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He was especially interested in how they could be applied to language understanding, which he wrote about in his 1999 paper “Connectionist Sentence. 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 Weight matrix.
Introduction Too much of anything is good for nothing! What happens when a data set has too many variables? Here are few possible situations which you might come across: • You find that most of the variables are correlated. • You lose patience and decide to run a model on whole data.
This returns poor accuracy and you feel terrible. • You become indecisive about what to do • You start thinking of some strategic method to find few important variables Trust me, dealing with such situations isn’t as difficult as it sounds.
Statistical techniques such as factor analysis, principal component analysis help to overcome such difficulties. In this post, I’ve explained the concept of principal component analysis in detail.
I’ve kept the explanation to be simple and informative. For practical understanding, I’ve also demonstrated using this technique in R with interpretations.
Note: Understanding this concept requires prior knowledge of statistics. Update (as on 28th July): Process of Predictive Modeling with PCA Components in R is added below. What is Principal Component Analysis? In simple words, principal component analysis is a method of extracting important variables (in form of components) from a large set of variables available in a data set. It extracts low dimensional set of features from a high dimensional data set with a motive to capture as much information as possible.
With fewer variables, visualization also becomes much more meaningful. PCA is more useful when dealing with 3 or higher dimensional data. It is always performed on a symmetric correlation or covariance matrix. This means the matrix should be numeric and have standardized data. Let’s understand it using an example: Let’s say we have a data set of dimension 300 ( n) × 50 ( p). N represents the number of observations and p represents number of predictors. Since we have a large p = 50, there can be p(p-1)/2 scatter plots i.e more than 1000 plots possible to analyze the variable relationship. Wouldn’t is be a tedious job to perform exploratory analysis on this data?
In this case, it would be a lucid approach to select a subset of p (p path setwd(path) #load train and test file >train test test$Item_Outlet_Sales combi combi$Item_Weight[is.na(combi$Item_Weight)] combi$Item_Visibility table(combi$Outlet_Size, combi$Outlet_Type) >levels(combi$Outlet_Size)[1] my_data colnames(my_data) Since PCA works on numeric variables, let’s see if we have any variable other than numeric. #check variable class >str(my_data) 'data.frame': 14204 obs. Of 9 variables: $ Item_Weight: num 9.3 5.92 17.5 19.2 8.93.
$ Item_Fat_Content: Factor w/ 5 levels 'LF','low fat'.: 3 5 3 5 3 5 5 3 5 5. $ Item_Visibility: num 0.016 0.0193 0.0168 0.054 0.054. $ Item_Type: Factor w/ 16 levels 'Baking Goods'.: 5 15 11 7 10 1 14 14 6 6.
$ Item_MRP: num 249.8 48.3 141.6 182.1 53.9. $ Outlet_Establishment_Year: int 1999 2009 1999 1998 1987 2009 1987 1985 2002 2007. $ Outlet_Size: Factor w/ 4 levels 'Other','High'.: 3 3 3 1 2 3 2 3 1 1. $ Outlet_Location_Type: Factor w/ 3 levels 'Tier 1','Tier 2'.: 1 3 1 3 3 3 3 3 2 2.
$ Outlet_Type: Factor w/ 4 levels 'Grocery Store'.: 2 3 2 1 2 3 2 4 2 2. Sadly, 6 out of 9 variables are categorical in nature. We have some additional work to do now. We’ll convert these categorical variables into numeric using one hot encoding. #load library >library(dummies) #create a dummy data frame >new_my_data str(new_my_data) And, we now have all the numerical values.
Let’s divide the data into test and train. #divide the new data >pca.train pca.test prin_comp names(prin_comp) [1] 'sdev' 'rotation' 'center' 'scale' 'x' The prcomp() function results in 5 useful measures: 1. Center and scale refers to respective mean and standard deviation of the variables that are used for normalization prior to implementing PCA #outputs the mean of variables prin_comp$center #outputs the standard deviation of variables prin_comp$scale 2. The rotation measure provides the principal component loading. Each column of rotation matrix contains the principal component loading vector. This is the most important measure we should be interested in. >prin_comp$rotation This returns 44 principal components loadings.
Is that correct? In a data set, the maximum number of principal component loadings is a minimum of (n-1, p). Let’s look at first 4 principal components and first 5 rows. >prin_comp$rotation[1:5,1:4] PC1 PC2 PC3 PC4 Item_Weight 0. -0.001285666 0.011246194 0.011887106 Item_Fat_ContentLF -0.
