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Classification - Machine Learning This is ‘Classification’ tutorial which is a part of the Machine Learning course offered by Simplilearn. We will learn Classification algorithms, types of classification algorithms, support vector machines(SVM), Naive Bayes, Decision Tree and Random Forest Classifier in this tutorial. Objectives Let us look at some of the objectives covered under this section of Machine Learning tutorial. Define Classification and list its algorithms Describe Logistic Regression and Sigmoid Probability Explain K-Nearest Neighbors and KNN classification Understand Support Vector Machines, Polynomial Kernel, and Kernel Trick Analyze Kernel Support Vector Machines with an example Implement the Naïve Bayes Classifier Demonstrate Decision Tree Classifier Describe Random Forest Classifier Classification: Meaning Classification is a type of supervised learning. It specifies the class to which data elements belong to and is best used when the output has finite and discrete values. It predicts a class for an input variable as well. There are 2 types of Classification: Binomial Multi-Class Classification: Use Cases Some of the key areas where classification cases are being used: To find whether an email received is a spam or ham To identify customer segments To find if a bank loan is granted To identify if a kid will pass or fail in an examination Classification: Example Social media sentiment analysis has two potential outcomes, positive or negative, as displayed by the chart given below. https://www.simplilearn.com/ice9/free_resources_article_thumb/classification-example-machine-learning.JPG This chart shows the classification of the Iris flower dataset into its three sub-species indicated by codes 0, 1, and 2. https://www.simplilearn.com/ice9/free_resources_article_thumb/iris-flower-dataset-graph.JPG The test set dots represent the assignment of new test data points to one class or the other based on the trained classifier model. Types of Classification Algorithms Let’s have a quick look into the types of Classification Algorithm below. Linear Models Logistic Regression Support Vector Machines Nonlinear models K-nearest Neighbors (KNN) Kernel Support Vector Machines (SVM) Naïve Bayes Decision Tree Classification Random Forest Classification Logistic Regression: Meaning Let us understand the Logistic Regression model below. This refers to a regression model that is used for classification. This method is widely used for binary classification problems. It can also be extended to multi-class classification problems. Here, the dependent variable is categorical: y ϵ {0, 1} A binary dependent variable can have only two values, like 0 or 1, win or lose, pass or fail, healthy or sick, etc In this case, you model the probability distribution of output y as 1 or 0. This is called the sigmoid probability (σ). If σ(θ Tx) > 0.5, set y = 1, else set y = 0 Unlike Linear Regression (and its Normal Equation solution), there is no closed form solution for finding optimal weights of Logistic Regression. Instead, you must solve this with maximum likelihood estimation (a probability model to detect the maximum likelihood of something happening). It can be used to calculate the probability of a given outcome in a binary model, like the probability of being classified as sick or passing an exam. https://www.simplilearn.com/ice9/free_resources_article_thumb/logistic-regression-example-graph.JPG Sigmoid Probability The probability in the logistic regression is often represented by the Sigmoid function (also called the logistic function or the S-curve): https://www.simplilearn.com/ice9/free_resources_article_thumb/sigmoid-function-machine-learning.JPG In this equation, t represents data values * the number of hours studied and S(t) represents the probability of passing the exam. Assume sigmoid function: https://www.simplilearn.com/ice9/free_resources_article_thumb/sigmoid-probability-machine-learning.JPG g(z) tends toward 1 as z -> infinity , and g(z) tends toward 0 as z -> infinity K-nearest Neighbors (KNN) K-nearest Neighbors algorithm is used to assign a data point to clusters based on similarity measurement. It uses a supervised method for classification. The steps to writing a k-means algorithm are as given below: https://www.simplilearn.com/ice9/free_resources_article_thumb/knn-distribution-graph-machine-learning.JPG Choose the number of k and a distance metric. (k = 5 is common) Find k-nearest neighbors of the sample that you want to classify Assign the class label by majority vote. KNN Classification A new input point is classified in the category such that it has the most number of neighbors from that category. For example: https://www.simplilearn.com/ice9/free_resources_article_thumb/knn-classification-machine-learning.JPG Classify a patient as high risk or low risk. Mark email as spam or ham. Keen on learning about Classification Algorithms in Machine Learning? Click here! Support Vector Machine (SVM) Let us understand Support Vector Machine (SVM) in detail below. SVMs are classification algorithms used to assign data to various classes. They involve detecting hyperplanes which segregate data into classes. SVMs are very versatile and are also capable of performing linear or nonlinear classification, regression, and outlier detection. Once ideal hyperplanes are discovered, new data points can be easily classified. https://www.simplilearn.com/ice9/free_resources_article_thumb/support-vector-machines-graph-machine-learning.JPG The optimization objective is to find “maximum margin hyperplane” that is farthest from the closest points in the two classes (these points are called support vectors). In the given figure, the middle line represents the hyperplane. SVM Example Let’s look at this image below and have an idea about SVM in general. Hyperplanes with larger margins have lower generalization error. The positive and negative hyperplanes are represented by: https://www.simplilearn.com/ice9/free_resources_article_thumb/positive-negative-hyperplanes-machine-learning.JPG Classification of any new input sample xtest : If w0 + wTxtest > 1, the sample xtest is said to be in the class toward the right of the positive hyperplane. If w0 + wTxtest < -1, the sample xtest is said to be in the class toward the left of the negative hyperplane. When you subtract the two equations, you get: https://www.simplilearn.com/ice9/free_resources_article_thumb/equation-subtraction-machine-learning.JPG Length of vector w is (L2 norm length): https://www.simplilearn.com/ice9/free_resources_article_thumb/length-of-vector-machine-learning.JPG You normalize with the length of w to arrive at: https://www.simplilearn.com/ice9/free_resources_article_thumb/normalize-equation-machine-learning.JPG SVM: Hard Margin Classification Given below are some points to understand Hard Margin Classification. The left side of equation SVM-1 given above can be interpreted as the distance between the positive (+ve) and negative (-ve) hyperplanes; in other words, it is the margin that can be maximized. Hence the objective of the function is to maximize with the constraint that the samples are classified correctly, which is represented as : https://www.simplilearn.com/ice9/free_resources_article_thumb/hard-margin-classification-machine-learning.JPG This means that you are minimizing ‖w‖. This also means that all positive samples are on one side of the positive hyperplane and all negative samples are on the other side of the negative hyperplane. This can be written concisely as : https://www.simplilearn.com/ice9/free_resources_article_thumb/hard-margin-classification-formula.JPG Minimizing ‖w‖ is the same as minimizing. This figure is better as it is differentiable even at w = 0. The approach listed above is called “hard margin linear SVM classifier.” SVM: Soft Margin Classification Given below are some points to understand Soft Margin Classification. To allow for linear constraints to be relaxed for nonlinearly separable data, a slack variable is introduced. (i) measures how much ith instance is allowed to violate the margin. The slack variable is simply added to the linear constraints. https://www.simplilearn.com/ice9/free_resources_article_thumb/soft-margin-calculation-machine-learning.JPG Subject to the above constraints, the new objective to be minimized becomes: https://www.simplilearn.com/ice9/free_resources_article_thumb/soft-margin-calculation-formula.JPG You have two conflicting objectives now—minimizing slack variable to reduce margin violations and minimizing to increase the margin. The hyperparameter C allows us to define this trade-off. Large values of C correspond to larger error penalties (so smaller margins), whereas smaller values of C allow for higher misclassification errors and larger margins. https://www.simplilearn.com/ice9/free_resources_article_thumb/machine-learning-certification-video-preview.jpg SVM: Regularization The concept of C is the reverse of regularization. Higher C means lower regularization, which increases bias and lowers the variance (causing overfitting). https://www.simplilearn.com/ice9/free_resources_article_thumb/concept-of-c-graph-machine-learning.JPG IRIS Data Set The Iris dataset contains measurements of 150 IRIS flowers from three different species: Setosa Versicolor Viriginica Each row represents one sample. Flower measurements in centimeters are stored as columns. These are called features. IRIS Data Set: SVM Let’s train an SVM model using sci-kit-learn for the Iris dataset: https://www.simplilearn.com/ice9/free_resources_article_thumb/svm-model-graph-machine-learning.JPG Nonlinear SVM Classification There are two ways to solve nonlinear SVMs: by adding polynomial features by adding similarity features Polynomial features can be added to datasets; in some cases, this can create a linearly separable dataset. https://www.simplilearn.com/ice9/free_resources_article_thumb/nonlinear-classification-svm-machine-learning.JPG In the figure on the left, there is only 1 feature x1. This dataset is not linearly separable. If you add x2 = (x1)2 (figure on the right), the data becomes linearly separable. Polynomial Kernel In sci-kit-learn, one can use a Pipeline class for creating polynomial features. Classification results for the Moons dataset are shown in the figure. https://www.simplilearn.com/ice9/free_resources_article_thumb/polynomial-kernel-machine-learning.JPG Polynomial Kernel with Kernel Trick Let us look at the image below and understand Kernel Trick in detail. https://www.simplilearn.com/ice9/free_resources_article_thumb/polynomial-kernel-with-kernel-trick.JPG For large dimensional datasets, adding too many polynomial features can slow down the model. You can apply a kernel trick with the effect of polynomial features without actually adding them. The code is shown (SVC class) below trains an SVM classifier using a 3rd-degree polynomial kernel but with a kernel trick. https://www.simplilearn.com/ice9/free_resources_article_thumb/polynomial-kernel-equation-machine-learning.JPG The hyperparameter coefθ controls the influence of high-degree polynomials. Kernel SVM Let us understand in detail about Kernel SVM. Kernel SVMs are used for classification of nonlinear data. In the chart, nonlinear data is projected into a higher dimensional space via a mapping function where it becomes linearly separable. https://www.simplilearn.com/ice9/free_resources_article_thumb/kernel-svm-machine-learning.JPG In the higher dimension, a linear separating hyperplane can be derived and used for classification. A reverse projection of the higher dimension back to original feature space takes it back to nonlinear shape. As mentioned previously, SVMs can be kernelized to solve nonlinear classification problems. You can create a sample dataset for XOR gate (nonlinear problem) from NumPy. 