Introduction to Neural Networks and PyTorch - IBM
Introduction to Neural Networks and PyTorch
Data Science > Machine Learning >
- There are 6 modules in this course:
- PyTorch is one of the top 10 highest paid skills in tech (Indeed). As the use of PyTorch for neural networks rockets, professionals with PyTorch skills are in high demand. This course is ideal for AI engineers looking to gain job-ready skills in PyTorch that will catch the eye of an employer.
- AI developers use PyTorch to design, train, and optimize neural networks to enable computers to perform tasks such as image recognition, natural language processing, and predictive analytics. During this course, you’ll learn about 2-D Tensors and derivatives in PyTorch. You’ll look at linear regression prediction and training and calculate loss using PyTorch. You’ll explore batch processing techniques for efficient model training, model parameters, calculating cost, and performing gradient descent in PyTorch. Plus, you’ll look at linear classifiers and logistic regression.
- Throughout, you’ll apply your new skills in hands-on labs, and at the end, you’ll complete a project you can talk about in interviews. If you’re an aspiring AI engineer with basic knowledge of Python and mathematical concepts, who wants to get hands-on with PyTorch, enroll today and get set to power your AI career forward!
- Module 1: Artificial Neural Networks
- Module 1: Artificial Neural Networks - Forward Propagation - Lab
- Labs.cogntiveclass.ai - Hyperlink
- Quiz 01:
- Module Quiz:
- Module 2: Basics of Deep Learning - Gradient Descent
- Module 2 - b- Backpropagation Algorithm
- Module 2 - c - Lab Backpropagation - Click (right) Github Hyperlinks
- Module 2 - d- Vanishing Gradient
- Module 2 - e:- Activation Functions
- Module 2 - f - Lab - Vanishing Gradient & Activation Functions - Click (right) Github Hyperlinks
- Module 2 - Practice Quiz
Your grade: 100%
Your latest: 100%•Your highest: 100%•To pass you need at least 60%. We keep your highest score.1.
_____________ is an iterative optimization algorithm for finding the minimum of a function.
Nice workGradient descent is an iterative optimization algorithm for finding the minimum of a function. To find the minimum of a function using gradient descent, we take steps proportional to the negative of the function's gradient at the current point.
1 / 1 point2.
How does the backpropagation training process start?
Nice workBackpropagation begins after forward propagation and computing a differentiable loss function (e.g., cross-entropy or mean squared error).
1 / 1 point3.
Which of the following types of activation functions can cause the vanishing gradient problem? Select three answers.
Nice workThe binary step activation function can cause the vanishing gradient problem.
Nice workThe sigmoid activation function can cause the vanishing gradient problem.
Nice workThe hyperbolic tangent activation function can cause the vanishing gradient problem.
1 / 1 point4.
Which of the following activation functions does not activate all neurons simultaneously?
Nice workIn addition to being nonlinear, the main advantage of using the ReLU function over the other activation functions is that it does not activate all the neurons simultaneously.
1 / 1 point5.
Which of the following activation functions is ideally used in the classifier’s output layer?
Nice workThe softmax function is ideally used in the classifier's output layer, where we try to get the probabilities to define each input's class.
- Module 2 - Graded Quiz
Your grade: 100%
Your latest: 100%•Your highest: 100%•To pass you need at least 70%. We keep your highest score.1.
Which of the following algorithms is used to optimize weights and biases in a neural network?
Nice workGradient descent is an iterative optimization algorithm that finds the minimum of a function by calculating gradients and updating parameters in the direction of steepest descent.
1 / 1 point2.
For a cost function, J = Σ(zi - wxi - b)², that we would like to minimize, which of the following expressions represent updating the parameter, w, using gradient descent?
Nice workThis expression correctly shows the gradient descent update rule where the parameter moves in the opposite direction of the gradient to minimize the cost function.
1 / 1 point3.
While reviewing a chart of an activation function, you see a function that outputs zero for all negative inputs and returns the input itself for positive values. What type of activation function is this?
Nice workReLU (Rectified Linear Unit) is defined as f(x) = max(0, x), creating a piecewise linear function that eliminates negative values while preserving positive ones.
1 / 1 point4.
You're analyzing an activation function that outputs values between -1 and 1 and has an S-shaped curve centered at the origin. Which activation function does this describe?
