Principal Component Analysis Explained with Examples

(Originally published on August 21, 2018)

Any machine learning model building task begins with a collection of data vectors wherein each vector consists of a fixed number of components. These components represent the measurements, known as attributes or features, deemed useful for the given machine learning task at hand. The number of components, i.e. the size of the vector, is termed as the dimensionality of the feature space. When the number of features is large, we are often interested in reducing their number to limit the number of training examples needed  to strike a proper balance with the number of model parameters.  One way to reduce the number of features is to look for a subset of original features via some suitable search technique. Another way to reduce the number of features or dimensionality is to map or transform the original features in to another feature space of smaller dimensionality. The Principal Component Analysis (PCA) is an example of this feature transformation approach where the new features are constructed by applying a linear transformation on the original set of features. The use of PCA does not require knowledge of the class labels associated with each data vector. Thus, PCA is characterized as a linear, unsupervised  technique for dimensionality reduction.

Basic Idea Behind PCA

The basic idea behind PCA is to exploit the correlations between the original features. To understand this, lets look at the following two plots showing how a pair of variables vary together. In the left plot, there is no relationship in how X-Y values are varying; the values seem to be varying randomly. On the other hand, the variations in the right side plot exhibits a pattern; the Y values are moving up in a linear fashion. In terms of correlation, we say that the values in the left plot show no correlation while the values in the right plot show good correlation. It is not hard to notice that given a X-value from the right plot, we can reasonably guess the Y-value; however this cannot be done for X-values in the left graph. This means that we can represent the data in the right plot with a good approximation lying along a line, that is we can reduce the original two-dimensional data in one dimensions. Thus achieving dimensionality reduction. Of course, such a reduction is not possible for data in the left plot where there is no correlation in X-Y pairs of values.

PCA Steps

Now that we know the basic idea behind the PCA, let's look at the steps needed to perform PCA. These are:

• We start with N d-dimensionaldata vectors, $\boldsymbol{x}_i, i= 1, \cdots,N$, and find the eigenvalues and eigenvectors of the sample covariance matrix of size d x d using the given data
• We select the top k eigenvalues, k ≤ d, and use the corresponding eigenvectors to define the linear transformation matrix A of size k x d for transforming original features into the new space.
• Obtain the transformed vectors, $\boldsymbol{y}_i, i= 1, \cdots,N$, using the following relationship. Note the transformation involves first shifting the origin of the original feature space using the mean of the input vectors as shown below, and then applying the transformation. 

$\boldsymbol{y}_i = \bf{A}(\bf{x}_i - \bf{m}_x)$

• The transformed vectors are the ones we then use for visualization and building our predictive model. We can also recover the original data vectors with certain error by using the following relationship

$\boldsymbol{\hat x}_i = \boldsymbol{A}^t\boldsymbol{y}_i + \boldsymbol{m}_x$

• The mean square error (mse) between the original and reconstructed vectors is the sum of the eigenvalues whose corresponding eigenvectors are not used in the transformation matrix A.

$ e_{mse} = \sum\limits_{j=k+1}\limits^d \lambda_j$

• Another way to look at how good the PCA is doing is by calculating the percentage variability, P, captured by the eigenvectors corresponding to top k eigenvalues. This is expressed by the following formula

$ P = \frac{\sum\limits_{j=1}^k \lambda_j}{\sum_{j=1}^d \lambda_j}$

A Simple PCA Example

Let's look at PCA computation in python using 10 vectors in three dimensions. The PCA calculations will be done following the steps given above. Lets first describe the input vectors, calculate the mean vector and the covariance matrix.

Next, we get the eigenvalues and eigenvectors. We are going to reduce the data to two dimensions. So we form the transformation matrix A using the eigenvectors of top two eigenvalues.

With the calculated A matrix, we transform the input vectors to obtain vectors in two dimensions to complete the PCA operation.

