It is, of course, possible to dummy code many columns both using the ifelse() function and the fastDummies package. PTIJ Should we be afraid of Artificial Intelligence? I'm working on a prediction problem and I'm building a decision tree in R, I have several categorical variables and I'd like to one-hot encode them consistently in my training and testing set. However, if we have many categories in our variables it may require many lines of code using the ifelse() function. Note, if you want to it is possible to rename the levels of a factor in R before making dummy variables. All the variables have freqRatio values close to 1. How to one hot encode several categorical variables in R, The open-source game engine youve been waiting for: Godot (Ep. Here's a simple solution to one-hot-encode your category using no packages. The next step in the data analysis pipeline (may) now be to analyze the data (e.g., regression or random forest modeling). A function determining what should be done with missing For instance, we could have used the model.matrix function, and the dummies package. Now, that youre done creating dummy variables, you might want to extract time from datetime. predict(object, newdata, na.action = na.pass, ), contr.ltfr(n, contrasts = TRUE, sparse = FALSE), The output of dummyVars is a list of class 'dummyVars' with Thanks for reading this and sign up for my newsletter at: Get full source code For instance, using the tibble package you can add empty column to the R dataframe or calculate/add new variables/columns to a dataframe in R. In this post, we have 1) worked with Rs ifelse() function, and 2) the fastDummies package, to recode categorical variables to dummy variables in R. In fact, we learned that it was an easy task with R. Especially, when we install and use a package such as fastDummies and have a lot of variables to dummy code (or a lot of levels of the categorical variable). WebThe predict function produces a data frame.. class2ind returns a matrix (or a vector if drop2nd = TRUE).. contr.ltfr generates a design matrix.. by encoding do you mean creating dummy variables? A logical indicating whether contrasts should be computed. In the previous section, we used the dummy_cols() method to make dummy variables from one column. Heres how to make indicator variables in R using the dummy_cols() function: Now, the neat thing with using dummy_cols() is that we only get two line of codes. The above output shows that the variable has been binned. Now it is time to use the findCorrelation function. Dealing with hard questions during a software developer interview. By clicking Accept all cookies, you agree Stack Exchange can store cookies on your device and disclose information in accordance with our Cookie Policy. Min. In the next section, we will go on and have a look at another approach for dummy coding categorical variables. WebGiven a formula and initial data set, the class dummyVars gathers all the information needed to produce a full set of dummy variables for any data set. Adding the zero specifies that you do not want an intercept or reference level and is equivalent to -1. Find centralized, trusted content and collaborate around the technologies you use most. How can I use dummy vars in caret without destroying my target variable? Launching the CI/CD and R Collectives and community editing features for Transform one column from categoric to binary, keep the rest, Reshape data in R (Split single column with multiple values into multiple colums with binary values), Converting a categorical variable to multiple binary variables, Create mutually exclusive dummy variables from categorical variable in R, Create variables in a for loop using character data, r - how to add columns dynamically based on numerical values sequences from another column, Convert categorical column to multiple binary columns, Want to assign the value of a dummy variable at one time in R, Generate a time dummy variable in R (panel data), Include trend variable from certain time period R, Creating a dummy with different arguments in R. How to create dummy variable based on the value of two columns in R? This is easy when you have two levels in the categorical variable, as with Credit_score. These can be encoded to 1 and 0, respectively. In the previous sections, we learned how to encode categorical variables. Before running the function, look for repeated words or sentences, only take the top 50 of them and replace the rest with 'others'. Once we are happy with our data, we need to split it into training and validation data sets - we will call these ml_penguin_train and ml_penguin_validate respectively. What are some tools or methods I can purchase to trace a water leak? Notice that in the first row, we have a value of 0 for sex.female and a value of 1 for sex.male - in other words, the data in the first row is for a male penguin. Step 1: Create the Data First, lets create the following data frame in R: #create data frame df <- data.frame(team=c ('A', 'A', 'B', 'B', 'B', 'B', 'C', 'C'), points=c (25, The matrix Ditself will be rank deficient if grouphas multiple columns. In the first column we created, we assigned a numerical value (i.e., 1) if the cell value in column discipline was A. But that's none of my business. International Administration, co-author of Monetizing Machine Learning and VP of Data Science at SpringML. The other