100 samples will be assigned the class sample 1, and 100 samples will be assigned the class label -1. https://www.simplilearn.com/ice9/free_resources_article_thumb/kernel-svm-graph-machine-learning.JPG As you can see, this data is not linearly separable. https://www.simplilearn.com/ice9/free_resources_article_thumb/kernel-svm-non-separable.JPG You now use the kernel trick to classify XOR dataset created earlier. https://www.simplilearn.com/ice9/free_resources_article_thumb/kernel-svm-xor-machine-learning.JPG Naïve Bayes Classifier What is Naive Bayes Classifier? Have you ever wondered how your mail provider implements spam filtering or how online news channels perform news text classification or even how companies perform sentiment analysis of their audience on social media? All of this and more are done through a machine learning algorithm called Naive Bayes Classifier. Naive Bayes Named after Thomas Bayes from the 1700s who first coined this in the Western literature. Naive Bayes classifier works on the principle of conditional probability as given by the Bayes theorem. Advantages of Naive Bayes Classifier Listed below are six benefits of Naive Bayes Classifier. Very simple and easy to implement Needs less training data Handles both continuous and discrete data Highly scalable with the number of predictors and data points As it is fast, it can be used in real-time predictions Not sensitive to irrelevant features Bayes Theorem We will understand Bayes Theorem in detail from the points mentioned below. According to the Bayes model, the conditional probability P(Y|X) can be calculated as: P(Y|X) = P(X|Y)P(Y) / P(X) This means you have to estimate a very large number of P(X|Y) probabilities for a relatively small vector space X. For example, for a Boolean Y and 30 possible Boolean attributes in the X vector, you will have to estimate 3 billion probabilities P(X|Y). To make it practical, a Naïve Bayes classifier is used, which assumes conditional independence of P(X) to each other, with a given value of Y. This reduces the number of probability estimates to 2*30=60 in the above example. Naïve Bayes Classifier for SMS Spam Detection Consider a labeled SMS database having 5574 messages. It has messages as given below: https://www.simplilearn.com/ice9/free_resources_article_thumb/naive-bayes-spam-machine-learning.JPG Each message is marked as spam or ham in the data set. Let’s train a model with Naïve Bayes algorithm to detect spam from ham. The message lengths and their frequency (in the training dataset) are as shown below: https://www.simplilearn.com/ice9/free_resources_article_thumb/naive-bayes-spam-spam-detection.JPG Analyze the logic you use to train an algorithm to detect spam: Split each message into individual words/tokens (bag of words). Lemmatize the data (each word takes its base form, like “walking” or “walked” is replaced with “walk”). Convert data to vectors using scikit-learn module CountVectorizer. Run TFIDF to remove common words like “is,” “are,” “and.” Now apply scikit-learn module for Naïve Bayes MultinomialNB to get the Spam Detector. This spam detector can then be used to classify a random new message as spam or ham. Next, the accuracy of the spam detector is checked using the Confusion Matrix. For the SMS spam example above, the confusion matrix is shown on the right. Accuracy Rate = Correct / Total = (4827 + 592)/5574 = 97.21% Error Rate = Wrong / Total = (155 + 0)/5574 = 2.78% https://www.simplilearn.com/ice9/free_resources_article_thumb/confusion-matrix-machine-learning.JPG Although confusion Matrix is useful, some more precise metrics are provided by Precision and Recall. https://www.simplilearn.com/ice9/free_resources_article_thumb/precision-recall-matrix-machine-learning.JPG Precision refers to the accuracy of positive predictions. https://www.simplilearn.com/ice9/free_resources_article_thumb/precision-formula-machine-learning.JPG Recall refers to the ratio of positive instances that are correctly detected by the classifier (also known as True positive rate or TPR). https://www.simplilearn.com/ice9/free_resources_article_thumb/recall-formula-machine-learning.JPG Precision/Recall Trade-off To detect age-appropriate videos for kids, you need high precision (low recall) to ensure that only safe videos make the cut (even though a few safe videos may be left out). The high recall is needed (low precision is acceptable) in-store surveillance to catch shoplifters; a few false alarms are acceptable, but all shoplifters must be caught. Learn about Naive Bayes in detail. Click here! Decision Tree Classifier Some aspects of the Decision Tree Classifier mentioned below are. Decision Trees (DT) can be used both for classification and regression. The advantage of decision trees is that they require very little data preparation. They do not require feature scaling or centering at all. They are also the fundamental components of Random Forests, one of the most powerful ML algorithms. Unlike Random Forests and Neural Networks (which do black-box modeling), Decision Trees are white box models, which means that inner workings of these models are clearly understood. In the case of classification, the data is segregated based on a series of questions. Any new data point is assigned to the selected leaf node. https://www.simplilearn.com/ice9/free_resources_article_thumb/decision-tree-classifier-machine-learning.JPG Start at the tree root and split the data on the feature using the decision algorithm, resulting in the largest information gain (IG). This splitting procedure is then repeated in an iterative process at each child node until the leaves are pure. This means that the samples at each node belonging to the same class. In practice, you can set a limit on the depth of the tree to prevent overfitting. The purity is compromised here as the final leaves may still have some impurity. The figure shows the classification of the Iris dataset. https://www.simplilearn.com/ice9/free_resources_article_thumb/decision-tree-classifier-graph.JPG IRIS Decision Tree Let’s build a Decision Tree using scikit-learn for the Iris flower dataset and also visualize it using export_graphviz API. https://www.simplilearn.com/ice9/free_resources_article_thumb/iris-decision-tree-machine-learning.JPG The output of export_graphviz can be converted into png format: https://www.simplilearn.com/ice9/free_resources_article_thumb/iris-decision-tree-output.JPG Sample attribute stands for the number of training instances the node applies to. Value attribute stands for the number of training instances of each class the node applies to. Gini impurity measures the node’s impurity. A node is “pure” (gini=0) if all training instances it applies to belong to the same class. https://www.simplilearn.com/ice9/free_resources_article_thumb/impurity-formula-machine-learning.JPG For example, for Versicolor (green color node), the Gini is 1-(0/54)2 -(49/54)2 -(5/54) 2 ≈ 0.168 https://www.simplilearn.com/ice9/free_resources_article_thumb/iris-decision-tree-sample.JPG Decision Boundaries Let us learn to create decision boundaries below. For the first node (depth 0), the solid line splits the data (Iris-Setosa on left). Gini is 0 for Setosa node, so no further split is possible. The second node (depth 1) splits the data into Versicolor and Virginica. If max_depth were set as 3, a third split would happen (vertical dotted line). https://www.simplilearn.com/ice9/free_resources_article_thumb/decision-tree-boundaries.JPG For a sample with petal length 5 cm and petal width 1.5 cm, the tree traverses to depth 2 left node, so the probability predictions for this sample are 0% for Iris-Setosa (0/54), 90.7% for Iris-Versicolor (49/54), and 9.3% for Iris-Virginica (5/54) CART Training Algorithm Scikit-learn uses Classification and Regression Trees (CART) algorithm to train Decision Trees. CART algorithm: Split the data into two subsets using a single feature k and threshold tk (example, petal length < “2.45 cm”). This is done recursively for each node. k and tk are chosen such that they produce the purest subsets (weighted by their size). The objective is to minimize the cost function as given below: https://www.simplilearn.com/ice9/free_resources_article_thumb/cart-training-algorithm-machine-learning.JPG The algorithm stops executing if one of the following situations occurs: max_depth is reached No further splits are found for each node Other hyperparameters may be used to stop the tree: min_samples_split min_samples_leaf min_weight_fraction_leaf max_leaf_nodes Gini Impurity or Entropy Entropy is one more measure of impurity and can be used in place of Gini. https://www.simplilearn.com/ice9/free_resources_article_thumb/gini-impurity-entrophy.JPG It is a degree of uncertainty, and Information Gain is the reduction that occurs in entropy as one traverses down the tree. Entropy is zero for a DT node when the node contains instances of only one class. Entropy for depth 2 left node in the example given above is: https://www.simplilearn.com/ice9/free_resources_article_thumb/entrophy-for-depth-2.JPG Gini and Entropy both lead to similar trees. DT: Regularization The following figure shows two decision trees on the moons dataset. https://www.simplilearn.com/ice9/free_resources_article_thumb/dt-regularization-machine-learning.JPG The decision tree on the right is restricted by min_samples_leaf = 4. The model on the left is overfitting, while the model on the right generalizes better. Random Forest Classifier Let us have an understanding of Random Forest Classifier below. A random forest can be considered an ensemble of decision trees (Ensemble learning). Random Forest algorithm: Draw a random bootstrap sample of size n (randomly choose n samples from the training set). Grow a decision tree from the bootstrap sample. At each node, randomly select d features. Split the node using the feature that provides the best split according to the objective function, for instance by maximizing the information gain. Repeat the steps 1 to 2 k times. (k is the number of trees you want to create, using a subset of samples) Aggregate the prediction by each tree for a new data point to assign the class label by majority vote (pick the group selected by the most number of trees and assign new data point to that group). Random Forests are opaque, which means it is difficult to visualize their inner workings. https://www.simplilearn.com/ice9/free_resources_article_thumb/random-forest-classifier-graph.JPG However, the advantages outweigh their limitations since you do not have to worry about hyperparameters except k, which stands for the number of decision trees to be created from a subset of samples. RF is quite robust to noise from the individual decision trees. Hence, you need not prune individual decision trees. The larger the number of decision trees, the more accurate the Random Forest prediction is. (This, however, comes with higher computation cost). Key Takeaways Let us quickly run through what we have learned so far in this Classification tutorial. Classification algorithms are supervised learning methods to split data into classes. They can work on Linear Data as well as Nonlinear Data. Logistic Regression can classify data based on weighted parameters and sigmoid conversion to calculate the probability of classes. K-nearest Neighbors (KNN) algorithm uses similar features to classify data. Support Vector Machines (SVMs) classify data by detecting the maximum margin hyperplane between data classes. Naïve Bayes, a simplified Bayes Model, can help classify data using conditional probability models. Decision Trees are powerful classifiers and use tree splitting logic until pure or somewhat pure leaf node classes are attained. Random Forests apply Ensemble Learning to Decision Trees for more accurate classification predictions. Conclusion This completes ‘Classification’ tutorial. In the next tutorial, we will learn 'Unsupervised Learning with Clustering.'

