Visualizing LLM Embeddings using HVT (Hierarchical Voronoi Tessellation)

Chepuri Gopi Krishna, Vishwavani

2024-11-08

1. Background

The HVT package is a collection of R functions to facilitate building topology preserving maps for rich multivariate data analysis. Tending towards a big data preponderance, a large number of rows. A collection of R functions for this typical workflow is organized below:

  1. Data Compression: Vector Quantization (VQ), HVQ (Hierarchical Vector Quantization) using means or medians. This step compresses the rows (long data frame) using a compression objective.

  2. Data Projection: Dimension projection of the compressed cells to 1D, 2D and Interactive surface plot with the Sammons Non-linear Algorithm. This step creates a topology preserving map (also called embeddings) coordinates into the desired output dimension.

  3. Tessellation: Create cells required for object visualization using the Voronoi Tessellation method, package includes heatmap plots for Hierarchical Voronoi Tessellations (HVT). This step enables data insights, visualization, and interaction with the topology preserving map. Useful for semi-supervised tasks.

  4. Scoring: Scoring data sets and recording their assignment using the map objects from the above steps, in a sequence of maps if required.

  5. Temporal Analysis and Visualization: A Collection of functions that leverages the capacity of the HVT package by analyzing time series data for its underlying patterns, calculation of transitioning probabilities and the visualizations for the flow of data over time.

What’s New?

Flow of clustHVT

Limitation of clustHVT

Abstract

2. Experimental Setup

Objective: To evaluate the effectiveness of the HVT package in compressing and visualizing high-dimensional LLM token embeddings while preserving the underlying structure and relationships between the embeddings.

Hypothesis:

Null Hypothesis (H0):

Alternate Hypothesis (H1):

Test Metrics:

Compression Ratio:

Visual Similarity Evaluation:


3. Notebook Requirements

This chunk verifies the installation of all the necessary packages to successfully run this vignette, if not, installs them and attach all the packages in the session environment.

list.of.packages <- c("HVT", "dplyr","magrittr", "ggplot2", "kableExtra", "gridExtra","grid", "httr", "stats", 
                      "factoextra", "cluster", "rmarkdown","xfun", "NbClust","plotly", "tidyr",
                      "dendextend","jsonlite","progress","reticulate","knitr", "tools", "reactable")

new.packages <-
  list.of.packages[!(list.of.packages %in% installed.packages()[, "Package"])]
if (length(new.packages))
  install.packages(new.packages, dependencies = TRUE, verbose = FALSE, repos='https://cloud.r-project.org/')
invisible(lapply(list.of.packages, library, character.only = TRUE))

3.1 User-defined Functions

In this section, we are establishing all the functions that will be utilized in this notebook. The detailed explanation follows below:

1. Function to Merge and Slice the data

This function combines multiple CSV files (tweet csvs of each celebrities or influencers) in a directory to a single data frame, and slice the first 3 tweets per celebrity or influencer. It’s used in section 4 as a first part of data processing.

merge_and_slice_data <- function(directory,output_path, num_users = 915) {
  csv_files <- list.files(directory, pattern = "*.csv", full.names = TRUE)
    read_csv_with_filename <- function(file_path) {
    df <- read.csv(file_path)
    filename <- file_path_sans_ext(basename(file_path))
    df$source_file <- filename
    return(df)
  }
    combined_data <- csv_files %>% lapply(read_csv_with_filename) %>% bind_rows()
  
    user_names <- unique(combined_data$source_file)
    select_users <- sample(user_names, num_users)
    sliced_data <- combined_data %>% filter(source_file %in% select_users) %>% group_by(source_file) %>%
                   slice_head(n = 3) %>% ungroup()
  
    write.csv(sliced_data, output_path, row.names = FALSE)
}

2. Function to clean the twitter data

This is a python function used in Twitter data processing notebook from kaggle to clean the data by removing the urls, retweet tags, convert the emoji to text and all the basic text cleaning process. It’s used in section 4 as a second part of data processing.

import demoji
import re
import ast
from typing import Union
import pandas as pd
from bs4 import BeautifulSoup

emoticon_data_path="./emoticon_dict.json"

class TwitterDataPrep:
    def __init__(self, emoticon_data_path=None):
        self.emoticons_dict = {}
        if emoticon_data_path:
            self.load_emoticons(emoticon_data_path)

    def load_emoticons(self, emoticon_data_path):
        """
        Load the emoticons dictionary from a JSON file.
        :param emoticon_data_path: Path to the emoticons JSON file.
        """
        import json
        with open(emoticon_data_path, 'r') as file:
            self.emoticons_dict = json.load(file)

    def clean_text(self, text: str) -> str:
       
        pat1 = r'@[^ ]+'  # remove username @
        pat2 = r'https?://[A-Za-z0-9./]+'  # remove urls
        pat3 = r'\'s'  # remove apostrophe
        pat4 = r'\#\w+'  # remove hashtag
        pat5 = r'&amp '  # remove unicode `&`
        pat7 = r'RT'  # remove RT / retweet
        pat8 = r'www\S+'  # remove link www
        combined_pat = r'|'.join((pat1, pat2, pat3, pat4, pat5, pat7, pat8))  # combine all patterns
        text = re.sub(combined_pat, "", text)
        text = re.sub(r'\s+', ' ', text)  # remove extra spaces
        return text.strip()

    def _parse_bytes(self, field: Union[str, ast.AST]) -> Union[str, ast.AST]:
      
        result = field
        try:
            result = ast.literal_eval(field)
        finally:
            return result.decode() if isinstance(result, bytes) else field

    def replace_emoticons(self, text: str) -> str:
      
        for emoticon, context in self.emoticons_dict.items():
            text = text.replace(emoticon, ' ' + context + ' ')
            text = re.sub(' +', ' ', text)
        return text

    def replace_emojis(self, text: str) -> str:
       
        for emoji, context in demoji.findall(text).items():
            text = text.replace(emoji, ' ' + context + ' ')
            text = re.sub(' +', ' ', text)
        return text

    def preprocess_data(self, df: pd.DataFrame) -> pd.DataFrame:
        
        df['tweet'] = df['tweet'].apply(self._parse_bytes)
        df['tweet'] = df['tweet'].str.normalize('NFKD')
        df['tweet'] = df['tweet'].map(self.clean_text)
        df['tweet'] = df['tweet'].apply(lambda x: self.replace_emoticons(x))
        df['tweet'] = df['tweet'].apply(lambda x: self.replace_emojis(x))
        return df

def preprocess_twitter_data(input_path: str = None, output_path: str = None, df: pd.DataFrame = None, emoticon_data_path: str = None):
  
    twitter_data_prep = TwitterDataPrep(emoticon_data_path=emoticon_data_path)
    
    if input_path:
        df = pd.read_csv(input_path)
    
    if df is not None:
        preprocessed_data = twitter_data_prep.preprocess_data(df)
        if output_path:
            preprocessed_data.to_csv(output_path, index=False)
        return preprocessed_data
    else:
        raise ValueError("'input_path' is not provided.")

3. Function for generating embeddings from textual data using OpenAI

This is the function used in section 4 to create embeddings from a textual tweets. This function uses Open-AI’s text-embedding-ada-002 model to create 1536 dimensional embedding space. The output numerical embeddings data will be stored as a csv file in the given ‘output_path’ argument.

create_embeddings <- function(input_path, output_path, api_key = Sys.getenv("OPENAI_API_KEY"), batch_size = 100) {
  df <- read.csv(input_path)
  get_embeddings <- function(texts) {
    tryCatch({
      response <- POST(
        url = "https://api.openai.com/v1/embeddings",
        add_headers(
          "Authorization" = paste("Bearer", api_key),
          "Content-Type" = "application/json"
        ),
        body = toJSON(list(
          model = "text-embedding-ada-002",
          input = texts
        ), auto_unbox = TRUE),
        encode = "json"
      )
      stop_for_status(response)
      content <- content(response, "parsed")
      return(lapply(content$data, function(item) item$embedding))
    }, error = function(e) {
      message("Error getting embeddings: ", e$message)
      return(NULL)
    })
  }
  
  # Create a progress bar
  num_batches <- ceiling(nrow(df) / batch_size)
  pb <- progress_bar$new(total = num_batches, format = "[:bar] :percent :eta")
  
  # Process tweets in batches using lapply
  batch_indices <- split(seq_len(nrow(df)), ceiling(seq_len(nrow(df))/batch_size))
  embeddings <- lapply(batch_indices, function(indices) {
    batch_embeddings <- get_embeddings(df$tweets[indices])
    pb$tick()
    return(batch_embeddings)
  })
  
  # Flatten the list of embeddings
  embeddings <- unlist(embeddings, recursive = FALSE)
  
  # Remove NULL values and convert embeddings list to dataframe
  embeddings <- Filter(Negate(is.null), embeddings)
  embeddings_df <- do.call(rbind, embeddings) %>% as.data.frame()
  colnames(embeddings_df) <- paste0("E", 1:ncol(embeddings_df))
  result_df <- cbind(df, embeddings_df)
  result_df <- result_df %>% mutate(across(everything(), as.character))
  write.csv(result_df, output_path, row.names = FALSE)
  message("Embeddings created and saved successfully.")
}

4. Function for generating Heatmaps for each column of a dataset

This is the function used in section 6.2 to plot the Heatmaps from the columns 1 to 50 of the dataset.

plot_columns <- function(dataset, no_of_cols, hvt.results) {
  cols_to_plot <- colnames(dataset)[no_of_cols]
  plots <- lapply(cols_to_plot, function(col_name) {
    print(plotHVT(hvt.results, hmap.cols = col_name, child.level = 1, 
                  plot.type = '2Dheatmap', centroid.size = 0.2))
  })
  invisible(plots)}

5. Function for writing clusters as .txt files

This is the function used in section 8 before making an API call for the cluster’s data. This function takes the tweets from each cluster and write it as .txt file externally which is later read while making the API call.

create_and_write_clusters <- function(cluster_numbers, cluster_method) {
  cluster_list <- lapply(cluster_numbers, function(i) {
    subset(cluster_method$names.column, cluster_method$clusters == i)
  })
  invisible(list2env(setNames(cluster_list, paste0("clust_", cluster_numbers)), envir = .GlobalEnv))
  write_cluster_to_file <- function(cluster, cluster_name) {
    processed_text <- paste0("'", 
                             gsub("[[:punct:]]", "", paste(cluster, collapse = " ")),
                             "'")
    filename <- file.path("./twitter_data/",paste0(cluster_name, ".txt"))
    writeLines(processed_text, filename)
  }
  cluster_names <- paste0("clust_", cluster_numbers)
  invisible(lapply(cluster_names, function(name) {
    cluster_data <- get(name)
    write_cluster_to_file(cluster_data, name)
  }))}

6. Function for making OpenAI-API Call to generate domain names

This is the function used in section 8 that runs the chat completions API to OpenAI’s gpt-3.5-turbo model where the content of each cluster is provided in the prompt, and the model is requested to generate a domain name for each cluster. This API call runs 10 iterations per cluster, to generate the domain names and provide the highly repeated value/domain name for that cluster.

get_mode <- function(v) {
  uniqv <- unique(v)
  uniqv[which.max(tabulate(match(v, uniqv)))]
}

get_cluster_domains <- function(k, iterations = 10) {
  url <- "http://172.25.1.85:31969/v1/chat/completions"
  read_file_content <- function(filename) {
 filepath <- file.path("./twitter_data/", filename)
     readLines(filepath, warn = FALSE)
 }
  
  cluster_names <- paste0("clust_", 1:k)
  content_list <- lapply(paste0(cluster_names, ".txt"), read_file_content)
  
  prompt <- "Suggest a single word/domain that best describes the topics in the following text. Your response should be just one word. This is for cluster {cluster_number}. Ensure that your suggestion is unique and not used for any other cluster. Previously used domains are: {used_domains}. Do not repeat any of these."
  
  send_request <- function(content, cluster_number, used_domains) {
    full_prompt <- gsub("\\{cluster_number\\}", cluster_number, prompt)
    full_prompt <- gsub("\\{used_domains\\}", paste(used_domains, collapse = ", "), full_prompt)
    full_content <- paste(full_prompt, content)
    
    request_body <- paste0('{
      "model": "gpt-3.5-turbo",
      "temperature":0,
      "messages": [
        {
          "role": "user",
          "content": "', full_content, '"
        }
      ]
    }')
    response <- POST(url,
                     body = request_body,
                     add_headers(.headers = c(
                       `Content-Type` = "application/json",
                       `Authorization` = "Bearer sk-musigma"
                     )),
                     content_type("application/json"))
    if (response$status_code == 200) {
      response_content <- content(response, as = "parsed")
      domain <- response_content$choices[[1]]$message$content
      domain
    } else {
      stop(paste("Error:", response$status_code, response$content))
    }
  }
  
  results <- matrix(nrow = iterations, ncol = k)
  colnames(results) <- paste0("Domain_name_for_cluster_", 1:k)
  
  results <- t(replicate(iterations, {
    used_domains <- character(0)
    unlist(lapply(seq_len(k), function(j) {
      domain <- send_request(content_list[[j]], j, used_domains)
      used_domains <<- c(used_domains, domain)
      domain
    }))
  }))
  
  results <- as.data.frame(results)
  colnames(results) <- paste0("Domain_name_for_cluster_", 1:k)  # Explicitly set column names

  mode_results <- sapply(results, get_mode)
  results <- rbind(results, Mode = mode_results)
  formatted_table <- displayTable(results)
  return(list(raw_data = results, formatted_table = formatted_table, mode = mode_results))
}

4. Dataset Understanding

4.1 Dataset Sourcing

  1. The textual corpus employed for generating the token embeddings in this notebook consists of tweets from top celebrities or Influencers on Twitter. This dataset is sourced from Kaggle

  2. The aforementioned link directs to Kaggle, where one can download a zip file containing individual CSV files of tweets from the top 915 celebrities.

Why Twitter Dataset?

The Twitter dataset used in this notebook is accessible to public (open source) that ensures transparency and reproducibility. Each tweet represents a discrete data point with complete contextual meaning and the dataset comprises unadulterated content generated by a diverse global user base, minimizing bias and enhancing the generalizability of our findings. Twitter’s platform, where users often share opinions on similar topics, presents an ideal scenario for evaluating the HVT package’s claim of grouping similar data points within the same cell.

4.2 Dataset Cleaning

  1. Using the Function to Merge and Slice the data from section 3.1, all the csvs are collated to a single csv and first 3 tweets of all the celebrities or influencers are extracted. This approach aims to diversify the dataset by including more celebrities or influencers, thus ensuring a broader range of content instead of repetitive contextual tweets.
merge_and_slice_data(directory = "./twitter_data/all_celebs_tweets_folder/" , output_path = "./twitter_data/sliced_data.csv")
  1. All the selected data from above step is given as an input to the Function to clean the twitter data from section 3.1 where the removal of encoded texts, tags and emoji to text conversion takes place. Returned will be the cleaned texts. The source of this function which is written in python traces back to the kaggle twitter data processing notebook.
preprocessed_data = preprocess_twitter_data( input_path='./twitter_data/sliced_data.csv',output_path='./twitter_data/cleaned_data.csv')
  1. In the subsequent step, we manually removed rows/tweets that are not in English, contain only emojis, and are incomplete. Below is a table displaying the first 10 rows of the manually selected data which will be used to get the embeddings data.
selected_csv <- read.csv("./twitter_data/selected_data.csv")
selected_csv <- selected_csv[,-1] #removing the s.no column
displayTable(selected_csv[1:10,])
tweets username
Metaverse VR creates own Ecosystem with P2E Games and NFT Technology. is high potential hidden gem project. Join the world now ! 10Ronaldinho
“HONEY” out now! 1LoganHenderson
Smile for the camera, guys! has released the music video for their new song “Fall.” 1LoganHenderson
Thank you to everyone for your continued support and I can’t wait to see you on tour! 1LoganHenderson
Happy July woman’s sandal woman’s sandal woman’s sandal wuv you 3gerardpique
Happy birthday, Curtis! Love, Marshall 4everBrandy
BMF fire Whole cast and crew. This is so dope, My legacy will be strong flexed biceps: medium skin tone eyes look at how many people i’m putting on. this is just 1 show GLG vertical traffic light GreenLightGang 4everBrandy
Joining tomorrow morning for a special performance blossom 7am local time on 50cent
You gotta do something evil to do something good. 50cent
Can’t wait to see the results! Let’s get the show on the road guys party popper 5SOS


4.3 Create Embeddings of the Dataset using OpenAI

  1. The cleaned data is then transformed into token embeddings using OpenAI’s text-embedding-ada-002 model, which generates 1536-dimensional data.

  2. The Function for generating embeddings from textual data mentioned in section 3.1 follows all the below given steps for the process of converting textual data to numerical embeddings.

Steps of Embeddings Generation

Step 1 Initialization and Data Loading:

The function starts by loading the input tweets dataset (.csv format) with the API key. (Users can store their API key in system environment and the same will be imported for the API call)

Step 2 Embedding Generation via API:

A helper function get_embeddings is used to interact with OpenAI via embeddings API url to processes the input dataset in batches, handles API errors (if any) and stores the returned numerical embeddings.

Step 3 Data Processing and Transformation:

Filters out any NULL responses from the API calls and converts the list of numerical embeddings into a dataframe corresponding to each tweet.