answers here offer direct routes to accomplish this taskone that many models (e.g. Maybe adding "fun= factor" in function dummy can help if that is the meaning of the variable. Your email address will not be published. I recommend using the dummyVars function in the caret package: You apply the same procedure to both the training and validation sets. The first line of code below imports the powerful caret package, while the second line uses the dummyVars() function to create a full set of dummy variables. And this has opened my eyes to the huge gap in educational material on applied data science. Here, we can see that as identified previously, none of the variables have zero or near zero variance (as shown in columns 3 and 4 of the output). It takes the base correlation matrix as its main input, and we use the cutoff argument to specify the maximum correlation value we are happy to allow between any pair of feature variables (the pair-wise correlation). by using the ifelse() function) you do not need to install any packages. If you have a survey question with 5 categorical values such as very unhappy, unhappy, neutral, happy and very happy. Using @zx8754's data, To make it work for data other than numeric we need to specify type as "character" explicitly. One of the additional benefits of binning is that it also takes care of the outliers. Thus, heres how we would convert, We can use this equation to find the estimated income for an individual based on their age and marital status. Your email address will not be published. model.matrix). You basically want to avoid highly correlated variables but it also save space. Creating dummy variables in R is a way to incorporate nominal variables into regression analysis It is quite easy to understand why we create dummy variables, once you understand the regression model. thanks for your contribution. Now, that I know how to do this, I can continue with my project. What capacitance values do you recommend for decoupling capacitors in battery-powered circuits? So what *is* the Latin word for chocolate? Note, recipes is a package that is part of the Tidyverse. Here are two examples of what you will do in this 1. Does the half-way point between two zip codes make geographical sense? Next, start creating the dummy variables in R using the ifelse() function: In this simple example above, we created the dummy variables using the ifelse() function. One-hot encoding is used to convert categorical variables into a format that can be used by machine learning algorithms. The dummyVars() method works on the categorical variables. Lets consider both options, using our ml_penguins_updated data set. For example, the 'Purpose' variable has six levels, as can be seen from the output below. 20 the random sampling employed by the createDataPartition function will occur within each class. The final representation will be, h (x) = sigmoid (Z) = (Z) or, And, after training a logistic regression model, we can plot the mapping of the output logits before (Z) and after the sigmoid function is applied ( (Z)). rank parameterization), # S3 method for default (2008) A balanced accuracy function for epistasis modeling in imbalanced datasets using multifactor dimensionality reduction., Sangamsh KS Owner at KS Analytical Consulting Author has 98 answers and 174.3K answer views 5 y Hey, Let me ease your life. It needs your categorical variable to be a factor. Factors can be ordered or unordered. For this example, we will set this limit to 0.8. elements, names Here is how to interpret the regression coefficients from the table: Since both dummy variables were not statistically significant, we could dropmarital statusas a predictor from the model because it doesnt appear to add any predictive value for income. I've updated Option 2 to provide a solution that avoids this, The open-source game engine youve been waiting for: Godot (Ep. Dont worry, we dont expect you to use them all! Heres the first 10 rows of the new dataframe with indicator variables: Notice how the column sex was automatically removed from the dataframe. The fourth line of code prints the structure of the resulting data, dat-transfored, which confirms that one-hot encoding is completed. Passing the dummyVars directly to the function is done by using the train(x = , y =, ) instead of a formula. How did Dominion legally obtain text messages from Fox News hosts? Here's an example using the iris dataset. Required fields are marked *. Or half single? The freqRatio column computes the frequency of the most prevalent value recorded for that variable, divided by the frequency of the second most prevalent value. WebThus, for a binomial logistic regression model with two parameters and , Z = + X. rev2023.3.1.43269. Let's start by loading the required libraries and the data. We will consider the Income variable as an example. WebdummyVars function from caret package colname1 + colname2, here are all the categorical variables in a data frame. 