Looking at the calculated mean square error, we find that it is not equal to the smallest eigenvalue (0.74992815) as expected. So what is the catch here? It turns out that the formula used in calculating the covariance matrix assums the number of examples, N, to be large. In our case, the number of examples is rather small, only 10. Thus, if we multiply the mse value by N/(N-1), known as the small sample correction, we will get the result identical to the smallest eigenvalue. As N becomes large, the ratio N/(N-1) approaches unity and no such correction is required.

The above PCA computation was deliberately done through a series of steps. In practice, the PCA can be easily done using the scikit-learn implementation as shown below.

Before wrapping up, let me summarize a few takeaways about PCA.

• You should not expect PCA to provide much reduction in dimensionality if the original features have little correlation with each other.
• It is often a good practice to perform data normalization prior to applying PCA. The normalization converts each feature to have a zero mean and unit variance. Without normalization, the features showing large variance tend to dominate the result. Such large variances could also be caused by the scales used for measuring different features. This can be easily done by using sklearn.preprocessing.StandardScalar class.
• Instead of performing PCA using the covariance matrix, we can also use the correlation matrix. The correlation matrix has a built-in normalization of features and thus the data normalization is not needed. Sometimes, the correlation matrix is referred to as the standardized covariance matrix.
• Eigenvalues and eigenvectors are typically calculated by the singular value decomposion (SVD) method of matrix factorization. Thus, PCA and SVD are often viewed the same. But you should remember that the starting point for PCA is a collection of data vectors that are needed to compute sample covariance/correlation matrices to perform eigenvector decomposition which is often done by SVD.

 

Gradient Boosted Regression and Classification Trees

 A long while ago I had posted about how a "A Bunch of Weak Classifiers Working Together Can Outperform a Single Strong Classifier." Such a bunch of classifiers/regressors is called an ensemble. As mentioned in that post, bagging and boosting are two techniques that are used to create an ensemble of classifiers/regressors. In this post, I will explain how to build an ensemble of decision trees using gradient boosting. Before going into the details, however, let us first understand boosting, gradient boosting, and why the decision tree classifiers and regression trees are the most suitable candidates for creating an ensemble.

The individual units in an ensemble are known as weak learners. Such learners are brittle in nature, i.e. a small change in the training data can have a major impact on their performance. This brittle nature helps to ensure that the weak learners are not well-correlated with each other. The choice of the tree model for weak learners is popular because tree models exhibit decent brittleness and such models are easily trained without much computation effort.

Boosting is one technique to combine weak learners to obtain a single strong learner or model. Bagging is another such technique. In bagging, all weak learners learn in parallel and independent of other learners. Every learner in bagging looks at only a part of the training data or a subset of features or both. In boosting, the weak learners are build sequentially using the full training data and the results from the preceding weak learners. Thus, boosting builds an additive model and the output of the ensemble is taken as the weighted sum of the predictions of the weak learners.

In the original boosting algorithm AdaBoost, the successive weak learners try to improve the accuracy by giving more importance to those training examples that are misclassified by the prior weak learners. In gradient boosting, the results from the preceding weak learners are compared with the desired output to obtain a measure of the error which the next weak learner tries to minimize. Thus, the focus in gradient boost is on the residuals, i.e. the difference between the desired output and the actual output, rather than on those specific examples that were misclassified by the earlier learners. The term gradient is used because the error minimization is carried out by using the gradient of the error function. The sequence of successive trees are of identical depth from 1 (stump) to 4.

A Walk Through Example to Build a Gradient Boosted Regression Tree

Let us work through an example to clearly understand how gradient boosted trees are build. We will take a regression example because building a gradient boosted regression tree is easier to understand than a gradient boosted classification tree. The example consists of two predictors, fertilizer input (x0) and insecticide input (x1), and the output variable is the crop yield (y). The goal is to build a gradient boosted regression tree to predict the crop yield given the input numbers for x0 and x1.

We will build a tree that minimizes the squared error (mse) between the actual output and the predicted output. We begin by building a base learner that yields the same prediction irrespective of the values of the predictor variables. This base prediction, let us denote it as y_hat_0, equals average of the output variable y, y_bar. Next, we calculate the residuals, the difference between the actual output y and the predicted output to obtain the residuals as shown below.