Nice workThe tanh function is symmetric around the origin and maps any real number to the range [-1, 1], making it zero-centered, unlike the sigmoid function.
1 / 1 point5.
For multi-class classification problems, which neural network component typically contains the softmax activation function?
Nice workSoftmax transforms the final layer's raw logits into a probability distribution, enabling the network to output confidence scores for each possible class.
1 / 1 point6.
What is the correct sequence of steps in the backpropagation training algorithm?
Step a: Calculate the error between the ground truth and the estimated or predicted output of the network. Step b: Update the weights and the biases through backpropagation. Step c: Calculate the network output using forward propagation. Step d: Repeat the previous steps until the error between the ground truth and the predicted output is below a predefined threshold.
Nice workThis sequence follows the logical flow of neural network training: forward pass to generate predictions, error calculation, backward pass to update parameters, and iteration until convergence.
1 / 1 point7.
You're working on a 15-layer neural network for image recognition, but the early layers seem to learn very slowly compared to the later layers. What is the most likely cause of this training issue?
Nice workIn deep networks, gradients can become exponentially smaller as they pass through many layers, especially when using activation functions whose derivatives approach zero. This causes earlier layers to learn extremely slowly.
1 / 1 point8.
In a simple neural network implementation, what is the correct mathematical relationship for computing the gradient of the loss function with respect to a weight connecting neuron i to neuron j?
Nice workThis correctly applies the chain rule where the gradient flows from the loss to the activation, then to the weighted sum, and finally to the weight parameter.
1 / 1 point9.
Which activation function is most effective at preventing the vanishing gradient problem in deep neural networks?
Nice workReLU has a constant derivative of 1 for positive inputs, which prevents the exponential decay of gradients that occurs with saturating activation functions like sigmoid and tanh.
1 / 1 point10.
You're coding a neural network from scratch and need to implement the forward propagation for a hidden layer neuron. What sequence of operations should your code perform?
Nice workThis follows the standard neuron computation: z = Σ(wi × xi) + b, then a = f(z), where f is the activation function.
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- Module 3: Deep Learning Libraries
- Module 3b - Regression Models with Keras
- Quiz 1
Your grade: 100%
Your latest: 100%•Your highest: 100%•To pass you need at least 60%. We keep your highest score.1.
Which of the following libraries is particularly suited for running machine learning algorithms on GPUs?
Nice workPyTorch is the cousin of the Torch framework, which is in Lua, and supports machine learning algorithms running on GPUs in particular.
1 / 1 point2.
Which of the following statements is correct?
Nice workKeras is a high-level API for building deep learning models, and it has gained favor for its ease of use and syntactic simplicity, facilitating fast development.
1 / 1 point3.
What are the two model classes in the Keras library? Select all that apply.
Nice workThere are two models in the Keras library. One is the Sequential model, and the other is the model class used with the functional API.
Nice workThere are two models in the Keras library. One is the Sequential model, and the other is the model class used with the functional API.
1 / 1 point4.
When adding a ‘dense’ layer to the first hidden layer in a regression model, what three things do we specify?
Nice workWhen adding a ‘dense’ layer to the first hidden layer in a regression model, we need to specify the number of neurons in each layer, the activation function, and the number of columns in our data set.
1 / 1 point5.
When building classification models with Keras, what do we need to do to the target column before we can use it?
Nice workWhen building classification models with Keras, we need to transform the target column into an array with binary values before we can use it.
1. Which of the following libraries is particularly suited for running machine learning algorithms on GPUs?
Correct Answer: PyTorch
Note: PyTorch and TensorFlow are designed with native support for CUDA to accelerate tensor computations on GPUs, whereas standard libraries like Pandas, NumPy, and Scikit-learn run primarily on the CPU.
2. Which of the following statements is correct?
Correct Answer: Keras is a high-level API that facilitates fast development and quick prototyping of deep learning models.
Note: Keras acts as a user-friendly interface built on top of lower-level frameworks (primarily TensorFlow) to streamline model building.
3. What are the two model classes in the Keras library? Select all that apply.
[X] The model class used with the functional API
[X] Sequential model
Note: Keras provides two primary ways to construct models: the straightforward
Sequentialclass for linear stacks of layers, and the flexibleModelclass used with the Functional API for complex architectures.4. When adding a ‘dense’ layer to the first hidden layer in a regression model, what three things do we specify?