542), How Intuit democratizes AI development across teams through reusability, We've added a "Necessary cookies only" option to the cookie consent popup. The dummyVars function breaks out unique values from a column into individual columns - if you have 1000 unique values in a column, dummying them will add 1000 new columns to your data set (be careful). Thus installing tidyverse, you can do a lot more than just creating dummy variables. There is a Next, we check for any extreme correlations close to 1 in magnitude: Next, we assess a summary of the correlation values: We can see that in general everything is ok, although we do have a high max correlation value of 0.873. It is also possible to create bin cut-offs automatically, as shown in the code below. Finally, we compare the original Income variable with the binned Income_New variable using the summary() function. Therefore, we have a multi-class classification problem, with the feature variables bill_length_mm, bill_depth_mm, flipper_length_mm, body_mass_g and sex, and the outcome variable species. Note: You can find the complete online documentation for the dummyVars() function here. Please note that much of the content in these notes has been developed from the caret package document (Kuhn 2019). want to make indicator variables from multiple columns. The initial code was suggested by Gabor Grothendieck on R-Help. Lets create three levels of the variable Income, which are Low for income levels lying below $380,000, High for income values above $760,000, and Mid50 for the middle 50 percentage values of the income distribution. By clicking Post Your Answer, you agree to our terms of service, privacy policy and cookie policy. To subscribe to this RSS feed, copy and paste this URL into your RSS reader. For instance, creating dummy variables this way will definitely make the R code harder to read. Convert your data to a data.table and use set by reference and row filtering, We can also use cSplit_e from splitstackshape. First, we are going to go into why we may need to dummy code some of our variables. It uses contr.ltfr as the Connect and share knowledge within a single location that is structured and easy to search. Bonus - The side panels are super customizable for uncovering complex relationships. If this is not set to TRUE, we only get one column. less than full You can do the following that will create a new df, trsf, but you could always reassign back to the original df: library(caret) Heres how to create dummy variables in R using the ifelse() function in two simple steps: In the first step, import the data (e.g., from a CSV file): In the code above, we need to make sure that the character string points to where our data is stored (e.g., our .csv file). Learn more about us. intercept and all the factor levels except the first level of the factor. Finally, it may be worth to mention that the recipes package is part of the tidyverse package. For building a machine learning model I used dummyVars () function to create the dummy variables for building a model. class2ind returns a matrix (or a vector if drop2nd = TRUE). Then, I can introduce this factor as a dummy variable in my models. Explain that part in a bit more detail so that we can use it for recoding the categorical variables (i.e., dummy code them). In the first section, of this post, you are going to learn when we need to dummy code our categorical variables. In other words, categorical variables, e.g.dummy variables, often have low percentUnique values. WebIn R, there are plenty of ways of translating text into numerical data. Also, if you want it to return character data then you can do so. Rscale() . Note, if you are planning on (also) doing Analysis of Variance, you can check the assumption of equal variances with the Brown-Forsythe Test in R. Now, before summarizing this R tutorial, it may be worth mentioning that there are other options to recode categorical data to dummy variables. For example, to see whether there is a long-term trend in a varible y : If you want to get K dummy variables, instead of K-1, try: The ifelse function is best for simple logic like this. WebHow to create a dummy variable in R - YouTube 0:00 / 3:00 How to create a dummy variable in R 20,952 views Apr 18, 2020 This video describes how to create a new Please note this limit choice is arbitrary - we could have picked a more or less extreme limit. From consulting in machine learning, healthcare modeling, 6 years on Wall Street in the financial industry, and 4 years at Microsoft, I feel like Ive seen it all. We observe that it is difficult to distinguish between Adelie and Chinstrap penguins when modelling body_mass_g against flipper_length_mm or bill_depth_mm. that the distance between all steps on the scale of the variable is the same length. If the data, we want to dummy code in R, is stored in Excel files, check out the post about how to read xlsx files in R. As we sometimes work with datasets with a lot of variables, using the ifelse() approach may not be the best way. 2020. In Computer Labs 9B-11B we will use the caret R package (Kuhn et al. It uses contr.ltfr as the Its best to create dummy variables or change to factors and then split the data into train-test. of all the factor variables in the model. This means that none of the feature variables have only a single unique value. It doesn't matter if some levels don't occur in your test set. For example, the Naive Bayes Algorithm requires all variables to be categorical, so encoding numerical variables is required. Should I include the MIT licence of a library which I use from a CDN? Reach me at amunategui@gmail.com. In this guide, you have learned methods of encoding data with R. You have applied these techniques on both quantitative and qualitative variables. In the case a specific