The next step is to adjust the predictions by relating the residuals with x0 and x1. We do this by fitting a regression tree to the residuals. We will use a tree of depth 1. To build the tree, we find the feature and the cut-off or the threshold value that best partitions the training data to minimize the mean squared error (mse). For the root node, such a combination consists of predictor x1 and the cut-off value of 11.5. The resulting tree is shown below. This tree was build using the DecisionTreeRegressor of the Sklearn library. Let's look at the meanings of different entries shown in the tree diagram. The mse value of 181.889 in the internal node of the tree is nothing but the error value if the residuals_0 column is approximated by the column average which is being shown by "value = 0." The number of examples associated with a node in the tree is given by "samples."

The above tree shows residual approximation by -11.667 for those training examples whose x1 value is less than or equal to 11.5 and the approximation by 11.667 when x1 is greater than 11.5. Assuming a learning rate of 0.75, we combine the above tree with the base tree (stump) to obtain updated predictions and the next set of residuals as shown below. To understand how the updated predictions were calculated, consider the predictor variables x0=6 and x1=4. The above tree tells us that for x1=4, the residual should be approximated by -11.667. The previous prediction for this particular x0,x1 combination is 57.667. Thus, the new prediction in this case becomes 57.667+0.75*(-11.667) which equals 48.916.

With the new set of residuals, we again build a tree to relate x0, x1 with residuals_1. This tree is shown below.

We update the predictions. For the first training example, the updating leads to the prediction 48.916+0.75*(-2.717) which equals 46.879. These updated predictions, y_hat2, and the new residuals, residuals_2 are shown below.

The tree building process continues in this fashion to generate either the specified number of weak learners or to the acceptable level of error.

Now that we know how gradient boosted trees are created, let us finish the above example using the GradientBoostingRegressor from the Sklearn library as shown below. The number of learners has been specified as 10 and the depth is set to 1.

import numpy as np from sklearn.ensemble import GradientBoostingRegressor from sklearn.metrics import mean_squared_error # Create data for illustration X = np.array([[6,4],[12,5], [16,9],[22,14],[24,20],[32,24]]) y = np.array([40, 46, 52, 60, 68,80]) regressor = GradientBoostingRegressor(     max_depth=1,     n_estimators=10,     learning_rate=0.75 ) gbt = regressor.fit(X, y) gbt.predict(X)

array([40.18852649, 46.37977649, 49.56846063, 62.91676214, 67.36073713,        79.58573713])

How about Gradient Boosted Classification Trees?

Let us convert the above regression example to a classification example by considering whether the crop yield was profitable or not based on fertilizer and insecticide costs. Thus, the output variable y now has two values, 0 (loss) and 1 (gain) as shown below.

Just like the procedure for building gradient boosted regression trees, we begin with a crude approximation for the output variable. The initial probability of class 1 over all the training examples, equal to 0.667, is taken as this approximation. The gradient boosted classifier is built using regression trees just like it is done while building gradient regression trees. The successive learners try to reduce the residuals, the difference between the y and the predicted probability.

Log Likelihood Loss Function and Log(odds)

The loss function minimized in the gradient boosted classification tree algorithm is the negative log likelihood. For a classification problem with two classes as we have, the negative log likelihood for the i-th training example is expressed by the following formula where p stands for the probability that the example is from class 1.

The loss is summed over all training examples to get the over all loss. The ratio p/(1-p) is called the odds or odds ratio, and often is expressed in terms of log(odds) by taking the natural log of the odds ratio. The probability p of an event and its log(odds) are related through the following equation:

Through some simple algebraic manipulation, we can also express the loss function as

Instead of calculating the gradient of the loss function with respect to p, it turns out that adjusting log(odds) offers a better way to minimize the loss function. Thus, taking the gradient of the loss function with respect to log(odd) results in the following expression:

The first term on the RHS of the equation is the negative of the desired output while the second term corresponds to the predicted probability (see the relationship between p and log(odds) above). The negative of the RHS in the above equation defines the residuals that we try to minimize by building successive trees via regression.