using Neural Networks (SSD) on Tensorflow. This repo documents steps and scripts used to train a hand detector using Tensorflow (Object Detection API). As with any DNN based task, the most expensive (and riskiest) part of the process has to do with finding or creating the right (annotated) dataset. I was interested mainly in detecting hands on a table (egocentric view point). I experimented first with the [Oxford Hands Dataset](http://www.robots.ox.ac.uk/~vgg/data/hands/) (the results were not good). I then tried the [Egohands Dataset](http://vision.soic.indiana.edu/projects/egohands/) which was a much better fit to my requirements. The goal of this repo/post is to demonstrate how neural networks can be applied to the (hard) problem of tracking hands (egocentric and other views). Better still, provide code that can be adapted to other uses cases. If you use this tutorial or models in your research or project, please cite [this](#citing-this-tutorial). Here is the detector in action. <img src="images/hand1.gif" width="33.3%"><img src="images/hand2.gif" width="33.3%"><img src="images/hand3.gif" width="33.3%"> Realtime detection on video stream from a webcam . <img src="images/chess1.gif" width="33.3%"><img src="images/chess2.gif" width="33.3%"><img src="images/chess3.gif" width="33.3%"> Detection on a Youtube video. Both examples above were run on a macbook pro **CPU** (i7, 2.5GHz, 16GB). Some fps numbers are: | FPS | Image Size | Device| Comments| | ------------- | ------------- | ------------- | ------------- | | 21 | 320 * 240 | Macbook pro (i7, 2.5GHz, 16GB) | Run without visualizing results| | 16 | 320 * 240 | Macbook pro (i7, 2.5GHz, 16GB) | Run while visualizing results (image above) | | 11 | 640 * 480 | Macbook pro (i7, 2.5GHz, 16GB) | Run while visualizing results (image above) | > Note: The code in this repo is written and tested with Tensorflow `1.4.0-rc0`. Using a different version may result in [some errors](https://github.com/tensorflow/models/issues/1581). You may need to [generate your own frozen model](https://pythonprogramming.net/testing-custom-object-detector-tensorflow-object-detection-api-tutorial/?completed=/training-custom-objects-tensorflow-object-detection-api-tutorial/) graph using the [model checkpoints](model-checkpoint) in the repo to fit your TF version. **Content of this document** - Motivation - Why Track/Detect hands with Neural Networks - Data preparation and network training in Tensorflow (Dataset, Import, Training) - Training the hand detection Model - Using the Detector to Detect/Track hands - Thoughts on Optimizations. > P.S if you are using or have used the models provided here, feel free to reach out on twitter ([@vykthur](https://twitter.com/vykthur)) and share your work! ## Motivation - Why Track/Detect hands with Neural Networks? There are several existing approaches to tracking hands in the computer vision domain. Incidentally, many of these approaches are rule based (e.g extracting background based on texture and boundary features, distinguishing between hands and background using color histograms and HOG classifiers,) making them not very robust. For example, these algorithms might get confused if the background is unusual or in situations where sharp changes in lighting conditions cause sharp changes in skin color or the tracked object becomes occluded.(see [here for a review](https://www.cse.unr.edu/~bebis/handposerev.pdf) paper on hand pose estimation from the HCI perspective) With sufficiently large datasets, neural networks provide opportunity to train models that perform well and address challenges of existing object tracking/detection algorithms - varied/poor lighting, noisy environments, diverse viewpoints and even occlusion. The main drawbacks to usage for real-time tracking/detection is that they can be complex, are relatively slow compared to tracking-only algorithms and it can be quite expensive to assemble a good dataset. But things are changing with advances in fast neural networks. Furthermore, this entire area of work has been made more approachable by deep learning frameworks (such as the tensorflow object detection api) that simplify the process of training a model for custom object detection. More importantly, the advent of fast neural network models like ssd, faster r-cnn, rfcn (see [here](https://github.com/tensorflow/models/blob/master/research/object_detection/g3doc/detection_model_zoo.md#coco-trained-models-coco-models) ) etc make neural networks an attractive candidate for real-time detection (and tracking) applications. Hopefully, this repo demonstrates this. > If you are not interested in the process of training the detector, you can skip straight to applying the [pretrained model I provide in detecting hands](#detecting-hands). Training a model is a multi-stage process (assembling dataset, cleaning, splitting into training/test partitions and generating an inference graph). While I lightly touch on the details of these parts, there are a few other tutorials cover training a custom object detector using the tensorflow object detection api in more detail[ see [here](https://pythonprogramming.net/training-custom-objects-tensorflow-object-detection-api-tutorial/) and [here](https://towardsdatascience.com/how-to-train-your-own-object-detector-with-tensorflows-object-detector-api-bec72ecfe1d9) ]. I recommend you walk through those if interested in training a custom object detector from scratch. ## Data preparation and network training in Tensorflow (Dataset, Import, Training) **The Egohands Dataset** The hand detector model is built using data from the [Egohands Dataset](http://vision.soic.indiana.edu/projects/egohands/) dataset. This dataset works well for several reasons. It contains high quality, pixel level annotations (>15000 ground truth labels) where hands are located across 4800 images. All images are captured from an egocentric view (Google glass) across 48 different environments (indoor, outdoor) and activities (playing cards, chess, jenga, solving puzzles etc). <img src="images/egohandstrain.jpg" width="100%"> If you will be using the Egohands dataset, you can cite them as follows: > Bambach, Sven, et al. "Lending a hand: Detecting hands and recognizing activities in complex egocentric interactions." Proceedings of the IEEE International Conference on Computer Vision. 2015. The Egohands dataset (zip file with labelled data) contains 48 folders of locations where video data was collected (100 images per folder). ``` -- LOCATION_X -- frame_1.jpg -- frame_2.jpg ... -- frame_100.jpg -- polygons.mat // contains annotations for all 100 images in current folder -- LOCATION_Y -- frame_1.jpg -- frame_2.jpg ... -- frame_100.jpg -- polygons.mat // contains annotations for all 100 images in current folder ``` **Converting data to Tensorflow Format** Some initial work needs to be done to the Egohands dataset to transform it into the format (`tfrecord`) which Tensorflow needs to train a model. This repo contains `egohands_dataset_clean.py` a script that will help you generate these csv files. - Downloads the egohands datasets - Renames all files to include their directory names to ensure each filename is unique - Splits the dataset into train (80%), test (10%) and eval (10%) folders. - Reads in `polygons.mat` for each folder, generates bounding boxes and visualizes them to ensure correctness (see image above). - Once the script is done running, you should have an images folder containing three folders - train, test and eval. Each of these folders should also contain a csv label document each - `train_labels.csv`, `test_labels.csv` that can be used to generate `tfrecords` Note: While the egohands dataset provides four separate labels for hands (own left, own right, other left, and other right), for my purpose, I am only interested in the general `hand` class and label all training data as `hand`. You can modify the data prep script to generate `tfrecords` that support 4 labels. Next: convert your dataset + csv files to tfrecords. A helpful guide on this can be found [here](https://pythonprogramming.net/creating-tfrecord-files-tensorflow-object-detection-api-tutorial/).For each folder, you should be able to generate `train.record`, `test.record` required in the training process. ## Training the hand detection Model Now that the dataset has been assembled (and your tfrecords), the next task is to train a model based on this. With neural networks, it is possible to use a process called [transfer learning](https://www.tensorflow.org/tutorials/image_retraining) to shorten the amount of time needed to train the entire model. This means we can take an existing model (that has been trained well on a related domain (here image