 The provided HTML code represents a web page for a fuzzy logic waveform detector, an interactive web application demonstrating the use of fuzzy logic in identifying different waveform shapes on the various noise levels. The web page includes meta tags and title, description and title for search engines and social media sharing, along with character set and viewport settings for responsive design. Starsheets, external CSS files for styling, including a flask-specific syntax for linking starsheets. JavaScript sources, external scripts including plot leaf for graphing functionality and the delayed script loading fads. Header section, contains a navigation bar with a logo and menu items, along with a unique menu toggle mechanism for mobile views. Control panels for selecting input waveforms and adjusting noise amplitude. Sections for displaying output wave details and statistical data. Graph containers for visualizing input and output waveforms. Informational content, detailed descriptions and insights about the application, its functionality, and the concept of fuzzy logic, particularly in signal processing. Technical details, mentions the use of HTML, CSS, JavaScript, SKFuzzyLibrary, plot leaf and flask. Closing note, a thank you note and invitation for feedback. Dynamic script loading, a script for loading Google AdSense is dynamically appended after a delay. The page is structured to be interactive and educational, focusing on the application of fuzzy logic in waveform detection and analysis. The provided JavaScript code is designed to work with the HTML page of the fuzzy logic waveform detector. It is executed when the DOM content is fully loaded. Here's an overview of its functionality. Element selection, it selects various DOM elements like drop-down for input wave selection, slider for noise amplitude, and spans for displaying detection error and unknown percentage. It also selects a div for the input graph. Update output, data, updates the text of elements to show the output wave, detection error, and unknown percentage based on the data received. Update input graph, signal, uses plot leaf to plot the input waveform graph using the signal data received. Retrieves values from the input wave drop-down and noise amplitude slider. Sends a post request to the server endpoint, analyze with the input wave and noise amplitude as JSON data. Upon receiving the response, it updates the output information and input graph using the received data. Handles any errors that occur during the fetch operation. Adds a vendlessness to the input wave selection and noise amplitude slider. Whenever there's a change or input event, the analyze waveform function is triggered. Initial analysis invocation, calls analyze waveform to perform an initial analysis when the page loads. This script essentially makes the web page interactive, allowing users to select different waveforms and noise levels, and then see the analysis results and graph updated in real time based on their selections. This Python script uses Flask to create a web server for the fuzzy logic waveform detector. It defines several key components and functionalities. Flask app initialization, the Flask class is used to create an instance of a web application. The home function is mapped to the root url. It serves the index, html page. Analyze root, analyze, post method. The analyze function handles post requests for analyze url. It receives JSON data from the request, extracts the input wave and noise amplitude, and processes them. The generate underscore signal function creates a signal based on the input wave type. The add underscore noise function adds noise to this signal. The analyze underscore signal function analyzes the noisy signal using fuzzy logic. The calculate underscore detection underscore error and calculate underscore unknown underscore percentage functions compute statistical metrics. The results, including the process wave information and statistics, are returned as JSON. Generate underscore signal creates different types of waveforms based on the input string. Add underscore noise adds Gaussian noise to the signal. Analyze underscore signal analyzes the noisy signal using fuzzy logic implemented in the fuzzy underscore logic function. Fuzzy underscore logic implements the fuzzy logic system using the sk fuzzy library. It uses membership functions and fuzzy rules to classify the signal into different wave types, square, triangle, trapezoid. Calculate underscore symmetry underscore array calculates symmetry of the signal. Calculate underscore peak underscore sharpness underscore array determines the sharpness of peaks in the signal. Calculate underscore detection underscore error computes the percentage error in detection. Calculate underscore unknown underscore percentage calculates the percentage of waves classified as unknown. The if underscore underscore name underscore underscore equals equals underscore underscore main underscore underscore block checks if the script is being run directly, not imported. If true, it starts the flask app with dbyg equals true for development purposes. This script is a back-end part of the Fuzzy Logic Waveform Detector web application. It handles data processing and analysis and communicates with the front-end, html, JavaScript, to provide interactive waveform detection functionality. The output in the Fuzzy underscore logic function is determined by the Fuzz, the Fuzz method, which computes the centroid of the aggregated membership functions. The final output waveform type, square, triangle, or trapezoid, is selected based on where this centroid value falls within certain ranges. If output less than one-third, the function returns, square. If output less than two-thirds, it returns, triangle. Otherwise, it returns, trapezoid. If you're always getting, square, as the output, it suggests that the centroid, output, is consistently falling below one-third. This could be due to several reasons. Input data characteristics, the characteristics of the input data might be said that they consistently lead to a low centroid value. Membership function definitions, the way membership functions are defined, mf underscore symmetry underscore low, mf underscore symmetry underscore mid, mf underscore symmetry underscore high, etc., might be influencing the centroid calculation. If these functions are not properly calibrated or if they don't accurately represent the characteristics of the different wave types, it might skew the results towards one particular category. Fuzzy logic rules, the rules, rule underscore square, rule underscore triangle, rule underscore trapezoid, and how they're aggregated could be biased towards producing lower centroid values. Signal processing, the methods calculate underscore symmetry underscore array and calculate underscore peak underscore sharpness underscore array might be processing the signal in a way that favors the conditions for a square classification. To diagnose this issue, you can check the values of symmetry and peak underscore sharpness arrays, as well as the outputs of membership functions and rules, to see if they're lying with your expectations. You can test with different input signals and noise levels to see if the issue persists across various scenarios. Debug the fuzzy underscore logic function by printing out intermediary values like the aggregated fuzzy set and the resulting centroid to understand how the final output is being computed. Graphically reviewing fuzzy parameters involves visualizing the membership functions, rules, and the resultant fuzzy sets. This can be done using libraries like matploblib in python. Here's a step-by-step guide to visualize these components in the context of your fuzzy logic waveform detector. Install matploblib. If you haven't already, you'll need to install the matploblib library. It can be installed via pip, copycode pip install matploblib. Define membership functions. Your script already defines membership functions. You can plot these functions to see their shape and range. Plot membership functions. Use matploblib to plot these functions. For each membership function, plot the universe x-axis against the membership value y-axis. Visualize rules. Rules are usually combinations of membership functions. You can visualize the result of these combinations. Show aggregated output. The aggregated output fuzzy set can also be plotted to see the combined effect of all rules. Plot defuzzified result. Finally, you can plot the centroid calculation on the aggregated output to see where the defuzzified result lands. Here is an example code snippet to illustrate how you could plot one of your membership functions. This code will plot the low-symmetry membership function. You would repeat similar steps for other membership functions, the rules, and the aggregated output. This graphical representation will help you better understand and debug your fuzzy logic system, giving insights into how different inputs affect the output. Continuing with the graphical representation of your fuzzy logic system, let's look at how to visualize the rest of the membership functions, the rules, and the aggregated output. For each fuzzy set, like symmetry and peak sharpness, plot all the corresponding membership functions. Plot the result of the fuzzy rules. These are combinations, like fmin or fmax, of the membership functions. Plot the aggregated output from all the rules. Finally, indicate the defuzzified result, centroid, on the aggregated output plot. Plot the result of the fuzzy rules. These are combinations, like fmin or fmax, of the membership functions. Continuing with the graphical representation of your fuzzy logic system, let's look at how to visualize the rest of the membership functions, the rules, and the aggregated output. One, plotting all membership functions, for each fuzzy set, like symmetry and peak sharpness, plot all the corresponding membership functions. Symmetry membership functions. To continue with the graphical representation, especially within the context of the fuzzy underscore logic function, you would integrate the plotting directly into your fuzzy logic processing. This allows you to visualize how the membership functions, rules, and aggregated output are working for each specific instance that the fuzzy underscore logic function is called. Here's how you can do this. Integrate plotting into fuzzy underscore logic function. Within the fuzzy underscore logic function, after defining membership functions and rules, you can add code to plot these elements. This way, each time fuzzy underscore logic is called, it will generate corresponding plots. Plot membership functions. Plot the membership functions for symmetry and peak sharpness. Visualize the activation of rules. Show how the rules are activated based on the input data. Display the aggregated output. Plot the aggregated output to see the combined effect of all activated rules. Plot the defuzzification process. Indicate the centroid value on the aggregated output plot to visualize the defuzzification result. Here's an example of how you can modify the fuzzy underscore logic function to include these plots. In this modification, after each step in the fuzzy logic process, corresponding plots are generated. This visual representation is immensely helpful for debugging and understanding how your fuzzy logic system behaves with different inputs. Remember that displaying these plots will be more useful during the development and testing phase. In a production environment, you might want to disable these plots or log them differently. In this modification, after each step in the fuzzy logic process, corresponding plots are generated. This visual representation is immensely helpful for debugging and understanding how your fuzzy logic system behaves with different inputs. Remember that displaying these plots will be more useful during the development and testing phase. In a production environment, you might want to disable these plots or log them differently. Fuzzy logic is a form of logic that allows for more flexible reasoning compared to traditional Boolean logic, which is strictly true or false. In fuzzy logic, truth values can be any real number between 0 and 1, representing degrees of truth. This approach is particularly useful in scenarios where the information is imprecise or uncertain, allowing for a more nuanced and realistic decision-making process. In the context of your fuzzy underscore logic function, it is returning an output based on the analysis of a noisy signal using fuzzy logic principles. Here's a breakdown of what the function is doing. Input analysis. The function receives a noisy signal as input and processes it to extract features like symmetry and peak sharpness. These features are used to evaluate the signal against defined fuzzy logic rules. Membership functions. The function defines several membership functions for different features of a signal, like low, mid, and high symmetry. Membership functions are curves that define how each point in the input space is mapped to a membership value or degree of truth between 0 and 1. Rule application. Fuzzy rules are applied to these membership values. These rules are logical statements that determine how the input features should be interpreted. The function uses the minimum, NP, FMIN, operator to combine the membership functions in these rules. Aggregation of rules. The results of these rules are then aggregated into a single fuzzy set. This is done using the maximum, NP, FMIN, operator which combines the outputs of individual rules. Defuzzification. Finally, the function uses the Fuzz. Defuzz method to perform defuzzification, converting the fuzzy result back into a single crisp output. This is typically done by finding the centroid of the aggregated fuzzy set, which represents the center of gravity or the balance point of this set. Return output. The function returns a string, square, triangle, or trapezoid, based on where the centroid value falls within predefined ranges. This output represents the classified shape of the waveform based on the fuzzy logic analysis. In essence, your fuzzy underscore logic function is using fuzzy logic to classify a waveform shape from a noisy signal, returning a value that best represents the characteristics of that waveform according to the defined fuzzy rules and membership functions. Fuzzy logic waveform detector. Welcome to fuzzy logic waveform detector, an application designed to demonstrate the capabilities of fuzzy logic in identifying various waveform shapes under different levels of noise. This tool allows users to select a waveform, adjust noise amplitude, and observe how these changes affect the detection accuracy of the system. Utilize the drop-down menus to choose the type of inward waveform you wish to analyze in the slider to simulate varying noise conditions. The application will display the identified waveform and statistical analysis, providing insights into the performance of the fuzzy logic detection algorithm. This is particularly useful for educational purposes, allowing students and professionals to experiment with signal processing concepts and understand the impact of noise on detection systems. Whether for academic learning or industry demonstration, this application serves as a valuable resource for exploring the intricacies of waveform analysis. How it works, the tool uses a fuzzy logic system to identify the inward waveform. The system consists of three inputs, the inward waveform, the noise amplitude, and the output waveform. The inward waveform is a crisp value while the noise amplitude and output waveform are fuzzy values. The system uses the inward waveform and noise amplitude to determine the output waveform. The output waveform is then compared to the inward waveform to determine the accuracy of the system. Getting started, select a waveform from the drop-down menu and adjust the noise amplitude using a slider. The application will display the identified waveform and statistical analysis, providing insights into the performance of the fuzzy logic detection algorithm. Features, interactive waveform selection, adjustable noise amplitude, real-time waveform detection, statistical analysis, key insights. Waveform detection is more accurate with lower noise amplitude. Waveform detection is more accurate with more distinct waveform shapes. Fuzzy logic. Fuzzy logic is a form of many-valued logic that deals with reasoning that is approximate rather than fixed and exact. In contrast with traditional binary logic, which uses the values of true or false, fuzzy logic uses the values of true, false and partially true. This allows for more nuanced reasoning and more accurate decision-making. Fuzzy logic is particularly useful in situations where the data is incomplete or imprecise, such as in signal processing. In biomedical data, fuzzy logic can be used to identify various waveform shapes on the different levels of noise. This is particularly useful in the field of signal processing, where it is often necessary to identify the shape of a waveform in order to determine the type of signal. For example, in electrocardiogram, ECG, data, it is often necessary to identify the shape of the waveform in order to determine the type of heartbeat. Fuzzy logic can be used to identify the shape of the waveform and determine the type of heartbeat with a high degree of accuracy. Technical details. The application is built using HTML, CSS, and JavaScript. The waveform detection system is implemented using the SK Fuzzy Fuzzy Logic Library in Python. The application uses the Plotli library to generate the graphs and the Flask library to host the application on a local server. Closing note. Thank you for using fuzzy logic waveform detector. If you have any questions or comments, please feel free to contact me using a form on the contact page. To display these plots on the front page of your web application, you need to generate the plots on the server side, convert them into images, send them to the client, Raza, and then display them on the web page. Here's a step-by-step process. Generate plots on server side. In your Flask backend, modify the fuzzy underscore logic function or any other relevant function to create and save the plots as images. You can use Matplotlib to save plots directly to a buffer. Convert plots to images. After generating a plot, instead of displaying it with PLT, show, save it to a bytes.io buffer as an image, e.g. PNG format. Encode images for web transfer. Convert the buffer content to a base 64 string. This makes it suitable for embedding directly into an HTML image tag or sending as JSON data. Send images to front end. In your Flask root, include these base 64 encoded strings in the JSON response sent back to the client. Display images on web page. In your front end JavaScript, modify the code to handle these image strings. When you receive the response from your Flask server, extract the image data and set it as the SRC attribute of an IMG tag in your HTML. Here's an example implementation in Python. If you prefer to send the plot data as JSON instead of images, you still need to convert the plots to a format that can be transmitted over HTTP and then rendered on the client side. One efficient way to do this is by converting the plots to a data URL format, base 64 encoded, and then embedding this data within JSON. Here's how to modify the process. Generate the plot in your Python code as before. Convert the plot to base 64 string as you did previously. Embed the base 64 string in your JSON response. This way, you're sending the image data as a string within your JSON structure. Fetch the data from your Flask root as you normally would. Extract the base 64 string from the JSON response. Set this string as the source for an image element in your HTML. Make sure you have an IMG tag in your HTML where you want the plot to be displayed. This method allows you to send the plot data as a part of your JSON response. The client side JavaScript then takes this data and uses it to display the image. Using PLT, savefig, buff, format equals png in this context, even though we're not transferring traditional image files, is a method to convert the plot generated by Matpoblib into a format that can be converted into a web page. Here's the reasoning behind it. Format conversion. Matpoblib creates plots in a graphical format. To incorporate these plots into a web page, they need to be converted into a web-friendly format. The png format is universally supported by web browsers and is suitable for high-quality images. Buffering. The bytes.io object acts as an in-memory buffer. When you use PLT, savefig, buff, format equals png. You're saving the plot as a png image into this buffer instead of an external file. This approach is useful for dynamically generated images that you don't need to store permanently. Base64 encoding, the contents of the buffer, which now contains the png image data, are then Base64 encoded. Base64 is a method of encoding binary data into a string format that can be easily transmitted over the internet. It's particularly useful for embedding binary data, like images, within text-based formats like JSON. Embedding in JSON, the Base64 encoded string can be included in your JSON response. This string represents the image data of your plot. Rendering on the client side. Once the client, web browser, receives this JSON response, it can decode the Base64 string back into binary format and render it as an image in the HTML document. Transferring the raw data required for plots from a Python backend to a JavaScript front-end for rendering is a viable approach. This method involves sending the numerical data points into the settings through JSON. The front-end, JavaScript, then uses this data to create the plots, typically using a JavaScript charting library like Chart, JS, Plotly, JS, or D3, JS. Here's a general outline of how to implement this. Prepare data points. Generate the necessary data points for your plots in your Python code. Create a JSON response. Package these data points into a JSON-friendly format. Fetch the data, and then age acts called to your Flask root and retrieve the JSON response. Use a charting library. Use a JavaScript library to render the plots using the received data. Include a canvas element in your HTML for the chart. Flexibility. You can modify the plot on the client side without needing to reprocess it on the server. Reduce server load. The server only needs to send raw data, not process and render images. Interactive plots. JavaScript charting libraries often allow you to create and hovering. There are several JavaScript charting libraries available. Your choice depends on your specific needs. Chart. JS, simple and easy to use, great for basic charts. Plotly. JS offers a wide range of chart types, including 3D charts, and is highly customizable. D3. JS, highly flexible and powerful, suitable for complex data visualizations, but has a steep learning curve. Each of these libraries has its own syntax process, so you'll need to refer to the respective documentation for implementation details. To include the sharpeners membership function in the plot and convert the entire plotting process into JavaScript, you'll need to follow these steps. Prepare the data. On the Python backend, calculate all the necessary values for your plots, such as the X and Y coordinates for the membership functions and any other relevant data. Create a JSON response. Structure the data in a JSON format. This should include all the data needed to recreate the plots on the client side. Send the JSON response. From your flask root, return the JSON response with the data to the front end. Write JavaScript code. On the front end, write JavaScript code that takes the JSON data and creates the plots using a charting library or custom drawing code. The JSON response from your back end, Python, might look something like this. On the front end, you would use a charting library or the HTML5 canvas API to draw the plots based on this data. Here is a solo code example of how you might do this with chart, JS. Remember to include less than canvas greater than elements in your HTML with corresponding IDs for each plot, e.g., id="symmetry chart". The charting library will use these canvas elements to draw the plots. To include the sharpener's membership function in the plot and convert the entire plotting process into JavaScript, you need to follow these steps. Prepare the data. On the Python backend, calculate all the necessary values for your plots, such as the X and Y coordinates for the membership functions and any other relevant data. Create a JSON response. Structure the data in a JSON format. This should include all the data needed to recreate the plots on the client side. Send the JSON response. From your flask root, return the JSON response with the data to the front end. Write JavaScript code. On the front end, write JavaScript code that takes the JSON data and creates the plots using a charting library or custom drawing code. The JSON response from your back end, Python, might look something like this. Python, copy code and generate underscore plot underscore data. Hash this function would normally calculate these values based on your input data return. Symmetry, X, list, range, minus 100, 101, low, fuzz, trimf, X, minus 100, minus 50, 0, for X in range. Minus 50, 0, 50, for X in range, minus 100, 101, high, fuzz, trimf, X, 0, 50, 1. On the front end, you would use a charting library or the HTML5 canvas API to draw the plots based on this data. Here is a pseudocode example of how you might do this with chart, JS. Remember to include less than canvas greater than elements in your HTML with corresponding IDs for each plot, e.g. it equals symmetry chart. The charting library will use these canvas elements to draw the plots. The HTML structure you've provided is for the fuzzy logic waveform detector web application. It already includes sections for interactive control elements like drop-down menus and sliders, placeholders for statistical information, and div containers where you can render the graphs. To display the fuzzy logic parameters, membership functions, rule activation, and aggregated output, using JavaScript, you need to modify the back end to send the necessary data to the front end and use a JavaScript library like Plotly, JS to render the graphs. Here's a simplified version of what you need to do. Back end, Flask, modify your Flask routes to send the data points for the membership functions, rule activation, and aggregated output as JSON. Front end, JavaScript, write JavaScript functions to request this data to send to the front end. Write JavaScript functions to request this data from the back end and plot it using Plotly, JS. Ensure you have Plotly, JS included in your HTML or loaded dynamically as you are doing with other scripts. HTML, adjust your div containerids to be unique and valid, and use them to target where the graphs should be displayed. Here is an example of how you might adjust the HTML and write JavaScript to use Plotly, JS for rendering the plots. In your Flask back end, you would have a route like, get fuzzy logic parameters that computes the required data and sends it back as JSON. This is a basic example to get you started. You would need to extend it to handle all the data points for the membership functions and include the logic for plotting the sharpness functions and rule activations. Also, the ins of the div elements in the HTML are updated to be more compliant with common conventions, using hyphens instead of spaces. Chat GPT. The HTML string. In your Flask back end, you have a route like, get fuzzy logic parameters that computes the required data and sends it back as JSON. This is a basic example to get you started. You would need to extend it to handle all the data points for the membership functions and include the logic for plotting the sharpness functions and rule activations. Also, the IDs of the div elements in the HTML are updated to be more compliant with common conventions, using hyphens instead of spaces. To update your Python Flask back end code to send the necessary values to the front end using JSON, you need to calculate the necessary data points for your plots based on the fuzzy logic operations. Structure these data points into a Python dictionary, which can be easily converted into JSON. Send the structured data as a JSON response to an Ajax request made by the front end. Here is a modified version of a Flask route that calculates the necessary data for the fuzzy logic parameters and returns them as a JSON response. This code assumes you have NumPy and SKFuzzy installed in your environment. It also assumes that you have the logic to calculate the real values for rule activation and aggregated output based on your fuzzy logic system, which you will replace where the random values are currently being generated. Once this Flask route is set up, your front end JavaScript code can make a get request to, get fuzzy logic parameters, receive the JSON response, and use it to plot the graphs as previously described. To update your existing Flask code to include the necessary values for the fuzzy logic plots you will need to make several additions. Specifically, you'll need to modify the Analyze route to also compute and return the data for the membership functions, rule activations, and aggregated output and response. Here's how you can update the Analyze route. With these changes, the Analyze route now returns all the necessary data for plotting the fuzzy logic parameters. The front end can then use this data to render the plots as previously described. Remember that this code assumes the rest of your Flask application and JavaScript front end does set up to handle this additional data. The front end will need to pass the fuzzy data object and use it to plot the graphs using a library like Plotly.js. Your updated fuzzy underscore logic function looks good. It now returns two things, the determined output underscore wave based on the centroid of the aggregated membership functions and the fuzzy underscore plot underscore data which is a dictionary containing all the data needed to plot the fuzzy logic parameters. Remember, this fuzzy underscore plot underscore data will be sent back to the front end as part of the JSON response. You'll need to make sure your front end JavaScript is ready to handle this data and plot it accordingly using a library like Plotly.js or any other plotting library Here's a recap of what you should ensure on the front end. Make a request to the analyze endpoint when you need to perform the analysis. Receive the response and extract the fuzzy underscore plot underscore data. Plot the data using the plotting library. For example, with Plotly.js you would use the extracted X and MF underscore asterisk arrays to create the traces for your plots. For example, pseudo code for Plotly.js on the front end could look like this. In your HTML ensure you have an element with the ID symmetry membership function to target for this plot and similarly for the plots. Lastly, make sure your web server is properly serving the necessary static files and that your JavaScript files are correctly included in your HTML so that when you make these requests everything is in place to handle the response and plot the data. Yes, in the provided fuzzy underscore logic function there is an issue with the return statement. The return keyword immediately exits the function which means any code after the return statement will not execute. The fuzzy underscore plot underscore data dictionary is being created after the return statements that conclude the function with the output wave type so fuzzy underscore plot underscore data will never be reached and returned. To fix this, you should ensure that the fuzzy underscore plot underscore data is assembled and returned within the scope of the function before any return statement that would exit the function. Here's how to adjust the code. With this change the fuzzy underscore logic function now properly returns both the identified wave type and the data needed for plotting the fuzzy parameters. The output underscore wave is determined first, followed by the creation of fuzzy underscore plot underscore data and finally, both return together. The JavaScript code snippet you've provided sets up an event listener that triggers once the DOM content is fully loaded. It performs the following actions. Selects elements from the DOM for manipulation including drop downs, sliders and divs for graphs. Defines update output, updates the displayed output wave, detection error and unknown percentage with the data received from the server. Defines update input graph, uses plotly to plot the input waveform graph. Defines analyze waveform, sends the selected waveform and noise amplitude to the servers, analyzes root and handles the response. Defines plot fuzzy logic parameters, plots fuzzy logic parameters using plotly but it needs correction to properly handle multiple traces for membership functions and rule activations. The plot fuzzy logic parameters function is incomplete and has a few issues that need to be addressed. The function attempts to plot the membership functions and rule activations but it doesn't correctly separate the traces for low, mid and high membership functions. Each membership function should be its own trace. The fetch call to get fuzzy logic parameters will not work as expected because the flask root get fuzzy logic parameters is not defined in the back-end code provided. This root needs to be implemented on the server side to respond with the fuzzy logic parameters. The plotting of rule activations assumes a single trace. However, you typically want to plot the activation as a separate trace for clarity. Here's how you could correct the plot fuzzy logic parameters function. You would also need to ensure that your HTML has div elements with corresponding IDs like symmetry membership function, sharpness membership function and rule activation for the plots. Lastly, make sure the analyze root in flask returns the necessary fuzzy logic plotting data within the JSON response and that the root get fuzzy logic parameters is defined and implemented to serve the fuzzy logic parameters. The existing analyze root should already be providing that information so you might not need a separate, yet fuzzy logic parameters root unless you want to load the fuzzy parameters independently of the waveform analysis. Your Python flask code is set up to handle post requests the analyze root, perform the analysis and return the results along with fuzzy logic data as a JSON response. Here's what you need to update to ensure that the code functions correctly. Ensure that the fuzzy underscore logic function returns the fuzzy logic plot data. You have already modified the fuzzy underscore logic function to return both the output underscore wave and fuzzy underscore plot underscore data, which is good. The analyze underscore signal function is correctly updated to return the fuzzy underscore data as well. Make sure to catch errors in your root. If there's any possibility of an error in your signal generation, noise addition or fuzzy logic analysis, you should handle these errors to prevent the server from crashing and to send a meaningful response back to the client. Here is an updated version of your flask root with error handling added. With this updated root, if an error occurs during the processing of the waveform analysis, the server will catch the exception and return a JSON object with an error message and a 500 status code, indicating a server error. Remember, in your front-end JavaScript, you should also handle the case where the server returns an error. Check the response for an error field and handle it appropriately, such as by showing an error message to the user. The error message unexpected token, less than, less than. Doctype is not valid.JSON usually indicates that the response from the server was not JSON, but HTML. This often happens when you make a request to a root that is not found, 404 error, or there's a server-side error that results in an HTML error page being sent instead of a JSON response. Here are some steps to troubleshoot and fix this issue. Check the server root. Make sure the root your requesting exists on the server and is set up to return JSON. In the code snippets you've provided, you're making a request to get fuzzy logic parameters, but this root isn't defined in your flask application. Ensure that you have a corresponding ad-app.root, get fuzzy logic parameters, methods equals, yet, in your flask app. Error handling on the server ensure that your flask roots are correctly handling errors and still returning JSON even when an error occurs. The flask error handler should catch exceptions and return a JSON response with an appropriate HTTP status code. Check the response before pausing as JSON. In your JavaScript code, before you attempt to pass the response as JSON, check the response as status code to ensure it's 200, OK? Handle the error appropriately. Here's an example of how you can do this. Content type header. Ensure the server is setting the content type header to application, JSON when returning the response. Flask's JSONified is this automatically, but if you're constructing responses manually, you need to set this header. Check the network response. Use the browser's developer tools to inspect the network response from your fetch request. Look at the preview or response tab to see what's being returned. If it's HTML, it will likely start with less than. Doctype HTML greater than, which will help you to understand what the issue might be. Server logs. Check the server logs for any unhandled exceptions or errors that may give clues as to why the server is returning HTML instead of JSON. Once you've ensured that the root exists and is returning the correct data, your front-end JavaScript should be able to pass the JSON without encountering the unexpected token. Less than, error. The 404 status code in the log message indicates that the root, get fuzzy logic parameters for the server. To resolve this issue, you need to define this root in your flask application. Here's how you can define the get fuzzy logic parameters root. Make sure to replace the placeholder comments with actual logic to compute the fuzzy logic parameters based on your application's needs. Once this root is properly defined, your front-end JavaScript's fetch call to get fuzzy logic parameters should receive a valid JSON response, assuming there are no other errors in the computation and response generation. You can use this application after making these changes so that the new root is registered and available for requests. Yes, you can certainly pass all the necessary data for plotting directly from your existing analyzed root without needing an extra root like your fuzzy logic parameters. This approach can simplify your application by reducing the number of HTTP requests and consolidating the logic in a single root. To do this, you should modify the analyzed root in your flask application to include the data required for plotting the fuzzy logic parameters in its response. The data for the membership functions, rule activations, and aggregated output are all included in the JSON response returned by the analyzed root. Here's how you can adjust your existing analyzed root. With this setup, when your front-end makes a post-request to analyze, it will receive not only the analysis results but also the fuzzy data necessary for plotting. Your JavaScript code can then use this data to render the plots using a library like plotly.js. Here's an example snippet of how the library has the advantage of keeping all related data in one response, reducing the complexity of your front-end code and potentially improving the responsiveness of your application by minimizing the number of server requests. The range of the symmetry functions, or any membership functions in fuzzy logic, typically depends on the context of your application and how you define the universe of discourse for the variables you're working with. Range between minus 1 to 1, if you're measuring symmetry in a way that allows for negative values, perhaps representing some form of asymmetry or inverse relationship, in a sense. This range is useful when you want to represent a variable that can have both positive and negative impacts or characteristics. Range between 0 to 1, this is more common in fuzzy logic, especially when dealing with probabilities, degrees of membership, or any other metric that doesn't go into negative territory. For instance, if you're measuring how closely a signal resembles a perfect symmetrical form, a range of 0 to 1 could be more appropriate, with 0 representing no symmetry and 1 representing perfect symmetry. For analysis in the context of waveform analysis, like in your application, you typically use a range from 0 to 1. This is because symmetry is usually measured as a degree of similarity between two sides of the waveform, where 0 would represent no symmetry, completely asymmetrical, and 1 would represent perfect symmetry. That said, if your analysis method or theoretical framework specifically requires the use of a minus 1 to 1 range to capture some form of inverse or negative symmetry, you should use that instead. The key is to ensure that the range represents the phenomena you're analyzing and fits the logic of your fuzzy system. In fuzzy logic, rule activation charts and the decision-making process involve evaluating how strongly different rules apply given certain inputs. Let's break down these concepts, especially in the context of your waveform detection system. Rule activation in fuzzy logic refers to the degree to which a particular fuzzy rule is applicable, based on the current input values. Each rule in a fuzzy logic system is typically a simple rule might be, if the symmetry is high and the sharpness is low, then the waveform is square. In your system, you likely have a set of rules like this to determine the type of waveform. Each rule's activation level is determined by the input values, in your case, symmetry and sharpness. The degree of membership in the respective fuzzy sets, low, medium, high, are these inputs as computed, and then these degrees are combined, usually using logical operators like and R to determine the rule's activation level. This visualizes the degree of activation for each rule across the range of input values. For instance, it could show how the activation level of the waveform is square, rule varies as the symmetry and sharpness change. This visualization helps understand which rules are most influential for different input conditions. The actual decision, in this case, identifying the waveform type, is derived through a process known as defuzzification. This process converts the fuzzy rule activations into a single crisp output. There are several methods for defuzzification, but a common one is the centroid method, which finds the center of gravity of the aggregated output fuzzy set. Here's a simplified outline of how the decision might be made in your system. Evaluate each rule, based on the input values, symmetry and sharpness, determine the activation level of each rule. For example, one rule might be significantly activated when symmetry is high and sharpness is low, suggesting a square waveform. Aggregate the activations, combine the activations of all rules to form a single fuzzy set. This aggregation represents the overall decision of the system in a fuzzy manner. Defuzzify, use a method like the centroid to find a crisp value