Step 4 Result Compilation and Storage:

Writes the combined dataframe row-wise (tweets + embeddings) to a new CSV file at the specified output path.

create_embeddings(input_path = "./twitter_data/selected_data.csv" output_path = "./twitter_data/final_embeddings.csv")

Here we load the generated embeddings dataset. Below is a table displaying the first 10 rows of the embeddings dataset generated from cleaned tweets.

data <- read.csv("./twitter_data/final_embeddings.csv")
displayTable(data[1:10,])
X tweets username E1 E2 E3 E4 E5 E6 E7 E8 E9 E10 E11 E12 E13 E14 E15 E16 E17 E18 E19 E20 E21 E22 E23 E24 E25 E26 E27 E28 E29 E30 E31 E32 E33 E34 E35 E36 E37 E38 E39 E40 E41 E42 E43 E44 E45 E46 E47 E48 E49 E50 E51 E52 E53 E54 E55 E56 E57 E58 E59 E60 E61 E62 E63 E64 E65 E66 E67 E68 E69 E70 E71 E72 E73 E74 E75 E76 E77 E78 E79 E80 E81 E82 E83 E84 E85 E86 E87 E88 E89 E90 E91 E92 E93 E94 E95 E96 E97 E98 E99 E100 E101 E102 E103 E104 E105 E106 E107 E108 E109 E110 E111 E112 E113 E114 E115 E116 E117 E118 E119 E120 E121 E122 E123 E124 E125 E126 E127 E128 E129 E130 E131 E132 E133 E134 E135 E136 E137 E138 E139 E140 E141 E142 E143 E144 E145 E146 E147 E148 E149 E150 E151 E152 E153 E154 E155 E156 E157 E158 E159 E160 E161 E162 E163 E164 E165 E166 E167 E168 E169 E170 E171 E172 E173 E174 E175 E176 E177 E178 E179 E180 E181 E182 E183 E184 E185 E186 E187 E188 E189 E190 E191 E192 E193 E194 E195 E196 E197 E198 E199 E200 E201 E202 E203 E204 E205 E206 E207 E208 E209 E210 E211 E212 E213 E214 E215 E216 E217 E218 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1 Metaverse VR creates own Ecosystem with P2E Games and NFT Technology. is high potential hidden gem project. 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6 Happy birthday, Curtis! 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7 BMF fire Whole cast and crew. 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10 Can’t wait to see the results! 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4.4 Datasets for HVT Analysis

This section showcases two datasets that are subsequently utilized in this notebook within the functions of the HVT package.

dataset_raw <- data[,-1] #removing the s.no column
dataset <- data[,-c(1:3)] #removing the categorical column
  1. dataset_raw - The embeddings dataset with all the columns that includes both categorical and numerical.
dim(dataset_raw)
## [1]  297 1538

The dimension of dataset_raw is 297 rows and 1538 columns (1536 numerical + 2 categorical columns).

Here we are displaying the dataset_raw with the first 10 rows and 1538 columns.

displayTable(dataset_raw[1:10,])
tweets username E1 E2 E3 E4 E5 E6 E7 E8 E9 E10 E11 E12 E13 E14 E15 E16 E17 E18 E19 E20 E21 E22 E23 E24 E25 E26 E27 E28 E29 E30 E31 E32 E33 E34 E35 E36 E37 E38 E39 E40 E41 E42 E43 E44 E45 E46 E47 E48 E49 E50 E51 E52 E53 E54 E55 E56 E57 E58 E59 E60 E61 E62 E63 E64 E65 E66 E67 E68 E69 E70 E71 E72 E73 E74 E75 E76 E77 E78 E79 E80 E81 E82 E83 E84 E85 E86 E87 E88 E89 E90 E91 E92 E93 E94 E95 E96 E97 E98 E99 E100 E101 E102 E103 E104 E105 E106 E107 E108 E109 E110 E111 E112 E113 E114 E115 E116 E117 E118 E119 E120 E121 E122 E123 E124 E125 E126 E127 E128 E129 E130 E131 E132 E133 E134 E135 E136 E137 E138 E139 E140 E141 E142 E143 E144 E145 E146 E147 E148 E149 E150 E151 E152 E153 E154 E155 E156 E157 E158 E159 E160 E161 E162 E163 E164 E165 E166 E167 E168 E169 E170 E171 E172 E173 E174 E175 E176 E177 E178 E179 E180 E181 E182 E183 E184 E185 E186 E187 E188 E189 E190 E191 E192 E193 E194 E195 E196 E197 E198 E199 E200 E201 E202 E203 E204 E205 E206 E207 E208 E209 E210 E211 E212 E213 E214 E215 E216 E217 E218 E219 E220 E221 E222 E223 E224 E225 E226 E227 E228 E229 E230 E231 E232 E233 E234 E235 E236 E237 E238 E239 E240 E241 E242 E243 E244 E245 E246 E247 E248 E249 E250 E251 E252 E253 E254 E255 E256 E257 E258 E259 E260 E261 E262 E263 E264 E265 E266 E267 E268 E269 E270 E271 E272 E273 E274 E275 E276 E277 E278 E279 E280 E281 E282 E283 E284 E285 E286 E287 E288 E289 E290 E291 E292 E293 E294 E295 E296 E297 E298 E299 E300 E301 E302 E303 E304 E305 E306 E307 E308 E309 E310 E311 E312 E313 E314 E315 E316 E317 E318 E319 E320 E321 E322 E323 E324 E325 E326 E327 E328 E329 E330 E331 E332 E333 E334 E335 E336 E337 E338 E339 E340 E341 E342 E343 E344 E345 E346 E347 E348 E349 E350 E351 E352 E353 E354 E355 E356 E357 E358 E359 E360 E361 E362 E363 E364 E365 E366 E367 E368 E369 E370 E371 E372 E373 E374 E375 E376 E377 E378 E379 E380 E381 E382 E383 E384 E385 E386 E387 E388 E389 E390 E391 E392 E393 E394 E395 E396 E397 E398 E399 E400 E401 E402 E403 E404 E405 E406 E407 E408 E409 E410 E411 E412 E413 E414 E415 E416 E417 E418 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E619 E620 E621 E622 E623 E624 E625 E626 E627 E628 E629 E630 E631 E632 E633 E634 E635 E636 E637 E638 E639 E640 E641 E642 E643 E644 E645 E646 E647 E648 E649 E650 E651 E652 E653 E654 E655 E656 E657 E658 E659 E660 E661 E662 E663 E664 E665 E666 E667 E668 E669 E670 E671 E672 E673 E674 E675 E676 E677 E678 E679 E680 E681 E682 E683 E684 E685 E686 E687 E688 E689 E690 E691 E692 E693 E694 E695 E696 E697 E698 E699 E700 E701 E702 E703 E704 E705 E706 E707 E708 E709 E710 E711 E712 E713 E714 E715 E716 E717 E718 E719 E720 E721 E722 E723 E724 E725 E726 E727 E728 E729 E730 E731 E732 E733 E734 E735 E736 E737 E738 E739 E740 E741 E742 E743 E744 E745 E746 E747 E748 E749 E750 E751 E752 E753 E754 E755 E756 E757 E758 E759 E760 E761 E762 E763 E764 E765 E766 E767 E768 E769 E770 E771 E772 E773 E774 E775 E776 E777 E778 E779 E780 E781 E782 E783 E784 E785 E786 E787 E788 E789 E790 E791 E792 E793 E794 E795 E796 E797 E798 E799 E800 E801 E802 E803 E804 E805 E806 E807 E808 E809 E810 E811 E812 E813 E814 E815 E816 E817 E818 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E1016 E1017 E1018 E1019 E1020 E1021 E1022 E1023 E1024 E1025 E1026 E1027 E1028 E1029 E1030 E1031 E1032 E1033 E1034 E1035 E1036 E1037 E1038 E1039 E1040 E1041 E1042 E1043 E1044 E1045 E1046 E1047 E1048 E1049 E1050 E1051 E1052 E1053 E1054 E1055 E1056 E1057 E1058 E1059 E1060 E1061 E1062 E1063 E1064 E1065 E1066 E1067 E1068 E1069 E1070 E1071 E1072 E1073 E1074 E1075 E1076 E1077 E1078 E1079 E1080 E1081 E1082 E1083 E1084 E1085 E1086 E1087 E1088 E1089 E1090 E1091 E1092 E1093 E1094 E1095 E1096 E1097 E1098 E1099 E1100 E1101 E1102 E1103 E1104 E1105 E1106 E1107 E1108 E1109 E1110 E1111 E1112 E1113 E1114 E1115 E1116 E1117 E1118 E1119 E1120 E1121 E1122 E1123 E1124 E1125 E1126 E1127 E1128 E1129 E1130 E1131 E1132 E1133 E1134 E1135 E1136 E1137 E1138 E1139 E1140 E1141 E1142 E1143 E1144 E1145 E1146 E1147 E1148 E1149 E1150 E1151 E1152 E1153 E1154 E1155 E1156 E1157 E1158 E1159 E1160 E1161 E1162 E1163 E1164 E1165 E1166 E1167 E1168 E1169 E1170 E1171 E1172 E1173 E1174 E1175 E1176 E1177 E1178 E1179 E1180 E1181 E1182 E1183 E1184 E1185 E1186 E1187 E1188 E1189 E1190 E1191 E1192 E1193 E1194 E1195 E1196 E1197 E1198 E1199 E1200 E1201 E1202 E1203 E1204 E1205 E1206 E1207 E1208 E1209 E1210 E1211 E1212 E1213 E1214 E1215 E1216 E1217 E1218 E1219 E1220 E1221 E1222 E1223 E1224 E1225 E1226 E1227 E1228 E1229 E1230 E1231 E1232 E1233 E1234 E1235 E1236 E1237 E1238 E1239 E1240 E1241 E1242 E1243 E1244 E1245 E1246 E1247 E1248 E1249 E1250 E1251 E1252 E1253 E1254 E1255 E1256 E1257 E1258 E1259 E1260 E1261 E1262 E1263 E1264 E1265 E1266 E1267 E1268 E1269 E1270 E1271 E1272 E1273 E1274 E1275 E1276 E1277 E1278 E1279 E1280 E1281 E1282 E1283 E1284 E1285 E1286 E1287 E1288 E1289 E1290 E1291 E1292 E1293 E1294 E1295 E1296 E1297 E1298 E1299 E1300 E1301 E1302 E1303 E1304 E1305 E1306 E1307 E1308 E1309 E1310 E1311 E1312 E1313 E1314 E1315 E1316 E1317 E1318 E1319 E1320 E1321 E1322 E1323 E1324 E1325 E1326 E1327 E1328 E1329 E1330 E1331 E1332 E1333 E1334 E1335 E1336 E1337 E1338 E1339 E1340 E1341 E1342 E1343 E1344 E1345 E1346 E1347 E1348 E1349 E1350 E1351 E1352 E1353 E1354 E1355 E1356 E1357 E1358 E1359 E1360 E1361 E1362 E1363 E1364 E1365 E1366 E1367 E1368 E1369 E1370 E1371 E1372 E1373 E1374 E1375 E1376 E1377 E1378 E1379 E1380 E1381 E1382 E1383 E1384 E1385 E1386 E1387 E1388 E1389 E1390 E1391 E1392 E1393 E1394 E1395 E1396 E1397 E1398 E1399 E1400 E1401 E1402 E1403 E1404 E1405 E1406 E1407 E1408 E1409 E1410 E1411 E1412 E1413 E1414 E1415 E1416 E1417 E1418 E1419 E1420 E1421 E1422 E1423 E1424 E1425 E1426 E1427 E1428 E1429 E1430 E1431 E1432 E1433 E1434 E1435 E1436 E1437 E1438 E1439 E1440 E1441 E1442 E1443 E1444 E1445 E1446 E1447 E1448 E1449 E1450 E1451 E1452 E1453 E1454 E1455 E1456 E1457 E1458 E1459 E1460 E1461 E1462 E1463 E1464 E1465 E1466 E1467 E1468 E1469 E1470 E1471 E1472 E1473 E1474 E1475 E1476 E1477 E1478 E1479 E1480 E1481 E1482 E1483 E1484 E1485 E1486 E1487 E1488 E1489 E1490 E1491 E1492 E1493 E1494 E1495 E1496 E1497 E1498 E1499 E1500 E1501 E1502 E1503 E1504 E1505 E1506 E1507 E1508 E1509 E1510 E1511 E1512 E1513 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Metaverse VR creates own Ecosystem with P2E Games and NFT Technology. is high potential hidden gem project. 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Smile for the camera, guys! has released the music video for their new song “Fall.” 1LoganHenderson -0.0007 -0.0086 0.0053 -0.0180 0.0166 0.0412 0.0013 -0.0157 -0.0012 -0.0128 0.0142 -0.0081 -0.0248 -0.0266 0.0238 -0.0068 0.0180 -0.0340 0.0141 -0.0166 0.0002 0.0109 0.0040 -0.0074 -0.0148 -0.0050 0.0219 -0.0149 0.0293 0.0117 -0.0025 -0.0018 -0.0220 -0.0281 -0.0229 -0.0050 -0.0303 -0.0111 -0.0013 -0.0146 0.0017 -0.0197 -0.0020 -0.0373 -0.0160 -0.0040 -0.0032 -0.0128 -0.0043 0.0037 -0.0077 -0.0021 -0.0243 -0.0274 0.0099 -0.0063 0.0148 0.0122 0.0152 0.0096 -0.0303 0.0090 0.0019 -0.0060 -0.0098 -0.0039 -0.0184 -0.0060 -0.0183 0.0035 -0.0064 0.0037 0.0198 -0.0100 0.0082 -0.0046 0.0041 -0.0177 0.0220 0.0292 0.0319 -0.0234 -0.0207 0.0445 -0.0007 0.0039 -0.0376 0.0255 0.0094 -0.0205 0.0064 0.0065 0.0260 -0.0146 -0.0033 0.0322 0.0104 0.0136 -0.0072 0.0011 -0.0014 0.0005 0.0099 0.0179 0.0059 -0.0329 -0.0121 -0.0095 0.0066 -0.0193 -0.0358 0.0027 -0.0137 -0.0399 -0.0043 0.0055 0.0032 -0.0091 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Happy birthday, Curtis! 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BMF fire Whole cast and crew. 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Can’t wait to see the results! 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  1. dataset - The embeddings dataset with only numerical columns.
dim(dataset)
## [1]  297 1536

The dimension of dataset is 297 rows and 1536 columns (only numerical).

Here we are displaying dataset with the first 10 rows and 1536 columns.

displayTable(dataset[1:10,])
E1 E2 E3 E4 E5 E6 E7 E8 E9 E10 E11 E12 E13 E14 E15 E16 E17 E18 E19 E20 E21 E22 E23 E24 E25 E26 E27 E28 E29 E30 E31 E32 E33 E34 E35 E36 E37 E38 E39 E40 E41 E42 E43 E44 E45 E46 E47 E48 E49 E50 E51 E52 E53 E54 E55 E56 E57 E58 E59 E60 E61 E62 E63 E64 E65 E66 E67 E68 E69 E70 E71 E72 E73 E74 E75 E76 E77 E78 E79 E80 E81 E82 E83 E84 E85 E86 E87 E88 E89 E90 E91 E92 E93 E94 E95 E96 E97 E98 E99 E100 E101 E102 E103 E104 E105 E106 E107 E108 E109 E110 E111 E112 E113 E114 E115 E116 E117 E118 E119 E120 E121 E122 E123 E124 E125 E126 E127 E128 E129 E130 E131 E132 E133 E134 E135 E136 E137 E138 E139 E140 E141 E142 E143 E144 E145 E146 E147 E148 E149 E150 E151 E152 E153 E154 E155 E156 E157 E158 E159 E160 E161 E162 E163 E164 E165 E166 E167 E168 E169 E170 E171 E172 E173 E174 E175 E176 E177 E178 E179 E180 E181 E182 E183 E184 E185 E186 E187 E188 E189 E190 E191 E192 E193 E194 E195 E196 E197 E198 E199 E200 E201 E202 E203 E204 E205 E206 E207 E208 E209 E210 E211 E212 E213 E214 E215 E216 E217 E218 E219 E220 E221 E222 E223 E224 E225 E226 E227 E228 E229 E230 E231 E232 E233 E234 E235 E236 E237 E238 E239 E240 E241 E242 E243 E244 E245 E246 E247 E248 E249 E250 E251 E252 E253 E254 E255 E256 E257 E258 E259 E260 E261 E262 E263 E264 E265 E266 E267 E268 E269 E270 E271 E272 E273 E274 E275 E276 E277 E278 E279 E280 E281 E282 E283 E284 E285 E286 E287 E288 E289 E290 E291 E292 E293 E294 E295 E296 E297 E298 E299 E300 E301 E302 E303 E304 E305 E306 E307 E308 E309 E310 E311 E312 E313 E314 E315 E316 E317 E318 E319 E320 E321 E322 E323 E324 E325 E326 E327 E328 E329 E330 E331 E332 E333 E334 E335 E336 E337 E338 E339 E340 E341 E342 E343 E344 E345 E346 E347 E348 E349 E350 E351 E352 E353 E354 E355 E356 E357 E358 E359 E360 E361 E362 E363 E364 E365 E366 E367 E368 E369 E370 E371 E372 E373 E374 E375 E376 E377 E378 E379 E380 E381 E382 E383 E384 E385 E386 E387 E388 E389 E390 E391 E392 E393 E394 E395 E396 E397 E398 E399 E400 E401 E402 E403 E404 E405 E406 E407 E408 E409 E410 E411 E412 E413 E414 E415 E416 E417 E418 E419 E420 E421 E422 E423 E424 E425 E426 E427 E428 E429 E430 E431 E432 E433 E434 E435 E436 E437 E438 E439 E440 E441 E442 E443 E444 E445 E446 E447 E448 E449 E450 E451 E452 E453 E454 E455 E456 E457 E458 E459 E460 E461 E462 E463 E464 E465 E466 E467 E468 E469 E470 E471 E472 E473 E474 E475 E476 E477 E478 E479 E480 E481 E482 E483 E484 E485 E486 E487 E488 E489 E490 E491 E492 E493 E494 E495 E496 E497 E498 E499 E500 E501 E502 E503 E504 E505 E506 E507 E508 E509 E510 E511 E512 E513 E514 E515 E516 E517 E518 E519 E520 E521 E522 E523 E524 E525 E526 E527 E528 E529 E530 E531 E532 E533 E534 E535 E536 E537 E538 E539 E540 E541 E542 E543 E544 E545 E546 E547 E548 E549 E550 E551 E552 E553 E554 E555 E556 E557 E558 E559 E560 E561 E562 E563 E564 E565 E566 E567 E568 E569 E570 E571 E572 E573 E574 E575 E576 E577 E578 E579 E580 E581 E582 E583 E584 E585 E586 E587 E588 E589 E590 E591 E592 E593 E594 E595 E596 E597 E598 E599 E600 E601 E602 E603 E604 E605 E606 E607 E608 E609 E610 E611 E612 E613 E614 E615 E616 E617 E618 E619 E620 E621 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0.0012 0.0382 0.0153 0.0007 0.0127 0.0534 0.0042 0.0102 0.0109 0.0036 -0.0039 0.0320 -0.0372 0.0041 0.0304 0.0078 -0.0048 0.0084 0.0065 0.0319 -0.0159 0.0140 0.0241 -0.0279 -0.0020 0.0012 -0.0312 0.0288 -0.0154 0.0024 0.0070 0.0201 0.0085 -0.0102 -0.0479 -0.0028 0.0133 0.0403 0.0207 -0.0004 0.0121 -0.0138 0.0083 0.0091 0.0425 -0.0023 -0.0250 0.0013 -0.0110 -0.0123 -0.1179 -0.0496 0.0055 0.0221 0.0136 0.0054 0.0206 0.0065 -0.0226 0.0325 0.0160 -0.0174 0.0175 -0.0159 0.0208 -0.0001 -0.0096 -0.0187 -0.0077 0.0159 -0.0339 -0.0008 0.0043 0.0090 -0.0170 0.0020 -0.0008 0.0235 0.0032 -0.0020 0.0113 -0.0121 0.0137 -0.0182 -0.0066 -0.0061 -0.0370 -0.0189 -0.0019 -0.0476 0.0039 -0.0111 -0.0065 -0.0198 -0.0212 -0.0170 -0.0096 0.0299 0.0074 -0.0275 -0.0352 0.0112 -0.0161 -0.0151 0.0077 -0.0307 0.0043 -0.0057 -0.0300 0.0127 -0.0002 -0.0001 -0.0027 0.0048 -0.0292 -0.0086 -0.0118 -0.0281 0.0091 -0.0067 -0.0423 0.0024 -0.0245 0.0359 -0.0313 0.0065 -0.0058 -0.0127 0.0278 0.0022 -0.0113 -0.0175 0.0350 -0.0097 0.0385 0.0216 0.0114 -0.0075 -0.0055 0.0022 -0.0247 0.0051 0.0123 0.0016 -0.0273 0.0102 0.0056 -0.0024 0.0013 0.0043 -0.0270 -0.0374 -0.0543 0.0249 -0.0093 -0.0018 0.0012 0.0060 0.0074 0.0035 0.0193 -0.0038 -0.0078 0.0141 -0.0256 -0.0074 -0.0206 -0.0158 0.0141 0.0080 -0.0077 -0.0288 0.0139 0.0036 0.0163 0.0110 -0.0098 0.0023 -0.0115 0.0091 -0.0122 -0.0272 0.0106 -0.0357 -0.0385 0.0150 0.0194 -0.0441 -0.0036 0.0216 -0.0013 -0.0216 -0.0213 -0.0171 -0.0149 -0.0194 -0.0006 -0.0069 -0.0059 -0.0139 0.0091 0.0164 0.0144 0.0297 0.0047 -0.0289 -0.0102 -0.0292 0.0072 0.0024 -0.0073 -0.0397 0.0310 0.0034 0.0219 0.0141 0.0331 -0.0044 0.0016 -0.0182 0.0056 -0.0050 -0.0017 -0.0036 0.0099 0.0277 0.0313 0.0130 -0.0197 -0.0196 -0.0179 0.0159 0.0003 -0.0067 -0.0240 0.0140 0.0115 0.0185 -0.0134 0.0116 -0.0276 0.0035 -0.0325 -0.0017 -0.0016 0.0123 0.0162 0.0314 0.0090 0.0115 0.0177 0.0312 0.0151 0.0193 -0.0187 -0.0213 -0.0087 0.0127 -0.0123 -0.0252 0.0035 0.0064 0.0175 -0.0197 -0.0137 0.0163 -0.0042 -0.0077 0.0001 -0.0134 -0.0067 0.0121 0.0188 -0.0254 0.0111 0.0129 0.0050 0.0147 0.0319 -0.0128 -0.0014 -0.0096 -0.0068 -0.0070 -0.0058 -0.0249 0.0242 -0.0078 -0.0010 0.0100 -0.0468 0.0679 0.0214 0.0031 0.0212 -0.0090 0.0113 -0.0017 0.0109 -0.0188 0.0279 0.0190 0.0020 -0.0004 -0.0349 -0.0086 -0.0203 -0.0061 0.0220 -0.0334 -0.0122 0.0230 -0.0114 0.0226 0.0159 -0.0149 -0.0371 0.0128 0.0027 0.0121 -0.0412 0.0011 -0.0003 -0.0056 -0.0135 -0.0173 -0.0097 0.0082 -0.0164 0.0127 -0.0098 -0.0069 0.0193 -0.0343 -0.0260 -0.0030 0.0146 -0.0194 0.0164 -0.0270

4.5 EDA Plots

This section displays four objects.

Variable Histograms: The histogram distribution of all the features in the dataset.

Box Plots: Box plots for all the features in the dataset. These plots will display the median and Interquartile range of each column at a panel level.

Correlation Matrix: This calculates the Pearson correlation which is a bivariate correlation value measuring the linear correlation between two numeric columns. The output plot is shown as a matrix.

Summary EDA: The table provides descriptive statistics for all the features in the dataset.

It uses an inbuilt function called edaPlots to display the above-mentioned four objects.

NOTE: Here we have used the dataset object which has 297 rows and 1536 numerical columns. The input dataset should be a data frame object and the columns should be only numeric type. For demo purposes, we are plotting the EDA plots only for the first 10 columns except for the summary table which will be displayed for all columns.


Histograms

edaPlots(dataset, output_type = "histogram", n_cols = 10)



Box Plots

edaPlots(dataset, output_type = "boxplot", n_cols = 10)



Correlation Plot

edaPlots(dataset, output_type = "correlation",n_cols = 10)



Summary Table

edaPlots(dataset, output_type = "summary", n_cols = 1536)



5. Model Training

The core function for compression in the workflow is hvq (hierarchical vector quantization), which is called within the trainHVT function. We have a parameter called ‘quantization error’. This parameter acts as a threshold and determines the number of levels in the hierarchy. It means that, if there are ‘n’ number of levels in the hierarchy, then all the clusters formed till this level will have quantization error equal to or less than the threshold quantization error. The user can define the number of clusters in the first level of the hierarchy, and then each cluster in the upcoming levels is subdivided into the same number of clusters. This process continues for all the clusters until the threshold quantization error is met. The output of this technique will be hierarchically arranged vector quantized data.

5.1 Comprehending trainHVT() and its parameters

trainHVT(
  dataset,min_compression_perc,n_cells,
  depth,quant.err,normalize,distance_metric,
  dim_reduction_method,error_metric,quant_method,
  scale_summary, diagnose, hvt_validation,
  train_validation_split_ratio
)

Each of the parameters of trainHVT function has been explained below:

The output of the trainHVT function (list of 7 elements) has been explained below, with an image attached for clear understanding.


NOTE: Here the attached ‘Figure:2’ is the example snapshot of the output list generated from trainHVT()

Figure 2: The Output list generated by trainHVT function.

Figure 2: The Output list generated by trainHVT function.

We will use the trainHVT function to compress our data while preserving essential features of the dataset. Our goal is to achieve data compression to at least 80%. In situations where the compression ratio does not meet the desired target, we can explore adjusting the model parameters as a potential solution. This involves making modifications to parameters such as the quantization error threshold or increasing the number of cells, and then rerunning the trainHVT function.

5.1.1 Selection Criteria of distance_metric

Figure 3: Pictorial representation of Euclidean and Manhattan distance. [img source](https://www.omnicalculator.com/math/manhattan-distance)

Figure 3: Pictorial representation of Euclidean and Manhattan distance. img source

Euclidean Distance: The Euclidean distance is the root of the squared sum of the differences between the coordinates of two points. It is the geometric straight-line distance between points in a continuous vector space, as denoted by the red line in the above ‘Figure:3’. Euclidean distance is well-suited for applications where the data distribution exhibits spherical or ellipsoidal patterns, as it accurately models the physical distance in these domains.

Manhattan Distance: The Manhattan distance, is the sum of the absolute differences between the coordinates of two points. It measures the distance between points along each dimension or axis, as denoted by the blue and green lines in the above ‘Figure:3’. The Manhattan distance is particularly useful when the underlying space is constrained to a grid-like structure, as it models the distance traveled along the axes rather than the straight-line path.

Distance Metric Selection:

  • The choice of distance metric depends on the underlying structure of the dataset and the specific application requirements.

  • In high-dimensional spaces, such as those encountered in natural language processing tasks involving LLM token embeddings, the Manhattan distance is often preferred over the Euclidean distance. This preference arises due to the “curse of dimensionality,” where the Euclidean distance becomes increasingly concentrated/lower, leading to a loss of meaning as the dimensionality increases. The Manhattan distance, by summing the absolute differences along each dimension, retains its ability to capture dissimilarities between points in high-dimensional spaces.

  • Additionally, the grid-like structure inherent in high-dimensional data aligns well with the Manhattan distance’s component-wise calculation approach. This structural alignment allows the Manhattan distance to effectively capture the differences between data points based on their individual feature values or dimensions, which is particularly beneficial in applications like text analysis or document similarity.

  • In low-dimensional spaces and applications where the data distribution is continuous, the Euclidean distance is often preferred. Its geometric interpretation as the straight-line distance between points aligns well with physical spaces and continuous data distributions, making it a natural choice in domains such as spatial data analysis and clustering algorithms where the underlying relationships between data points are better represented by the Euclidean metric.

  • In summary, the selection of the appropriate distance metric is contingent upon the dimensionality of the data, the underlying data distribution and the geometric constraints of the problem domain.

5.1.2 Selection Criteria of error_metric

The error metric can be mean or max. For a specified number of cells, each cell will contain n data points. The centroid of each cell is computed as the mean of these n points. For each data point, the distance to the centroid is calculated using a chosen distance metric, resulting in n distances. The error metric for the cell is then determined based on these distances.

  • mean: If the mean is used as the input, the error metric or quantization error for the cell is the mean of the n distances.

  • max: If the max is used as the input, the error metric or quantization error for the cell is the maximum value of the n distances.

5.2 Model Training using trainHVT()

Input Parameters

hvt.results <- trainHVT(dataset, n_cells = 110, depth = 1, quant.err = 0.6, 
                        distance_metric = "L1_Norm", error_metric = "mean",
                        normalize = TRUE, quant_method="kmeans", dim_reduction_method = "sammon")
## Initial stress        : 0.54918
## stress after  10 iters: 0.34110, magic = 0.018
## stress after  20 iters: 0.27290, magic = 0.043
## stress after  30 iters: 0.21914, magic = 0.098
## stress after  40 iters: 0.13601, magic = 0.500
## stress after  50 iters: 0.13455, magic = 0.500
## stress after  60 iters: 0.13438, magic = 0.500
## stress after  70 iters: 0.13435, magic = 0.500

Let’s look at the summary table.

displayTable(data = hvt.results[[3]]$compression_summary)
segmentLevel noOfCells noOfCellsBelowQuantizationError percentOfCellsBelowQuantizationErrorThreshold parameters
1 110 93 0.85 n_cells: 110 quant.err: 0.6 distance_metric: L1_Norm error_metric: mean quant_method: kmeans

NOTE: Based on the above table, the value of ‘percentOfCellsBelowQuantizationErrorThreshold’ indicates that 85% of compression has taken place for the specified number of cells (110) and the quantization error (0.6)

Lets look at the dimensionality reduction evaluation metrics.

displayTable(data= hvt.results$model_info$distance_measures)
L1_Metrics L2_Metrics Value
Structure Preservation Metrics Trustworthiness 0.7246
Continuity 0.8510
Sammon’s Stress 0.0357
Distance Preservation Metrics RMSE 369.5849
Interpretive Quality Metrics Silhouette Score 0.3566
KNN Retention Score 0.1830
Computational Efficiency Metrics Execution Duration(sec) 0.0936

According to the values above we can conclude that the model has achieved a good balance between preserving the local neighborhood relationships, global structure, and pairwise distances in the low-dimensional embedding, with moderate reconstruction accuracy and cluster separation.


6. Visualization

This section focuses on projecting the plots that used Sammon’s dimensionality reduction algorithm to compress the dataset from higher to lower dimensions.

6.1 Comprehending plotHVT() and its parameters

plotHVT <-(hvt.results, line.width, color.vec, pch1, centroid.size ,
           title, maxDepth, child.level, hmap.cols, cell_id,
           quant.error.hmap, n_cells.hmap,
           label.size , sepration_width, layer_opacity,
           dim_size, plot.type) 

6.2 Visualizing using plotHVT()

6.2.1 plotHVT - 1D

Input parameters

  • hvt.results = trainHVT() results
  • plot.type = 1D plot
plotHVT(hvt.results,plot.type  = '1D')

6.2.2 plotHVT - 2D Projection

Input parameters

  • hvt.results = trainHVT() results
  • plot.type = 2D projection plot
plotHVT(hvt.results, plot.type = '2Dproj')

6.2.3 plotHVT - 2D HVT

Input parameters

  • hvt.results = trainHVT() results
  • plot.type = 2D Hierarchical voronoi tessellation plot
  • cell_id = TRUE (to display the cell id’s in plot)
plotHVT(hvt.results, plot.type  = '2Dhvt', cell_id = TRUE)

6.2.4 plotHVT - 2D HEATMAP

Input parameters

  • hvt.results = trainHVT() results
  • plot.type = 2D Heatmap
  • hmap.cols = ‘n’ (The no.of.entities/datapoints in each cell based on which the color gradient will be filled)
plotHVT(hvt.results,  hmap.cols = "n", plot.type  = '2Dheatmap', cell_id = TRUE)


Heatmaps for the columns

NOTE: We have 1536 columns displaying all of them will blow up the notebook hence we are displaying the Heatmaps for only the first 50 columns

The Function for generating Heatmaps for each column of a dataset from section 3.1 is used to generate the below Heatmaps.

Heatmaps for columns 1 to 17

Input parameters

  • dataset = The dataset with 297 rows and 1536 numerical columns
  • no_of_cols = the columns to be plotted[from 1 to 17]
  • hvt.results = trainHVT() results
no_of_cols <- 1:17
plot_columns(dataset, no_of_cols, hvt.results)

Heatmaps for columns 18 to 34

Input parameters

  • dataset = The dataset with 297 rows and 1536 numerical columns
  • no_of_cols = the columns to be plotted[from 18 to 34]
  • hvt.results = trainHVT() results
no_of_cols <- 18:34
plot_columns(dataset, no_of_cols, hvt.results)

Heatmaps for columns 35 to 50

Input parameters

  • dataset = The dataset with 297 rows and 1536 numerical columns
  • no_of_cols = the columns to be plotted[from 35 to 50]
  • hvt.results = trainHVT() results
no_of_cols <- 35:50
plot_columns(dataset, no_of_cols, hvt.results)

6.2.5 plotHVT - 2D Interactive surface plot

Input parameters

  • hvt.results = trainHVT() results
  • plot.type = Interactive surface plot
  • hmap.cols = ‘n’ (The no.of.entities/datapoints in each cell based on which the color gradient will be filled)
plotHVT(hvt.results, hmap.cols = "n", plot.type  = 'surface_plot')

7. Scoring

Once the model is built, the next step is to score each data point in the dataset, assigning it to a specific cell within the model using the scoreHVT function. This section of the vignette will elucidate how the dataset and the results from model training are utilized to determine the cell assignment of each data point of scoring dataset, identify anomalies, and calculate quantization errors.

Moreover, we will delve into the visualization plots that offer a comprehensive view of the contents within each cell, including the number of data points and their distribution. These insights are crucial for understanding the model’s structure and evaluating its performance in capturing the underlying patterns of the high-dimensional LLM token embeddings.

7.1 Comprehending scoreHVT() and its parameters

scoreHVT(dataset, hvt.results.model, child.level, mad.threshold, line.width, 
         color.vec, normalize, yVar, distance_metric, error_metric, 
         analysis.plots, names.column)

The parameters are explained below:

7.2 Model Scoring using scoreHVT()

Input Parameters

scoring <- scoreHVT(dataset_raw, hvt.results, mad.threshold = 0.7,normalize = T,
                    analysis.plots  = TRUE,names.column = dataset_raw[,1])

Let’s look at the scored model summary.

The model info displays five attributes which are explained below:

scoring$model_info$scored_model_summary
## $input_dataset
## [1] "297 Rows & 1536 Columns"
## 
## $scored_qe_range
## [1] "0 to 0.7145"
## 
## $mad.threshold
## [1] 0.7
## 
## $no_of_anomaly_datapoints
## [1] 1
## 
## $no_of_anomaly_cells
## [1] 1

Let’s look at the dimensions of scored table..

dim(scoring$scoredPredictedData)
## [1]  297 1545

It has 297 rows and 1545 columns (standard output 9 columns + 1536 input dataset columns)

Below are the newly added 9 columns after scoring using scoreHVT

Let’s look at the scored data table. For the sake of brevity we are displaying the first 10 rows and all the 1545 columns.

displayTable(scoring[["scoredPredictedData"]], value = 0.7) 
Segment.Level Segment.Parent Segment.Child n Cell.ID Quant.Error centroidRadius diff anomalyFlag E1 E2 E3 E4 E5 E6 E7 E8 E9 E10 E11 E12 E13 E14 E15 E16 E17 E18 E19 E20 E21 E22 E23 E24 E25 E26 E27 E28 E29 E30 E31 E32 E33 E34 E35 E36 E37 E38 E39 E40 E41 E42 E43 E44 E45 E46 E47 E48 E49 E50 E51 E52 E53 E54 E55 E56 E57 E58 E59 E60 E61 E62 E63 E64 E65 E66 E67 E68 E69 E70 E71 E72 E73 E74 E75 E76 E77 E78 E79 E80 E81 E82 E83 E84 E85 E86 E87 E88 E89 E90 E91 E92 E93 E94 E95 E96 E97 E98 E99 E100 E101 E102 E103 E104 E105 E106 E107 E108 E109 E110 E111 E112 E113 E114 E115 E116 E117 E118 E119 E120 E121 E122 E123 E124 E125 E126 E127 E128 E129 E130 E131 E132 E133 E134 E135 E136 E137 E138 E139 E140 E141 E142 E143 E144 E145 E146 E147 E148 E149 E150 E151 E152 E153 E154 E155 E156 E157 E158 E159 E160 E161 E162 E163 E164 E165 E166 E167 E168 E169 E170 E171 E172 E173 E174 E175 E176 E177 E178 E179 E180 E181 E182 E183 E184 E185 E186 E187 E188 E189 E190 E191 E192 E193 E194 E195 E196 E197 E198 E199 E200 E201 E202 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E403 E404 E405 E406 E407 E408 E409 E410 E411 E412 E413 E414 E415 E416 E417 E418 E419 E420 E421 E422 E423 E424 E425 E426 E427 E428 E429 E430 E431 E432 E433 E434 E435 E436 E437 E438 E439 E440 E441 E442 E443 E444 E445 E446 E447 E448 E449 E450 E451 E452 E453 E454 E455 E456 E457 E458 E459 E460 E461 E462 E463 E464 E465 E466 E467 E468 E469 E470 E471 E472 E473 E474 E475 E476 E477 E478 E479 E480 E481 E482 E483 E484 E485 E486 E487 E488 E489 E490 E491 E492 E493 E494 E495 E496 E497 E498 E499 E500 E501 E502 E503 E504 E505 E506 E507 E508 E509 E510 E511 E512 E513 E514 E515 E516 E517 E518 E519 E520 E521 E522 E523 E524 E525 E526 E527 E528 E529 E530 E531 E532 E533 E534 E535 E536 E537 E538 E539 E540 E541 E542 E543 E544 E545 E546 E547 E548 E549 E550 E551 E552 E553 E554 E555 E556 E557 E558 E559 E560 E561 E562 E563 E564 E565 E566 E567 E568 E569 E570 E571 E572 E573 E574 E575 E576 E577 E578 E579 E580 E581 E582 E583 E584 E585 E586 E587 E588 E589 E590 E591 E592 E593 E594 E595 E596 E597 E598 E599 E600 E601 E602 E603 E604 E605 E606 E607 E608 E609 E610 E611 E612 E613 E614 E615 E616 E617 E618 E619 E620 E621 E622 E623 E624 E625 E626 E627 E628 E629 E630 E631 E632 E633 E634 E635 E636 E637 E638 E639 E640 E641 E642 E643 E644 E645 E646 E647 E648 E649 E650 E651 E652 E653 E654 E655 E656 E657 E658 E659 E660 E661 E662 E663 E664 E665 E666 E667 E668 E669 E670 E671 E672 E673 E674 E675 E676 E677 E678 E679 E680 E681 E682 E683 E684 E685 E686 E687 E688 E689 E690 E691 E692 E693 E694 E695 E696 E697 E698 E699 E700 E701 E702 E703 E704 E705 E706 E707 E708 E709 E710 E711 E712 E713 E714 E715 E716 E717 E718 E719 E720 E721 E722 E723 E724 E725 E726 E727 E728 E729 E730 E731 E732 E733 E734 E735 E736 E737 E738 E739 E740 E741 E742 E743 E744 E745 E746 E747 E748 E749 E750 E751 E752 E753 E754 E755 E756 E757 E758 E759 E760 E761 E762 E763 E764 E765 E766 E767 E768 E769 E770 E771 E772 E773 E774 E775 E776 E777 E778 E779 E780 E781 E782 E783 E784 E785 E786 E787 E788 E789 E790 E791 E792 E793 E794 E795 E796 E797 E798 E799 E800 E801 E802 E803 E804 E805 E806 E807 E808 E809 E810 E811 E812 E813 E814 E815 E816 E817 E818 E819 E820 E821 E822 E823 E824 E825 E826 E827 E828 E829 E830 E831 E832 E833 E834 E835 E836 E837 E838 E839 E840 E841 E842 E843 E844 E845 E846 E847 E848 E849 E850 E851 E852 E853 E854 E855 E856 E857 E858 E859 E860 E861 E862 E863 E864 E865 E866 E867 E868 E869 E870 E871 E872 E873 E874 E875 E876 E877 E878 E879 E880 E881 E882 E883 E884 E885 E886 E887 E888 E889 E890 E891 E892 E893 E894 E895 E896 E897 E898 E899 E900 E901 E902 E903 E904 E905 E906 E907 E908 E909 E910 E911 E912 E913 E914 E915 E916 E917 E918 E919 E920 E921 E922 E923 E924 E925 E926 E927 E928 E929 E930 E931 E932 E933 E934 E935 E936 E937 E938 E939 E940 E941 E942 E943 E944 E945 E946 E947 E948 E949 E950 E951 E952 E953 E954 E955 E956 E957 E958 E959 E960 E961 E962 E963 E964 E965 E966 E967 E968 E969 E970 E971 E972 E973 E974 E975 E976 E977 E978 E979 E980 E981 E982 E983 E984 E985 E986 E987 E988 E989 E990 E991 E992 E993 E994 E995 E996 E997 E998 E999 E1000 E1001 E1002 E1003 E1004 E1005 E1006 E1007 E1008 E1009 E1010 E1011 E1012 E1013 E1014 E1015 E1016 E1017 E1018 E1019 E1020 E1021 E1022 E1023 E1024 E1025 E1026 E1027 E1028 E1029 E1030 E1031 E1032 E1033 E1034 E1035 E1036 E1037 E1038 E1039 E1040 E1041 E1042 E1043 E1044 E1045 E1046 E1047 E1048 E1049 E1050 E1051 E1052 E1053 E1054 E1055 E1056 E1057 E1058 E1059 E1060 E1061 E1062 E1063 E1064 E1065 E1066 E1067 E1068 E1069 E1070 E1071 E1072 E1073 E1074 E1075 E1076 E1077 E1078 E1079 E1080 E1081 E1082 E1083 E1084 E1085 E1086 E1087 E1088 E1089 E1090 E1091 E1092 E1093 E1094 E1095 E1096 E1097 E1098 E1099 E1100 E1101 E1102 E1103 E1104 E1105 E1106 E1107 E1108 E1109 E1110 E1111 E1112 E1113 E1114 E1115 E1116 E1117 E1118 E1119 E1120 E1121 E1122 E1123 E1124 E1125 E1126 E1127 E1128 E1129 E1130 E1131 E1132 E1133 E1134 E1135 E1136 E1137 E1138 E1139 E1140 E1141 E1142 E1143 E1144 E1145 E1146 E1147 E1148 E1149 E1150 E1151 E1152 E1153 E1154 E1155 E1156 E1157 E1158 E1159 E1160 E1161 E1162 E1163 E1164 E1165 E1166 E1167 E1168 E1169 E1170 E1171 E1172 E1173 E1174 E1175 E1176 E1177 E1178 E1179 E1180 E1181 E1182 E1183 E1184 E1185 E1186 E1187 E1188 E1189 E1190 E1191 E1192 E1193 E1194 E1195 E1196 E1197 E1198 E1199 E1200 E1201 E1202 E1203 E1204 E1205 E1206 E1207 E1208 E1209 E1210 E1211 E1212 E1213 E1214 E1215 E1216 E1217 E1218 E1219 E1220 E1221 E1222 E1223 E1224 E1225 E1226 E1227 E1228 E1229 E1230 E1231 E1232 E1233 E1234 E1235 E1236 E1237 E1238 E1239 E1240 E1241 E1242 E1243 E1244 E1245 E1246 E1247 E1248 E1249 E1250 E1251 E1252 E1253 E1254 E1255 E1256 E1257 E1258 E1259 E1260 E1261 E1262 E1263 E1264 E1265 E1266 E1267 E1268 E1269 E1270 E1271 E1272 E1273 E1274 E1275 E1276 E1277 E1278 E1279 E1280 E1281 E1282 E1283 E1284 E1285 E1286 E1287 E1288 E1289 E1290 E1291 E1292 E1293 E1294 E1295 E1296 E1297 E1298 E1299 E1300 E1301 E1302 E1303 E1304 E1305 E1306 E1307 E1308 E1309 E1310 E1311 E1312 E1313 E1314 E1315 E1316 E1317 E1318 E1319 E1320 E1321 E1322 E1323 E1324 E1325 E1326 E1327 E1328 E1329 E1330 E1331 E1332 E1333 E1334 E1335 E1336 E1337 E1338 E1339 E1340 E1341 E1342 E1343 E1344 E1345 E1346 E1347 E1348 E1349 E1350 E1351 E1352 E1353 E1354 E1355 E1356 E1357 E1358 E1359 E1360 E1361 E1362 E1363 E1364 E1365 E1366 E1367 E1368 E1369 E1370 E1371 E1372 E1373 E1374 E1375 E1376 E1377 E1378 E1379 E1380 E1381 E1382 E1383 E1384 E1385 E1386 E1387 E1388 E1389 E1390 E1391 E1392 E1393 E1394 E1395 E1396 E1397 E1398 E1399 E1400 E1401 E1402 E1403 E1404 E1405 E1406 E1407 E1408 E1409 E1410 E1411 E1412 E1413 E1414 E1415 E1416 E1417 E1418 E1419 E1420 E1421 E1422 E1423 E1424 E1425 E1426 E1427 E1428 E1429 E1430 E1431 E1432 E1433 E1434 E1435 E1436 E1437 E1438 E1439 E1440 E1441 E1442 E1443 E1444 E1445 E1446 E1447 E1448 E1449 E1450 E1451 E1452 E1453 E1454 E1455 E1456 E1457 E1458 E1459 E1460 E1461 E1462 E1463 E1464 E1465 E1466 E1467 E1468 E1469 E1470 E1471 E1472 E1473 E1474 E1475 E1476 E1477 E1478 E1479 E1480 E1481 E1482 E1483 E1484 E1485 E1486 E1487 E1488 E1489 E1490 E1491 E1492 E1493 E1494 E1495 E1496 E1497 E1498 E1499 E1500 E1501 E1502 E1503 E1504 E1505 E1506 E1507 E1508 E1509 E1510 E1511 E1512 E1513 E1514 E1515 E1516 E1517 E1518 E1519 E1520 E1521 E1522 E1523 E1524 E1525 E1526 E1527 E1528 E1529 E1530 E1531 E1532 E1533 E1534 E1535 E1536
1 1 5 1 25 0.627 963.0354 962.4084 0 1.9939 -0.9525 -0.9958 -1.4879 0.9279 -1.1706 -0.6112 -0.6424 -0.6077 -0.5544 -1.4186 0.3833 -0.0678 -1.8291 1.7712 0.6467 -1.0899 -1.2759 0.0598 0.8509 1.6249 0.5913 -0.6148 -1.6117 0.9092 2.3868 -1.6836 0.7008 0.3531 1.7399 -0.6863 -0.6318 0.8735 0.0191 0.9682 0.7636 1.9564 0.7763 0.0908 1.4630 0.6363 0.4422 -1.2549 0.3111 -0.8109 -1.1731 -0.9796 2.2764 1.1180 -1.5735 -1.7979 0.5459 1.5403 0.0858 -1.4565 -0.1067 0.5849 -1.4224 -0.5522 -0.6734 0.9785 -1.7093 -0.7581 0.0642 -1.5196 0.4309 0.3774 1.4723 -0.2181 -0.5304 -0.2088 0.2279 1.9701 0.3995 0.7389 2.1273 1.4518 0.0223 0.2587 -0.2608 -1.4780 -0.1892 0.8062 -1.5261 -0.2343 0.7924 1.4024 -0.3458 1.6508 -0.7769 1.4809 -0.6827 0.5224 0.8780 -1.9926 1.0107 1.3894 -1.2760 0.1026 -1.2943 1.6325 1.3942 -0.5487 -1.0794 -0.7162 0.3986 0.2656 0.7592 0.6671 -0.4131 1.1164 0.9112 -0.8288 0.8028 1.0356 2.1030 0.5454 1.8007 0.5637 1.8298 1.3039 -1.6847 -1.7467 0.0491 -0.6109 -1.4081 -1.0580 -0.9059 0.4326 -1.9458 1.2313 -1.0798 0.4070 -0.2651 -0.1219 -0.5988 0.7297 -0.6376 -0.5023 -0.0690 -0.6161 -0.3863 1.0278 -0.2656 -1.1672 -0.5129 -0.3532 0.5845 0.4965 0.2798 -0.3602 -0.0516 -1.0773 0.0154 -0.6679 -2.4940 1.4639 0.3860 0.6970 0.2845 -1.4509 0.7407 -2.1837 0.5964 1.1560 1.7412 0.7906 0.3195 -0.4006 -0.6938 -1.7109 -0.4080 -1.5790 -0.3310 -1.9592 -0.7918 -1.5801 0.3975 1.7684 -0.7750 1.1966 -3.1887 -0.2716 0.2111 -0.2695 0.0797 -0.6632 0.2730 0.2083 -0.8074 1.5023 -1.2169 -0.0237 0.2682 0.7720 0.4555 -0.0507 0.1826 -2.5265 -1.6751 1.1685 0.8931 0.8691 1.9041 0.2850 -0.7561 -0.8368 -0.3392 0.7994 -1.0747 0.9417 -1.0685 1.0632 -0.7240 -1.2910 1.8690 -1.0843 -1.3279 -0.9254 0.3458 1.0333 1.8552 -0.0528 -0.6659 -0.3361 1.0563 0.0102 0.3505 -0.5161 -0.1175 0.5958 1.2358 -0.4791 0.2772 1.8568 1.1674 -0.4480 0.4703 0.6985 -0.2046 2.3334 -1.1279 -1.0123 -0.8698 -0.3437 -1.2090 -0.0642 0.5481 0.7448 -1.0056 -0.6956 0.5427 0.0700 0.0299 2.0278 -0.4660 -0.2423 -1.6114 -1.8488 0.3389 -0.3802 0.4043 -2.6851 -0.2688 -1.0027 1.4305 -0.5181 -0.2489 0.2050 0.2096 1.1381 0.2656 -1.0585 0.6680 -0.1325 -0.6548 -0.2437 -1.1160 -0.7753 -0.0926 0.6748 -1.1873 1.6926 1.2563 -0.5092 -0.1244 -2.2110 0.5883 -0.6845 -0.1907 0.5190 1.3420 1.7831 -0.3185 -1.2819 -1.9612 0.5659 -0.9883 1.1244 -0.0914 -0.6375 -2.7295 0.7003 -0.5602 0.5440 -0.6892 -0.6365 -0.5434 -1.2042 -0.7278 1.0224 -2.1150 0.6951 -1.0923 0.3102 1.0817 0.7767 0.1376 0.7288 -1.1017 -0.8704 -0.5636 0.0990 0.5918 -0.9162 -0.0013 -0.8629 1.2031 -0.5002 -1.0356 0.2565 1.9734 -1.4445 0.4578 1.8482 0.4079 0.3802 0.2294 -0.0130 -0.2252 0.3694 1.7141 0.7403 -2.2699 -1.1069 -0.4658 0.1281 1.7153 -1.4169 -0.2055 0.4792 2.2040 0.1845 -0.7076 1.1233 -1.4996 -1.0856 -1.1476 -1.0638 0.9775 -1.4070 -0.0228 0.6085 0.1096 0.7369 0.8368 -0.0804 1.5700 -1.5237 1.1909 -0.1451 -1.0855 0.0660 0.3631 0.3704 -0.2094 0.0316 -0.0225 -0.6966 -1.4740 3.0086 0.9183 -0.3612 0.8050 -0.1048 3.0252 -1.3715 -0.1645 0.1449 -1.3335 0.1318 0.7365 0.5278 2.2307 0.2602 0.2156 -2.3057 -0.2448 -0.0544 0.1475 -1.1961 0.3929 -2.2793 0.1180 0.2921 1.1358 0.8090 0.6872 -0.8128 0.1647 0.1645 -0.1446 -0.7089 0.5836 1.0647 -2.4462 0.2605 -0.7428 0.4847 0.1497 0.4262 1.2027 -0.5077 -1.5590 -1.4856 0.9456 0.7870 -1.3809 0.4521 0.8675 2.1659 1.3413 -1.3280 2.1162 1.0171 1.6202 -0.2365 -0.1494 1.3739 1.0670 1.1105 -0.8430 -1.4505 -0.6821 -0.1910 0.0847 1.0256 0.0803 0.0134 -0.7860 0.4701 -0.3147 0.0834 -0.4911 -0.0680 -1.1613 -1.0054 2.1686 -1.5006 -1.5797 -0.1782 -0.2196 0.9841 -1.1245 -0.3925 -0.2960 0.4670 -0.3263 -1.9496 0.7341 -0.8636 -0.2926 -0.3087 0.2412 -0.5759 0.2146 -0.0321 -0.5497 0.8545 -1.6918 -0.9354 -0.7420 0.7906 -0.6545 -0.3574 -2.8319 0.2685 0.6816 0.1801 -0.7932 -0.8501 0.6159 -1.3215 -0.1515 0.6192 1.1751 -0.9561 -0.7330 0.6745 -0.7303 -0.2171 1.2629 -0.4335 0.0781 -0.4060 0.6786 1.2688 -1.7939 -0.5014 0.6841 -0.1429 -0.3078 -0.7164 2.1484 0.8995 0.2965 0.1883 -1.3961 -1.7746 -1.4087 0.1125 0.5721 -2.0556 -1.3405 1.1908 -0.2741 -0.8984 0.2074 0.0038 0.7218 -0.3475 0.0439 -1.3953 -0.1737 -0.7191 -0.0408 -1.5627 -0.4876 2.7874 -0.7094 0.7423 -0.2223 0.0017 1.1458 0.4369 -0.0992 -1.7706 0.1825 0.0852 -1.4554 -0.7676 0.6269 -1.2349 0.0774 -1.3619 0.7236 -0.9227 -0.4530 -0.3424 3.2467 -1.3819 -0.8504 -0.0257 1.6579 0.2111 0.2592 1.5201 0.2760 -0.2712 -0.3128 -0.9214 -1.7198 -2.5074 -0.0309 2.0289 -0.2150 -1.3598 -3.0582 -0.9279 0.6305 -0.6269 -0.7269 -0.8982 0.3904 1.1951 -1.4198 1.3313 -0.0420 -0.8002 1.0216 -0.4517 -1.0293 -1.6067 -2.4049 0.5178 0.3311 -0.1415 0.0969 0.5140 0.2075 0.1132 -0.6472 0.0168 -0.2821 -1.7945 -1.4203 -0.4142 -0.5159 -0.2441 -2.1249 1.1501 1.7403 1.3873 -0.9509 -1.4477 -1.8168 1.6215 -0.0964 -1.5909 -0.7816 0.5364 0.0116 0.4653 -0.1471 -1.3962 1.2724 0.6847 0.8630 -2.4664 0.3710 1.0720 -0.7346 -1.5074 0.0489 -0.8777 -2.3603 1.4464 -1.7417 1.4250 0.3711 -0.6886 -0.0808 0.5316 -0.7693 1.5158 -0.5488 1.5974 1.6569 0.1458 -0.1943 -0.4931 0.4679 -0.4516 -0.2736 0.8217 -1.3072 0.0973 -0.4582 -0.4306 -0.4344 0.1154 0.6286 0.3265 -0.1735 -1.0871 1.3017 -0.5369 -0.2293 -0.7718 -0.6276 -0.0724 -2.2960 -0.8940 -1.4030 -1.5159 -0.8977 1.4410 0.4867 -0.2230 -0.4034 0.3116 1.0222 1.1797 -0.4309 -0.1510 -0.8971 -1.3978 0.7843 0.8600 -0.1972 1.0497 -0.7054 -1.5760 -0.9532 0.1803 1.7803 -1.3860 -0.7846 -1.3864 0.2143 0.0004 -0.2903 -0.8812 -0.4112 -0.2560 1.6837 1.2923 -0.3671 1.2785 -2.1541 0.3812 -0.5932 1.6608 -0.9946 2.4268 -0.6675 -1.2797 -0.0631 0.0727 1.2162 -0.3682 -0.5412 0.1408 0.3906 -0.1192 -0.3408 0.5938 1.1687 0.4033 0.1354 0.5842 0.2721 -0.0571 0.8177 0.9364 2.0835 0.2476 -0.1106 2.4043 0.3860 -0.4375 1.1038 -0.4714 -0.3062 0.5382 0.7249 0.8222 -0.2765 1.4902 -0.1682 -2.1238 0.2042 -2.3363 2.3950 -2.2303 -0.9923 -0.2275 -1.8826 0.3291 0.9612 0.2217 -0.4978 0.3724 -2.2857 -1.3106 0.6345 -0.4385 -0.6085 -0.3052 1.9157 -0.8970 0.3090 -1.0711 0.6653 2.0762 0.8674 0.4189 1.9130 -0.1576 0.8173 0.5022 0.1075 0.8130 -1.3175 0.5317 -1.0142 -0.6452 0.9029 0.8420 -0.4822 0.9195 -0.5667 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-2.8550 2.4067 -1.5533 1.2555 -0.1707 -1.1241 0.4595 0.5826 1.3664 -0.7132 2.0730 0.6148 1.5810 0.1031 -0.1277 -0.1722 -1.1653 -0.0608 -0.4905 0.0307 1.2835 -0.1699 0.7283 -0.4643 0.1538 -0.3577 -0.6239 -0.9713 0.2107 -0.3548 -1.0704 0.4297 0.3755 -0.9931 -0.9646 -1.2636 -0.9218 -1.1266 -2.0753 -2.8728 -1.6011 1.9429 0.2419 0.0072 0.4208 -0.5990 0.2986 -1.0247 1.5394 -0.3003 2.2169 -0.7293 0.2384 -0.9444 1.1510 -0.4144 0.2110 -0.1123 0.2012 0.4927 1.7232 0.6260 0.3475 0.9340 -0.1263 -0.1900 -0.3815 -0.3919 -0.5391 1.6779 1.1790 -0.7912 -1.1634 1.5449 -0.1870 -0.0055 -1.0548 -0.0737 -1.1947 -0.8840 -0.6869 0.7526 -0.0162 -1.5721 -0.9423 0.3499 -0.6591 -0.8427 -0.2834 0.0428 -2.5144 0.9698 0.3073 -0.3052 1.1022 -0.6583 0.6997 0.5401 1.1379 -1.6790 0.3277 -1.2187 -1.1899 -0.5687 -0.3015 -1.4676 0.9660 -0.8659 1.0336 -0.1457 0.1775 -1.2571 1.5282 2.0721 -0.4704 -1.0663 -0.9936 1.2358 -0.1740 0.8559 0.8141 0.7302 -1.3922 -0.0250 -1.9068 0.0252 1.2824 -1.3654 0.5917 1.8015 0.2109 0.4859 1.7976 0.5186 0.8927 0.5935 0.4541 0.8945 0.0419 -1.0285 -0.5438 0.7670 0.0680 1.1565 -2.0597 -2.3358 0.4439 -1.6068 -1.9098 -1.9469 -1.7364 0.3993 1.2326 -0.9673 0.5803 1.6427 1.6638 -0.0124 0.4303 0.1598 -0.4033 1.6297 0.5768 1.5458 -0.3078 0.9835 -0.8964 -1.5891 -0.8222 -1.1993 0.5218 0.0171 2.5688 -0.3013 -0.8888 0.7240 0.8042 -0.7827 0.4332 1.0224 -0.6365 0.4978 1.2178 -0.7651 0.1670 1.8603 -0.3704 -1.5399 1.4635 0.2015 0.5088 -0.5887 0.4129 0.5042 0.6916 1.2492 -0.5543 1.5060 0.2080 0.8679 1.5884 0.1171 -0.5517 -1.7817 0.1913 0.4937 -0.5158 0.3556 -1.8362 -1.7836 -0.5436 -0.8756 0.3234 -1.0435 -0.0954 0.1709 1.2908 0.6649 -1.5003 0.6164 -0.8059 -0.5183 0.5656 0.1921 1.2364 1.2139 -0.7131 -1.6446 -0.5588 -1.1762 -0.9845 0.1340 0.8617 1.1480 0.2357 -0.5657 0.2787 1.2835 -0.7256 -1.1094 2.3771 -1.2268 -0.4022 0.0047 0.6419 -0.5392 -0.7231 0.2312 -1.3050 -0.4128 -0.4546 -1.3555 2.0356 1.0538 0.6231 -0.7057 -1.2172 -1.3760 0.9522 0.9952 -0.2127 -0.0402 -0.6350 3.1097 -1.3817 -0.0749 -0.1689 -0.9225 0.0442 1.2940 0.6094 -0.5626 0.4811 0.2935 0.2748 -0.0423 0.3105 -1.1938 0.0130 -0.0690 -0.7388 1.1169 1.0721 -2.0318 1.1088 1.5503 -0.5236 -0.5725 0.0667 -1.0150 1.1561 0.7148 -0.8843 -1.4659 -2.5561 0.9145 0.5563 -1.0443 -0.8354 2.0437 -1.2273 0.3785 1.5159 1.5808 0.3486 0.4225 -0.0351 1.1205 0.6446 0.7771 -0.5575 0.4965 0.9165 0.7077 0.7004 0.8057 -0.1106 0.6559 2.3655 -1.6857 0.9696 1.1206 0.2527 -0.7786 -0.4683 -0.5818 -0.9229 0.2870 2.5222 -1.5560 0.5537 -1.7690 1.0684 0.1101 0.0921 1.8283 1.6807 -1.9063 0.0169 0.2152 -0.3468 1.2138 -0.5055 0.6705 -0.0100 0.3000 -0.1964 0.2875 -1.2921 -1.5049 -0.3304 -0.4463 -0.7462 0.1255 -0.1515 1.4238 0.8358 1.2622 1.2784 -1.3832 -0.8024 0.5000 1.8198 1.2400 -1.6889 -1.0757 0.5313 -2.3261 -2.6402 -0.6729 3.5569 -0.9029 1.1040 -1.2882 1.6916 -0.0081 0.0363 -0.1187 0.8779 -0.1987 0.8788 0.2066 -1.3576 0.8886 0.6310 -0.4490 -0.1906 -0.4355 0.2340 0.2401 -0.9350 -0.7175 0.9394 -2.4049 -2.4579 -0.0996 0.0403 0.9441 -0.5810 -0.8433 0.6837 0.1379 1.1703 0.6335 -1.0476 -1.3763 0.0171 2.1393 -1.1178 -0.4193 0.3092 2.1258 2.4012 -0.8086 -0.5782 -0.2242 0.0058 -1.4630 1.4553 0.5454 0.7038 0.8738 -1.4878 -1.4112 -1.4691 -0.0849 0.5941 -0.0708 0.1594 0.3040 0.2058 -1.0712 1.0128 -1.6570 -1.2216 0.3359 -0.0206 0.5205 -0.9254 0.1743 0.5035 -0.1359 -0.0931 -0.2487 -0.6191 -0.7249 0.1585 -0.4601 -0.8329 -0.0812 0.7252 -0.1104 -0.3469 -1.3269 -2.0176 1.1899 0.3058 0.8447 -1.6987 -1.4110 -1.1499 0.1568 1.6242 1.4607 -0.9578 0.7566 0.2988 1.2633 1.1422 -0.1386 0.0157 1.0157 1.1080 1.5256 1.0672