classification) and retrain its final layer(s) to detect hands for us. Sweet!. Given that neural networks sometimes have thousands or millions of parameters that can take weeks or months to train, transfer learning helps shorten training time to possibly hours. Tensorflow does offer a few models (in the tensorflow [model zoo](https://github.com/tensorflow/models/blob/master/research/object_detection/g3doc/detection_model_zoo.md#coco-trained-models-coco-models)) and I chose to use the `ssd_mobilenet_v1_coco` model as my start point given it is currently (one of) the fastest models (read the SSD research [paper here](https://arxiv.org/pdf/1512.02325.pdf)). The training process can be done locally on your CPU machine which may take a while or better on a (cloud) GPU machine (which is what I did). For reference, training on my macbook pro (tensorflow compiled from source to take advantage of the mac's cpu architecture) the maximum speed I got was 5 seconds per step as opposed to the ~0.5 seconds per step I got with a GPU. For reference it would take about 12 days to run 200k steps on my mac (i7, 2.5GHz, 16GB) compared to ~5hrs on a GPU. > **Training on your own images**: Please use the [guide provided by Harrison from pythonprogramming](https://pythonprogramming.net/training-custom-objects-tensorflow-object-detection-api-tutorial/) on how to generate tfrecords given your label csv files and your images. The guide also covers how to start the training process if training locally. [see [here] (https://pythonprogramming.net/training-custom-objects-tensorflow-object-detection-api-tutorial/)]. If training in the cloud using a service like GCP, see the [guide here](https://github.com/tensorflow/models/blob/master/research/object_detection/g3doc/running_on_cloud.md). As the training process progresses, the expectation is that total loss (errors) gets reduced to its possible minimum (about a value of 1 or thereabout). By observing the tensorboard graphs for total loss(see image below), it should be possible to get an idea of when the training process is complete (total loss does not decrease with further iterations/steps). I ran my training job for 200k steps (took about 5 hours) and stopped at a total Loss (errors) value of 2.575.(In retrospect, I could have stopped the training at about 50k steps and gotten a similar total loss value). With tensorflow, you can also run an evaluation concurrently that assesses your model to see how well it performs on the test data. A commonly used metric for performance is mean average precision (mAP) which is single number used to summarize the area under the precision-recall curve. mAP is a measure of how well the model generates a bounding box that has at least a 50% overlap with the ground truth bounding box in our test dataset. For the hand detector trained here, the mAP value was **0.9686@0.5IOU**. mAP values range from 0-1, the higher the better. <img src="images/accuracy.jpg" width="100%"> Once training is completed, the trained inference graph (`frozen_inference_graph.pb`) is then exported (see the earlier referenced guides for how to do this) and saved in the `hand_inference_graph` folder. Now its time to do some interesting detection. ## Using the Detector to Detect/Track hands If you have not done this yet, please following the guide on installing [Tensorflow and the Tensorflow object detection api](https://github.com/tensorflow/models/blob/master/research/object_detection/g3doc/installation.md). This will walk you through setting up the tensorflow framework, cloning the tensorflow github repo and a guide on - Load the `frozen_inference_graph.pb` trained on the hands dataset as well as the corresponding label map. In this repo, this is done in the `utils/detector_utils.py` script by the `load_inference_graph` method. ```python detection_graph = tf.Graph() with detection_graph.as_default(): od_graph_def = tf.GraphDef() with tf.gfile.GFile(PATH_TO_CKPT, 'rb') as fid: serialized_graph = fid.read() od_graph_def.ParseFromString(serialized_graph) tf.import_graph_def(od_graph_def, name='') sess = tf.Session(graph=detection_graph) print("> ====== Hand Inference graph loaded.") ``` - Detect hands. In this repo, this is done in the `utils/detector_utils.py` script by the `detect_objects` method. ```python (boxes, scores, classes, num) = sess.run( [detection_boxes, detection_scores, detection_classes, num_detections], feed_dict={image_tensor: image_np_expanded}) ``` - Visualize detected bounding detection_boxes. In this repo, this is done in the `utils/detector_utils.py` script by the `draw_box_on_image` method. This repo contains two scripts that tie all these steps together. - detect_multi_threaded.py : A threaded implementation for reading camera video input detection and detecting. Takes a set of command line flags to set parameters such as `--display` (visualize detections), image parameters `--width` and `--height`, videe `--source` (0 for camera) etc. - detect_single_threaded.py : Same as above, but single threaded. This script works for video files by setting the video source parameter videe `--source` (path to a video file). ```cmd # load and run detection on video at path "videos/chess.mov" python detect_single_threaded.py --source videos/chess.mov ``` > Update: If you do have errors loading the frozen inference graph in this repo, feel free to generate a new graph that fits your TF version from the model-checkpoint in this repo. Use the [export_inference_graph.py](https://github.com/tensorflow/models/blob/master/research/object_detection/export_inference_graph.py) script provided in the tensorflow object detection api repo. More guidance on this [here](https://pythonprogramming.net/testing-custom-object-detector-tensorflow-object-detection-api-tutorial/?completed=/training-custom-objects-tensorflow-object-detection-api-tutorial/). ## Thoughts on Optimization. A few things that led to noticeable performance increases. - Threading: Turns out that reading images from a webcam is a heavy I/O event and if run on the main application thread can slow down the program. I implemented some good ideas from [Adrian Rosebuck](https://www.pyimagesearch.com/2017/02/06/faster-video-file-fps-with-cv2-videocapture-and-opencv/) on parrallelizing image capture across multiple worker threads. This mostly led to an FPS increase of about 5 points. - For those new to Opencv, images from the `cv2.read()` method return images in [BGR format](https://www.learnopencv.com/why-does-opencv-use-bgr-color-format/). Ensure you convert to RGB before detection (accuracy will be much reduced if you dont). ```python cv2.cvtColor(image_np, cv2.COLOR_BGR2RGB) ``` - Keeping your input image small will increase fps without any significant accuracy drop.(I used about 320 x 240 compared to the 1280 x 720 which my webcam provides). - Model Quantization. Moving from the current 32 bit to 8 bit can achieve up to 4x reduction in memory required to load and store models. One way to further speed up this model is to explore the use of [8-bit fixed point quantization](https://heartbeat.fritz.ai/8-bit-quantization-and-tensorflow-lite-speeding-up-mobile-inference-with-low-precision-a882dfcafbbd). Performance can also be increased by a clever combination of tracking algorithms with the already decent detection and this is something I am still experimenting with. Have ideas for optimizing better, please share! <img src="images/general.jpg" width="100%"> Note: The detector does reflect some limitations associated with the training set. This includes non-egocentric viewpoints, very noisy backgrounds (e.g in a sea of hands) and sometimes skin tone. There is opportunity to improve these with additional data. ## Integrating Multiple DNNs. One way to make things more interesting is to integrate our new knowledge of where "hands" are with other detectors trained to recognize other objects. Unfortunately, while our hand detector can in fact detect hands, it cannot detect other objects (a factor or how it is trained). To create a detector that classifies multiple different objects would mean a long involved process of assembling datasets for each class and a lengthy training process. > Given the above, a potential strategy is to explore structures that allow us **efficiently** interleave output form multiple pretrained models for various object classes and have them detect multiple objects on a single image. An example of this is with my primary use case where I am interested in understanding the position of objects on a table with respect to hands on same table. I am currently doing some work on a threaded application that loads multiple detectors and outputs bounding boxes on a single image. More on this soon.