Back to Example

Coming back to our example, since the initial probability attached to every training example is 0.667 (ratio of class 1 labels and class 0 labels), the residuals are:

-0.667,  0.333,  0.333,  0.333,  0.333, -0.667

Let us try to fit a regression tree to the residuals at hand. Again, we will fit a tree of unit depth using the DecisionTreeRegressor from the Sklearn. The resulting tree shown below.

While in the case of the gradient regression trees, the "value" from the leaf nodes were directly used in the calculations of the updated residuals, we need to map "value" into log(odds) for calculating a new set of residuals in the present case. This mapping is done using the following relationship:

Let us try to understand the quantities in the RHS of the above expression. We will do so by referring to the right leaf node of the above tree. This leaf node has 5 samples that refer to the index "i" in the above expression. The item "value" in the leaf nodes is the residual for each example in the leaf node. There are five examples associated with the right leaf node; these are all but the first example of our data set. The previous probability is 0.667 for each of these examples. Thus, the mapped_log(odds) for these five examples are 5*0.133/(5*0.667*0.333), which equals 0.599. With 0.75 as the learning rate, the new log(odds) values for these five examples become 0.693(old log(odds)) + 0.75*(0.599) resulting in 1.142 for each of the five examples of the data set for which x0<9. Converting these log(odds) to probabilities, we see that 0.758 is the probability for the last five examples in the right leaf node to be from class 1. Carrying out similar calculations for the left leaf node, we end up with -1.559 as the updated log(odds) which translates to 0.174 as the probability that our first example in the data set is from class 1 or from the positive class.

Subtracting the updated probabilities from column y, we get a new set of residuals as shown below:

[-0.174,  0.242,  0.242,  0.242,  0.242, -0.758]

The regression tree built to fit these residuals is shown below.

We need to calculate the mapped_log(odds) again. The example associated with the right leaf in this case is the last example of the data set. The mapped log(odds) value for this example comes out as -4.132 following the formula for mapping shown above. The updated log(odds) value is then the 1.142(prior value) + 0.75* (-4.132). Converting the result into the probability, we get 0.123 as the probability that the last example in our data set comes from the positive class. Thus, it is classified as belonging to the negative (label 0) class. Let us now map the left leaf node value to log(odds) using the mapping formula.

= 5*0.159/((4*0.758*(1-0.758)+0.174*(1-0.174)))
= 0.906

This results in 0.906. Out of the five examples associated with the left leaf node, the four examples(all with y = 1) were together in the previous tree and had the updated log(odds) value 1.142 at that instance. The current log(odds) value for these four examples is then 1.142+0.75*0.906, equal to 1.821 which yields a probability value of 0.860. Thus, these four examples are correctly assigned the probability of 0.860 for the positive class. The remaining fifth example in the left node has a prior log(odds) value of -1.559. Thus, its updated log(odds) becomes -1.559+0.75*0.906, equal to -0.879. This gives 0.293 as the probability of this example from the positive class. Since the probability for the negative class is higher, this example is also correctly classified. At this stage, there is no need to add anymore learner because all examples have been correctly classified.

The above example was done to illustrate how the gradient boosted classifier builds its learners. Let us apply the GradientBoostingClassifier from the Sklearn library to check our step by step assembly of the learners explained above. We will print out the probability values to compare against our calculations.

from sklearn.ensemble import GradientBoostingClassifier clf = GradientBoostingClassifier(     criterion='mse', max_depth=1,     n_estimators=2,     learning_rate=0.75 ) gbdt = clf.fit(X, y) list(gbdt.predict_proba(X))

[array([0.70657313, 0.29342687]),
 array([0.13928973, 0.86071027]),
 array([0.13928973, 0.86071027]),
 array([0.13928973, 0.86071027]),
 array([0.13928973, 0.86071027]),
 array([0.87645946, 0.12354054])]

You can see that these probabilities are identical to those calculated earlier. Thus, you now know how to build a gradient boosted classifier.

Before You Leave

Let us summarize the post before you leave. Boosting relies on an ensemble of trees to perform regression and classification. The gradient boosted trees for regression and classification give excellent performance. