPy Tutorial SciPy Lecture Notes A matlab, R, IDL, NumPy/SciPy dictionary © Copyright 2008-2020, The SciPy community. Last updated on Jun 29, 2020. Created using Sphinx 2.4.4.