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# Liberty House Club **A Parallel Binance Chain to Enable Smart Contracts** _NOTE: This document is under development. Please check regularly for updates!_ ## Table of Contents - [Motivation](#motivation) - [Design Principles](#design-principles) - [Consensus and Validator Quorum](#consensus-and-validator-quorum) * [Proof of Staked Authority](#proof-of-staked-authority) * [Validator Quorum](#validator-quorum) * [Security and Finality](#security-and-finality) * [Reward](#reward) - [Token Economy](#token-economy) * [Native Token](#native-token) * [Other Tokens](#other-tokens) - [Cross-Chain Transfer and Communication](#cross-chain-transfer-and-communication) * [Cross-Chain Transfer](#cross-chain-transfer) * [BC to BSC Architecture](#bc-to-bsc-architecture) * [BSC to BC Architecture](#bsc-to-bc-architecture) * [Timeout and Error Handling](#timeout-and-error-handling) * [Cross-Chain User Experience](#cross-chain-user-experience) * [Cross-Chain Contract Event](#cross-chain-contract-event) - [Staking and Governance](#staking-and-governance) * [Staking on BC](#staking-on-bc) * [Rewarding](#rewarding) * [Slashing](#slashing) - [Relayers](#relayers) * [BSC Relayers](#bsc-relayers) * [Oracle Relayers](#oracle-relayers) - [Outlook](#outlook) # Motivation After its mainnet community [launch](https://www.binance.com/en/blog/327334696200323072/Binance-DEX-Launches-on-Binance-Chain-Invites-Further-Community-Development) in April 2019, [Binance Chain](https://www.binance.org) has exhibited its high speed and large throughput design. Binance Chain’s primary focus, its native [decentralized application](https://en.wikipedia.org/wiki/Decentralized_application) (“dApp”) [Binance DEX](https://www.binance.org/trade), has demonstrated its low-latency matching with large capacity headroom by handling millions of trading volume in a short time. Flexibility and usability are often in an inverse relationship with performance. The concentration on providing a convenient digital asset issuing and trading venue also brings limitations. Binance Chain's most requested feature is the programmable extendibility, or simply the [Smart Contract](https://en.wikipedia.org/wiki/Smart_contract) and Virtual Machine functions. Digital asset issuers and owners struggle to add new decentralized features for their assets or introduce any sort of community governance and activities. Despite this high demand for adding the Smart Contract feature onto Binance Chain, it is a hard decision to make. The execution of a Smart Contract may slow down the exchange function and add non-deterministic factors to trading. If that compromise could be tolerated, it might be a straightforward idea to introduce a new Virtual Machine specification based on [Tendermint](https://tendermint.com/core/), based on the current underlying consensus protocol and major [RPC](https://docs.binance.org/api-reference/node-rpc.html) implementation of Binance Chain. But all these will increase the learning requirements for all existing dApp communities, and will not be very welcomed. We propose a parallel blockchain of the current Binance Chain to retain the high performance of the native DEX blockchain and to support a friendly Smart Contract function at the same time. # Design Principles After the creation of the parallel blockchain into the Binance Chain ecosystem, two blockchains will run side by side to provide different services. The new parallel chain will be called “**Binance Smart Chain**” (short as “**BSC**” for the below sections), while the existing mainnet remains named “**Binance Chain**” (short as “**BC**” for the below sections). Here are the design principles of **BSC**: 1. **Standalone Blockchain**: technically, BSC is a standalone blockchain, instead of a layer-2 solution. Most BSC fundamental technical and business functions should be self-contained so that it can run well even if the BC stopped for a short period. 2. **Ethereum Compatibility**: The first practical and widely-used Smart Contract platform is Ethereum. To take advantage of the relatively mature applications and community, BSC chooses to be compatible with the existing Ethereum mainnet. This means most of the **dApps**, ecosystem components, and toolings will work with BSC and require zero or minimum changes; BSC node will require similar (or a bit higher) hardware specification and skills to run and operate. The implementation should leave room for BSC to catch up with further Ethereum upgrades. 3. **Staking Involved Consensus and Governance**: Staking-based consensus is more environmentally friendly and leaves more flexible option to the community governance. Expectedly, this consensus should enable better network performance over [proof-of-work](https://en.wikipedia.org/wiki/Proof_of_work) blockchain system, i.e., faster blocking time and higher transaction capacity. 4. **Native Cross-Chain Communication**: both BC and BSC will be implemented with native support for cross-chain communication among the two blockchains. The communication protocol should be bi-directional, decentralized, and trustless. It will concentrate on moving digital assets between BC and BSC, i.e., [BEP2](https://github.com/binance-chain/BEPs/blob/master/BEP2.md) tokens, and eventually, other BEP tokens introduced later. The protocol should care for the minimum of other items stored in the state of the blockchains, with only a few exceptions. # Consensus and Validator Quorum Based on the above design principles, the consensus protocol of BSC is to fulfill the following goals: 1. Blocking time should be shorter than Ethereum network, e.g. 5 seconds or even shorter. 2. It requires limited time to confirm the finality of transactions, e.g. around 1-min level or shorter. 3. There is no inflation of native token: BNB, the block reward is collected from transaction fees, and it will be paid in BNB. 4. It is compatible with Ethereum system as much as possible. 5. It allows modern [proof-of-stake](https://en.wikipedia.org/wiki/Proof_of_stake) blockchain network governance. ## Proof of Staked Authority Although Proof-of-Work (PoW) has been recognized as a practical mechanism to implement a decentralized network, it is not friendly to the environment and also requires a large size of participants to maintain the security. Ethereum and some other blockchain networks, such as [MATIC Bor](https://github.com/maticnetwork/bor), [TOMOChain](https://tomochain.com/), [GoChain](https://gochain.io/), [xDAI](https://xdai.io/), do use [Proof-of-Authority(PoA)](https://en.wikipedia.org/wiki/Proof_of_authority) or its variants in different scenarios, including both testnet and mainnet. PoA provides some defense to 51% attack, with improved efficiency and tolerance to certain levels of Byzantine players (malicious or hacked). It serves as an easy choice to pick as the fundamentals. Meanwhile, the PoA protocol is most criticized for being not as decentralized as PoW, as the validators, i.e. the nodes that take turns to produce blocks, have all the authorities and are prone to corruption and security attacks. Other blockchains, such as EOS and Lisk both, introduce different types of [Delegated Proof of Stake (DPoS)](https://en.bitcoinwiki.org/wiki/DPoS) to allow the token holders to vote and elect the validator set. It increases the decentralization and favors community governance. BSC here proposes to combine DPoS and PoA for consensus, so that: 1. Blocks are produced by a limited set of validators 2. Validators take turns to produce blocks in a PoA manner, similar to [Ethereum’s Clique](https://eips.ethereum.org/EIPS/eip-225) consensus design 3. Validator set are elected in and out based on a staking based governance ## Validator Quorum In the genesis stage, a few trusted nodes will run as the initial Validator Set. After the blocking starts, anyone can compete to join as candidates to elect as a validator. The staking status decides the top 21 most staked nodes to be the next validator set, and such an election will repeat every 24 hours. **BNB** is the token used to stake for BSC. In order to remain as compatible as Ethereum and upgradeable to future consensus protocols to be developed, BSC chooses to rely on the **BC** for staking management (Please refer to the below “[Staking and Governance](#staking-and-governance)” section). There is a **dedicated staking module for BSC on BC**. It will accept BSC staking from BNB holders and calculate the highest staked node set. Upon every UTC midnight, BC will issue a verifiable `ValidatorSetUpdate` cross-chain message to notify BSC to update its validator set. While producing further blocks, the existing BSC validators check whether there is a `ValidatorSetUpdate` message relayed onto BSC periodically. If there is, they will update the validator set after an **epoch period**, i.e. a predefined number of blocking time. For example, if BSC produces a block every 5 seconds, and the epoch period is 240 blocks, then the current validator set will check and update the validator set for the next epoch in 1200 seconds (20 minutes). ## Security and Finality Given there are more than ½\*N+1 validators are honest, PoA based networks usually work securely and properly. However, there are still cases where certain amount Byzantine validators may still manage to attack the network, e.g. through the “[Clone Attack](https://arxiv.org/pdf/1902.10244.pdf)”. To secure as much as BC, BSC users are encouraged to wait until receiving blocks sealed by more than ⅔\*N+1 different validators. In that way, the BSC can be trusted at a similar security level to BC and can tolerate less than ⅓\*N Byzantine validators. With 21 validators, if the block time is 5 seconds, the ⅔\*N+1 different validator seals will need a time period of (⅔\*21+1)*5 = 75 seconds. Any critical applications for BSC may have to wait for ⅔\*N+1 to ensure a relatively secure finality. However, besides such arrangement, BSC does introduce **Slashing** logic to penalize Byzantine validators for **double signing** or **inavailability**, which will be covered in the “Staking and Governance” section later. This Slashing logic will expose the malicious validators in a very short time and make the “Clone Attack” very hard or extremely non-beneficial to execute. With this enhancement, ½\*N+1 or even fewer blocks are enough as confirmation for most transactions. ## Reward All the BSC validators in the current validator set will be rewarded with transaction **fees in BNB**. As BNB is not an inflationary token, there will be no mining rewards as what Bitcoin and Ethereum network generate, and the gas fee is the major reward for validators. As BNB is also utility tokens with other use cases, delegators and validators will still enjoy other benefits of holding BNB. The reward for validators is the fees collected from transactions in each block. Validators can decide how much to give back to the delegators who stake their BNB to them, in order to attract more staking. Every validator will take turns to produce the blocks in the same probability (if they stick to 100% liveness), thus, in the long run, all the stable validators may get a similar size of the reward. Meanwhile, the stakes on each validator may be different, so this brings a counter-intuitive situation that more users trust and delegate to one validator, they potentially get less reward. So rational delegators will tend to delegate to the one with fewer stakes as long as the validator is still trustful (insecure validator may bring slashable risk). In the end, the stakes on all the validators will have less variation. This will actually prevent the stake concentration and “winner wins forever” problem seen on some other networks. Some parts of the gas fee will also be rewarded to relayers for Cross-Chain communication. Please refer to the “[Relayers](#relayers)” section below. # Token Economy BC and BSC share the same token universe for BNB and BEP2 tokens. This defines: 1. The same token can circulate on both networks, and flow between them bi-directionally via a cross-chain communication mechanism. 