Correct Answer: The number of neurons in each layer, the activation function, and the number of columns in our data set.
Note: For the very first hidden layer, Keras requires the input shape (
input_dimorinput_shape), which corresponds to the number of feature columns in your dataset. The number of targets is only defined later in the final output layer.5. When building classification models with Keras, what do we need to do to the target column before we can use it?
Correct Answer: Transform it into an array of binary values
Note: Categorical labels must typically be converted into a one-hot encoded format (an array of binary vectors via tools like
to_categorical) so the neural network can compute categorical cross-entropy loss effectively.- Module 3 d - Classification Models with Keras
- Module 3 - Lab - Classification with Keras
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- Quiz 2 - Graded Quiz - Classification with Keras
Your grade: 100%
Your latest: 100%•Your highest: 100%•To pass you need at least 70%. We keep your highest score.1.
Imagine you’re rapidly building a deep learning model and want to minimize boilerplate code. Which of the following would best support fast development and easy prototyping?
Nice workKeras is a high-level API for building deep learning models. It has gained favor for its ease of use and syntactic simplicity, facilitating fast development.
1 / 1 point2.
Which library is known for its high-level, user-friendly API that makes deep learning more accessible to beginners?
Nice workKeras is designed with a focus on user experience, providing a simple and intuitive API that makes deep learning accessible to beginners and enables rapid prototyping.
1 / 1 point3.
You’re using Keras to build deep-learning models. Which model types does the Keras library support?
Nice workThe Sequential model simplifies neural network construction by allowing layers to be added individually, perfect for beginners learning model design.
1 / 1 point4.
For classification models in Keras, which activation function is most appropriate for the output layer when dealing with multi-class problems?
Nice workThe softmax activation function is ideal for multi-class classification as it converts raw output scores into probability distributions that sum to 1, making it suitable for categorical outputs.
1 / 1 point5.
For regression problems involving continuous target variables, which output layer activation function provides the most flexibility?
Nice workLinear activation enables the network to output any real number without bounds, providing the flexibility to predict continuous values across the entire range of possible outcomes.
1 / 1 point6.
For a neural network to predict categorical outputs, which loss function optimally measures prediction accuracy?
Nice workCategorical cross-entropy calculates the logarithmic difference between predicted probabilities and actual class labels, providing efficient gradient information for optimizing classification models.
1 / 1 point7.
For continuous outcome prediction models, which evaluation metric best captures prediction accuracy?
Nice workMSE penalizes larger prediction errors more heavily than smaller ones, providing a comprehensive measure of how well the regression model predicts continuous values.
1 / 1 point8.
Which Keras compilation parameters should be used for optimal performance in regression model implementation?
Nice workThis combination provides an appropriate loss function for continuous outcomes, efficient optimization, and meaningful performance metrics for regression assessment.
1 / 1 point9.
You’ve developed a Keras model to classify customer reviews as positive, negative, or neutral. Which metric best indicates your model’s effectiveness?
Nice workAccuracy provides a clear percentage of correctly classified reviews, making it easy to understand how well the model performs across all three sentiment categories.
1 / 1 point10.
Why is Keras particularly suitable for deep learning beginners compared to lower-level frameworks?
Nice workKeras simplifies neural network development by providing high-level APIs that hide low-level complexities while allowing beginners to experiment with different architectures and parameters.
- Module 4 - Deep Learning Models
- Module 4 - Shallow Versus Deep Neural Networks
- Module 4 - Deep Learn Models - 4b - Convolutional Neural Networks
- Module - 4d - Recurrent Neural Networks (RNNs)
- Module 4 -e - autoencoders Unsupervised Deep Learning Networks
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- Module 4 - h - Using Pre-trained Models
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- Practice Quiz - Module 4
- Graded Quiz - Module 4
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- Introduction to Neural Networks & PyTorch - IBM
- Intro - Module #1 - Exploring Tensors
Course Overview
2:54/5:53Hello, and welcome to this course!
In Introduction to Neural Networks with PyTorch, you’ll build a strong foundation in machine learning by working with tensors, data, and predictive models. You’ll learn how data is represented and manipulated using one- and two-dimensional tensors, along with essential mathematical concepts like vectors, matrices, and derivatives that power model training.
