NLP – Word Embeddings – GloVe

What are word embeddings?

Word embeddings are a type of representation for text data, which allows words with similar meaning to have a similar representation in a neural network model. Word embeddings are trained such that words that are used in similar contexts will have similar vectors in the embedding space. This is useful because it allows the model to generalize better and makes it easier to learn from smaller amounts of data. Word embeddings can be trained using a variety of techniques, such as word2vec and GloVe, and are commonly used as input to deep learning models for natural language processing tasks.

So are they represented as arrays of numbers?

Yes, word embeddings are typically represented as arrays of numbers. The length of the array will depend on the size of the embedding space, which is a parameter that is chosen when the word embeddings are created. For example, if the size of the embedding space is 50, each word will be represented as a vector of length 50, with each element of the vector representing a dimension in the embedding space.

In a neural network model, these word embedding vectors are typically fed into the input layer of the model, and the rest of the layers in the model are then trained to perform some task, such as language translation or sentiment analysis. The model learns to combine the various dimensions of the word embedding vectors in order to make predictions or decisions based on the input data.

How are word embeddings determined?

There are a few different techniques for determining word embeddings, but the most common method is to use a neural network to learn the embeddings from a large dataset of text. The basic idea is to train a neural network to predict a word given the words that come before and after it in a sentence, using the output of the network as the embedding for the input word. The network is trained on a large dataset of text, and the weights of the network are used to determine the embeddings for each word.

There are a few different variations on this basic approach, such as using a different objective function or incorporating additional information into the input to the network. The specific details of how word embeddings are determined will depend on the specific method being used.

What are the specific methods for generating word embeddings?

Word embeddings are a type of representation for natural language processing tasks in which words are represented as numerical vectors in a high-dimensional space. There are several algorithms for generating word embeddings, including:

  1. Word2Vec: This algorithm uses a neural network to learn the vector representations of words. It can be trained using two different techniques: continuous bag-of-words (CBOW) and skip-gram.
  2. GloVe (Global Vectors): This algorithm learns word embeddings by factorizing a matrix of word co-occurrence statistics.
  3. FastText: This is an extension of Word2Vec that learns word embeddings for subwords (character n-grams) in addition to full words. This allows the model to better handle rare and out-of-vocabulary words.
  4. ELMo (Embeddings from Language Models): This algorithm generates word embeddings by training a deep bi-directional language model on a large dataset. The word embeddings are then derived from the hidden state of the language model.
  5. BERT (Bidirectional Encoder Representations from Transformers): This algorithm is a transformer-based language model that generates contextual word embeddings. It has achieved state-of-the-art results on a wide range of natural language processing tasks.

What is the word2vec CBOW model?

The continuous bag-of-words (CBOW) model is one of the two main techniques used to train the Word2Vec algorithm. It predicts a target word based on the context words, which are the words surrounding the target word in a text.

The CBOW model takes a window of context words as input and predicts the target word in the center of the window. The input to the model is a one-hot vector representation of the context words, and the output is a probability distribution over the words in the vocabulary. The model is trained to maximize the probability of predicting the correct target word given the context words.

During training, the model adjusts the weights of the input-to-output connections in order to minimize the prediction error. Once training is complete, the model can be used to generate word embeddings for the words in the vocabulary. These word embeddings capture the semantic relationships between words and can be used for various natural language processing tasks.

What is the word2vec skip-gram model?

The skip-gram model is the other main technique used to train the Word2Vec algorithm. It is the inverse of the continuous bag-of-words (CBOW) model, which predicts a target word based on the context words. In the skip-gram model, the target word is used to predict the context words.

Like the CBOW model, the skip-gram model takes a window of context words as input and predicts the target word in the center of the window. The input to the model is a one-hot vector representation of the target word, and the output is a probability distribution over the words in the vocabulary. The model is trained to maximize the probability of predicting the correct context words given the target word.

During training, the model adjusts the weights of the input-to-output connections in order to minimize the prediction error. Once training is complete, the model can be used to generate word embeddings for the words in the vocabulary. These word embeddings capture the semantic relationships between words and can be used for various natural language processing tasks.

What are the steps for the GloVe algorithm?

GloVe learns word embeddings by factorizing a matrix of word co-occurrence statistics, which can be calculated from a large corpus of text.

The main steps of the GloVe algorithm are as follows:

  1. Calculate the word co-occurrence matrix: Given a large corpus of text, the first step is to calculate the co-occurrence matrix, which is a symmetric matrix X where each element X_ij represents the number of times word i appears in the context of word j. The context of a word can be defined as a window of words around the word, or it can be the entire document.
  2. Initialize the word vectors: The next step is to initialize the word vectors, which are the columns of the matrix W. The word vectors are initialized with random values.
  3. Calculate the pointwise mutual information (PMI) matrix: The PMI matrix is calculated as follows:

PMI_ij = log(X_ij / (X_i * X_j))

where X_i is the sum of all the elements in the ith row of the co-occurrence matrix, and X_j is the sum of all the elements in the jth column of the co-occurrence matrix. The PMI matrix is a measure of the association between words and reflects the strength of the relationship between them.

  1. Factorize the PMI matrix: The PMI matrix is then factorized using singular value decomposition (SVD) or another matrix factorization technique to obtain the word vectors. The word vectors are the columns of the matrix W.
  2. Normalize the word vectors: Finally, the word vectors are normalized to have unit length.

Once the GloVe algorithm has been trained, the word vectors can be used to represent words in a high-dimensional space. The word vectors capture the semantic relationships between words and can be used for various natural language processing tasks.

How is the matrix factorization performed in GloVe? What is the goal?

The goal of matrix factorization in GloVe is to find two matrices, called the word matrix and the context matrix, such that the dot product of these matrices approximates the co-occurrence matrix. The word matrix contains the word vectors for each word in the vocabulary, and the context matrix contains the context vectors for each word in the vocabulary.

To find these matrices, GloVe minimizes the difference between the dot product of the word and context matrices and the co-occurrence matrix using a least-squares optimization method. This results in word vectors that capture the relationships between words in the corpus.

In GloVe, the objective function that is minimized during matrix factorization is the least-squares error between the dot product of the word and context matrices and the co-occurrence matrix. More specifically, the objective function is given by:


How is the objective function minimized?

In each iteration of SGD, a mini-batch of co-occurrence pairs (i, j) is selected from the co-occurrence matrix, and the gradients of the objective function with respect to the parameters are computed for each pair. The parameters are then updated using these gradients and a learning rate, which determines the step size of the updates.

This process is repeated until the objective function has converged to a minimum or a preset number of iterations has been reached. The process of selecting mini-batches and updating the parameters is often referred to as an epoch. SGD is an efficient method for minimizing the objective function in GloVe because it does not require computing the Hessian matrix, which is the matrix of second-order partial derivatives of the objective function.

When should GloVe be used instead of Word2Vec?

GloVe (Global Vectors) and Word2Vec are two widely used methods for learning word vectors from a large corpus of text. Both methods learn vector representations of words that capture the semantics of the words and the relationships between them, and they can be used in various natural language processing tasks, such as language modeling, information retrieval, and machine translation.

GloVe and Word2Vec differ in the way they learn word vectors. GloVe learns word vectors by factorizing a co-occurrence matrix, which is a matrix that contains information about how often words co-occur in a given corpus. Word2Vec, on the other hand, learns word vectors using a shallow neural network with a single hidden layer.

One advantage of GloVe is that it is computationally efficient, as it does not require training a neural network. This makes it well suited for use with large corpora. However, Word2Vec has been shown to perform better on some tasks, such as syntactic analogies and named entity recognition.

How is the co-occurrence matrix reduced to lower dimensions in GloVe?

In GloVe (Global Vectors), the co-occurrence matrix is not directly reduced to lower dimensions. Instead, the co-occurrence matrix is used to learn word vectors, which are then reduced to lower dimensions using dimensionality reduction techniques, such as principal component analysis (PCA) or t-distributed stochastic neighbor embedding (t-SNE).

To learn word vectors from the co-occurrence matrix in GloVe, the matrix is factorized into two matrices, called the word matrix and the context matrix, using a least-squares optimization method. The word matrix contains the word vectors for each word in the vocabulary, and the context matrix contains the context vectors for each word in the vocabulary.