2. The total circulation of the same token should be managed across the two networks, i.e. the total effective supply of a token should be the sum of the token’s total effective supply on both BSC and BC. 3. The tokens can be initially created on BSC in a similar format as ERC20 token standard, or on BC as a BEP2, then created on the other. There are native ways on both networks to link the two and secure the total supply of the token. ## Native Token BNB will run on BSC in the same way as ETH runs on Ethereum so that it remains as “native token” for both BSC and BC. This means, in addition to BNB is used to pay most of the fees on Binance Chain and Binance DEX, BNB will be also used to: 1. pay “fees“ to deploy smart contracts on BSC 2. stake on selected BSC validators, and get corresponding rewards 3. perform cross-chain operations, such as transfer token assets across BC and BSC ### Seed Fund Certain amounts of BNB will be burnt on BC and minted on BSC during its genesis stage. This amount is called “Seed Fund” to circulate on BSC after the first block, which will be dispatched to the initial BC-to-BSC Relayer(described in later sections) and initial validator set introduced at genesis. These BNBs are used to pay transaction fees in the early stage to transfer more BNB from BC onto BSC via the cross-chain mechanism. The BNB cross-chain transfer is discussed in a later section, but for BC to BSC transfer, it is generally to lock BNB on BC from the source address of the transfer to a system-controlled address and unlock the corresponding amount from special contract to the target address of the transfer on BSC, or reversely, when transferring from BSC to BC, it is to lock BNB from the source address on BSC into a special contract and release locked amount on BC from the system address to the target address. The logic is related to native code on BC and a series of smart contracts on BSC. ## Other Tokens BC supports BEP2 tokens and upcoming [BEP8 tokens](https://github.com/binance-chain/BEPs/pull/69), which are native assets transferrable and tradable (if listed) via fast transactions and sub-second finality. Meanwhile, as BSC is Ethereum compatible, it is natural to support ERC20 tokens on BSC, which here is called “**BEP2E**” (with the real name to be introduced by the future BEPs,it potentially covers BEP8 as well). BEP2E may be “Enhanced” by adding a few more methods to expose more information, such as token denomination, decimal precision definition and the owner address who can decide the Token Binding across the chains. BSC and BC work together to ensure that one token can circulate in both formats with confirmed total supply and be used in different use cases. ### Token Binding BEP2 tokens will be extended to host a new attribute to associate the token with a BSC BEP2E token contract, called “**Binder**”, and this process of association is called “**Token Binding**”. Token Binding can happen at any time after BEP2 and BEP2E are ready. The token owners of either BEP2 or BEP2E don’t need to bother about the Binding, until before they really want to use the tokens on different scenarios. Issuers can either create BEP2 first or BEP2E first, and they can be bound at a later time. Of course, it is encouraged for all the issuers of BEP2 and BEP2E to set the Binding up early after the issuance. A typical procedure to bind the BEP2 and BEP2E will be like the below: 1. Ensure both the BEP2 token and the BEP2E token both exist on each blockchain, with the same total supply. BEP2E should have 3 more methods than typical ERC20 token standard: * symbol(): get token symbol * decimals(): get the number of the token decimal digits * owner(): get **BEP2E contract owner’s address.** This value should be initialized in the BEP2E contract constructor so that the further binding action can verify whether the action is from the BEP2E owner. 2. Decide the initial circulation on both blockchains. Suppose the total supply is *S*, and the expected initial circulating supply on BC is *K*, then the owner should lock S-K tokens to a system controlled address on BC. 3. Equivalently, *K* tokens is locked in the special contract on BSC, which handles major binding functions and is named as **TokenHub**. The issuer of the BEP2E token should lock the *K* amount of that token into TokenHub, resulting in *S-K* tokens to circulate on BSC. Thus the total circulation across 2 blockchains remains as *S*. 4. The issuer of BEP2 token sends the bind transaction on BC. Once the transaction is executed successfully after proper verification: * It transfers *S-K* tokens to a system-controlled address on BC. * A cross-chain bind request package will be created, waiting for Relayers to relay. 5. BSC Relayers will relay the cross-chain bind request package into **TokenHub** on BSC, and the corresponding request and information will be stored into the contract. 6. The contract owner and only the owner can run a special method of TokenHub contract, `ApproveBind`, to verify the binding request to mark it as a success. It will confirm: * the token has not been bound; * the binding is for the proper symbol, with proper total supply and decimal information; * the proper lock are done on both networks; 10. Once the `ApproveBind` method has succeeded, TokenHub will mark the two tokens are bounded and share the same circulation on BSC, and the status will be propagated back to BC. After this final confirmation, the BEP2E contract address and decimals will be written onto the BEP2 token as a new attribute on BC, and the tokens can be transferred across the two blockchains bidirectionally. If the ApproveBind fails, the failure event will also be propagated back to BC to release the locked tokens, and the above steps can be re-tried later. # Cross-Chain Transfer and Communication Cross-chain communication is the key foundation to allow the community to take advantage of the dual chain structure: * users are free to create any tokenization, financial products, and digital assets on BSC or BC as they wish * the items on BSC can be manually and programmingly traded and circulated in a stable, high throughput, lighting fast and friendly environment of BC * users can operate these in one UI and tooling ecosystem. ## Cross-Chain Transfer The cross-chain transfer is the key communication between the two blockchains. Essentially the logic is: 1. the `transfer-out` blockchain will lock the amount from source owner addresses into a system controlled address/contracts; 2. the `transfer-in` blockchain will unlock the amount from the system controlled address/contracts and send it to target addresses. The cross-chain transfer package message should allow the BSC Relayers and BC **Oracle Relayers** to verify: 1. Enough amount of token assets are removed from the source address and locked into a system controlled addresses/contracts on the source blockchain. And this can be confirmed on the target blockchain. 2. Proper amounts of token assets are released from a system controlled addresses/contracts and allocated into target addresses on the target blockchain. If this fails, it can be confirmed on source blockchain, so that the locked token can be released back (may deduct fees). 3. The sum of the total circulation of the token assets across the 2 blockchains are not changed after this transfer action completes, no matter if the transfer succeeds or not.  The architecture of cross-chain communication is as in the above diagram. To accommodate the 2 heteroid systems, communication handling is different in each direction. ## BC to BSC Architecture BC is a Tendermint-based, instant finality blockchain. Validators with at least ⅔\*N+1 of the total voting power will co-sign each block on the chain. So that it is practical to verify the block transactions and even the state value via **Block Header** and **Merkle Proof** verification. This has been researched and implemented as “**Light-Client Protocol**”, which are intensively discussed in [the Ethereum](https://github.com/ethereum/wiki/wiki/Light-client-protocol) community, studied and implemented for [Cosmos inter-chain communication](https://github.com/cosmos/ics/blob/a4173c91560567bdb7cc9abee8e61256fc3725e9/spec/ics-007-tendermint-client/README.md). BC-to-BSC communication will be verified in an “**on-chain light client**” implemented via BSC **Smart Contracts** (some of them may be **“pre-compiled”**). After some transactions and state change happen on BC, if a transaction is defined to trigger cross-chain communication,the Cross-chain “**package**” message will be created and **BSC Relayers** will pass and submit them onto BSC as data into the "build-in system contracts". The build-in system contracts will verify the package and execute the transactions if it passes the verification. The verification will be guaranteed with the below design: 1. BC blocking status will be synced to the light client contracts on BSC from time to time, via block header and pre-commits, for the below information: * block and app hash of BC that are signed by validators * current validatorset, and validator set update 2. the key-value from the blockchain state will be verified based on the Merkle Proof and information from above #1. After confirming the key-value is accurate and trustful, the build-in system contracts will execute the actions corresponding to the cross-chain packages. Some examples of such packages that can be created for BC-to-BSC are: 1. Bind: bind the BEP2 tokens and BEP2E 2. Transfer: transfer tokens after binding, this means the circulation will decrease (be locked) from BC and appear in the target address balance on BSC 3. Error Handling: to handle any timeout/failure event for BSC-to-BC communication 4. Validatorset update of BSC To ensure no duplication, proper message sequence and timely timeout, there is a “Channel” concept introduced on BC to manage any types of the communication. For relayers, please also refer to the below “Relayers” section. ## BSC to BC Architecture BSC uses Proof of Staked Authority consensus protocol, which has a chance to fork and requires confirmation of more blocks. One block only has the signature of one validator, so that it is not easy to rely on one block to verify data from BSC. To take full advantage of validator quorum of BC, an idea similar to many [Bridge ](https://github.com/poanetwork/poa-bridge)or Oracle blockchains is adopted: 1. The cross-chain communication requests from BSC will be submitted and executed onto BSC as transactions. The execution of the transanction wil emit `Events`, and such events can be observed and packaged in certain “**Oracle**” onto BC. Instead of Block Headers, Hash and Merkle Proof, this type of “Oracle” package directly contains the cross-chain information for actions, such as sender, receiver and amount for transfer. 2. To ensure the security of the Oracle, the validators of BC will form anothe quorum of “**Oracle Relayers**”. Each validator of the BC should run a **dedicated process** as the Oracle Relayer. These Oracle Relayers will submit and vote for the cross-chain communication package, like Oracle, onto BC, using the same validator keys. Any package signed by more than ⅔\*N+1 Oracle Relayers’ voting power is as secure as any block signed by ⅔\*N+1 of the same quorum of validators’ voting power. By using the same validator quorum, it saves the light client code on BC and continuous block updates onto BC. Such Oracles also have Oracle IDs and types, to ensure sequencing and proper error handling. ## Timeout and Error Handling There are scenarios that the cross-chain communication fails. For example, the relayed package cannot be executed on BSC due to some coding bug in the contracts. **Timeout and error handling logics are** used in such scenarios. For the recognizable user and system errors or any expected exceptions, the two networks should heal themselves. For example, when BC to BSC transfer fails, BSC will issue a failure event and Oracle Relayers will execute a refund on BC; when BSC to BC transfer fails, BC will issue a refund package for Relayer to relay in order to unlock the fund. However, unexpected error or exception may still happen on any step of the cross-chain communication. In such a case, the Relayers and Oracle Relayers will discover that the corresponding cross-chain channel is stuck in a particular sequence. After a Timeout period, the Relayers and Oracle Relayers can request a “SkipSequence” transaction, the stuck sequence will be marked as “Unexecutable”. A corresponding alerts will be raised, and the community has to discuss how to handle this scenario, e.g. payback via the sponsor of the validators, or event clear the fund during next network upgrade. ## Cross-Chain User Experience Ideally, users expect to use two parallel chains in the same way as they use one single chain. It requires more aggregated transaction types to be added onto the cross-chain communication to enable this, which will add great complexity, tight coupling, and maintenance burden. Here BC and BSC only implement the basic operations to enable the value flow in the initial launch and leave most of the user experience work to client side UI, such as wallets. E.g. a great wallet may allow users to sell a token directly from BSC onto BC’s DEX order book, in a secure way. ## Cross-Chain Contract Event Cross-Chain Contract Event (CCCE) is designed to allow a smart contract to trigger cross-chain transactions, directly through the contract code. This becomes possible based on: 1. Standard system contracts can be provided to serve operations callable by general smart contracts; 2. Standard events can be emitted by the standard contracts; 3. Oracle Relayers can capture the standard events, and trigger the corresponding cross-chain operations; 4. Dedicated, code-managed address (account) can be created on BC and accessed by the contracts on the BSC, here it is named as **“Contract Address on BC” (CAoB)**. Several standard operations are implemented: 1. BSC to BC transfer: this is implemented in the same way as normal BSC to BC transfer, by only triggered via standard contract. The fund can be transferred to any addresses on BC, including the corresponding CAoB of the transfer originating contract. 