aggregation function is needed for dcast and the result of of dcast need to be merged back to the original: which gives (note that the result is ordered according to the by column): 3) use the spread-function from tidyr (with mutate from dplyr). Read on to learn how to create dummy variables for categorical variables in R. In this section, before answering some frequently asked questions, you are briefly going to learn what you need to follow this post. It contains well written, well thought and well explained computer science and programming articles, quizzes and practice/competitive programming/company interview Questions. We can download, install and load the caret package in RStudio as follows: To illustrate an example application of the caret package, we will use the familiar penguins data set from the palmerpenguins R package (Horst, Hill, and Gorman 2020). Pandas: Use Groupby to Calculate Mean and Not Ignore NaNs. Often however, correlated feature variables can be problematic. 1st Qu. For example, an individual who is 35 years old and married is estimated to have an income of$68,264: Income = 14,276.2 + 1,471.7*(35) + 2,479.7*(1) 8,397.4*(0) = $68,264. How does the NLT translate in Romans 8:2? By clicking Accept all cookies, you agree Stack Exchange can store cookies on your device and disclose information in accordance with our Cookie Policy. Option 2 below avoid this, be standardizing the data before calling train(). Installing packages can be done using the install.packages() function. Well, these are some situations when we need to use dummy variables. You can make linear regression with marginal distributions using histograms, densities, box plots, and more. To subscribe to this RSS feed, copy and paste this URL into your RSS reader. How to Create Dummy Variables in R in Two Steps: ifelse() example, 2) Create the Dummy Variables with the ifelse() Function, Three Steps to Create Dummy Variables in R with the fastDummies Package, How to Create Dummy Variables for More than One Column, How to Make Dummy Variables in R with the step_dummy() Function, How to Generate a Sequence of Numbers in R with :, seq() and rep(), R to conditionally add a column to the dataframe based on other columns, calculate/add new variables/columns to a dataframe in R, Categorical Variables in Regression Analysis:A Comparison of Dummy and Effect Coding, No More: Effect Coding as an Alternative to Dummy Coding With Implications for Higher Education Researchers, Random Forests, Decision Trees, and Categorical Predictors:The Absent Levels Problem, How to Make a Residual Plot in R & Interpret Them using ggplot2, Psychomotor Vigilance Task (PVT) in PsychoPy (Free Download), How to Remove/Delete a Row in R Rows with NA, Conditions, Duplicated, Python Scientific Notation & How to Suppress it in Pandas & NumPy, How to Create a Matrix in R with Examples empty, zeros, How to Convert a List to a Dataframe in R dplyr. I get the following error:Error in loadNamespace(j <- i[[1L]], c(lib.loc, .libPaths()), versionCheck = vI[[j]]) : there is no package called ggvis In addition: Warning message: package mlr was built under R version 3.2.5 Error: package or namespace load failed for mlr, the resulting table cannot be used as a data.frame. A vector of levels for a factor, or the number of levels. Evil Carrot by Brettf is licensed under CC BY 2.0. To learn more, see our tips on writing great answers. Running the above code will generate 5 new columns containing the dummy coded variables. Partner is not responding when their writing is needed in European project application. Because that is how a regression model would use it. But hopefully our machine learning model will be able to use the data for these variables to make accurate predictions. How to Transform Data in Python (Log, Square Root, Cube Root), How to Calculate Polychoric Correlation in R. Your email address will not be published. If we are, for example, interested in the impact of different educational approaches on political attitudes, it is not possible to assume that science education is twice as much as social science education, or that a librarian education is half the education in biomedicine. This will include an intercept column (all ones) and one column for each of the years in your data set except one, which will be the "default" or intercept value. WebUse the rbind () function to add new rows in a Data Frame: Example Data_Frame <- data.frame ( Training = c ("Strength", "Stamina", "Other"), Pulse = c (100, 150, 120), Duration = c (60, 30, 45) ) # Add a new row New_row_DF <- rbind (Data_Frame, c ("Strength", 110, 110)) # Print the new row New_row_DF Try it Yourself Add Columns In R, there are plenty of ways of translating text into numerical data. On the right, of the arrow we take our dataframe and create a recipe for preprocessing our data (i.e., this is what this function is for). Is variance swap long volatility of volatility. Glad you appreciated the tutorial. In other words, if we have a data set comprised roughly 50% Adelie penguin data, 20% Chinstrap data and 30% Gentoo data, the createDataPartition sampling will preserve this overall class distribution of 50/20/30. Theoretically Correct vs Practical Notation. Things to keep in mind, Hi there, this is Manuel Amunategui- if you're enjoying the content, find more at ViralML.com, Get full source code and video If we would now like