After the word vectors have been learned, they can be reduced to lower dimensions using dimensionality reduction techniques. For example, PCA can be used to project the word vectors onto a lower-dimensional space, while t-SNE can be used to embed the word vectors in a two-dimensional space for visualization.

It is worth noting that reducing the dimensionality of the word vectors may result in some loss of information, as some of the relationships between words may be lost in the lower-dimensional space. Therefore, it is important to consider the trade-off between the dimensionality of the word vectors and their representational power.

Interpreting GloVe from the Ratio of Co-occurrence Probabilities

GloVe uses the ratio of co-occurrence probabilities to learn the word vectors and context vectors. Specifically, it minimizes the difference between the dot product of the word and context vectors and the log of the ratio of co-occurrence probabilities. This allows GloVe to learn word vectors that capture the meanings and relationships between words in the language.

Chatting with GPT – Natural Language Processing

Let’s talk about natural language processing. How does input text data get transformed to numerical vectors?

In natural language processing (NLP), input text data is often transformed into numerical vectors in order to make it more suitable for machine learning algorithms to work with. There are several methods that can be used to transform text data into numerical vectors, including:

  1. One-hot encoding: This is a simple method that converts each word in the text into a unique integer index, and then represents each word as a binary vector, with a “1” in the position corresponding to the index of the word and “0”s in all other positions. One-hot encoding is a common method for converting text data into numerical vectors, but it can result in very large and sparse vectors, which may not be efficient to work with.
  2. Word embeddings: Word embeddings are numerical vectors that represent the meaning of a word in a continuous, low-dimensional space. Word embeddings can be trained on large datasets of text data using techniques such as word2vec or GloVe, and they can capture the contextual relationships between words and the similarity between words. Word embeddings are a popular method for representing text data in NLP, as they can capture the meaning of words in a compact and dense form.
  3. Bag of words: The bag of words model represents text data as a fixed-length vector, with each element in the vector representing the number of times a particular word appears in the text. The bag of words model is a simple and commonly used method for representing text data, but it does not capture the order or context of the words in the text.

There are many other methods for converting text data into numerical vectors, and the specific method that is used may depend on the specific NLP task and the goals of the model. It is important to choose an appropriate method for representing text data in order to effectively capture the meaning and context of the text.

What are word2vec word embeddings? How are they trained?

Word2vec is a method for training word embeddings, which are numerical vectors that represent the meaning of a word in a continuous, low-dimensional space. Word embeddings are a popular method for representing text data in natural language processing (NLP), as they can capture the meaning of words in a compact and dense form and can be used as input to machine learning algorithms.

Word2vec is a type of neural network model that is trained to predict the context of a word given a set of surrounding words, or to predict a target word given a set of context words. During training, the model is presented with a large dataset of text data and learns to predict the context of a word based on the words that appear around it. The learned model can then be used to generate word embeddings by taking the weights of the model’s hidden layer as the word vectors.

There are two main variants of the word2vec model: continuous bag of words (CBOW) and skip-gram. In the CBOW model, the goal is to predict the target word given a set of context words, while in the skip-gram model, the goal is to predict the context words given a target word.

Word2vec word embeddings can be trained on large datasets of text data using techniques such as stochastic gradient descent (SGD) or negative sampling.

Provide the exact steps for the Word2Vec algorithm, include equations.

Word2Vec is a technique for learning vector representations of words, also known as “word embeddings”, in a way that captures the context and meaning of the words. The basic idea behind Word2Vec is to use the context of words in a text corpus to predict a target word, based on a sliding window of surrounding words. The prediction task is accomplished using a shallow neural network with a single hidden layer.

Here are the steps for the Word2Vec algorithm:

  1. Preprocess the text data: This involves cleaning and formatting the text data, such as lowercasing, stemming, and removing stop words.
  2. Tokenize the text: This involves dividing the text into individual words or tokens.
  3. Build a vocabulary: This involves creating a list of all the unique words in the text data and assigning each word a unique index.
  4. Create word-context pairs: For each word in the text, create a word-context pair by considering a window of surrounding words. For example, if the window size is 2, the word-context pairs for the word “cat” in the sentence “The cat sat on the mat” would be (“cat”, “The”), (“cat”, “sat”), (“cat”, “on”), and (“cat”, “the”).
  5. Train the model: Use the word-context pairs to train a shallow neural network with a single hidden layer. The input layer consists of the context words, and the output layer consists of the target word. The hidden layer contains a set of “neurons” that represent the word embeddings.

The training process involves minimizing the error between the predicted output (the target word) and the true output (the context words) using stochastic gradient descent. The error is measured using a loss function, such as the negative log likelihood loss.

Here is the equation for the negative log likelihood loss:

Loss = -log P(w | c)

where w is the target word and c is the context.

  1. Extract the word embeddings: Once the model is trained, the word embeddings can be extracted from the hidden layer of the neural network. These word embeddings can then be used for various natural language processing tasks, such as text classification, machine translation, and information retrieval.

You mentioned that word2vec can be trained using negative sampling. How does that work?

Yes, that’s correct! In addition to the traditional Word2Vec algorithm that uses the negative log likelihood loss function, there is also a variant called “negative sampling” that can be used to train the model. Negative sampling is a simplified training method that can be used to learn word embeddings more efficiently, especially for large text corpora.

Here’s how negative sampling works:

  1. Preprocess and tokenize the text data as described in the previous steps.
  2. Build a vocabulary and create word-context pairs as described in the previous steps.
  3. Train the model: For each word-context pair, randomly sample a number of negative examples (i.e., words that are not the target word). The objective is to maximize the probability of the target word given the context words, while minimizing the probability of the negative examples given the context words.

The training process involves minimizing the following loss function:

Loss = -log(sigmoid(u_w * v_c)) – ∑log(sigmoid(-u_n * v_c))

where u_w and v_c are the embeddings for the target word and the context words, respectively, and u_n is the embedding for a negative example. The sigmoid function is used to squash the output to a value between 0 and 1.

  1. Extract the word embeddings: Once the model is trained, the word embeddings can be extracted from the hidden layer of the neural network as described in the previous steps.

So when word2vec is trained using negative sampling, that means that just the loss function changes, otherwise stochastic gradient descent is still used?

Yes, that’s correct. When training word2vec using negative sampling, the loss function changes, but the underlying training algorithm, stochastic gradient descent (SGD), remains the same.

In word2vec, the goal is to learn a set of word vectors that can predict the surrounding words in a text. Negative sampling is a variant of the word2vec model that uses a different loss function to achieve this goal. Specifically, it uses a simplified version of the traditional word2vec loss function that only considers a small, randomly-selected subset of the negative examples (words that are not the target word) at each training step, rather than considering all of the negative examples. This allows the model to be trained more efficiently, as it doesn’t need to compute the loss for all of the negative examples at each training step.

SGD is still used to optimize the model’s parameters (the word vectors) by updating them in the direction that reduces the loss. The key difference with negative sampling is that the loss function being optimized is different from the traditional word2vec loss function.

But basically word2vec is trained using a pretty simple neural network, with just one hidden layer

Yes, that’s correct. Word2vec is typically implemented as a shallow neural network with just one hidden layer. The input to the network is a one-hot encoded word, and the hidden layer is a dense (fully-connected) layer with a fixed number of units. The output of the hidden layer is passed through a softmax activation function to produce a probability distribution over the vocabulary, which represents the model’s prediction of the probability of each word in the vocabulary given the input word.

You mentioned that there are two variants of word2vec – CBOW and skip-gram, what is the difference between the two?

In the Continuous Bag-of-Words (CBOW) model, the goal is to predict the current word given the surrounding context. For example, given the context “the cat sat on,” the model would try to predict the word “mat.” CBOW uses a sliding window approach to build up the context from the surrounding words.

On the other hand, the Skip-Gram model does the opposite: it tries to predict the surrounding context given a current word. So in the same example, given the word “mat,” the model would try to predict the context “the cat sat on.” Skip-gram works well with small amounts of data and is faster to train than CBOW.

What is the algorithms for the CBOW model?

The CBOW model uses a neural network with a single hidden layer to predict the current word based on the surrounding context. The input to the model is a one-hot encoded vector representing the context words, and the output is a probability distribution over the vocabulary, with the goal of predicting the target word with the highest probability.

The model is trained using the negative sampling loss function, which tries to maximize the probability of the target word given the context and minimize the probability of randomly sampled negative examples (words that are not the target word).