2. Transfer on BC: this is implemented as a special cross-chain transfer, while the real transfer is from **CAoB** to any other address (even another CAoB). 3. BC to BSC transfer: this is implemented as two-pass cross-chain communication. The first is triggered by the BSC contract and propagated onto BC, and then in the second pass, BC will start a normal BC to BSC cross-chain transfer, from **CAoB** to contract address on BSC. A special note should be paid on that the BSC contract only increases balance upon any transfer coming in on the second pass, and the error handling in the second pass is the same as the normal BC to BSC transfer. 4. IOC (Immediate-Or-Cancel) Trade Out: the primary goal of transferring assets to BC is to trade. This event will instruct to trade a certain amount of an asset in CAoB into another asset as much as possible and transfer out all the results, i.e. the left the source and the traded target tokens of the trade, back to BSC. BC will handle such relayed events by sending an “Immediate-Or-Cancel”, i.e. IOC order onto the trading pairs, once the next matching finishes, the result will be relayed back to BSC, which can be in either one or two assets. 5. Auction Trade Out: Such event will instruct BC to send an auction order to trade a certain amount of an asset in **CAoB** into another asset as much as possible and transfer out all the results back to BSC at the end of the auction. Auction function is upcoming on BC. There are some details for the Trade Out: 1. both can have a limit price (absolute or relative) for the trade; 2. the end result will be written as cross-chain packages to relay back to BSC; 3. cross-chain communication fees may be charged from the asset transferred back to BSC; 4. BSC contract maintains a mirror of the balance and outstanding orders on CAoB. No matter what error happens during the Trade Out, the final status will be propagated back to the originating contract and clear its internal state. With the above features, it simply adds the cross-chain transfer and exchange functions with high liquidity onto all the smart contracts on BSC. It will greatly add the application scenarios on Smart Contract and dApps, and make 1 chain +1 chain > 2 chains. # Staking and Governance Proof of Staked Authority brings in decentralization and community involvement. Its core logic can be summarized as the below. You may see similar ideas from other networks, especially Cosmos and EOS. 1. Token holders, including the validators, can put their tokens “**bonded**” into the stake. Token holders can **delegate** their tokens onto any validator or validator candidate, to expect it can become an actual validator, and later they can choose a different validator or candidate to **re-delegate** their tokens¹. 2. All validator candidates will be ranked by the number of bonded tokens on them, and the top ones will become the real validators. 3. Validators can share (part of) their blocking reward with their delegators. 4. Validators can suffer from “**Slashing**”, a punishment for their bad behaviors, such as double sign and/or instability. 5. There is an “**unbonding period**” for validators and delegators so that the system makes sure the tokens remain bonded when bad behaviors are caught, the responsible will get slashed during this period. ## Staking on BC Ideally, such staking and reward logic should be built into the blockchain, and automatically executed as the blocking happens. Cosmos Hub, who shares the same Tendermint consensus and libraries with Binance Chain, works in this way. BC has been preparing to enable staking logic since the design days. On the other side, as BSC wants to remain compatible with Ethereum as much as possible, it is a great challenge and efforts to implement such logic on it. This is especially true when Ethereum itself may move into a different Proof of Stake consensus protocol in a short (or longer) time. In order to keep the compatibility and reuse the good foundation of BC, the staking logic of BSC is implemented on BC: 1. The staking token is BNB, as it is a native token on both blockchains anyway 2. The staking, i.e. token bond and delegation actions and records for BSC, happens on BC. 3. The BSC validator set is determined by its staking and delegation logic, via a staking module built on BC for BSC, and propagated every day UTC 00:00 from BC to BSC via Cross-Chain communication. 4. The reward distribution happens on BC around every day UTC 00:00. ## Rewarding Both the validator update and reward distribution happen every day around UTC 00:00. This is to save the cost of frequent staking updates and block reward distribution. This cost can be significant, as the blocking reward is collected on BSC and distributed on BC to BSC validators and delegators. (Please note BC blocking fees will remain rewarding to BC validators only.) A deliberate delay is introduced here to make sure the distribution is fair: 1. The blocking reward will not be sent to validator right away, instead, they will be distributed and accumulated on a contract; 2. Upon receiving the validator set update into BSC, it will trigger a few cross-chain transfers to transfer the reward to custody addresses on the corresponding validators. The custody addresses are owned by the system so that the reward cannot be spent until the promised distribution to delegators happens. 3. In order to make the synchronization simpler and allocate time to accommodate slashing, the reward for N day will be only distributed in N+2 days. After the delegators get the reward, the left will be transferred to validators’ own reward addresses. ## Slashing Slashing is part of the on-chain governance, to ensure the malicious or negative behaviors are punished. BSC slash can be submitted by anyone. The transaction submission requires **slash evidence** and cost fees but also brings a larger reward when it is successful. So far there are two slashable cases. ### Double Sign It is quite a serious error and very likely deliberate offense when a validator signs more than one block with the same height and parent block. The reference protocol implementation should already have logic to prevent this, so only the malicious code can trigger this. When Double Sign happens, the validator should be removed from the Validator **Set** right away. Anyone can submit a slash request on BC with the evidence of Double Sign of BSC, which should contain the 2 block headers with the same height and parent block, sealed by the offending validator. Upon receiving the evidence, if the BC verifies it to be valid: 1. The validator will be removed from validator set by an instance BSC validator set update Cross-Chain update; 2. A predefined amount of BNB would be slashed from the **self-delegated** BNB of the validator; Both validator and its delegators will not receive the staking rewards. 3. Part of the slashed BNB will allocate to the submitter’s address, which is a reward and larger than the cost of submitting slash request transaction 4. The rest of the slashed BNB will allocate to the other validators’ custody addresses, and distributed to all delegators in the same way as blocking reward. ### Inavailability The liveness of BSC relies on everyone in the Proof of Staked Authority validator set can produce blocks timely when it is their turn. Validators can miss their turn due to any reason, especially problems in their hardware, software, configuration or network. This instability of the operation will hurt the performance and introduce more indeterministic into the system. There can be an internal smart contract responsible for recording the missed blocking metrics of each validator. Once the metrics are above the predefined threshold, the blocking reward for validator will not be relayed to BC for distribution but shared with other better validators. In such a way, the poorly-operating validator should be gradually voted out of the validator set as their delegators will receive less or none reward. If the metrics remain above another higher level of threshold, the validator will be dropped from the rotation, and this will be propagated back to BC, then a predefined amount of BNB would be slashed from the **self-delegated** BNB of the validator. Both validators and delegators will not receive their staking rewards. ### Governance Parameters There are many system parameters to control the behavior of the BSC, e.g. slash amount, cross-chain transfer fees. All these parameters will be determined by BSC Validator Set together through a proposal-vote process based on their staking. Such the process will be carried on BC, and the new parameter values will be picked up by corresponding system contracts via a cross-chain communication. # Relayers Relayers are responsible to submit Cross-Chain Communication Packages between the two blockchains. Due to the heterogeneous parallel chain structure, two different types of Relayers are created. ## BSC Relayers Relayers for BC to BSC communication referred to as “**BSC Relayers**”, or just simply “Relayers”. Relayer is a standalone process that can be run by anyone, and anywhere, except that Relayers must register themselves onto BSC and deposit a certain refundable amount of BNB. Only relaying requests from the registered Relayers will be accepted by BSC. The package they relay will be verified by the on-chain light client on BSC. The successful relay needs to pass enough verification and costs gas fees on BSC, and thus there should be incentive reward to encourage the community to run Relayers. ### Incentives There are two major communication types: 1. Users triggered Operations, such as `token bind` or `cross chain transfer`. Users must pay additional fee to as relayer reward. The reward will be shared with the relayers who sync the referenced blockchain headers. Besides, the reward won't be paid the relayers' accounts directly. A reward distribution mechanism will be brought in to avoid monopolization. 2. System Synchronization, such as delivering `refund package`(caused by failures of most oracle relayers), special blockchain header synchronization(header contains BC validatorset update), BSC staking package. System reward contract will pay reward to relayers' accounts directly. If some Relayers have faster networks and better hardware, they can monopolize all the package relaying and leave no reward to others. Thus fewer participants will join for relaying, which encourages centralization and harms the efficiency and security of the network. Ideally, due to the decentralization and dynamic re-election of BSC validators, one Relayer can hardly be always the first to relay every message. But in order to avoid the monopolization further, the rewarding economy is also specially designed to minimize such chance: 1. The reward for Relayers will be only distributed in batches, and one batch will cover a number of successful relayed packages. 2. The reward a Relayer can get from a batch distribution is not linearly in proportion to their number of successful relayed packages. Instead, except the first a few relays, the more a Relayer relays during a batch period, the less reward it will collect. ## Oracle Relayers Relayers for BSC to BC communication are using the “Oracle” model, and so-called “**Oracle Relayers**”. Each of the validators must, and only the ones of the validator set, run Oracle Relayers. Each Oracle Relayer watches the blockchain state change. Once it catches Cross-Chain Communication Packages, it will submit to vote for the requests. After Oracle Relayers from ⅔ of the voting power of BC validators vote for the changes, the cross-chain actions will be performed. Oracle Replayers should wait for enough blocks to confirm the finality on BSC before submitting and voting for the cross-chain communication packages onto BC. The cross-chain fees will be distributed to BC validators together with the normal BC blocking rewards. Such oracle type relaying depends on all the validators to support. As all the votes for the cross-chain communication packages are recorded on the blockchain, it is not hard to have a metric system to assess the performance of the Oracle Relayers. The poorest performer may have their rewards clawed back via another Slashing logic introduced in the future. # Outlook It is hard to conclude for Binance Chain, as it has never stopped evolving. The dual-chain strategy is to open the gate for users to take advantage of the fast transferring and trading on one side, and flexible and extendable programming on the other side, but it will be one stop along the development of Binance Chain. Here below are the topics to look into so as to facilitate the community better for more usability and extensibility: 1. Add different digital asset model for different business use cases 2. Enable more data feed, especially DEX market data, to be communicated from Binance DEX to BSC 3. Provide interface and compatibility to integrate with Ethereum, including its further upgrade, and other blockchain 4. Improve client side experience to manage wallets and use blockchain more conveniently ------ [1]: BNB business practitioners may provide other benefits for BNB delegators, as they do now for long term BNB holders.