7.2.1 Scored Plot

Let’s look at the ‘scoredPlotly’ which visualizes the 2D Heatmap, hover on it will display the no. of observations and the tweets in a cell with its cell ID.

scoring$scoredPlotly

This plot visualizes a 2D Heatmap using a Voronoi diagram where each region represents a cell and the color intensity indicates the number of tweets within that cell. The color scale on the right shows the range of observations, with darker colors representing fewer observations and lighter colors indicating higher observations.

Key observations:

Central Density: The central region exhibits a higher density of observations, marked by lighter colors, implying more tweets.

Peripheral Sparsity: The outer regions are predominantly darker, signifying fewer observations.

Cell Specifics: Hovering over a cell, as demonstrated, reveals detailed information such as Cell ID, number of observations, and tweets.


8. Hierarchical Clustering

8.1 Comprehending clustHVT and its parameters

In this section, we will explore the newly added clustHVT function in the HVT package.

Let’s look at the function parameters.

clustHVT(data,
         trainHVT_results,
         scoreHVT_results,
         indices,
         clustering_method,
         clusters_k)

Each of the parameters of clustHVT function has been explained below:

The Hyperparameters of NbClust& hclust functions which are used in clustHVT function has been explained below:

For more information, visit here.

Let’s look at the output list from clustHVT function.

  1. Optimal number of clusters summary: The output of the NbClust function comprises four plots, two corresponding to the D index and Hubert index, respectively. It also includes an interpretation of these indices, a summary table displaying the number of clusters recommended by all indices, and a conclusion table providing a comprehensive breakdown of the cluster suggestions across each index.

  2. Dendogram plot: A dendrogram plot is a tree-like diagram that visually depicts the hierarchical relationships among clusters, with branch heights reflecting similarity or dissimilarity, and is used to determine the number of clusters through size-based visual comparison.

  3. Cluster HVT plot: A 2D HVT Plotly visualization that colors cells according to clusters derived from AGNES clustering results. It is interactive, allowing users to view cell contents by hovering over them.

8.2 hclust on 2D data

In this section, we are performing Hierarchical Clustering (hclust) on 2D dataset.

8.2.1 About dataset

In this section, we are using Sammon’s coordinates data which is 2 Dimensional (x and y coordinates) of the centroids of 110 cells is taken for the analysis.

centroid_data <- scoring$centroidData
hclust_data_1 <- centroid_data[,2:3]
dim(hclust_data_1)
## [1] 110   2

The data has 110 rows and 2 columns (x and y coordinates). Let’s have a look at the data.

displayTable(hclust_data_1)
x y
-389.8128 -218.3527
-453.2892 43.1680
-46.5469 465.2851
-299.4868 -333.3965
-156.5955 405.2584
-401.3678 -102.3605
-89.7634 -392.6072
-365.4588 127.2700
-281.9396 272.9182
-315.8800 193.5009
-221.5164 105.5817
-191.6421 146.5122
-248.6823 74.2845
-219.0094 223.3941
-290.2938 123.7106
-325.5442 27.4215
-339.2483 -63.1875
-225.7471 174.5218
-73.7163 205.8910
-126.5352 215.6485
-153.5729 271.2678
12.0816 345.3534
-286.5778 -65.0788
-159.9992 64.8302
-194.8970 -55.2059
-244.7368 -89.1437
-226.6281 -167.7565
-85.3478 262.3173
-265.1166 24.3438
-178.1197 -224.2079
-287.4686 -165.0773
-162.3493 -288.7577
-129.5616 -165.7233
-172.1562 220.2929
-89.3060 317.4978
-34.2148 323.8360
-93.6005 20.9949
-101.8810 -98.8031
-46.1317 -25.5600
-91.9465 80.7582
-115.2567 -31.6807
-75.8149 137.1793
-224.5265 -15.1111
49.4653 274.6778
-2.8999 271.8199
-27.6630 -274.3004
-24.8589 33.4035
-168.7345 -120.4750
-45.2263 -90.3633
-110.2804 -266.6740
-34.8578 90.2205
283.0589 -164.7684
-223.3585 -239.8115
-65.6729 -178.1384
82.0589 335.0828
118.7024 278.0067
4.5072 -329.2800
296.2824 92.7242
29.4273 -177.1057
9.8946 -95.5097
36.2561 192.9018
21.9640 -38.9666
-17.6602 -174.0566
35.9215 60.8590
11.9455 149.4490
49.2005 7.8142
-56.8258 -292.5697
208.5524 -118.9474
107.4657 -343.6376
140.1493 310.5073
65.7953 -211.2189
77.9247 -140.3272
49.1393 106.5359
118.0087 87.3626
133.5958 -130.7762
87.9879 168.5436
126.7972 -8.8858
146.4528 -66.2908
96.2006 -56.6190
110.0233 36.2924
199.4336 -232.6893
241.5640 72.3343
225.2003 -54.4538
134.9822 203.5282
153.3562 139.1026
179.1013 17.6462
202.6922 126.4800
171.5489 -297.6205
238.3001 -207.7709
296.9595 -113.0700
170.2799 -214.3528
54.7577 -300.4614
191.8482 233.3803
273.3562 -30.4607
124.2423 -257.7405
299.1853 7.5282
269.7236 165.8641
227.5844 196.6262
308.9365 50.8174
334.7487 -68.2903

8.2.2 Clustering Analysis

In this section, we will use clustHVT function to conduct clustering analysis. Let’s look at the NbClust function’s output for determining the number of optimal clusters for the dataset.

clust.results <- clustHVT(data = hclust_data_1, 
                          trainHVT_results = hvt.results,
                          scoreHVT_results = scoring, 
                          clusters_k = 'champion')

## *** : The Hubert index is a graphical method of determining the number of clusters.
##                 In the plot of Hubert index, we seek a significant knee that corresponds to a 
##                 significant increase of the value of the measure i.e the significant peak in Hubert
##                 index second differences plot. 
## 

## *** : The D index is a graphical method of determining the number of clusters. 
##                 In the plot of D index, we seek a significant knee (the significant peak in Dindex
##                 second differences plot) that corresponds to a significant increase of the value of
##                 the measure. 
##  
## ** Among all indices:
## ** * 1 proposed 1 as the best number of clusters
## ** * 3 proposed 2 as the best number of clusters
## ** * 6 proposed 3 as the best number of clusters
## ** * 1 proposed 4 as the best number of clusters
## ** * 2 proposed 12 as the best number of clusters
## ** * 5 proposed 15 as the best number of clusters
## *******************************************************************
## ***** Conclusion *****
## ** * According to the majority rule, the best number of clusters is 3
## *******************************************************************
##                          Index     Number_clusters    Value_Index
## -------------------------------------------------------------------
##                       kl           2.0000           6.8713
##                       ch          12.0000          78.3943
##                 hartigan           3.0000          32.0431
##                   cindex          12.0000           0.2501
##                       db          15.0000           0.8624
##               silhouette           3.0000           0.3252
##                ratkowsky           3.0000           0.4339
##                     ball           3.0000     1780468.0797
##                   hubert               NA               NA
##                   dindex               NA               NA
##               ptbiserial           3.0000           0.5264
##                      gap           2.0000          -0.6991
##                     frey           1.0000               NA
##                  mcclain           2.0000           0.6399
##                    gamma          15.0000           0.9158
##                    gplus          15.0000          16.1623
##                      tau           3.0000         887.7868
##                     dunn          15.0000           0.1610
##                  sdindex           4.0000           0.0105
##                     sdbw          15.0000           0.0996
## *******************************************************************

From the above results, we can clearly see that highest recommendation is 3 as the optimal number of clusters. Let’s look at the dendogram plot for the same.

clust.results$dendogram()

Though the highest recommendation is 3, the corresponding dendogram results in large clusters. However, for our specific use case, we prefer to have smaller clusters. So we will select the next highest recommendation from the NbClust summary. With 5 indices supporting this choice, we will proceed with ‘challenger’ to achieve a more granular clustering solution.

clust.results <- clustHVT(data = hclust_data_1, 
              trainHVT_results = hvt.results,
              scoreHVT_results = scoring, 
              clusters_k = 'challenger')

Let’s look at the dendogram for challenger number of clusters.

clust.results$dendogram()

Let’s look at the Cluster HVT plot for the 15 clusters.

clust.results$clusterplot

8.2.3 OpenAI API call for generating domain names

  • Now, we will pass all the data from each cluster to OpenAI’s gpt-3.5-turbo model to generate a keyword/domain name of that particular cluster.