to remove this feature variable from our data set, we need to be careful - column 3 in the base_cor matrix does not correspond to column 3 in our data set (since we assessed columns 2 to 5). Introduction to Statistics is our premier online video course that teaches you all of the topics covered in introductory statistics. rev2023.3.1.43269. document.getElementById( "ak_js_1" ).setAttribute( "value", ( new Date() ).getTime() ); This site uses Akismet to reduce spam. Webr r; r r; r- r; r-/ r Most of the contrasts functions in R produce full rank Depending on the objective of your project, you can apply any or all of these encoding techniques. If that's a problem, use, https://www.rdocumentation.org/packages/mlr/versions/2.9/topics/createDummyFeatures, The open-source game engine youve been waiting for: Godot (Ep. Is Hahn-Banach equivalent to the ultrafilter lemma in ZF. The second parameter are set to TRUE so that we get a column for male and a column for female. class2ind is most useful for converting a factor outcome vector to a This code will create two new columns where, in the column Male you will get the number 1 when the subject was a male and 0 when she was a female. Is, of course, possible to create the dummy coded variables no packages clicking Post Answer! Panels are super customizable for uncovering complex relationships on and have a look at another for... Then, I can dummyvars in r to trace a water leak variables into a format that can be to. And easy to search quizzes and practice/competitive programming/company interview questions dont worry, learned. Making dummy variables for building a machine learning and VP of data at! Make accurate predictions are two examples of what you will do in guide... Kuhn et al our premier online video course that teaches you all the! Can help if that is part of the variable is the meaning of the benefits..., respectively in European project application find centralized, trusted content and collaborate around the technologies you use.... What * is * the Latin word for chocolate the dummyVars ( ) function below avoid this, can! Above code will generate 5 new columns containing the dummy variables Chinstrap penguins when modelling body_mass_g against flipper_length_mm bill_depth_mm! To 1 should I include the MIT licence of a factor, or the number of.! Using no packages set by reference and row filtering, we only get one column well written, well and... ) you do not want an intercept or reference level and is equivalent to ultrafilter! Learning model will be able to use the findCorrelation function variable to be factor! For female indicator variables: Notice how the column sex was automatically removed from the caret package: apply... To encode categorical variables or a vector if drop2nd = TRUE ) often however correlated... News hosts often have low percentUnique values tips on writing great answers employed the! = + X. rev2023.3.1.43269 heres the first level of the new dataframe with indicator dummyvars in r: how... Model I used dummyVars ( ) function lemma in ZF the complete documentation. Situations when we need to use them all set by reference and row,... Possible to dummy code our categorical variables in R before making dummy variables or change to factors and split. //Www.Rdocumentation.Org/Packages/Mlr/Versions/2.9/Topics/Createdummyfeatures, the Naive Bayes Algorithm requires all variables to be categorical so... Data science of this Post, you agree to our terms of service, privacy policy and policy! Introduce this factor as a dummy variable in my models you apply the same length returns a (... Often have low percentUnique values factor, or the number of levels for a binomial logistic regression would. Several categorical variables, often have low percentUnique values means that none of the tidyverse.! The Latin word for chocolate expect you to use the data before train. It does n't matter if some levels do n't occur in your set. Dummyvars ( ) function benefits of binning is that it is difficult to distinguish between Adelie and Chinstrap when! Without destroying my target variable correlated feature variables can be seen from the dataframe variables... Data science the dummies package be encoded to 1 make geographical sense a package that is of... But it also save space point between two zip codes make geographical sense will! Answers here offer direct routes to accomplish this taskone that many models e.g! N'T occur in your test set want to extract time from datetime you are going to learn we... Fun= factor '' in function dummy can help if that is the meaning of the resulting data, dat-transfored which. Writing is needed in European project application findCorrelation function other answers here offer routes! And, Z = + X. rev2023.3.1.43269 your Answer, you agree to our terms of service, policy. For uncovering complex relationships to convert categorical variables Computer science and programming articles quizzes. Do in this 1 online video course that teaches you all of the variable has been.. The Its best to create bin cut-offs automatically, as shown in the first,! The ultrafilter lemma in ZF target variable introduce this factor as a dummy variable in my models is part the... The resulting data, dat-transfored, which confirms that one-hot encoding is used to convert categorical variables dummy categorical! Meaning of the resulting data, dat-transfored, which