Here’s a summary of the CBOW algorithm:

  1. Initialize the model weights and biases.
  2. Iterate through each training example, which consists of a target word and a context of surrounding words.
  3. Convert the context words and target word to one-hot encoded vectors.
  4. Calculate the hidden layer activations by taking the dot product of the input vector (one-hot encoded context words) and the input weight matrix, and adding the bias.
  5. Apply the activation function (e.g. sigmoid or ReLU) to the hidden layer activations to get the hidden layer output.
  6. Calculate the output layer activations by taking the dot product of the hidden layer output and the output weight matrix, and adding the bias.
  7. Apply the softmax function to the output layer activations to get the predicted probability distribution over the vocabulary.
  8. Calculate the loss using the negative sampling loss function, which compares the predicted probability of the target word to the actual label (1 for the target word, 0 for the negative samples).
  9. Backpropagate the error and update the model weights and biases using gradient descent.
  10. Repeat steps 2-9 until the model has converged (i.e. the loss is minimized).

The CBOW algorithm uses the negative sampling loss function to train the model. The negative sampling loss function is defined as:

Loss = -(log(sigmoid(u_o * v_c)) + ∑ log(sigmoid(-u_i * v_c)))
where:

  • u_o is the output weight vector for the target word.
  • v_c is the input weight vector for the context words.
  • u_i is the output weight vector for a negative sample (a word that is not the target word).

The first term in the loss function maximizes the probability of the target word given the context, while the second term minimizes the probability of the negative samples.

The sigmoid function is applied to the dot product of the weight vectors to squash the output to a value between 0 and 1, which can be interpreted as a probability. The log function is applied to the sigmoid output to make the loss function smoother and easier to optimize.

The negative sampling loss function is used because it is computationally more efficient than the full softmax loss function, which involves calculating the probabilities of all words in the vocabulary.

Why is the dot product used? What does it represent?

In the word2vec algorithm, the dot product of the input and output weight vectors is used to measure the similarity between the context words and the target word. The dot product is a measure of the angle between two vectors, and it is defined as the product of the magnitudes of the vectors and the cosine of the angle between them.

In the case of word2vec, the input and output weight vectors are learned embeddings for the context words and the target word, respectively. These embeddings are learned during training, and they represent the underlying relationships between the words in the vocabulary.

The dot product of the input and output weight vectors is then used to predict the probability of the target word given the context. If the dot product is high, it means that the angle between the vectors is small, indicating that the context and target words are similar. This results in a high probability for the target word. On the other hand, if the dot product is low, it means that the angle between the vectors is large, indicating that the context and target words are dissimilar. This results in a low probability for the target word.

Overall, the dot product is used in the word2vec algorithm as a measure of similarity between the context words and the target word, which is used to predict the probability of the target word given the context.

What is the algorithm for the skip-gram model?

The Skip-Gram model is a neural network that uses a single hidden layer to predict the surrounding context given a current word. The input to the model is a one-hot encoded vector representing the current word, and the output is a probability distribution over the vocabulary, with the goal of predicting the context words with the highest probability.

The Skip-Gram model is trained using the negative sampling loss function, which tries to maximize the probability of the context words given the current word and minimize the probability of randomly sampled negative examples (words that are not in the context).

Here’s a summary of the Skip-Gram algorithm:

  1. Initialize the model weights and biases.
  2. Iterate through each training example, which consists of a current word and a context of surrounding words.
  3. Convert the current word and context words to one-hot encoded vectors.
  4. Calculate the hidden layer activations by taking the dot product of the input vector (one-hot encoded current word) and the input weight matrix, and adding the bias.
  5. Apply the activation function (e.g. sigmoid or ReLU) to the hidden layer activations to get the hidden layer output.
  6. Calculate the output layer activations by taking the dot product of the hidden layer output and the output weight matrix, and adding the bias.
  7. Apply the softmax function to the output layer activations to get the predicted probability distribution over the vocabulary.
  8. Calculate the loss using the negative sampling loss function, which compares the predicted probabilities of the context words to the actual labels (1 for the context words, 0 for the negative samples).
  9. Backpropagate the error and update the model weights and biases using gradient descent.
  10. Repeat steps 2-9 until the model has converged (i.e. the loss is minimized).

The negative sampling loss function used in the Skip-Gram model is defined as:

Loss = -(∑ log(sigmoid(u_c * v_o)) + ∑ log(sigmoid(-u_i * v_o)))

where:

  • u_c is the output weight vector for a context word.
  • v_o is the input weight vector for the current word.
  • u_i is the output weight vector for a negative sample (a word that is not in the context).

The first term in the loss function maximizes the probability of the context words given the current word, while the second term minimizes the probability of the negative samples.

The sigmoid function is applied to the dot product of the weight vectors to squash the output to a value between 0 and 1, which can be interpreted as a probability. The log function is applied to the sigmoid output to make the loss function smoother and easier to optimize.

The negative sampling loss function is used because it is computationally more efficient than the full softmax loss function, which involves calculating the probabilities of all words in the vocabulary.

In which cases is it better to use the CBOW model vs. Skip-Gram for word2vec?

The Continuous Bag-of-Words (CBOW) model and the Skip-Gram model are two approaches for learning vector representations of words, also known as word embeddings, in natural language processing tasks. These models are trained to predict a target word given the context of surrounding words, using a large corpus of text as input.

The main difference between the CBOW and Skip-Gram models is the way in which they define the context of a word. The CBOW model predicts the target word based on the context provided by the surrounding words, while the Skip-Gram model predicts the surrounding context words based on the target word.

In general, the CBOW model is faster to train and performs better on smaller datasets, while the Skip-Gram model is slower to train but can provide better results on larger datasets. The CBOW model is also better at handling rare words, as it takes into account the context provided by multiple surrounding words, while the Skip-Gram model only considers the context provided by a single target word.

In practice, the choice between the CBOW and Skip-Gram models will depend on the specific application and the available resources. If you have a large dataset and the time and computational resources to train a model, the Skip-Gram model may provide better results. If you have a smaller dataset and need to train a model quickly, the CBOW model may be a better choice.

Chatting with GPT – Neural Networks Cont’d

The information provided about neural networks has been quite useful. I would like to learn more about loss functions. Please explain loss functions in the context of a neural networks – what are some examples of loss functions? How are they used?

Loss functions are a key component of neural networks, as they help to evaluate how well a model is performing. They do this by calculating the difference between the model’s predicted output and the true target output, and then generating a scalar value that represents the model’s overall error.

There are many different loss functions that can be used in neural networks, depending on the specific task at hand. Some common examples include:

  1. Mean Squared Error (MSE) loss: This is a common loss function for regression tasks, where the goal is to predict a continuous numerical value. MSE loss is calculated by taking the average of the squared differences between the predicted output and the true output.
    MSE Loss = (1/n) * ∑(ŷ – y)^2
    A lower MSE loss indicates that the model is performing better and making more accurate predictions.
  2. Cross-Entropy Loss (CEL): This loss function is commonly used for classification tasks, where the goal is to predict a class label. CEL is calculated by taking the negative log of the predicted probability of the true class.
    CEL = – ∑(y * log(ŷ))
    A lower CEL indicates that the model is performing better and making more accurate predictions.
  3. Hinge Loss: This loss function is often used for support vector machines (SVMs) and is suitable for binary classification tasks. Hinge loss is calculated by taking the maximum of 0 and the difference between the predicted score for the true class and the predicted score for the false class, plus a margin.
    Hinge Loss = max(0, 1 – y * (w * x + b))
    where:
    y is the true class label (1 if the example belongs to the true class, -1 otherwise)
    w and b are the model’s weights and bias, respectively
    x is the input example

    The value 1 in the equation represents the margin. The margin is a hyperparameter that can be adjusted to control the influence of the loss function. A larger margin results in a more “relaxed” loss function, while a smaller margin results in a more “strict” loss function.
  4. Kullback-Leibler Divergence (KLD) Loss: This loss function is often used in generative models, where the goal is to model the distribution of the data. KLD loss is calculated by taking the difference between the predicted distribution and the true distribution.
    KLD Loss = ∑(p(x) * log(p(x)/q(x)))
    where q(x) is the predicted distribution and p(x) is the true distribution

In general, loss functions are used to guide the training process in a neural network. During training, the model’s weights and biases are adjusted in an attempt to minimize the loss function. This process is known as backpropagation, and it helps the model to learn patterns in the data and make more accurate predictions.