As you progress, you’ll explore how to prepare and structure datasets, including preprocessing techniques and handling different data types such as images. You’ll then dive into core modeling concepts like linear regression and gradient descent, understanding how models make predictions, measure error using loss functions, and improve through parameter updates. You’ll also learn how to evaluate models effectively using training, validation, and test data.
Finally, you’ll extend your knowledge to more complex scenarios, including multiple inputs and outputs, as well as classification tasks. You’ll apply everything in a real-world project by building a model to predict League of Legends match outcomes, combining data preparation, training, and evaluation into a complete machine learning workflow.
This course is part of the IBM Deep Learning with PyTorch, Keras, and Tensorflow Professional Certificate. Consider enrolling in this professional certificate to continue building your skills and deepen your understanding of machine learning, neural networks, and deep learning with PyTorch.
Prerequisites
To get the most out of this course, prior experience with Python programming is required. A solid understanding of basic mathematical concepts, particularly matrices and gradients, is also strongly recommended.
Course objectives:
After completing this course, you will be able to:
Perform tensor operations in PyTorch
Implement and train linear regression models from scratch using PyTorch functions
Explain concepts of logistic regression and apply them to classification problems
Implement regression and classification models in PyTorch using gradient-based optimization
Course outline:
Module 1: Tensors In this module, you'll build your foundation in PyTorch by working directly with tensors. You'll explore one- and two-dimensional tensors, common operations, and key attributes, including shape, data type, and the total number of elements. You'll also examine basic differentiation concepts and see how PyTorch tracks and computes gradients automatically. Through guided practice, you’ll connect linear algebra concepts to practical implementation.
Module 2: Datasets In this module, you'll learn how to structure and prepare data for training in PyTorch. You'll create custom dataset structures, define how data is accessed, and apply preprocessing steps using transformations. You'll also work with image datasets and standard data-handling patterns. By the end, you'll understand how data flows into a model during training.
Module 3: Linear Regression and Gradient Descent In this module, you'll learn how to build and train linear regression models in PyTorch. You'll explore how models are defined, how parameters are stored, and how loss functions measure prediction error. You'll examine cost surfaces, gradient descent, learning rates, and stopping criteria. Through hands-on training, you'll observe how model parameters update over time as the model minimizes loss.
Module 4: Linear Regression the PyTorch Way In this module, you'll discover how to implement efficient training workflows using PyTorch tools for data handling and optimization. You'll compare batch, stochastic, and mini-batch gradient descent, and examine how batch size, number of training cycles, and learning rate affect convergence. You'll also learn how to structure complete training processes with forward passes, error calculation, backward updates, and parameter adjustments. Finally, you'll explore training, validation, and test splits to evaluate performance and detect overfitting.
Module 5: Multiple Input-Output Linear Regression In this module, you'll explore how to extend linear regression to handle multiple input features and multiple outputs. You'll learn how to build higher-dimensional models and understand how parameters expand from single values to vectors and matrices. You'll work with vectorized loss calculations, gradient descent, and structured training workflows. Through hands-on labs, you'll build, train, and evaluate multi-dimensional regression models step by step.
Module 6: Logistic Regression for Classification In this module, you'll explore how to move from regression to classification. You'll learn how logistic regression models generate probabilities and how those probabilities are used to make class predictions. You'll examine the Bernoulli distribution and maximum likelihood estimation and understand why cross-entropy loss is preferred over Mean Squared Error for classification tasks. You'll also explore optimization and regularization techniques that improve classification performance.
Module 7: Final Project In this module, you'll apply what you've learned throughout the course in a hands-on classification project. You'll build a model to predict the outcomes of League of Legends matches. Using various in-game statistics, you'll combine your knowledge of PyTorch, logistic regression, and data handling to create a robust predictive system.
Tools/software used
You will need a computer with internet access and a modern web browser. All labs are conducted online, so no local software installation is required.
Congratulations on taking this step toward building your career in neural networks and deep learning.