confirms that one-hot encoding is to. These are some situations when we need to use the caret R package ( et! Code will generate 5 new columns containing the dummy coded variables ( ) method on. Variable in my models specifies that you do not want an intercept or level... For the dummyVars ( ) function ) you do not need to use the caret document... Code using the ifelse ( ) function factor levels except the first section, of course, possible to the... The Naive Bayes Algorithm requires all variables to make dummy variables or change to and! Meaning of the topics covered in introductory Statistics 9B-11B we will go on and have a survey with... Location that is part of the outliers you might want to it is time to the... Create the dummy coded variables that much of the feature variables can be used by machine model! Can also use cSplit_e from splitstackshape to encode categorical variables, e.g.dummy variables, e.g.dummy variables, you do! The same procedure to both the training and validation sets be categorical, so encoding variables. It does n't matter if some levels do n't occur in your set. Variables or change to factors and then split the data before calling (. We need to use them all we will go on and have a survey question 5! Are some tools or methods I can introduce this factor as a dummy in. First, we could have used the dummy_cols ( ) function function ) you do not need to code..., quizzes and practice/competitive programming/company interview questions 1 and 0, respectively water leak messages from Fox News hosts battery-powered... Want it to return character data then you can do a lot more than just creating dummy.. Is not responding when their writing is needed in European project application to! These techniques on both quantitative and qualitative variables automatically removed from the dataframe are! An example problem, use, https: //www.rdocumentation.org/packages/mlr/versions/2.9/topics/createDummyFeatures, the 'Purpose ' variable been. May need to use them all applied data science at SpringML resulting,! Done with missing for instance, we are going to go into we. N'T matter if some levels do n't occur in your test set variable has six,! Want it to return character data then you can make linear regression with marginal distributions using histograms,,! Can do a lot more than just creating dummy variables you use most introduce factor. My target variable to both the training and validation sets resulting data, dat-transfored, which confirms one-hot. Make dummy variables this way will definitely make the R code harder to read within a single location that structured! Webin R, there are plenty of ways of translating text into numerical data,! Find centralized, trusted content and collaborate around the technologies you use.... And is equivalent to the ultrafilter lemma in ZF new columns containing dummy! Same procedure to both the training and validation sets can continue with my project Hahn-Banach equivalent to.. Create dummy variables split the data into train-test Adelie and Chinstrap penguins when modelling body_mass_g against flipper_length_mm bill_depth_mm... Programming/Company interview questions get a column for female in function dummy can help if that 's a problem,,! Random sampling employed by the createDataPartition function will occur within each class, co-author of machine! Sections, we dont expect you to use them all dummy variables, variables... Legally obtain text messages from Fox News hosts or change to factors and then split the.... Change to factors and then split the data before calling train ( function! Unhappy, neutral, happy and very happy factor as a dummy variable in my models only. Works on the scale of the feature variables have freqRatio values close to 1 is required, is! For dummy coding categorical variables, e.g.dummy variables, you agree to our terms of,... Of what you will do in this guide, you agree to our terms of service privacy... Find centralized, trusted content and collaborate around the technologies you use most will definitely make the R harder! Low percentUnique values by machine learning algorithms column dummyvars in r female code was by. The resulting data, dat-transfored, which confirms that one-hot encoding is used to convert categorical variables R... Are all the variables have only a single location that is the same procedure to both the and! Structure of the resulting data, dat-transfored, which confirms that one-hot encoding is completed could have used dummy_cols. Of service, privacy policy and cookie policy text into numerical data variable is same. To create dummy variables word for chocolate go on and have a survey question with 5 categorical such... Brettf is licensed under CC by 2.0 two parameters and, Z = X.., if we have many categories in our variables to this RSS feed copy. Our ml_penguins_updated data set, unhappy, unhappy, unhappy, unhappy, unhappy, unhappy unhappy. The data before calling train ( ) worth to mention that the distance between all steps on the scale the! To install any packages dont expect you to use dummy variables for building a model //www.rdocumentation.org/packages/mlr/versions/2.9/topics/createDummyFeatures. We could have used the dummy_cols ( ) function here is easy when you have a look at approach! True ) the content in these notes has been developed from the below!

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