So once I choose a loss function for my neural network, how will it be used to estimate the weights?

Once you have chosen a loss function for your neural network, it will be used to guide the training process and estimate the weights of the model.

During training, the neural network is presented with a series of input examples, along with the corresponding true output (also known as the target output). The model processes the input and generates a predicted output. The loss function is then used to calculate the difference between the predicted output and the true output, which is known as the error.

The error is then used to update the model’s weights and biases in an attempt to minimize the loss function. This process is known as backpropagation, and it helps the model to learn patterns in the data and make more accurate predictions.

The weights and biases are updated using an optimization algorithm, such as stochastic gradient descent (SGD). During each training iteration, the optimization algorithm adjusts the weights and biases based on the error calculated by the loss function. This process is repeated until the model reaches a satisfactory level of accuracy, or until a specified number of training iterations has been reached.

Provide the steps of how the loss function and errors are used in backpropagation, include equations.

Backpropagation is a process used to train neural networks by adjusting the model’s weights and biases in an attempt to minimize the loss function. The following are the steps involved in backpropagation:

  1. Initialize the model’s weights and biases: Before training can begin, the model’s weights and biases must be initialized. This can be done using a variety of methods, such as setting all weights and biases to 0, or using random values drawn from a normal distribution.
  2. Feed forward: During each training iteration, the model is presented with a series of input examples, along with the corresponding true output (also known as the target output). The model processes the input and generates a predicted output, ŷ, using the following equation:

ŷ = f(w * x + b)

where:

  • f is the activation function
  • w and b are the model’s weights and bias, respectively
  • x is the input example
  1. Calculate the error: The loss function is then used to calculate the difference between the predicted output, ŷ, and the true output, y, which is known as the error. This can be expressed as follows:

Error = Loss(ŷ, y)

where Loss is the chosen loss function.

  1. Backpropagate the error: The error is then used to update the model’s weights and biases in an attempt to minimize the loss function. This is done using an optimization algorithm, such as stochastic gradient descent (SGD).
  2. Adjust the weights and biases: During each training iteration, the optimization algorithm adjusts the weights and biases based on the error calculated by the loss function. This can be expressed as follows:

w’ = w – α * ∂Loss/∂w b’
b’ = b – α * ∂Loss/∂b

where:

  • w’ and b’ are the updated weights and biases, respectively
  • α is the learning rate
  • ∂Loss/∂w and ∂Loss/∂b are the partial derivatives of the loss function with respect to the weights and biases, respectively
  1. Repeat until convergence: This process is repeated until the model reaches a satisfactory level of accuracy, or until a specified number of training iterations has been reached.

How do I choose the data on which the model should be trained? How do I split training vs. test data? And what is the difference between training, test, and validation data sets?

When training a neural network, it is important to choose a representative and diverse dataset to ensure that the model is able to learn patterns in the data that generalize well to new, unseen examples.

Once you have chosen a dataset, it is common practice to split it into three sets: training, validation, and test.

  • Training data is used to fit the model and learn the patterns in the data. The model is presented with a series of input examples, along with the corresponding true output (also known as the target output). The model processes the input and generates a predicted output, and the loss function is used to calculate the difference between the predicted output and the true output. This error is then used to update the model’s weights and biases in an attempt to minimize the loss function. This process is repeated until the model reaches a satisfactory level of accuracy, or until a specified number of training iterations has been reached.
  • Validation data is used to tune the model’s hyperparameters, such as the learning rate and the regularization coefficients. The model is presented with a series of input examples from the validation set, and the loss function is used to calculate the error between the predicted output and the true output. The hyperparameters are then adjusted in an attempt to minimize the validation loss.
  • Test data is used to evaluate the model’s generalization performance. The model is presented with a series of input examples from the test set, and the loss function is used to calculate the error between the predicted output and the true output. This error is used to measure the model’s accuracy on new, unseen data.

It is important to keep the training, validation, and test sets separate, as mixing them could result in a model that performs well on the training and validation sets, but poorly on the test set. This could lead to overfitting, which is when a model is overly optimized for the training and validation data, but fails to generalize well to new, unseen examples.

There are a few different strategies for splitting a dataset into training, validation, and test sets. One common approach is to use a 70/15/15 split, where 70% of the data is used for training, 15% is used for validation, and 15% is used for testing. Another approach is to use a 80/10/10 split, where 80% of the data is used for training, 10% is used for validation, and 10% is used for testing. The specific split will depend on the size and diversity of the dataset, as well as the specific requirements of the task at hand.

Can I use a neural network for both regression and classification problem types?

Yes, neural networks can be used for both regression and classification problems.

In a regression problem, the output of the model is a continuous value, such as a price or a probability. The goal is to predict the output value based on the input features. For example, you might use a neural network to predict the price of a house based on its size, location, and other features.

In a classification problem, the output of the model is a categorical value, such as a class label. The goal is to predict the class label of an input sample based on its features. For example, you might use a neural network to predict whether an email is spam or not based on the contents of the email.

Neural networks are very flexible and can be used to solve a wide range of problems. They can be used for both regression and classification, as well as other types of problems such as clustering, generation, and reinforcement learning.

What is meant by regularization in regards to a neural network model?

Regularization is a technique that is used to prevent overfitting in a neural network model. Overfitting occurs when a model is too complex and has too many parameters, which leads to poor generalization to new, unseen data.

There are several ways to regularize a neural network model, including:

  1. L1 regularization: This involves adding a penalty term to the loss function that is proportional to the absolute value of the weights. This has the effect of pushing the weights towards 0, which can help reduce overfitting.
  2. L2 regularization: This involves adding a penalty term to the loss function that is proportional to the square of the weights. This has the effect of pushing the weights towards 0, which can help reduce overfitting.
  3. Dropout: This involves randomly dropping out units (neurons) from the network during training, which can help prevent overfitting by reducing the number of parameters in the model.
  4. Early stopping: This involves stopping the training process early if the model starts to overfit, based on the performance on a validation set.

Regularization is an important technique for improving the generalization performance of a neural network model. It helps the model to generalize better to new, unseen data, rather than just memorizing the training data.

Chatting with GPT – Neural Networks

Chat GPT link

*NOTE: I have manually modified some of the equations produced by ChatGPT, such as adding a t-1 subscript

Hello, please tell me what is ChatGPT?

ChatGPT is a variant of the GPT (Generative Pre-training Transformer) language model that was specifically designed for use in chatbots and conversational AI systems. It is trained on a large dataset of human-human conversations and is able to generate human-like responses to user input. ChatGPT can be used to build chatbots that can hold natural language conversations with users, helping businesses and organizations automate customer service, sales, and other types of interactions.

Like other GPT models, ChatGPT is a transformer-based language model that uses attention mechanisms to process and generate text. It is trained using a combination of supervised learning, in which the model is given labeled examples of conversation, and unsupervised learning, in which the model is trained to predict the next word in a sequence based on the context of the words that come before it. This allows ChatGPT to learn the patterns and structure of human conversation and generate appropriate responses to user input.

I would like to learn more about machine learning, let’s start with the concept of neural networks. Please explain the concept of a neural network.

A neural network is a type of machine learning model that is inspired by the structure and function of the human brain. It is composed of a large number of interconnected “neurons,” which are simple processing units that receive input, transform it, and produce output.

In a neural network, the input data is fed through a series of hidden layers, each of which is made up of a large number of neurons. These hidden layers process the input data using weighted connections between the neurons. The weights of these connections are adjusted during the training process, allowing the neural network to “learn” from the input data.

The output of the neural network is produced by the output layer, which is also made up of a number of neurons. The output layer takes the processed data from the hidden layers and produces the final output of the neural network.

Neural networks can be used for a wide variety of machine learning tasks, including classification, regression, and clustering. They are particularly powerful for tasks that require the processing of large amounts of data or the detection of patterns and relationships in data that are not immediately apparent.

Overall, the concept of a neural network is a key element of machine learning and is used in many different applications and industries.

How does a neural network look in terms of mathematical equations? What is the architecture?

The architecture of a neural network refers to the way that the neurons are organized and connected within the network. A typical neural network consists of an input layer, one or more hidden layers, and an output layer. The input layer receives the raw input data, and the output layer produces the final output of the network. The hidden layers are located between the input and output layers and are responsible for processing the input data and producing intermediate output that is passed on to the output layer.

In terms of mathematical equations, the computation performed by a neural network can be represented by a series of matrix multiplications and element-wise non-linear functions (also called activation functions).

For example, consider a neural network with one hidden layer and an input layer with m inputs and an output layer with n outputs. Let’s say that the input data is represented by a matrix X with dimensions m x N (where N is the number of data points) and the weights between the input layer and the hidden layer are represented by a matrix W1 with dimensions m x p (where p is the number of neurons in the hidden layer). The output of the hidden layer is then given by:

h = f(XW1 + b1)

where f is the activation function and b1 is a bias term.

The weights between the hidden layer and the output layer are represented by a matrix W2 with dimensions p x n, and the final output of the neural network is given by:

y = f(hW2 + b2)

where b2 is a bias term.

This is just a simple example, but it illustrates the basic idea of how a neural network can be represented mathematically. In practice, neural networks can have many more layers and a much larger number of neurons, and the equations can become quite complex. However, the basic idea of using matrix multiplications and non-linear activation functions to transform the input data and produce output remains the same.

What exactly do you mean by activation function? What are some examples of such functions?

An activation function is a mathematical function that is applied to the output of a neuron in a neural network. It determines whether the neuron will “fire” and produce an output, or whether it will remain inactive. Activation functions are a key component of neural networks because they allow the network to learn non-linear relationships in the data.

There are many different activation functions that can be used in neural networks, each with its own properties and characteristics. Some common examples include:

Sigmoid: The sigmoid function is a smooth, S-shaped curve that maps any input value to a value between 0 and 1. It is often used in the output layer of a binary classification neural network because it can produce a probability-like output.

Tanh: The tanh (hyperbolic tangent) function is also a smooth, S-shaped curve, but it maps input values to a range of -1 to 1. It is similar to the sigmoid function, but it allows for negative output values as well.

ReLU (Rectified Linear Unit): The ReLU function is a piecewise linear function that maps negative input values to 0 and positive input values to the same value. It is a simple and efficient activation function that has been widely adopted in many neural network architectures.

Leaky ReLU: The leaky ReLU function is a variant of the ReLU function that allows a small negative slope for negative input values. This can help prevent the “dying ReLU” problem, in which some neurons in the network become inactive and stop producing output.

Softmax: The softmax function is a generalization of the sigmoid function that is often used in the output layer of a neural network for multi-class classification. It maps the input values to a probability distribution over the different classes, such that the sum of the probabilities is 1.

These are just a few examples of the many activation functions that are used in neural networks. The choice of activation function can have a significant impact on the performance of the network, and different activation functions may be better suited for different types of tasks and data.

So how would I choose an activation function then? Would each layer have a different activation function?

The choice of activation function can have a significant impact on the performance of a neural network, and it is an important design decision that can affect the network’s ability to learn from the data and generalize to new examples.

There is no one-size-fits-all activation function that is best for all situations, and it is often necessary to experiment with different activation functions to find the one that works best for a particular task. In general, it can be helpful to consider the properties of the activation function and how they align with the characteristics of the data and the goals of the task.

For example, if you are building a neural network for a binary classification task, you may want to use a sigmoid or a tanh activation function in the output layer, since these functions can produce a probability-like output between 0 and 1. On the other hand, if you are building a neural network for a multi-class classification task, you may want to use a softmax activation function in the output layer, since it can produce a probability distribution over the different classes.

As for the hidden layers of the network, ReLU and its variants (such as Leaky ReLU) are often a good choice, since they are simple and efficient and have been shown to work well in a wide range of tasks. However, other activation functions may also be suitable, depending on the characteristics of the data and the specific goals of the task.

In general, it is possible to use different activation functions in different layers of a neural network, and this can sometimes be helpful to improve the network’s performance. However, it is also important to be mindful of the complexity of the network and to avoid adding unnecessary complexity that could harm the network’s ability to learn effectively.

So once I have my data inputs, my data outputs, and I choose my activation functions, how do I choose the number of layers in my neural network?

The number of layers in a neural network, as well as the number of neurons in each layer, are important design choices that can impact the network’s ability to learn from the data and generalize to new examples. In general, the more layers and neurons a network has, the more capacity it has to learn from the data. However, too many layers and neurons can also make the network more prone to overfitting, which is when the network performs well on the training data but poorly on new, unseen examples.

There is no hard and fast rule for choosing the number of layers and neurons in a neural network, and it is often necessary to experiment with different configurations to find the one that works best for a particular task. However, there are a few guidelines that you can follow to help you make informed decisions:

  • Start with a small number of layers and neurons and gradually increase them until you see improvements in the network’s performance. This can help you avoid the problem of overfitting, since a smaller network is less likely to overfit the data.
  • Consider the complexity of the task and the amount of data you have available. For more complex tasks or larger datasets, you may need a deeper network (i.e., one with more layers) to learn effectively. On the other hand, for simpler tasks or smaller datasets, a shallower network may be sufficient.
  • Experiment with different architectures and compare the performance of the different models. This can help you identify the best-performing configuration for your task.

Ultimately, the choice of the number of layers and neurons in a neural network is a trade-off between model capacity and generalization. You want to choose a configuration that has enough capacity to learn from the data, but not so much that it overfits the data. Finding the right balance can require some experimentation and trial and error, but with the right approach, you can build a neural network that performs well on your task.

Once I will choose the number of layers, the number of neurons in each layers, the activation functions, how do I train the model then? How will the weights be calculated?

To train a neural network, you need to adjust the weights of the connections between the neurons in such a way that the output of the network is as close as possible to the desired output for a given set of inputs. The process of adjusting the weights to minimize the difference between the predicted output and the desired output is known as optimization.

There are several optimization algorithms that can be used to train a neural network, each with its own pros and cons. Some common optimization algorithms include:

  • Stochastic Gradient Descent (SGD): SGD is a simple and widely-used optimization algorithm that involves updating the weights of the network based on the gradient of the loss function with respect to the weights. The loss function is a measure of how well the network is doing at predicting the desired output given the input data, and the gradient is a vector that points in the direction of the steepest increase in the loss function. SGD involves repeatedly updating the weights in the opposite direction of the gradient, in an attempt to minimize the loss function and improve the performance of the network.
  • Adam (Adaptive Moment Estimation): Adam is a variant of SGD that uses an adaptive learning rate, which means that the step size for updating the weights is adjusted based on the past gradient information. Adam tends to be more efficient and require fewer hyperparameter tuning compared to SGD.
  • Mini-batch gradient descent: Mini-batch gradient descent is a variant of SGD that involves dividing the training data into small batches and updating the weights based on the average of the gradients of the loss function for each batch. This can be more efficient than SGD, especially for large datasets, since it allows the network to learn from the data more quickly.
  • RProp (Resilient Backpropagation): RProp is a variant of SGD that uses a heuristic approach to adjust the learning rate for each weight individually, based on the past gradient information. It is generally faster and more efficient than SGD, and it is often used in large-scale neural network training.

Please explain stochastic gradient descent further, include equations

Stochastic Gradient Descent (SGD) is a widely-used optimization algorithm for training neural networks and other machine learning models. It involves updating the weights of the model based on the gradient of the loss function with respect to the weights. The loss function is a measure of how well the model is doing at predicting the desired output given the input data, and the gradient is a vector that points in the direction of the steepest increase in the loss function.

SGD works by iteratively updating the weights of the model in the opposite direction of the gradient, in an attempt to minimize the loss function and improve the performance of the model. The update rule for SGD can be expressed as follows:

w = w – alpha * gradient

where w is the weight of the model, alpha is the learning rate (a hyperparameter that controls the step size for updating the weights), and gradient is the gradient of the loss function with respect to the weight.

The learning rate is an important hyperparameter in SGD, and it determines the step size for updating the weights. A larger learning rate can lead to faster convergence, but it can also make the optimization process more unstable. On the other hand, a smaller learning rate can lead to slower convergence, but it can also make the optimization process more stable.

SGD can be implemented in a number of ways, and it is often used in conjunction with other techniques, such as mini-batch gradient descent or momentum, to improve the efficiency and stability of the optimization process.

You also mentioned ADAM – Adaptive Moment Estimation, how does this algorithm differ from SGD? Please provide equations.

Adam (Adaptive Moment Estimation) is a variant of Stochastic Gradient Descent (SGD) that uses an adaptive learning rate, which means that the step size for updating the weights is adjusted based on the past gradient information. Adam is a popular optimization algorithm that is often used to train neural networks and other machine learning models.

One of the key differences between Adam and SGD is that Adam uses an exponentially weighted average of the past gradients to scale the learning rate, rather than a fixed learning rate as in SGD. This allows Adam to automatically adjust the learning rate based on the characteristics of the data and the optimization process, which can lead to faster convergence and better performance.

The update rule for Adam can be expressed as follows:

m = beta1 * m_t-1 + (1 – beta1) * gradient

v = beta2 * v_t-1 + (1 – beta2) * gradient^2

w = w_t-1 – alpha * m / (sqrt(v) + epsilon)

where w is the weight of the model, alpha is the learning rate (a hyperparameter that controls the step size for updating the weights), m and v are the exponentially weighted averages of the past gradients and squared gradients, respectively, beta1 and beta2 are hyperparameters that control the decay rate of the moving averages, and epsilon is a small value that is used to prevent division by zero.

Adam uses two moving averages, one for the gradients and one for the squared gradients, which allows it to incorporate both the mean and the variance of the past gradients into the learning rate. This can help to stabilize the optimization process and prevent oscillations or divergences.

You also listed RProp (Resilient Backpropagation) as an optimization algorithms, please explain RProp with equations.

Resilient backpropagation (RProp) is an optimization algorithm used in artificial neural networks for training supervised learning models. It is an extension of the backpropagation algorithm that aims to improve the convergence speed of the training process by adapting the learning rate for each weight in the network during training.

Like backpropagation, RProp uses the gradient of the loss function to update the weights of the network. The gradient is calculated using the chain rule and the weights are updated using the following equation:

weight(t+1) = weight(t) – learning_rate * gradient(t)

In backpropagation, the learning rate is a fixed hyperparameter that is set before training. In RProp, the learning rate is not fixed and is instead adapted for each weight based on the sign of the gradient. In RProp, the learning rate for each weight in the network is updated individually, based on the sign of the gradient of the loss function with respect to that weight. This means that the learning rate for each weight is updated individually based on the convergence of that weight, rather than being updated based on the convergence of all weights in the network. This can improve the convergence speed of the training process, as the learning rate can be adjusted according to the convergence of each weight.

The learning rate is updated using the following equation:

learning_rate(i)_t+1 = learning_rate(i)_t – step_size(i)_t-1 * sng (dE_t-1 / dw(i)_t-1 )

I have also often heard the AdaGrad optimizer mentioned, please explain that to me

In AdaGrad, the learning rate is not fixed and is instead adapted for each weight based on the history of the gradients of the loss function with respect to that weight. Specifically, the learning rate is adjusted based on the sum of the squares of the gradients of the loss function with respect to the weight.

The learning rate for each weight is updated using the following equation:

learning_rate(t+1) = learning_rate(t) / sqrt(sum(gradient(t)^2))

where sum(gradient(t)^2) is the sum of the squares of the gradients of the loss function with respect to the weight up to time t.

This means that the learning rate for each weight is inversely proportional to the sum of the squares of the gradients of the loss function with respect to that weight. This has the effect of decreasing the learning rate for weights that have had large gradients in the past, and increasing the learning rate for weights that have had small gradients in the past.

I have also heard of gradient boosting, explain that

Gradient boosting is a machine learning technique for building predictive models by creating an ensemble of weak learners and combining their predictions to create a stronger, more accurate model. The general idea of gradient boosting is to train a sequence of weak learners (e.g. decision trees) in a stage-wise manner, with each subsequent learner trying to correct the errors made by the previous learner. At each stage, the gradient boosting algorithm fits a new weak learner to the residual errors made by the previous learner. The residual errors are the difference between the true values and the predicted values of the target variable.

In gradient boosting, weights are assigned to the incorrectly classified samples in order to give more importance to the samples that are difficult to classify. This helps the weak learners in the ensemble to focus more on the samples that are most likely to improve the overall performance of the model.

The steps of the gradient boosting algorithm can be summarized in the following equations:

  1. Initialize the ensemble with a single weak learner, such as a decision tree, and make predictions using this learner:

prediction(1) = weak_learner_prediction(1)

  1. Calculate the residual errors between the true values and the predicted values of the target variable:

residual_error(t) = true_value(t) – prediction(t)

  1. Train a new weak learner to predict the residual errors from the previous step:

weak_learner_prediction(t+1) = train(weak_learner, residual_error(t))

  1. Update the predictions of the ensemble by adding the predictions of the new weak learner to the previous predictions:

prediction(t+1) = prediction(t) + weak_learner_prediction(t+1)

  • Repeat steps 2-4 until the desired number of weak learners has been trained.

What is AdaBoost and what are the steps of the algorithm? Include equations.

AdaBoost works by weighting the observations in the training set and adjusting the weight of each observation at each iteration of the algorithm based on the performance of the previous models. The idea behind AdaBoost is to train a sequence of weak learners in an iterative manner, with each learner trying to correct the mistakes of the previous learner. The final model is a weighted average of all the weak models.

Here are the steps of the AdaBoost algorithm:

  1. Initialize the weights of the observations in the training set:
    • At the beginning, all the observations in the training set are given the same weight.
  2. Train a weak learner:
    • Train a weak learner (e.g., a decision tree) on the training set using the current weights of the observations (My note: sample with replacement).
  3. Calculate the error of the weak learner:
    • Calculate the error (my note: total error) on the training set using the following equation:error = sum(weights[i] * I[prediction[i] != y[i]])
    where I is the indicator function that returns 1 if the prediction is incorrect and 0 if the prediction is correct, weights is the array of weights for the observations, prediction is the array of predictions made by the weak learner, and y is the array of true labels for the observations.
  4. Calculate the weight of the weak learner:
    • Calculate the weight of the weak learner using the following equation :
      alpha = 0.5 * log((1 – total error) / total error)
  5. Adjust the weights of the observations:
    • Adjust the weights of the observations using the following equation:
      weights[i+1] = weights[i] * exp(-alpha * I[prediction[i] == y[i]])
  6. Normalize the weights:
    • Normalize the weights so that they sum up to 1.
  7. Repeat steps 2-6:
    • Repeat steps 2-6 a predetermined number of times or until the error of the model becomes small enough.
  8. Return the final model:
    • Return the final model as a weighted average of all the weak models.

My note:

Short note on exp(-alpha * I[prediction[i] == y[i]]) i.e. for misclassification

When the performance is relatively large the last stump did a good job in classifying the records now the new sample weight will be much larger than the old one. When the performance is relatively low the last stump did not do a good job in classifying the records now the new sample weight will only be little larger than the old one.

No misclassification: When the performance is relatively large the last stump did a good job in classifying the records now the new sample weight will be very small than the old one. When the performance is relatively small the last stump did not do a good job in classifying the records now the new sample weight will only be little smaller than the old one.

What is the difference between gradient boosting algorithm and adaptive boosting?

Gradient boosting and adaptive boosting are both ensemble learning algorithms that can be used to improve the performance of a weak learner (a model that performs only slightly better than random guessing) by combining the predictions of multiple weak learners.

The main difference between gradient boosting and adaptive boosting is the way in which the weak learners are trained and combined.

In gradient boosting, the weak learners are trained sequentially, with each learner trying to correct the mistakes of the previous learner. In adaptive boosting (also known as AdaBoost), the weak learners are also trained sequentially, but the training process is different. Each weak learner is trained on a different subset of the training data, and the weights of the training examples are adjusted after each learner is trained to give more weight to the examples that were misclassified. The final prediction is made by combining the predictions of all of the learners using a weighted sum, where the weight of each learner is determined by its accuracy on the training data.

Advent of Code Day 5 – my bonus question

I am doing the Advent of Code. So far I have solved all the questions for the four previous days and part one of the question for day five. I have also created my own question for fun, the question is below:

After many hours of walking, the Elves come to a forest glade. They are quite tired and hungry, one of the elves suddenly notices that the glade is full of mushrooms. The Elves are familiar with this mushrooms species – they are edible and quite tasty. The Elves pick all of the mushrooms and are almost ready to make mushroom soup, when they remember about one tricky problem – there is a poisonous mushroom species that looks very similar and often a poisonous mushroom will grow right among the edible mushrooms.