Helpful Tips for Course Completion
1. Familiarize yourself with the course content
- Browse module overviews and objectives
- Understand topics and associated assets
- Familiarize yourself with the content order
- Identify upcoming topics
- Connect ideas to create a completion plan for the course
2. Form your plan and make a rough timeline for course completion
- Review completion time estimates for module assets
- Set a reasonable time goal for each module
- Determine the course completion deadline
- Schedule daily study time
3. Actively manage your learning
Complete your independent tasks:
- Take notes during the course
- Download transcripts for reference
- Highlight important parts in transcripts
- Complete all labs
- Review terms using glossaries
Get support:
- Actively participate in discussion forums
Pass your quizzes:
- Review study notes
- Complete practice quizzes and review feedback
- Complete graded quizzes
- Review related videos or readings for incorrect answers
- Review video/transcript/reading for correct answers
- Retake quizzes until you pass them
4. Talk with your friends and family about the course
- Stay accountable, commit to the course
- Talk to friends and family about it
- Engage in conversations about interesting topics
- Seek beneficial perspectives from others
5. Follow your plan
- Stay motivated with your plan
- Set achievable goals in step 2
- Reward yourself upon achieving goals
- C2- Module 1 - Lesson 1 - Working 1D Tensors - Intro Modern Neural Network
- Quiz:
- Module 1 -Lesson 2: Intro to 2D Tensors in PyTorch
- Module 1 - Lesson 2 - 2D Tensor Operations in PyTorch
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- Module 1 - Lesson 2 - Understanding Differentiation in PyTorch
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Imagine training a neural network in PyTorch to recognize handwritten digits. After the model makes a prediction, it calculates how wrong it was using a loss function. But how does the model know which weights to adjust to improve its predictions? Automatic differentiation computes gradients that show exactly how each parameter affects the error, allowing the model to update itself and gradually learn from its mistakes. - (00:50) The autograd system allows PyTorch to compute derivatives automatically. When tensors are created with gradient tracking enabled, PyTorch records every operation applied to them. For example, a tensor x can be created with gradient tracking enabled, and a new value y can be computed using operations such as squaring x and adding 3x. PyTorch records these operations in a computational graph, which represents how the output is derived from earlier calculations. This graph later allows PyTorch to compute derivatives automatically during the backward pass.
- (01:30) A computational graph represents how values depend on earlier computations. In the previous example, the tensor x is used to compute two operations, x² and 3x, and the results are combined to produce y. This relationship can be visualized as a graph, where x is the starting node, the operations x² and 3x form intermediate nodes, and y is the final output. Each node represents an operation, and the connections show how data flows between them. During training, PyTorch traverses this graph in reverse order to compute gradients.
- (02:15) Gradient tracking allows PyTorch to monitor how tensors participate in differentiable computations. When a tensor is created with requires_grad=True, all operations involving that tensor are recorded. For example, if a tensor x is used to compute y = 2x + 1, PyTorch tracks how y depends on x. This tracking allows gradients to be computed later during model training. Gradient tracking is typically enabled for model parameters, such as weights and biases, because these values must be updated during training.
- (02:45) After computing a value such as a loss, gradients are obtained by calling the backward method. For example, if a tensor x is used to compute an output, such as y = x² + 3x, the backward pass is performed through the computational graph. PyTorch then computes the derivative of y with respect to x, storing the result as the gradient of x. (03:09) This gradient represents how much the output changes with respect to the input. In neural networks, these gradients are used by optimization algorithms to update model parameters.
- (03:20) Neural networks usually involve many parameters, meaning the loss function depends on multiple variables. In these cases, gradients are computed using partial derivatives. A partial derivative measures how a function changes with respect to one variable while holding other variables constant. For example, if an output y is computed using two variables, such as x and w, PyTorch can compute the derivative of y with respect to x and the derivative of y with respect to w. These partial derivatives show how each parameter contributes to the output.
- (04:03) Neural networks consist of multiple layers and operations, so computing gradients through such structures require the chain rule. The chain rule states that the derivative of a composed function is obtained by multiplying the derivatives of intermediate functions. For example, an input x may pass through functions such as f(x) and then g(f(x)) to produce an output. During the backward pass, gradients propagate through each operation in reverse order from the output through g′(f(x)) and then f′(x) until reaching x. Autograd automatically applies the chain rule as it traverses the computational graph, allowing gradients to be computed efficiently even for deep neural networks with many layers.
- (04:45) In this video, you learned that differentiation helps train machine learning and deep learning models by guiding parameter updates. Models adjust their parameters to minimize a loss function using gradients. PyTorch uses the autograd system to record tensor operations and build a computational graph. The computational graph shows how outputs depend on earlier computations. The backward pass computes and stores gradients for tensors that track gradients. Partial derivatives and the chain rule allow gradients to propagate through multiple layers of a neural network.