0. 350z Coilover Install here. 003768557 -0.009790094 -0.016789483 Item_Fat_Contentlow fat -0. 0.001866905 -0.003066415 -0.018396143 Item_Fat_ContentLow Fat 0. -0.002234328 0.028309811 0.056822747 Item_Fat_Contentreg 0. 0.001120931 0.009033254 -0.001026615 3. In order to compute the principal component score vector, we don’t need to multiply the loading with data. Rather, the matrix x has the principal component score vectors in a 8523 × 44 dimension.
>dim(prin_comp$x) [1] 8523 44 Let’s plot the resultant principal components. >biplot(prin_comp, scale = 0) The parameter scale = 0 ensures that arrows are scaled to represent the loadings. To make inference from image above, focus on the extreme ends (top, bottom, left, right) of this graph.
We infer than first principal component corresponds to a measure of Outlet_TypeSupermarket, Outlet_Establishment_Year 2007. Similarly, it can be said that the second component corresponds to a measure of Outlet_Location_TypeTier1, Outlet_Sizeother. For exact measure of a variable in a component, you should look at rotation matrix(above) again. The prcomp() function also provides the facility to compute standard deviation of each principal component.
Sdev refers to the standard deviation of principal components. #compute standard deviation of each principal component >std_dev pr_var pr_var[1:10] [1] 4.563615 3.217702 2.744726 2.541091 2.198152 2.015320 1.932076 1.256831 [9] 1.203791 1.168101 We aim to find the components which explain the maximum variance. This is because, we want to retain as much information as possible using these components. So, higher is the explained variance, higher will be the information contained in those components. To compute the proportion of variance explained by each component, we simply divide the variance by sum of total variance.
This results in: #proportion of variance explained >prop_varex prop_varex[1:20] [1] 0.10371853 0.07312958 0.06238014 0.05775207 0.04995800 0.04580274 [7] 0.04391081 0.02856433 0.02735888 0.02654774 0.02559876 0.02556797 [13] 0.02549516 0.02508831 0.02493932 0.02490938 0.02468313 0.02446016 [19] 0.02390367 0.02371118 This shows that first principal component explains 10.3% variance. Second component explains 7.3% variance. Third component explains 6.2% variance and so on. So, how do we decide how many components should we select for modeling stage? The answer to this question is provided by a scree plot.
A scree plot is used to access components or factors which explains the most of variability in the data. It represents values in descending order. #scree plot >Winoptimizer 2010 Advanced Serial Monitor. plot(prop_varex, xlab = 'Principal Component', ylab = 'Proportion of Variance Explained', type = 'b') The plot above shows that ~ 30 components explains around 98.4% variance in the data set. In order words, using PCA we have reduced 44 predictors to 30 without compromising on explained variance.
This is the power of PCA>Let’s do a confirmation check, by plotting a cumulative variance plot. This will give us a clear picture of number of components. #cumulative scree plot >plot(cumsum(prop_varex), xlab = 'Principal Component', ylab = 'Cumulative Proportion of Variance Explained', type = 'b') This plot shows that 30 components results in variance close to ~ 98%. Therefore, in this case, we’ll select number of components as 30 [PC1 to PC30] and proceed to the modeling stage. This completes the steps to implement PCA on train data. For modeling, we’ll use these 30 components as predictor variables and follow the normal procedures. Predictive Modeling with PCA Components After we’ve calculated the principal components on training set, let’s now understand the process of predicting on test data using these components.
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The process is simple. Just like we’ve obtained PCA components on training set, we’ll get another bunch of components on testing set. Finally, we train the model.
But, few important points to understand: • We should not combine the train and test set to obtain PCA components of whole data at once. Because, this would violate the entire assumption of generalization since test data would get ‘leaked’ into the training set. In other words, the test data set would no longer remain ‘unseen’. Eventually, this will hammer down the generalization capability of the model. • We should not perform PCA on test and train data sets separately. Because, the resultant vectors from train and test PCAs will have different directions ( due to unequal variance). Due to this, we’ll end up comparing data registered on different axes.
Therefore, the resulting vectors from train and test data should have same axes. So, what should we do? We should do exactly the same transformation to the test set as we did to training set, including the center and scaling feature. Let’s do it in R: #add a training set with principal components >train.data train.data install.packages('rpart') >library(rpart) >rpart.model rpart.model #transform test into PCA >test.data test.data test.data rpart.prediction sample final.sub write.csv(final.sub, 'pca.csv',row.names = F) That’s the complete modeling process after PCA extraction. I’m sure you wouldn’t be happy with your leaderboard rank after you upload the solution. Try using random forest!
For Python Users: To implement PCA in python, simply import PCA from sklearn library. The interpretation remains same as explained for R users above. Ofcourse, the result is some as derived after using R. The data set used for Python is a cleaned version where missing values have been imputed, and categorical variables are converted into numeric. The modeling process remains same, as explained for R users above.
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