At this point the elves have determined the molecular structure of each mushrooms that they picked. The structure always consists of five segments and each segment consists of a number and a letter.

Example: 0.9H 0.08G 0.27L 0.57M 0.84P

Each letter molecule (A – Z) has a corresponding weight, from 0 to 25. The numbers also represent additional weight units. It is therefore possible to calculate the molecular weight of each mushroom. In the above example the weight would be 0.9 + 7 + 0.08 + 6 + 0.27 + 11 + 0.57 + 12 + 0.84 + 15 = 53.66

If the structure had a negative number, such as if it would be 0.9H -0.08G 0.27L 0.57M 0.84P, then the negative segment would need to be subtracted. The weight then would be 0.9 + 7 – 0.08 – 6 + 0.27 + 11 + 0.57 + 12 + 0.84 + 15 = 41.5

The Elves are aware that the value of each segment of a mushroom comes from a process generated by ~N(12.5, 4.5) and there is no correlation between the segments. (The value of the segment is number + letter, for example 0.9H is 7.9, while -0.08G is -6.08).

The mushroom that is poisonous is definitely tricky to find for the Elves because it looks exactly the same as the edible mushrooms. BUT! The molecular structure of this mushroom gives it away! It is very unlikely that such structure would be generated by the same process as for the edible mushrooms. Find the poisonous mushroom from the input list so that the Elves can start cooking their soup.

The list of mushrooms is in the link below:

Advent of Code Day 5 bonus question input

Postpartum psychosis stories

I think it’s very important to talk about this condition that not everyone is aware of – postpartum psychosis. Postpartum psychosis is when psychosis occurs sometime after a woman gives birth, it’s a separate condition from postpartum depression. Below I have posted links to several stories of postpartum psychosis. One important message stated was that a woman would likely not be aware that she is suffering from psychosis. The delusions/hallucinations that occur during psychosis feel very real to the patient. If a woman had never experienced psychosis before and maybe even was not aware of existence of postpartum psychosis, it would be quite difficult for her to understand in that state that what she is experiencing is not real and is a serious medical condition.

I think it’s very useful for everyone to hear these stories below and to be aware what are the symptoms. During psychosis the person can have auditory hallucinations, hear voices, they can also feel that objects/people are sending them messages. In one of the stories a woman described how during her psychotic episode she heard voices coming from the radiator and thought that different colours were sending her messages. Insomnia can also be a symptom of psychosis and can exacerbate the lack of sleep caused by having a new born, and lack of sleep in turn can exacerbate psychotic symptoms. Another woman tells her story of how she started to see her mom, who actually passed away. She also hallucinated a man in her house who she thought was planning to kill her. Manic symptoms can also occur such as believing that there is a divine intervention and you are now being able to know things that others can’t, believing that you can achieve anything such as learning a language overnight, as well as rapid speech and sleeping only a few hours a day.

Postpartum psychosis – Katy’s story

Postpartum psychosis – story #2

Postpartum psychosis – story #3

Off-topic, using NLP to summarize Russian propaganda articles on the invasion of Ukraine

I have a pretty simple article summarizer in python that I use for my blog. It’s not machine learning, it’s pretty basic – collecting N articles based on a search query, cleaning up the text, tokenizing, removing stop words. Then word frequencies are calculated and each sentence is scored based on those frequencies. Top N sentences are selected for the summary.

I usually use it to summarize articles on some specific supplements, but today for fun I decided to try a different search query. My query was “путин россия украина спецоперация”. I think that even though this summarizer is pretty simple, the summary turned out to be pretty meaningful. It gives a pretty good overview of what sort of info you would get if you were reading Russian news. I selected the top 15 sentences for the summary.

Что касается слов президента о спецоперации на Донбассе, то они показывают, что наши военнослужащие снова спасают мир от коричневой чумы нацизма, и все поставленные задачи, вне всякого сомнения, будут выполнены. Верховный главнокомандующий еще раз подтвердил, спецоперация на Украине будет доведена до своего логического завершения по тем задачам, которые он озвучил 24 февраля, в день начала СВО», — сказал полковник. И президент в своей речи указал на союзников из Европы, Азии, Африки и Латинской Америки, которые не прогибаются перед так называемым гегемоном, выбирают суверенный путь развития и хотят коллективно решать вопросы безопасности и сформировать многополярный мир.

Украинские издания утверждают, что СНБО одобрил санкции против патриарха Кирилла, а также Сергея Кириенко, Евгения Пригожина, Романа Абрамовича, Олега Дерипаски, Михаила Фридмана, Виктора Медведчука, а также Виктора Януковича. Наш долг перед памятью миллионов жертв Второй мировой войны — жестко реагировать на попытки фальсификации истории, противодействовать распространению любых форм неонацизма, русофобии и расизма», — призвал президент. Все указывает на то, что временной, человеческий, материальный, дипломатический ресурс «спецоперации» близится к исчерпанию, и Путин делает решительный шаг, чтобы закончить как можно скорее, зафиксировав прибыли и убытки. “Наша задача, наша миссия, солдат, ополченцев Донбасса – эту войну прекратить, защитить людей и, конечно, защитить саму Россию”, – подчеркнул Путин.

Выступление Путина демонстрирует готовность Вооруженных сил России довести специальную военную операцию на Украине до победного конца, пояснил URA.RU военный эксперт, ветеран разведки, полковник Анатолий Матвийчук. Война, развязанная Западом и киевской хунтой, будет закончена», — заключил Красов. Российское высокоточное оружие опережает иностранные аналоги на годы и десятилетия, при этом значительно превосходит их по тактико-техническим характеристикам, а оборонная промышленность лежит в основе суверенитета России, отметил президент.

Говоря о шагах со стороны Запада, ранее председатель комиссии Госдумы по расследованию фактов вмешательства иностранных государств во внутренние дела России Василий Пискарев заявлял о том, что зарубежные НКО насаждают радикальные идеологии в российском обществе, сообщает РАПСИ. «Сперва коллективный Запад изощрялся доказать, что якобы „разорвал в клочья“ российскую экономику, теперь там думают, что изолировали нашу страну от всего остального мира. Вооруженные силы России надежно защищают свою страну и несут свободу другим народам, заявил Путин, открывая работу форума «Армия-2022» и Армейских международных игр — 2022. «У России в запасе есть несколько военных вариантов, и для США и их союзников по НАТО эти сценарии — серьезный повод для беспокойства», — пишет журнал. Несмотря на то, что Запад был воодушевлен последними «успехами» ВСУ, генерал США Марк Милли призвал опасаться непредсказуемости России.

Dealing with depression after encephalitis. After many years of trials, this is my current depression regimen, just wanted to share.

Hello everyone, I just wanted to share my current depression regimen and some situation info, in case anyone has similar health issues. I have experienced many hospitalizations since 2015, including involuntary psychiatric hospitalizations. Finally in 2017 I was diagnosed with autoimmune encephalitis (brain inflammation), as well as autoimmune thyroiditis. I was treated with intravenous corticosteroids and that led to some improvement. I continue to experience health issues, but I have made several life style changes that have helped me and that I wanted to share. Again, I was diagnosed with autoimmune disease, and my neuropsychiatrist believes that the encephalitis greatly contributed to my depression. Clearly it’s not the case for everyone, so I am not stating that this should work for all. I have been doing better since these changes, I was able to complete a graduate degree, get back to painting, and started writing and playing guitar again. These were huge improvements for me as I was not able to enjoy any hobbies when I had severe depression and was not able to pursue graduate courses.