  • Subsetting the data in each cluster by the below code and processing it without punctuation marks to be given as input to API call. Data from each cluster will be saved as a .txt file and the files will be read during the API call.

  • Let’s call the Function for writing clusters as .txt files which is explained in section 3.1 for this process.

create_and_write_clusters(cluster_numbers = 1:15, cluster_method = clust.results$cluster_data)
  • The below chunk makes a call to OpenAI’s gpt-3.5-Turbo via chat completions API and print the responses for clusters below.

  • Let’s call the Function for making OpenAI-API Call to generate domain names which is explained in section 3.1. This function makes 10 API calls for each cluster and prints all 10 responses. The domain name that appears most frequently is selected as the domain name for that cluster.

result <- get_cluster_domains(15)
result$formatted_table
Domain_name_for_cluster_1 Domain_name_for_cluster_2 Domain_name_for_cluster_3 Domain_name_for_cluster_4 Domain_name_for_cluster_5 Domain_name_for_cluster_6 Domain_name_for_cluster_7 Domain_name_for_cluster_8 Domain_name_for_cluster_9 Domain_name_for_cluster_10 Domain_name_for_cluster_11 Domain_name_for_cluster_12 Domain_name_for_cluster_13 Domain_name_for_cluster_14 Domain_name_for_cluster_15
Conservation Socialmedia Parenting Entertainment Spirituality Inspiration Lifestyle Sports Literature Food Travel Celebrity Empowerment Music Celebration
Conservation Socialmedia Parenting Entertainment Spirituality Inspiration Lifestyle Sports Literature Food Travel Celebrity Empowerment Music Celebration
Conservation Socialmedia Parenting Entertainment Spirituality Inspiration Lifestyle Sports Literature Food Travel Celebrity Empowerment Music Celebration
Conservation Socialmedia Parenting Entertainment Spirituality Inspiration Lifestyle Sports Literature Food Travel Celebrity Empowerment Music Celebration
Conservation Socialmedia Parenting Entertainment Spirituality Inspiration Lifestyle Sports Literature Food Travel Celebrity Empowerment Music Celebration
Conservation Socialmedia Parenting Entertainment Spirituality Inspiration Lifestyle Sports Literature Food Travel Celebrity Empowerment Music Celebration
Conservation Socialmedia Parenting Entertainment Spirituality Inspiration Lifestyle Sports Literature Food Travel Celebrity Empowerment Music Celebration
Conservation Socialmedia Parenting Entertainment Spirituality Inspiration Lifestyle Sports Literature Food Travel Celebrity Empowerment Music Celebration
Conservation Socialmedia Parenting Entertainment Spirituality Inspiration Lifestyle Sports Literature Food Travel Celebrity Empowerment Music Celebration
Conservation SocialMedia Parenting Entertainment Spirituality Inspiration Fashion Sports Literature Food Travel Celebrity Empowerment Music Celebration
Conservation Socialmedia Parenting Entertainment Spirituality Inspiration Lifestyle Sports Literature Food Travel Celebrity Empowerment Music Celebration
print(result$mode)
##  Domain_name_for_cluster_1  Domain_name_for_cluster_2 
##             "Conservation"              "Socialmedia" 
##  Domain_name_for_cluster_3  Domain_name_for_cluster_4 
##                "Parenting"            "Entertainment" 
##  Domain_name_for_cluster_5  Domain_name_for_cluster_6 
##             "Spirituality"              "Inspiration" 
##  Domain_name_for_cluster_7  Domain_name_for_cluster_8 
##                "Lifestyle"                   "Sports" 
##  Domain_name_for_cluster_9 Domain_name_for_cluster_10 
##               "Literature"                     "Food" 
## Domain_name_for_cluster_11 Domain_name_for_cluster_12 
##                   "Travel"                "Celebrity" 
## Domain_name_for_cluster_13 Domain_name_for_cluster_14 
##              "Empowerment"                    "Music" 
## Domain_name_for_cluster_15 
##              "Celebration"

Let’s replace the cluster numbers to above generated domain names in the dataset.

cluster_data <- clust.results$cluster_data
invisible(lapply(1:15, function(i) {
  cluster_data$cluster_name[cluster_data$clusters == i] <<- result$mode[[i]]}))

8.2.4 Cluster Domain Naming: OpenAI vs Manual Approach

In this section, we will generate the Heatmaps of OpenAI and Manual Approach for the Sammon’s coordinates (2D - x and y coordinates) of the centroids of 110 cells.

OpenAI Approach

  • The OpenAI Approach is a 2D Heatmap where colors represent different domains assigned to the clusters. These domain names are generated using OpenAI’s GPT-3.5-turbo via the chat completions API. Tweets from each cluster are provided as input, along with a system prompt requesting domain names that accurately reflect the context of the data. To ensure consistent results, this process is repeated 10 times for each cluster, and the most frequently occurring name is selected as the domain name for that cluster.
centroid_cluster_data <- cluster_data %>% dplyr::select('Cell.ID', 'cluster_name','names.column')
hclust_plot_1 <- clustHVT(dataset = centroid_cluster_data, hvt.results = hvt.results, domains.column = "cluster_name", type = "plot")

Manual Approach

  • The Manual Approach is a 2D Heatmap, features colors representing domains assigned to clusters, but in this case, the domain names are determined manually. This involves a detailed examination of the tweets within each cluster, allowing for a more nuanced assignment of domain names based on Human interpretation and context.
clust_1 <- subset(cluster_data$names.column, cluster_data$clusters == 1)
cluster_names <- c("Activism *", "Engagement *", "Perspectives *", 
                   "Upcoming Shows *", "Social Ideology *", "Heartwarming *", 
                   "Excitement *", "Television Media *", "Governance *", 
                   "Exhilaration *", "Media Posts *", "Gratitude *", 
                   "Appreciation *", "Promotion *", "Wishes *")

cluster_data$cluster_name <- cluster_names[cluster_data$clusters]
centroid_cluster_data <- cluster_data %>% dplyr::select('Cell.ID', 'cluster_name','names.column')
manual_plot_1 <-clustHVT(dataset = centroid_cluster_data, hvt.results = hvt.results, domains.column = "cluster_name", type = "plot")

8.2.5 Visualization of plots: OpenAI vs Manual Approach

In this section, we are visualizing the plots of both OpenAI where the domain names are generated by OpenAI via API call and the Manual where the domain names are manually read and assigned.

plotly::subplot(hclust_plot_1, manual_plot_1) %>%
plotly::layout(annotations = list(
      list(x = 0.15, y = 1.05, text = "OpenAI Approach", showarrow = F, xref='paper',yref='paper', width = 150),
      list(x = 0.9, y = 1.05, text = "Manual Approach", showarrow = F, xref='paper', yref='paper', width = 165))) 

(*) Manually assigned domain names (to differentiate the domain names from OpenAI generated)

Inference

  • A visual comparison between the plots of OpenAI Approach and the Manual Approach indicates that the domain names assigned to clusters exhibit complete uniqueness across both methods.

  • The labeling methodology utilized by OpenAI largely depends on the content of a restricted sample of tweets, generally one or two per cluster. For example, the ‘sports’ domain in the OpenAI plot is represented by only a single tweet pertaining to sports, whereas the majority of the remaining tweets discuss media shows. This distinction was more accurately captured by the Manual Approach, which categorized it as ‘Television Media’.

8.2.6 Calculation of Average Silhouette Width(ASW)

  • The Average Silhouette Width (ASW) is a measure to evaluate the quality of clusters in a clustering analysis. It assess how similar each point in a cluster is to other points in the same cluster compared to points in different clusters. Higher ASW indicates better-defined and more separated clusters.
clusters <- clust.results$clusters
ward_sil <- silhouette(clusters, dist(hclust_data_1))
ward_asw <- round(mean(ward_sil[, "sil_width"]), digits = 2)
print(ward_asw)
## [1] 0.27
  • For Sammon’s coordinates data (2D - x and y coordinates) of the centroids of 110 cells, the ASW for ward.D2 is 0.27 .

  • Refer this paper for more information on ASW calculation for evaluating Cluster Analysis.

8.3 hclust on 1538D data

In this section, we are performing Hierarchical Clustering (hclust) on 1538 dimensional dataset.

8.3.1 About Dataset

In this section, we are taking the Sammon’s coordinates data which is 2 Dimensional (x and y coordinates) with the centroid’s embedding data (1536D) of the 110 cells.

centroid_data_1 <- hvt.results[[3]]$centroid_data
centroid_data <- scoring$centroidData
centroid_data_2 <- centroid_data[,2:3]

hclust_data_2 <- cbind(centroid_data_1,centroid_data_2)
dim(hclust_data_2)
## [1]  110 1538

The dimension of the centroid data is 110 rows and 1538 columns (1536 columns + 2D coordinates)

Let’s have a look at the data. For the sake of brevity, we are displaying only the first 10 rows and all the 1538 columns.