This paper proposes a deep learning-based method to detect multiple people from a single overhead depth image with high precision. Our neural network, called DPDnet, is composed by two fully-convolutional encoder-decoder blocks built with residual layers. The main block takes a depth image as input and generates a pixel-wise confidence map, where each detected person in the image is represented by a Gaussian-like distribution, The refinement block combines the depth image and the output from the main block, to refine the confidence map. Both blocks are simultaneously trained end-to-end using depth images and ground truth head position labels. The paper provides a rigorous experimental comparison with some of the best methods of the state-of-the-art, being exhaustively evaluated in different publicly available datasets. DPDnet proves to outperform all the evaluated methods with statistically significant differences, and with accuracies that exceed 99%. The system was trained on one of the datasets (generated by the authors and available to the scientific community) and evaluated in the others without retraining, proving also to achieve high accuracy with varying datasets and experimental conditions. Additionally, we made a comparison of our proposal with other CNN-based alternatives that have been very recently proposed in the literature, obtaining again very high performance. Finally, the computational complexity of our proposal is shown to be independent of the number of users in the scene and runs in real time using conventional GPUs.

Multiple-precision GPU accelerated linear algebra routines (dense and sparse) based on residue number system

Analytic Number Theory high-precision GnuMP routines

a babel-macro for number-precision to simplify code

Using a more modern and concise, object-oriented approach, it is more convenient and intuitive to solve the super large number and floating point precision problems in PHP

A blazing-fast, accurate Fibonacci number calculator using the GMP (GNU Multiple Precision Arithmetic) library for arbitrary-precision integer math with optional OpenMP parallelization*.

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.

Bioinformatics analysis pipeline for the functional annotation and translation of somatic SNVs/InDels and copy number abberations for precision cancer medicine using Personal Cancer Genome Reporter (PCGR). The pipeline offers germline SNVs/INDELS intepretation and annotation using Cancer Predisposition Sequencing Reporter (CPSR).

Pascal library for cross-compiler consistent and exact conversion between double-precision floating point number values and strings

# Cervical_cancer_detection_using_ML # Introduction According to World Health Organisation (WHO), when detected at an early stage, cervical cancer is one of the most curable cancers. Hence, the main motive behind this project is to detect the cancer in its early stages so that it can be treated and managed in the patients effectively. # Flow of project is as explained below: This project is divided into 5 parts: 1. Data Cleaning 2. Exploratory Data Analysis 3. Baseline model: Logistic Regression 4. Ensemble Models: Bagging with Decision Trees, Random forest and Boosting 5. Model Comparison and results # Refer below for References: Link to basic information regarding cervical cancer : https://www.cdc.gov/cancer/cervical/basic_info/index.htm The dataset for tackling the problem is supplied by the UCI repository for Machine Learning. Link to Dataset : https://archive.ics.uci.edu/ml/datasets/Cervical+cancer+%28Risk+Factors%29 The dataset contains a list of risk factors that lead up to the Biopsy examination. The generation of the predictor variable is taken care of in part 2 (Exploratory data analysis) of this report. We will try to predict the 'biopsy' variable from the dataset using Logistic Regression, Random Forest, Bagging with Decision Trees and Boosting with XGBoost Classifier. # Results: Based on our Base model and The Ensemble Models we used, we observed - 1. After the entire process of training, hyperparameter tuning and tackling class imbalance was complete , we obtained the results as depicted through the graphics. 2. We observe that Bagging and Random Forest gives the highest accuracy and precision of 97.09 and 80% resp. 3. Plotting the Confusion matrix showed us that Random Forest using upsampling and class weights gives us 2 false positives and 3 false negatives with auc of 0.87 # Why random forest is the best model?? 1. So as we see, while comparing all of our models,RF has maximum f1_score and accuracy along with Bagging i.e. 76.2 n 97.09% resp. 2. And it also produces the same amount of false negatives with a recall of 72.73% just like all the other models. 3. But we still consider RF better coz of its added advantage that, the decision trees are decorrelated as compared to bagging leading to lesser variance and greater ability to generalize. # Conclusion: On observing the feature importance of the best model i.e random forest, we can see that the most important features are Schiller, Hinselmann, HPV, Citology, etc. This also makes sense because Schiller and Hinselmann are actually the tests used to detect cervical cancer. # Problems Faced: A major problem encountered while training the model was that it had too little data to train. On collaborating with all the hospitals in India, we can have enough data points to train a model with a higher recall, thus making the model better. # Scope of Improvement As next steps I would want to do exactly that, to deploy the model and refine it. We may also modify the number of the predictor variables, as it may well turn out that there are other predictors which may not be present in our current dataset. This can only be found by practical implementation of our predictions.

A library for Erlang/OTP (and Elixir language) of numerical routines on single and double-precision real and complex number vectors and matrices

Program that calculates the number Pi with big precision

high-precision Rust-based synthesizer that decomposes single-qubit Z-axis rotations into Clifford+T gate sequences, using number-theoretic algorithms and geometry-of-numbers methods.