from the aggregated fuzzy set. This value represents the final decision. In your system, this might be a value that determines whether the waveform is more like a square, triangle or trapezoid. If the centroid falls in a range associated with square, the system decides the waveform is square. If it falls in a range for triangle, it decides on a triangle. Similarly for trapezoid. In summary, the rule activation chart shows how applicable each rule is under different input conditions, and the decision output waveform is determined by aggregating these activations and then defuzzifying the aggregate to a single crisp value that indicates the type of waveform. To enhance the understanding of the decision-making process in your physiological waveform detection system using fuzzy logic, you can plot additional elements beyond the symmetry and sharpness membership functions and the rule activation chart. Here are some suggestions. The aggregated output fuzzy set. After individual rule activations are determined, they're typically combined into a single fuzzy set representing the overall output decision before defuzzification. Plotting this aggregated output shows how the various rules contribute collectively to the final decision. Defuzzification process visualization illustrate the defuzzification process, such as the centroid method. This could involve plotting the aggregated output fuzzy set and indicating the center of gravity, centroid, on this plot, which is the crisp value used as the final output. Input signal and noisy signal plot the original input signal alongside the noisy signal. This visual comparison can help in understanding how noise affects the waveform and the challenges in detecting the correct waveform type. Symmetry and sharpness over time plot the calculated symmetry and sharpness values over time or over the signal. This can help in understanding how these metrics vary across the signal and how they influence the fuzzy logic decision. Temporal evolution of rule activation if your system's inputs change over time like in a time series, plotting the evolution of rule activations over time can provide insights into the dynamic decision-making process. Confidence level or degree of membership if available, plot the degree of membership or confidence level for the final decision. This can help in understanding the certainty of the decision made by the fuzzy logic system. Comparison of different waveforms if your system can handle multiple waveform types a comparative plot showing how different types of waveforms are processed differently by the system can be insightful. Aerometrics over time if you're calculating detection error or unknown percentages plotting these metrics over time or against various parameters can provide insights into the performance of the system under different conditions. Each of these plots contributes to a more comprehensive understanding of how the fuzzy logic system processes input data and arrives at a decision. They can be particularly useful for debugging, fine-tuning the system and for educational purposes to illustrate the concepts of fuzzy logic in a practical application. To add both the aggregated output fuzzy set plot and the defuzzification process visualization you'll need to modify the back-end to compute and send the necessary data and then update your front-end JavaScript to render these plots. Here's a guide on how to do this. Aggregated output fuzzy set this is the combined result of all the rule activations. You should already have this data in your fuzzy underscore logic function referred to as aggregated in your existing code. Defuzzification process visualization typically this involves showing where the centroid falls within the aggregated output. You should calculate the centroid as you do for determining the output wave and send this value along with the aggregated output data. Here's how you can modify the fuzzy underscore logic function. Plotting the aggregated output fuzzy set use the aggregated output data from the response to plot this fuzzy set. Visualizing the defuzzification process mark the centroid on the aggregated output plot. Here's an example of how you might modify your JavaScript. Ensure that your HTML contains elements with the ID's aggregated output and defuzzification for these new plots. These editions will provide a visual representation of the aggregated output of your fuzzy logic system and clearly show the process of defuzzification, enhancing the understanding of how the final decision is made. The image you've uploaded shows two plots, both appear to be the aggregated output fuzzy set with the second plot intended to include a centroid line, which seems to be missing or not visible. To address these issues, here's what you can do. Ensure the centroid line is visible. If the centroid line is not visible, it might be because the line's y-values are not visible. If the centroid line is not visible, it might be because the line's y-values do not span the full range of the y-axis, or the color of the line is not contrasting enough against the background. Ensure the plots are not repeated. If the same plot is being shown twice, it might be because the plotting function is being called twice or the same div element is being targeted twice. Ensure each plot has a unique div ID and the plotting function is called once per plot. Here is an updated JavaScript snippet with some potential fixes. In this snippet, we can add a marker to ensure it's visible. We've added a shapes property in the layout to draw a line, which is an alternative to adding the line as a separate trace. We've ensured plotly. New plot is called once for the aggregated output plot. Ensure that the back-end sends the correct centroid value and that it is appropriately formatted for plotlet to interpret. If the centroid line is still not visible after these changes, it might be due to it coinciding with a peak or trough in the aggregated fuzzy set, making it blend into the plot. Adjusting the Z-index, which is not directly available in plotly, but can be mimicked by plotting order or the color to something more contrasting can help. You may also consider adding text annotations or markers to highlight the centroid more clearly.