  1. I cut out all refined carbs and processed foods. There is sufficient evidence indicating that these foods contribute to inflammation. I am not doing keto or low carb, I am not trying to be very strict with myself, I enjoy all sorts of complex carbs such as baked plantains, potatoes, oatmeal, fruits, berries, etc.
  2. Switched to low glycemic foods – this related to #1, as cutting out refined simple carbs in general does leave one with complex carbs that have lower glycemic index.
  3. Foods that cause an immune reaction – this clearly does not occur for most people, but some do react to certain foods. I noticed that I feel physically and emotionally worse after eating gluten, dairy, or soy, so I had to drop these from my diet.
  4. I go to sleep earlier and stay away from my laptop/phone screen after 9pm. I used to stay up late, but now I go to bed around 11pm. After 9pm I usually dim the lights in the room a bit and I read on my Kindle. Kindle Paperwhite does not emit a high amount of blue light. I also installed blackout curtains so that I spend the night sleeping in the dark.
  5. Sleep is very important – so when I really can’t fall asleep, I do use a cannabis oil (NightNight CBN + CBD oil). But changing my diet, losing weight, and going to bed earlier, did reduce my insomnia, so I don’t need the oil every day.
  6. Significantly decreasing my caffeine intake – personally for me it did lower my anxiety and the occurrence of panic attacks, I now only have green tea in the afternoon, otherwise I drink rooibos tea, water, kefir, decaf tea.
  7. Intermittent fasting – I do fell less brain fog and more clear headed when I am not eating the whole day. I used to surf the internet at 1am eating Sweet & Salty bars. Then my mind would go into dark places and I would start reading about serial killers. Now I eat two to three meals a day between 9am and 5pm, I fast for 16-18 hours a day.
  8. Seeing a psychologist – going through CBT and DBT did help, and this related to #5. I still experience racing thoughts, anxiety, and other issues, but I can now more easily choose to not follow my thoughts. For example – I did used to read a lot about US serial killers and then I would freak myself out and I would start to think that someone could climb through the window. Now I choose more what I read – should I keep reading about mass murders? What is the point of that for me? Will that change anything for the better?
  9. Sunlight – I try to get some sunlight each morning, if I have no energy to come out, I still stick out of the window and get some sunlight on my face.
  10. Exercise – I experience certain pains due to autoimmune disease, and fatigue, so I don’t do extensive exercise, but I do yoga at home. And by exercise I don’t mean that I do a whole hour after work, I do certain yoga poses occasionally throughout the day. I think that’s still better than no exercise.
  11. Shrooms – I did several shroom trips, at home alone, after I was treated for encephalitis. I haven’t done shrooms for a while due to pregnancy and breastfeeding, but the positive antidepressant effects of the trips still remain for me.
  12. CBT, again – accepting that some days are better than others, some are worse, but also seeing the positive – in general I am doing much much better now than in 2016. I am female, hormones fluctuate, I do feel worse during the luteal phase, but I experience a lot more enjoyable moments than before my steroids treatment and this lifestyle change.

Eggshells – a cheap non-dairy source of calcium

If you are going to adhere to a dairy free diet, you are going to need to find another source of calcium. Humans require calcium for muscle contraction, blood clotting, normal heart rhythm, as well as nerve functions. The recommended dietary allowance (RDA) for calcium is around 1000mg per day. The RDA is elevated to 1,300 milligrams per day during adolescence because this is the life stage with accelerated bone growth.

For women above age fifty and men older than seventy-one, the RDAs are also a bit higher for several reasons including that as we age, calcium absorption in the gut decreases, vitamin D3 activation is reduced, and maintaining adequate blood levels of calcium is important to prevent an acceleration of bone tissue loss (especially during menopause). Results of some large trials found that higher calcium intakes (usually achieved with a supplement) was associated with improved bone density and slightly lower risk of hip fractures.

Now in regards to eggshells – most people probably don’t eat them, but eggshells are an excellent source of calcium. Eggshells are also a natural source of other elements such as strontium and fluorine. Approximately half an eggshell from a 42g egg would provide 750 mg of calcium, which is 75% of the daily recommended value of 1000mg. Clinical and experimental studies showed that eggshell powder has positive effects on bone and cartilage and that it is suitable in the prevention and treatment of osteoporosis.

How can you eat an eggshell though?

My process is simple – I often eat boiled eggs for breakfast in the morning. Once I peel the egg, I am left with an eggshell that has already been sanitized by boiling (it’s important to sanitize the eggshells due to the possibility of the Salmonella bacteria being on the shell). I then bake the eggshells at a low temperature of 225F in order to dry them. After they cool down, I simply grind the shells into fine powder using a coffee grinder. The powder can be consumed with a spoon, added to smoothies, oatmeal, dough, etc.

Celiac disease and dairy proteins – summarization of articles

I want to address the issue of whether dairy could be an issue for those with celiac disease. I don’t think that I will be able to arrive at an exact answer with this post, but I do wish to summarize existing articles and evidence on this topic. From my personal experience, I get all the same symptoms from dairy products as from foods containing gluten. The symptoms include pains in the lower abdomen, bloating, constipation, fatigue, inflammation of the eyelids, as well as psychiatric symptoms including panic attacks, anxiety, and depression. Gathering anecdotal evidence by speaking to reddit users in the gluten-free subreddit, multiple individuals have also expressed the same experience with dairy causing similar symptoms to gluten. Also these individuals noticed that the same symptoms were caused by lactose-free products, therefore likely the culprit is not the sugar (lactose), but the proteins in dairy (casein). Below I will summarize several articles addressing the consumption of casein by individuals with celiac disease.

The first study that I found looking at the correlation between gluten and casein is from 2007, Mucosal reactivity to cow’s milk protein in coeliac disease. This article discusses the fact that some celiac patients on a gluten-free diet still experience gastrointestinal symptoms. The authors then examine whether these patients have an inflammatory immune response to the protein in cow’s milk. The results of this study indicated that in fact in a fraction of celiac patients did experience a similar reaction to the milk protein as to gluten. As usual, I used python to create article summaries, including this one.

Summary:
On clinical grounds cow’s milk (CM) protein sensitivity may be suspected. Here, using rectal protein challenge, we investigated the local inflammatory reaction to gluten and CM protein in adult patients with CD in remission.
In 18 of 20 patients gluten challenge induced neutrophil activation defined as increased MPO release and increased NO synthesis.
A mucosal inflammatory response similar to that elicited by gluten was produced by CM protein in about 50% of the patients with coeliac disease.

Summary using LexRank (graph-based method for computing relative importance of sentences):

Mean rectal ΔMPO was 303 ± 27 µg/l after casein challenge and 16 ± 27 µg/l after challenge with α-lactalbumin.
Compared to healthy controls, patients with CD showed significant increases in rectal NO and MPO concentrations measured 15 h after challenge with both CM and gluten (P < 0·001), while ECP was increased to a similar extent in the two groups ( ).
The major finding in this study is that rectal challenge with CM protein frequently induced a local inflammatory mucosal reaction in patients with CD but not in healthy controls.
Our patients with CD had normal serum levels of IgA, IgG and IgE against casein and α-lactalbumin, which might be explained by the fact that they were on a gluten-free diet and therefore had improved the mucosal integrity.
Our finding that, in a fraction of coeliac patients, CM protein challenge may induce an inflammatory reaction of the same magnitude, as did gluten challenge, may also suggest an innate as well as adaptive immune response to CM, and casein in particular.

There were several other studies on the topic of gluten-free and casein-diet, but they all investigated whether this diet would help patients on the autism spectrum, which is not the topic of my post. I did find another short article on gluten-free and casein-free diet helping with psychotic symptoms. Personally I have a similar experience, as consuming any gluten or dairy increases my paranoia, panic attacks, and intrusive thoughts. The authors claim that there is a following mechanism for psychosis:

“In autism and schizophrenia, incomplete digestion of certain proteins, gluten and casein, cause an autoimmune response as indicated by elevated levels of IgA and IgG antibodies. This intestinal malabsorption also causes pathogenic elements (peptide fractions), which bind to opioid receptors by crossing the blood-brain barrier. This releases exorphins (opiate-like substances, similar to certain drugs) that cause psychotic symptoms.”

Evidence-Based Practice: Introduction of a Gluten-Free and Casein-Free Diet to Alleviate Psychotic Symptoms
A case review of a young boy yielded an unexpected resolution of psychotic symptoms after the introduction of a gluten-free, casein-free (GFCF) diet.
The purpose of this paper is to show that health care professionals may use a gluten-free and casein-free diet (GFCF) as an additional element to standard treatment methods, to alleviate psychotic symptoms.
Additionally noted were similarities between autism and schizophrenia.
Introduction of a GFCF diet helps reduce psychotic symptoms, and gives another option for patients resistant to traditional treatment methods, especially adolescents and young adults.
Keywords: autism, gluten-free, casein-free diet (GFCF), psychosis, schizophrenia