displayTable(hclust_data_2[1:10,])
E1 E2 E3 E4 E5 E6 E7 E8 E9 E10 E11 E12 E13 E14 E15 E16 E17 E18 E19 E20 E21 E22 E23 E24 E25 E26 E27 E28 E29 E30 E31 E32 E33 E34 E35 E36 E37 E38 E39 E40 E41 E42 E43 E44 E45 E46 E47 E48 E49 E50 E51 E52 E53 E54 E55 E56 E57 E58 E59 E60 E61 E62 E63 E64 E65 E66 E67 E68 E69 E70 E71 E72 E73 E74 E75 E76 E77 E78 E79 E80 E81 E82 E83 E84 E85 E86 E87 E88 E89 E90 E91 E92 E93 E94 E95 E96 E97 E98 E99 E100 E101 E102 E103 E104 E105 E106 E107 E108 E109 E110 E111 E112 E113 E114 E115 E116 E117 E118 E119 E120 E121 E122 E123 E124 E125 E126 E127 E128 E129 E130 E131 E132 E133 E134 E135 E136 E137 E138 E139 E140 E141 E142 E143 E144 E145 E146 E147 E148 E149 E150 E151 E152 E153 E154 E155 E156 E157 E158 E159 E160 E161 E162 E163 E164 E165 E166 E167 E168 E169 E170 E171 E172 E173 E174 E175 E176 E177 E178 E179 E180 E181 E182 E183 E184 E185 E186 E187 E188 E189 E190 E191 E192 E193 E194 E195 E196 E197 E198 E199 E200 E201 E202 E203 E204 E205 E206 E207 E208 E209 E210 E211 E212 E213 E214 E215 E216 E217 E218 E219 E220 E221 E222 E223 E224 E225 E226 E227 E228 E229 E230 E231 E232 E233 E234 E235 E236 E237 E238 E239 E240 E241 E242 E243 E244 E245 E246 E247 E248 E249 E250 E251 E252 E253 E254 E255 E256 E257 E258 E259 E260 E261 E262 E263 E264 E265 E266 E267 E268 E269 E270 E271 E272 E273 E274 E275 E276 E277 E278 E279 E280 E281 E282 E283 E284 E285 E286 E287 E288 E289 E290 E291 E292 E293 E294 E295 E296 E297 E298 E299 E300 E301 E302 E303 E304 E305 E306 E307 E308 E309 E310 E311 E312 E313 E314 E315 E316 E317 E318 E319 E320 E321 E322 E323 E324 E325 E326 E327 E328 E329 E330 E331 E332 E333 E334 E335 E336 E337 E338 E339 E340 E341 E342 E343 E344 E345 E346 E347 E348 E349 E350 E351 E352 E353 E354 E355 E356 E357 E358 E359 E360 E361 E362 E363 E364 E365 E366 E367 E368 E369 E370 E371 E372 E373 E374 E375 E376 E377 E378 E379 E380 E381 E382 E383 E384 E385 E386 E387 E388 E389 E390 E391 E392 E393 E394 E395 E396 E397 E398 E399 E400 E401 E402 E403 E404 E405 E406 E407 E408 E409 E410 E411 E412 E413 E414 E415 E416 E417 E418 E419 E420 E421 E422 E423 E424 E425 E426 E427 E428 E429 E430 E431 E432 E433 E434 E435 E436 E437 E438 E439 E440 E441 E442 E443 E444 E445 E446 E447 E448 E449 E450 E451 E452 E453 E454 E455 E456 E457 E458 E459 E460 E461 E462 E463 E464 E465 E466 E467 E468 E469 E470 E471 E472 E473 E474 E475 E476 E477 E478 E479 E480 E481 E482 E483 E484 E485 E486 E487 E488 E489 E490 E491 E492 E493 E494 E495 E496 E497 E498 E499 E500 E501 E502 E503 E504 E505 E506 E507 E508 E509 E510 E511 E512 E513 E514 E515 E516 E517 E518 E519 E520 E521 E522 E523 E524 E525 E526 E527 E528 E529 E530 E531 E532 E533 E534 E535 E536 E537 E538 E539 E540 E541 E542 E543 E544 E545 E546 E547 E548 E549 E550 E551 E552 E553 E554 E555 E556 E557 E558 E559 E560 E561 E562 E563 E564 E565 E566 E567 E568 E569 E570 E571 E572 E573 E574 E575 E576 E577 E578 E579 E580 E581 E582 E583 E584 E585 E586 E587 E588 E589 E590 E591 E592 E593 E594 E595 E596 E597 E598 E599 E600 E601 E602 E603 E604 E605 E606 E607 E608 E609 E610 E611 E612 E613 E614 E615 E616 E617 E618 E619 E620 E621 E622 E623 E624 E625 E626 E627 E628 E629 E630 E631 E632 E633 E634 E635 E636 E637 E638 E639 E640 E641 E642 E643 E644 E645 E646 E647 E648 E649 E650 E651 E652 E653 E654 E655 E656 E657 E658 E659 E660 E661 E662 E663 E664 E665 E666 E667 E668 E669 E670 E671 E672 E673 E674 E675 E676 E677 E678 E679 E680 E681 E682 E683 E684 E685 E686 E687 E688 E689 E690 E691 E692 E693 E694 E695 E696 E697 E698 E699 E700 E701 E702 E703 E704 E705 E706 E707 E708 E709 E710 E711 E712 E713 E714 E715 E716 E717 E718 E719 E720 E721 E722 E723 E724 E725 E726 E727 E728 E729 E730 E731 E732 E733 E734 E735 E736 E737 E738 E739 E740 E741 E742 E743 E744 E745 E746 E747 E748 E749 E750 E751 E752 E753 E754 E755 E756 E757 E758 E759 E760 E761 E762 E763 E764 E765 E766 E767 E768 E769 E770 E771 E772 E773 E774 E775 E776 E777 E778 E779 E780 E781 E782 E783 E784 E785 E786 E787 E788 E789 E790 E791 E792 E793 E794 E795 E796 E797 E798 E799 E800 E801 E802 E803 E804 E805 E806 E807 E808 E809 E810 E811 E812 E813 E814 E815 E816 E817 E818 E819 E820 E821 E822 E823 E824 E825 E826 E827 E828 E829 E830 E831 E832 E833 E834 E835 E836 E837 E838 E839 E840 E841 E842 E843 E844 E845 E846 E847 E848 E849 E850 E851 E852 E853 E854 E855 E856 E857 E858 E859 E860 E861 E862 E863 E864 E865 E866 E867 E868 E869 E870 E871 E872 E873 E874 E875 E876 E877 E878 E879 E880 E881 E882 E883 E884 E885 E886 E887 E888 E889 E890 E891 E892 E893 E894 E895 E896 E897 E898 E899 E900 E901 E902 E903 E904 E905 E906 E907 E908 E909 E910 E911 E912 E913 E914 E915 E916 E917 E918 E919 E920 E921 E922 E923 E924 E925 E926 E927 E928 E929 E930 E931 E932 E933 E934 E935 E936 E937 E938 E939 E940 E941 E942 E943 E944 E945 E946 E947 E948 E949 E950 E951 E952 E953 E954 E955 E956 E957 E958 E959 E960 E961 E962 E963 E964 E965 E966 E967 E968 E969 E970 E971 E972 E973 E974 E975 E976 E977 E978 E979 E980 E981 E982 E983 E984 E985 E986 E987 E988 E989 E990 E991 E992 E993 E994 E995 E996 E997 E998 E999 E1000 E1001 E1002 E1003 E1004 E1005 E1006 E1007 E1008 E1009 E1010 E1011 E1012 E1013 E1014 E1015 E1016 E1017 E1018 E1019 E1020 E1021 E1022 E1023 E1024 E1025 E1026 E1027 E1028 E1029 E1030 E1031 E1032 E1033 E1034 E1035 E1036 E1037 E1038 E1039 E1040 E1041 E1042 E1043 E1044 E1045 E1046 E1047 E1048 E1049 E1050 E1051 E1052 E1053 E1054 E1055 E1056 E1057 E1058 E1059 E1060 E1061 E1062 E1063 E1064 E1065 E1066 E1067 E1068 E1069 E1070 E1071 E1072 E1073 E1074 E1075 E1076 E1077 E1078 E1079 E1080 E1081 E1082 E1083 E1084 E1085 E1086 E1087 E1088 E1089 E1090 E1091 E1092 E1093 E1094 E1095 E1096 E1097 E1098 E1099 E1100 E1101 E1102 E1103 E1104 E1105 E1106 E1107 E1108 E1109 E1110 E1111 E1112 E1113 E1114 E1115 E1116 E1117 E1118 E1119 E1120 E1121 E1122 E1123 E1124 E1125 E1126 E1127 E1128 E1129 E1130 E1131 E1132 E1133 E1134 E1135 E1136 E1137 E1138 E1139 E1140 E1141 E1142 E1143 E1144 E1145 E1146 E1147 E1148 E1149 E1150 E1151 E1152 E1153 E1154 E1155 E1156 E1157 E1158 E1159 E1160 E1161 E1162 E1163 E1164 E1165 E1166 E1167 E1168 E1169 E1170 E1171 E1172 E1173 E1174 E1175 E1176 E1177 E1178 E1179 E1180 E1181 E1182 E1183 E1184 E1185 E1186 E1187 E1188 E1189 E1190 E1191 E1192 E1193 E1194 E1195 E1196 E1197 E1198 E1199 E1200 E1201 E1202 E1203 E1204 E1205 E1206 E1207 E1208 E1209 E1210 E1211 E1212 E1213 E1214 E1215 E1216 E1217 E1218 E1219 E1220 E1221 E1222 E1223 E1224 E1225 E1226 E1227 E1228 E1229 E1230 E1231 E1232 E1233 E1234 E1235 E1236 E1237 E1238 E1239 E1240 E1241 E1242 E1243 E1244 E1245 E1246 E1247 E1248 E1249 E1250 E1251 E1252 E1253 E1254 E1255 E1256 E1257 E1258 E1259 E1260 E1261 E1262 E1263 E1264 E1265 E1266 E1267 E1268 E1269 E1270 E1271 E1272 E1273 E1274 E1275 E1276 E1277 E1278 E1279 E1280 E1281 E1282 E1283 E1284 E1285 E1286 E1287 E1288 E1289 E1290 E1291 E1292 E1293 E1294 E1295 E1296 E1297 E1298 E1299 E1300 E1301 E1302 E1303 E1304 E1305 E1306 E1307 E1308 E1309 E1310 E1311 E1312 E1313 E1314 E1315 E1316 E1317 E1318 E1319 E1320 E1321 E1322 E1323 E1324 E1325 E1326 E1327 E1328 E1329 E1330 E1331 E1332 E1333 E1334 E1335 E1336 E1337 E1338 E1339 E1340 E1341 E1342 E1343 E1344 E1345 E1346 E1347 E1348 E1349 E1350 E1351 E1352 E1353 E1354 E1355 E1356 E1357 E1358 E1359 E1360 E1361 E1362 E1363 E1364 E1365 E1366 E1367 E1368 E1369 E1370 E1371 E1372 E1373 E1374 E1375 E1376 E1377 E1378 E1379 E1380 E1381 E1382 E1383 E1384 E1385 E1386 E1387 E1388 E1389 E1390 E1391 E1392 E1393 E1394 E1395 E1396 E1397 E1398 E1399 E1400 E1401 E1402 E1403 E1404 E1405 E1406 E1407 E1408 E1409 E1410 E1411 E1412 E1413 E1414 E1415 E1416 E1417 E1418 E1419 E1420 E1421 E1422 E1423 E1424 E1425 E1426 E1427 E1428 E1429 E1430 E1431 E1432 E1433 E1434 E1435 E1436 E1437 E1438 E1439 E1440 E1441 E1442 E1443 E1444 E1445 E1446 E1447 E1448 E1449 E1450 E1451 E1452 E1453 E1454 E1455 E1456 E1457 E1458 E1459 E1460 E1461 E1462 E1463 E1464 E1465 E1466 E1467 E1468 E1469 E1470 E1471 E1472 E1473 E1474 E1475 E1476 E1477 E1478 E1479 E1480 E1481 E1482 E1483 E1484 E1485 E1486 E1487 E1488 E1489 E1490 E1491 E1492 E1493 E1494 E1495 E1496 E1497 E1498 E1499 E1500 E1501 E1502 E1503 E1504 E1505 E1506 E1507 E1508 E1509 E1510 E1511 E1512 E1513 E1514 E1515 E1516 E1517 E1518 E1519 E1520 E1521 E1522 E1523 E1524 E1525 E1526 E1527 E1528 E1529 E1530 E1531 E1532 E1533 E1534 E1535 E1536 x y
0.4643 -2.8683 -0.7107 1.4731 -0.0203 0.0010 -0.0091 0.2571 -0.0667 2.1139 0.8241 -0.3020 -1.1907 -0.6759 0.0461 -2.0725 0.2006 -0.3851 0.1768 -0.2012 0.3038 -0.6143 0.3690 0.2766 1.0433 0.0912 -0.3806 -0.1665 -0.3630 0.6875 0.4664 0.6663 0.8690 -1.5760 -0.5661 -1.2149 0.6084 -0.2790 0.7234 1.3158 -0.5018 2.1917 0.9883 -0.6966 -0.0853 1.6447 0.0896 1.7469 0.9972 1.2382 0.2562 0.2359 -0.8010 1.3886 -0.4358 -1.8316 -1.4931 1.2630 -0.7770 0.2393 -0.2088 -1.8545 0.0879 0.3344 -1.3762 -0.0045 -1.0758 1.0930 0.3206 0.0706 -0.5454 -0.1614 0.7389 -0.4109 -0.0723 -0.3578 0.3844 -0.4767 -0.4487 0.2363 -0.3705 0.2100 -0.3094 0.3107 0.0989 1.7768 0.3702 -1.0599 0.3011 -0.0905 -0.0038 0.4161 -0.5660 -0.3774 0.1835 0.8757 -0.4152 -0.6486 -1.0231 0.8835 0.5129 -0.8705 -1.2460 -1.0722 -0.9210 0.0180 -1.1488 1.6273 1.0168 0.3947 1.6627 0.5068 1.4818 -0.2724 -0.3345 -0.4471 -1.2437 -0.1666 -0.1183 -0.6987 0.2623 1.0335 -0.4197 -0.7551 -0.4956 1.5315 -1.1099 -1.5058 0.2528 0.6581 -0.7031 -0.0995 -0.0960 1.0630 0.6820 0.8853 0.0415 0.0873 -0.8802 -0.7703 -0.1777 0.1269 -0.8103 1.1126 -0.4779 0.6589 0.4097 -0.2678 0.5904 0.9873 0.8244 -0.0771 0.5360 0.1265 -0.0125 -1.3471 -1.0338 0.8644 0.4521 0.3006 -0.9664 1.4741 0.0137 0.5582 0.8244 -0.0313 -0.1239 0.4361 -0.8383 -1.0921 -1.2599 1.1447 0.3330 -1.2516 0.2354 3.2173 0.2340 -1.7048 -0.4347 1.0954 -0.9925 -0.0021 -1.1413 -1.0255 -0.1399 0.2128 0.0781 -0.4176 -0.9311 -0.3195 -0.2693 -1.8969 0.1880 0.3522 -0.8986 0.7886 -0.9433 0.3690 -0.9749 -0.6916 1.7559 0.5015 -0.9576 0.5556 0.2228 -0.0555 -0.4951 -0.1568 1.4618 0.0248 -0.2059 -0.4547 0.4252 -0.0306 0.9836 1.1019 0.6151 -0.2594 1.4904 0.3463 0.4617 0.6222 0.3744 -2.1251 -0.0299 0.9670 0.6855 -1.0208 -1.0533 -0.2584 -0.3358 0.4565 -0.0756 0.4409 1.0528 0.5440 -0.9335 -0.3886 1.3861 -0.2224 -0.7283 -0.8117 -0.2126 -0.9798 -0.1677 0.4724 0.4821 0.1423 0.2251 -1.2394 0.0986 1.0292 0.4859 -1.0068 -1.0225 -2.0873 0.2163 0.7194 0.7308 -1.4089 0.1580 -0.3134 0.8279 0.0943 -0.2118 0.1075 0.6507 -0.7380 -0.0480 0.5020 -0.2834 -1.1981 -0.4616 0.2888 -1.0599 0.9985 0.1439 0.2454 -0.3743 -0.3642 -2.5295 0.6977 1.9479 1.3103 0.1749 0.8499 0.0935 -0.5228 -1.0679 0.3763 0.4880 -1.1025 -1.4539 -0.6494 0.3933 0.1098 1.6478 0.7207 1.8138 -0.5247 0.3714 -0.2079 -0.6920 0.6550 -1.4005 0.2178 -0.4272 1.7589 -0.7347 1.6220 0.1924 -0.4466 -0.9513 -0.8204 1.0725 -0.2272 -0.0710 0.0632 -0.4515 -0.0677 0.0509 0.1423 -0.8812 0.6136 -0.0821 0.1057 0.1987 0.0367 -1.3311 1.4420 -0.6532 0.0741 -0.0008 -0.0700 -0.5091 -0.5719 -0.4068 -0.4581 -0.2912 -0.4543 1.6040 -0.4844 0.6436 -0.7421 -0.5124 1.0246 0.6207 0.3262 1.3067 0.0317 1.1350 1.1425 -0.9422 -1.0427 1.9537 -1.4899 1.4926 0.7919 0.1050 0.9606 -0.0860 0.6469 0.0388 0.9087 -1.1305 0.0279 -0.2071 0.1036 0.7280 -0.8720 0.5950 -0.2441 1.1257 0.2488 0.9532 -0.8787 0.1615 0.4184 -0.4958 -0.1434 0.5887 0.1375 -1.4626 0.2942 -0.6659 -0.3229 -0.2480 -0.8594 -0.5793 0.7209 -0.6412 -1.3812 -0.9546 0.5488 0.6718 0.8078 0.1487 0.1791 -1.5609 -0.4435 -0.1984 0.4939 2.1561 -0.6221 0.1687 -1.5599 -1.0748 1.7723 -0.1002 1.1270 0.0728 0.7573 -0.1372 -0.4266 0.6007 0.2573 -1.7275 -1.0788 -1.4058 1.0911 -0.2676 0.3960 -1.2186 0.3054 -0.5470 -1.0756 -0.2485 0.4105 -0.4917 -1.2966 0.1815 -0.2768 -0.1276 -0.1266 0.9115 0.4896 0.6624 0.6443 -0.3108 0.3944 -0.8046 1.3261 1.1910 -0.6285 -1.5555 -0.3608 0.4683 -1.0602 0.2772 -0.8460 -0.2147 -0.5886 0.4805 0.1497 -0.9900 0.5245 0.9596 0.1222 -1.7414 -0.5101 -1.6570 -0.0252 -1.0184 -0.0836 0.9099 -1.0729 0.5005 0.4064 0.4953 -0.2586 -1.1715 -0.1492 0.5162 1.2317 -0.8438 -1.2229 -1.3533 1.9666 0.5708 -0.3524 1.4603 0.7322 -0.4399 -0.2970 0.5191 1.7209 0.2859 -1.3815 -0.0891 -0.5197 0.2856 1.0929 -0.9225 0.9306 -0.7559 0.6814 1.2570 -1.6103 0.3132 -0.2643 -0.8408 -1.3130 -0.4543 0.0492 1.4008 2.1190 -1.0414 -0.4571 -1.4246 0.2425 -0.0913 1.2039 -1.2493 -1.4453 0.1149 -0.7728 -0.4869 -0.3084 -2.2948 -1.2915 0.4004 -1.6816 -0.0738 0.2473 1.3345 1.3174 -1.5588 1.5225 -0.6613 -0.8619 -0.6382 -0.8472 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-0.2330 0.5915 0.8059 -0.1984 -0.1534 -0.2054 0.4200 0.3075 0.0250 0.2951 1.1252 0.1633 0.2962 -0.2316 -1.1475 -0.7171 0.2148 -0.3692 -0.1346 1.2222 -0.4018 0.0025 -0.5623 -0.3806 -0.1476 0.0121 0.3431 -0.1397 -0.8222 -0.9623 -0.1831 -1.3157 -0.8124 0.0148 -0.0562 -0.8256 0.0938 1.0827 0.0354 0.3454 0.1137 -0.1082 0.2482 -1.5746 0.2121 -0.0157 -0.4830 -0.1900 0.3699 -0.3458 0.0627 -0.2261 -0.2278 0.2566 -0.0039 0.9686 -0.6021 -1.2727 -0.6457 -0.1416 -0.2257 0.2958 0.1326 -0.3533 -0.4545 -0.4589 0.8796 0.2725 -1.1509 -0.0673 0.1969 0.3101 -0.2445 -0.3265 0.2806 -0.0998 -0.2126 -0.6209 -0.0718 -0.1380 -0.1137 0.3377 -0.5057 0.2019 0.6761 0.1151 -0.9056 0.2921 0.6693 0.6859 -0.8302 0.0009 -0.5387 -0.8786 -0.1741 0.1001 -1.6090 0.2116 -0.3601 0.2533 0.8749 0.9539 -0.1388 -0.4396 -0.3125 -0.4006 0.7176 0.5378 -0.1373 -1.7482 -0.3101 -0.0570 0.2737 -0.4189 -0.0231 0.1494 -0.1967 -0.0324 0.4894 -0.0030 -1.0520 0.1819 0.6210 0.0778 -0.8981 -0.4976 0.5410 -0.2743 -0.4007 1.3149 -0.8367 -0.2518 0.3991 0.4979 -0.0618 0.2811 0.0851 -0.0496 -0.2049 -0.0549 -0.0757 0.5405 -0.6092 -0.2424 -0.7623 -0.4873 0.2231 0.1231 -0.9098 0.3523 0.2294 0.1868 0.0089 0.2068 0.6586 -0.9632 -0.3818 0.3449 -0.3687 -0.2416 -0.1505 0.3093 0.3693 0.7946 1.1725 0.0804 1.2139 0.3815 0.2952 -0.1218 0.0445 0.8324 0.1077 -0.6054 -0.6352 0.2208 0.3014 -0.1517 -0.7507 -0.5717 -0.0603 -0.1553 -0.4294 -0.4453 -0.0671 0.4035 -0.1703 0.7761 0.6854 0.1075 -0.8188 -0.7898 0.0531 -0.8428 0.0475 0.0124 0.8078 0.0523 -0.4907 0.2191 -0.7883 -0.0536 0.0397 0.8959 -0.3298 0.3230 0.3807 0.2000 -0.4661 0.3623 0.1243 -0.3461 0.5410 0.4378 -0.0280 -0.2849 0.6641 -0.3540 1.1468 0.3799 0.0557 0.5292 -0.4112 0.5507 0.0788 0.2136 -0.7291 0.3129 -0.0140 -0.2115 -0.2304 0.6696 0.8725 -0.8690 0.8515 0.4728 0.2257 -0.5406 -0.4875 -0.9776 -0.1459 -0.6330 0.0336 -0.5823 0.1018 -0.4699 0.7592 -0.5641 -0.1920 -0.2880 0.2808 -0.6755 -0.3260 0.6853 0.2598 -0.9229 0.5870 -0.5789 0.0348 -0.1100 -1.0048 0.3206 0.3504 -0.6427 -0.8310 -0.3966 0.7021 -0.7455 0.4071 0.4448 0.0406 0.2414 -0.8036 -0.7069 0.3162 -0.3515 -0.0775 -0.1862 0.0476 0.0267 -0.5973 -0.5531 1.2127 -0.1877 0.0824 -0.2113 0.1371 -0.9975 1.4309 0.3014 -0.7752 -0.0305 0.1309 -0.7921 1.2258 -0.3964 -0.1663 -0.6567 -0.4173 0.2829 -0.1759 -0.7378 -0.8767 -1.5051 -1.1013 0.3726 -0.0372 -0.1164 -0.4287 -0.1195 0.6422 0.3731 -0.1593 0.6835 -0.1880 -0.6711 -0.6325 1.2693 -0.9027 -0.3945 -0.5389 -0.7204 -0.9040 -0.4708 1.6979 -0.6192 -0.3220 0.3215 0.4562 -0.3845 -1.1448 0.9386 -0.0386 0.5052 -0.3518 0.7457 0.2551 0.8068 -0.1957 -0.2196 -0.6797 -0.1301 -0.7506 -0.1442 0.8491 -0.0176 0.1099 -0.1546 -0.6008 0.2790 0.1675 0.1281 0.2137 1.0523 -0.2251 0.0875 -0.1522 0.6604 1.2878 0.4176 1.0334 0.2566 -0.3409 0.6173 0.6311 -0.4252 0.8364 0.7163 0.5390 0.4057 -0.0030 -0.1613 -0.1977 -0.1611 0.7233 0.4363 -0.1138 -0.5350 0.7761 -0.3409 1.2091 -0.4036 0.3850 -0.5141 -0.0305 0.2952 -0.2820 -0.6935 -0.4008 0.2821 0.9495 -0.5890 -0.0296 1.1236 -1.3733 0.0157 0.5327 0.9084 -0.3883 0.1085 -0.4160 -0.2297 -0.7947 0.2571 -0.4104 -0.0005 -0.2060 0.3379 -0.4533 0.3575 0.3898 0.5388 -0.4609 0.0760 -0.4370 -0.9403 0.0955 0.9415 -0.0777 -0.3399 0.9634 -0.0492 0.2741 0.1067 -0.0591 0.5976 -1.1804 0.6054 -0.5202 0.7518 -0.1690 0.0683 0.2379 -0.6995 -1.0307 -0.7092 0.0056 0.8613 -0.1479 0.6674 0.3399 0.4193 0.0332 0.1953 -0.6722 0.1502 -0.0774 1.1618 0.3993 -0.8882 0.1119 -1.2605 -0.2380 0.8364 -0.4925 -0.8593 0.3956 0.6525 1.2177 0.7917 -0.1170 -0.6578 1.3352 0.4804 -0.1500 -0.0451 0.3251 -0.2931 0.9296 0.4193 -0.0633 -0.1764 0.0648 0.5827 0.6949 0.2474 -0.3225 -0.2988 -0.0517 0.2406 0.0429 -0.2312 0.0215 -0.9473 -0.0364 -0.6942 -0.1010 -1.8175 -0.1515 0.0901 0.0489 -0.1105 -0.2769 -0.4172 -0.1891 0.2032 -0.1518 -0.7274 0.1016 0.7403 -0.5018 -1.3518 0.5367 -0.4152 -0.0975 0.2139 0.6236 -0.5668 1.0098 0.0763 0.5935 -0.1946 -0.6588 0.7083 -1.1382 0.0822 0.7198 -1.1517 0.2749 0.0012 -0.3026 -0.9519 -0.1796 0.2621 0.5058 -0.5301 -0.0270 0.6564 -0.4504 0.0072 -0.2321 0.4870 -1.0573 0.2456 -0.3963 0.3917 0.0938 -0.7463 -0.6658 0.3736 -0.0514 -0.7389 -0.1103 -0.7404 1.0269 0.5915 -0.0808 -1.0173 -0.3741 0.1558 -0.2179 0.7938 0.9166 -0.6038 0.2551 0.3717 -0.5685 -0.0903 0.6966 -1.1826 0.0083 -0.7434 -0.7433 0.3353 -0.3252 0.1658 0.5697 -0.5250 0.2369 0.7018 0.5884 0.0558 0.5488 -0.2999 -0.3916 1.5221 0.1680 -0.0693 -0.3742 0.4022 0.5813 0.2201 0.4420 0.6637 -0.6584 -0.1178 0.0528 -0.1118 0.1764 0.5666 -0.4417 0.5159 -0.7556 -0.7664 -0.4497 0.0925 -0.3253 0.1431 -0.2621 -0.2757 0.7839 -1.0998 0.0253 1.1190 -0.4090 -0.6982 -315.8800 193.5009