- Module 1 - Lesson 2 - Understanding Differentiation in PyTorch
- Module 1 - Lesson 2 - Practical Quiz
1. Which of the following commands would return the total number of elements contained in a tensor?
Answer:
tensor_2d.numel()Note: The
.numel()method explicitly returns the total element count..shapeand.size()return the dimensions (e.g., 2x3), and.ndimreturns the number of axes.
2. When performing indexing and slicing on a 2D tensor, which of the following would you use to extract only the second row of the 2D tensor?
Answer:
A[1,:]Note: PyTorch uses zero-based indexing. The
1targets the second row, and the:indicates that you want to include all columns in that row.
3. Which of the following 2D tensor operations uses the dot product calculations of rows and columns rather than using element-wise multiplication?
Answer: Matrix multiplication
Note: Standard matrix multiplication relies on the dot product of rows and columns. Conversely, the Hadamard product specifically refers to element-wise multiplication.
4. Which differentiation feature allows PyTorch to monitor how tensors participate in differentiable computations?
Answer: Gradient tracking
Note: This is the feature (enabled by setting
requires_grad=True) that tells PyTorch to actively monitor and record every operation a specific tensor participates in.
5. Which differentiation feature in PyTorch measures how a function changes with respect to one variable while holding other variables constant?
Answer: Partial derivatives
Note: This is the mathematical definition of a partial derivative. When you call
.backward()on a function with multiple independent variables, PyTorch computes the partial derivative for each one.
- Module 1 - Lesson 3 - Graded Quiz
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Question 1: What is the primary role of matrices in the context of linear transformations as described in the course?
Answer: Matrices act as operators that transform vectors by multiplying them to produce new vectors.
Question 2: How does the Hadamard product differ from the dot product when applied to 1D tensors in PyTorch?
Answer: The Hadamard product performs element-wise multiplication, while the dot product computes the sum of the products of corresponding elements.
Question 3: Which tensor operation is crucial for combining input tensors with weight tensors in fully connected layers of a neural network in PyTorch?
Answer: Matrix multiplication
Question 4: In the context of PyTorch, what is the primary function of the numel() method when applied to tensors?
Answer: The numel() method returns the number of elements in a tensor.
Question 5: In PyTorch, how can you determine the number of dimensions in a 2D tensor?
Answer: By using the .ndim method
Question 6: In PyTorch, what is the primary difference between element-wise multiplication and matrix multiplication in 2D tensors?
Answer: Element-wise multiplication is performed between tensors of the same size, while matrix multiplication involves dot products between rows and columns.
Question 7: In PyTorch, what is the role of the autograd system in training neural networks?
Answer: The autograd system in PyTorch records all operations on tensors with gradient tracking enabled, allowing for automatic computation of gradients during the backward pass.
Correct answers:
- Matrices act as operators that transform vectors by multiplying them to produce new vectors.
- The Hadamard product performs element-wise multiplication, while the dot product computes the sum of the products of corresponding elements.
- Matrix multiplication
- The numel() method returns the number of elements in a tensor.
- By using the .ndim method
- Element-wise multiplication is performed between tensors of the same size, while matrix multiplication involves dot products between rows and columns.
- The autograd system in PyTorch records all operations on tensors with gradient tracking enabled, allowing for automatic computation of gradients during the backward pass.
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- Module 2 - Lesson#1 - Practical Quiz
1.
What is the purpose of implementing the __getitem__ method in a custom dataset?
Try againThis method controls how data is retrieved when indexing is used.
0 / 1 point2.
Why are transform classes used in dataset workflows?
Nice workThese are commonly used to adjust or prepare data during access.
3.
What is the role of using transforms.Compose in a dataset pipeline?
Nice workThis helps organize multiple preprocessing steps into a single pipeline.
1 / 1 point4.
What is the main advantage of using built-in datasets from Torchvision, like Fashion-MNIST?
Nice workThese datasets simplify setup, allowing developers to focus on modeling.
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5.
What does each sample returned from a dataset like Fashion-MNIST typically contain?
Nice workThis structure supports training by pairing inputs with targets.
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1.
What is the purpose of implementing the __getitem__ method in a custom dataset?
This does not relate to the method described in the question.Nice work
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