8.3.2 Clustering Analysis

In this section, we will use clustHVT function to conduct clustering analysis. Let’s look at the NbClust function’s output for determining the number of optimal clusters for the dataset.

clust.results <- clustHVT(data = hclust_data_2, 
              trainHVT_results = hvt.results,
              scoreHVT_results = scoring, 
              clusters_k = 'champion')

## *** : The Hubert index is a graphical method of determining the number of clusters.
##                 In the plot of Hubert index, we seek a significant knee that corresponds to a 
##                 significant increase of the value of the measure i.e the significant peak in Hubert
##                 index second differences plot. 
## 

## *** : The D index is a graphical method of determining the number of clusters. 
##                 In the plot of D index, we seek a significant knee (the significant peak in Dindex
##                 second differences plot) that corresponds to a significant increase of the value of
##                 the measure. 
##  
## ** Among all indices:
## ** * 1 proposed 1 as the best number of clusters
## ** * 2 proposed 2 as the best number of clusters
## ** * 2 proposed 3 as the best number of clusters
## ** * 4 proposed 4 as the best number of clusters
## ** * 1 proposed 5 as the best number of clusters
## ** * 1 proposed 6 as the best number of clusters
## ** * 1 proposed 7 as the best number of clusters
## ** * 1 proposed 9 as the best number of clusters
## ** * 1 proposed 12 as the best number of clusters
## ** * 1 proposed 13 as the best number of clusters
## ** * 3 proposed 15 as the best number of clusters
## *******************************************************************
## ***** Conclusion *****
## ** * According to the majority rule, the best number of clusters is 4
## *******************************************************************
##                          Index     Number_clusters    Value_Index
## -------------------------------------------------------------------
##                       kl           4.0000           2.4230
##                       ch           7.0000          72.3145
##                 hartigan           4.0000          23.4170
##                   cindex           6.0000           0.2628
##                       db           9.0000           0.9109
##               silhouette           4.0000           0.2924
##                ratkowsky          12.0000           0.0934
##                     ball           3.0000     1665598.0961
##                   hubert               NA               NA
##                   dindex               NA               NA
##               ptbiserial           4.0000           0.5193
##                      gap           2.0000          -0.6952
##                     frey           1.0000               NA
##                  mcclain           2.0000           0.6980
##                    gamma          15.0000           0.9173
##                    gplus          15.0000          15.9381
##                      tau           3.0000         823.6557
##                     dunn          13.0000           0.2003
##                  sdindex           5.0000           0.0100
##                     sdbw          15.0000           0.0800
## *******************************************************************

From the above results, we can clearly see that highest recommendation is 4 as the optimal number of clusters. Let’s look at the dendogram plot for the same.

clust.results$dendogram()

Though the highest recommendation is 4, the corresponding dendogram results in large clusters. However, for our specific use case, we prefer to have smaller clusters. So we will select the next highest recommendation from the NbClust summary. With 3 indices supporting this choice, we will proceed with ‘challenger’ to achieve a more granular clustering solution.

clust.results <- clustHVT(data = hclust_data_2, 
              trainHVT_results = hvt.results,
              scoreHVT_results = scoring, 
              clusters_k = 'challenger')

Let’s look at the dendogram for challenger number of clusters.

clust.results$dendogram()

Let’s look at the Cluster HVT plot for the 15 clusters.

clust.results$clusterplot

8.3.3 OpenAI API call for generating domain names

Now, we will pass all the data from each cluster to OpenAI’s gpt-3.5-turbo model to generate a keyword/domain name of that particular cluster.

Subsetting the data in each cluster by the below code and processing it without punctuation marks to be given as input in API call. Data from each cluster will be saved as a .txt file and the files will be read during the API call.

Let’s call the Function for writing clusters as .txt files which is explained in section 3.1 for this process.

cluster_data <- clust.results$cluster_data
create_and_write_clusters(cluster_numbers = 1:15, cluster_method = cluster_data)

The below chunk makes a call to OpenAI’s gpt-3.5-Turbo via chat completions API and print the responses for clusters below.

result <- get_cluster_domains(15)
result$formatted_table
Domain_name_for_cluster_1 Domain_name_for_cluster_2 Domain_name_for_cluster_3 Domain_name_for_cluster_4 Domain_name_for_cluster_5 Domain_name_for_cluster_6 Domain_name_for_cluster_7 Domain_name_for_cluster_8 Domain_name_for_cluster_9 Domain_name_for_cluster_10 Domain_name_for_cluster_11 Domain_name_for_cluster_12 Domain_name_for_cluster_13 Domain_name_for_cluster_14 Domain_name_for_cluster_15
Activism Engagement Culinary Entertainment Spirituality Childcare Appreciation Wellness Memory Cuisine Achievement Gratitude Collectibles Celebration Blessings
Activism Engagement Culinary Entertainment Spirituality Parenting Inspiration Empowerment Memory Cuisine Sports Fashion Collectibles Celebrity Celebration
Activism Engagement Culinary Entertainment Spirituality Childcare Appreciation Empowerment Memory Cuisine Achievement Gratitude Collectibles Celebration Blessings
Activism Engagement Culinary Entertainment Spirituality Parenting Inspiration Empowerment Memory Cuisine Sports Celebrity Collectibles Celebrity Celebration
Activism Engagement Culinary Entertainment Spirituality Childcare Appreciation Wellness Memory Cuisine Achievement Gratitude Collectibles Celebrity Blessings
Activism Engagement Culinary Entertainment Spirituality Parenting Inspiration Wellness Memory Cuisine Sports Fashion Collectibles Celebrity Celebration
Activism Engagement Culinary Entertainment Spirituality Parenting Inspiration Wellness Memory Cuisine Sports Celebrity Collectibles Celebrity Celebration
Activism Engagement Culinary Entertainment Spirituality Childcare Appreciation Empowerment Memory Cuisine Achievement Celebration Collectibles Journey Blessings
Activism Engagement Culinary Entertainment Spirituality Parenting Inspiration Wellness Memory Cuisine Sports Fashion Collectibles Celebrity Celebration
Activism Influence Culinary Entertainment Spirituality Safety Appreciation Wellness Memory Cuisine Achievement Gratitude Collectibles Connection Blessings
Activism Engagement Culinary Entertainment Spirituality Parenting Appreciation Wellness Memory Cuisine Achievement Gratitude Collectibles Celebrity Blessings
print(result$mode)
##  Domain_name_for_cluster_1  Domain_name_for_cluster_2 
##                 "Activism"               "Engagement" 
##  Domain_name_for_cluster_3  Domain_name_for_cluster_4 
##                 "Culinary"            "Entertainment" 
##  Domain_name_for_cluster_5  Domain_name_for_cluster_6 
##             "Spirituality"                "Parenting" 
##  Domain_name_for_cluster_7  Domain_name_for_cluster_8 
##             "Appreciation"                 "Wellness" 
##  Domain_name_for_cluster_9 Domain_name_for_cluster_10 
##                   "Memory"                  "Cuisine" 
## Domain_name_for_cluster_11 Domain_name_for_cluster_12 
##              "Achievement"                "Gratitude" 
## Domain_name_for_cluster_13 Domain_name_for_cluster_14 
##             "Collectibles"                "Celebrity" 
## Domain_name_for_cluster_15 
##                "Blessings"

Let’s replace the cluster numbers to above generated domain names in the dataset.

invisible(lapply(1:15, function(i) {
  cluster_data$cluster_name[cluster_data$clusters == i] <<- result$mode[[i]]}))

8.3.4 Cluster Domain Naming: OpenAI vs Manual Approach

In this section, we will generate the Heatmaps of OpenAI and Manual Approach for the Sammon’s coordinates (2D - x and y coordinates) with the embeddings (1536D) of the centroids of 110 cells.

OpenAI Approach

  • The OpenAI Approach is a 2D Heatmap where colors represent different domains assigned to the clusters. These domain names are generated using OpenAI’s GPT-3.5-turbo via the chat completions API. Tweets from each cluster are provided as input, along with a system prompt requesting domain names that accurately reflect the context of the data. To ensure consistent results, this process is repeated 10 times for each cluster, and the most frequently occurring name is selected.
centroid_cluster_data <- cluster_data %>% dplyr::select('Cell.ID', 'cluster_name','names.column')
hclust_plot_2 <- clustHVT(dataset = centroid_cluster_data, hvt.results = hvt.results, domains.column = "cluster_name", type = "plot")

Manual Approach

  • The Manual Approach is a 2D Heatmap, features colors representing domains assigned to clusters, but in this case, the domain names are determined manually. This involves a detailed examination of the tweets within each cluster, allowing for a more nuanced assignment of domain names based on Human interpretation and context.
clust_1 <- subset(cluster_data$names.column, cluster_data$clusters == 1)
cluster_names <- c("Announcements *", "Engagement *", "Personal Freedom *", 
                   "Upcoming Events *", "Social Ideology *", "Thoughts *", 
                   "Heartwarming *", "Promotion *", "Governance *", 
                   "Exhilaration *", "Media Posts *", "Gratitude *", 
                   "Excitement *", "Appreciation *", "Wishes *")

cluster_data$cluster_name <- cluster_names[cluster_data$clusters]
centroid_cluster_data <- cluster_data %>% dplyr::select('Cell.ID', 'cluster_name','names.column')
manual_plot_2 <- clustHVT(dataset = centroid_cluster_data, hvt.results = hvt.results, domains.column = "cluster_name", type = "plot")

8.3.5 Visualization of plots: OpenAI vs Manual Approach

In this section, we are visualizing the OpenAI plot where the domain names are generated by OpenAI via API call and the Manual plot where the domain names are manually read and assigned.

plotly::subplot(hclust_plot_2, manual_plot_2) %>%
plotly::layout(annotations = list(
      list(x = 0.15, y = 1.05, text = "OpenAI Approach", showarrow = F, xref='paper',yref='paper', width = 150),
      list(x = 0.9, y = 1.05, text = "Manual Approach", showarrow = F, xref='paper', yref='paper', width = 165))) 

(*) Manually assigned domain names (to differentiate the domain names from OpenAI generated)

Inference

  • A visual comparison between the plots of OpenAI Approach and the Manual Approach indicates that the labels assigned to clusters exhibit complete uniqueness across both methods.

  • The labeling methodology utilized by OpenAI largely depends on the content of a restricted sample of tweets, generally one or two per cluster. For example, the ‘safety’ domain in the OpenAI plot is represented by only a single tweet pertaining to be precautious, whereas the majority of the remaining tweets discuss personal opinions. This distinction was more accurately captured by the Manual Approach, which categorized it as ‘Thoughts’.

8.3.6 Calculation of Average Silhouette Width(ASW)

  • The Average Silhouette Width (ASW) is a measure to evaluate the quality of clusters in a clustering analysis. It assesses how similar each point in a cluster is to other points in the same cluster compared to points in different clusters. Higher ASW indicates better-defined and more separated clusters.
clusters <- clust.results$clusters
ward_sil_2 <- silhouette(clusters, dist(hclust_data_2))
ward_asw_2 <- round(mean(ward_sil_2[, "sil_width"]), digits = 2)
print(ward_asw_2)
## [1] 0.24
  • For Sammon’s coordinates data (2D - x and y coordinates) with the embeddings (1536D) of the centroids of 110 cells, the ASW for ward is 0.24 .

8.4 Comparative Visualizations of OpenAI and Manual Approach: 2D vs 1538D Datasets

This section provides a side-by-side visual examination of the OpenAI plots and the manual plots for both datasets. We will compare and analyze the differences and similarities between these visual representations.

8.4.1 Comparison of OpenAI Approach

Here we are comparing the plots of OpenAI approach where the domain names are generated by OpenAI via API call for both (2D and 1538D) datasets.

plotly::subplot(hclust_plot_1, hclust_plot_2) %>%
plotly::layout(annotations = list(
      list(x = 0.15, y = 1.05, text = "OpenAI plot - 2D", showarrow = F, xref='paper',yref='paper', width = 150),
      list(x = 0.9, y = 1.05, text = "OpenAI plot - 1538D", showarrow = F, xref='paper', yref='paper', width = 165)))
  • Visual comparison of the plots reveals significant structural changes of the clusters, even with consistent number of clusters k = 15, due to variations in the dataset dimensions (2D & 1538D).

  • From a visual inspection, only a few domain names are repeated, while the rest have changed. The keen observation suggests that OpenAI’s attempt to capture the full context was partially successful, as many domain names seem to reflect only one or two sentences or tweets from a cluster, rather than providing a domain name that comprehensively represents the entire context of the cluster.

8.4.2 Comparison of Manual plots

Here we are comparing the plots of Manual approach where the domain names are generated by human interpretation (2D and 1538D) datasets.

plotly::subplot(manual_plot_1, manual_plot_2) %>%
plotly::layout(annotations = list(
      list(x = 0.15, y = 1.05, text = "Manual plot - 2D", showarrow = F, xref='paper',yref='paper', width = 150),
      list(x = 0.9, y = 1.05, text = "Manual plot - 1538D", showarrow = F, xref='paper', yref='paper', width = 165)))
  • Visual comparison of the plots reveals significant structural changes of the clusters, even with consistent number of clusters k = 15, due to variations in the dataset dimensions (2D & 1538D).

  • From visual inspection, the majority of domain names are repeated, with only a few changing. This observation indicates that the manual approach was largely effective in capturing the full context, as many domain names appear to reflect the overall idea of the content.

8.5 Tanglegram Comparison of 2D and 1538D datasets

A tanglegram is a visual comparison of two hierarchical structures or dendograms side by side, with lines connecting matching elements to highlight differences and similarities. This is useful for comparing different clustering results.

Let’s look at the tanglegram of

  1. Sammon’s coordinates data (2D - x and y coordinates) of the centroids of 110 cells
  2. Sammon’s coordinates data (2D - x and y coordinates) with the embeddings (1536D) of the centroids of 110 cells
dend1 <- as.dendrogram(hc_1)
dend2 <- as.dendrogram(hc_2)
labels(dend1) <- as.character(labels(dend1))
labels(dend2) <- as.character(labels(dend2))
tanglegram(dend1, dend2,   main_left = "Sammon's coordinates(2D)", main_right = "Sammon's coordinates with embeddings(1536D + 2D)", cex_main_left = 1.5, cex_main_right = 1.5)


Insights

  1. The colored lines connecting the two dendograms indicate relationship between clusters on both sides. Lines of the same color tend to stay together, indicating that clusters are largely preserved between the two dendograms.

  2. Points connected by shorter lines are more stable in their representation across both datasets, implying that their characteristics are consistently captured in both cases. The data points represented by shorter lines have undergone less dramatic shifts in their relative positions.

  3. The crossing patterns of these lines suggest that while overall cluster structure is preserved, there have been some reorganizations within and between clusters.

9. Conclusion

1) Hierarchical Voronoi Tessellation


2) Hierarchical Clustering

In our comparative analysis of hierarchical clustering, we evaluated the performance of the Ward.D2 method using two distinct datasets:

The Average Silhouette Width (ASW), a metric that quantifies clustering quality on a scale from -1 to 1, was employed to assess the clustering efficacy. Higher ASW values close to 1 indicate more distinct and well-separated clusters, while those values close to -1 indicate poor clusters.


3) Insights on OpenAI and Manual Approaches in generating domain names for clusters

By evaluating the domain names produced by OpenAI’s gpt-3.5-turbo against those assigned manually, the following conclusions can be drawn:

Reasoning:


10. References

  1. Topology Preserving Maps

  2. Vector Quantization

  3. K-means

  4. Sammon’s Projection

  5. Voronoi Tessellations

  6. LLM Embeddings Explained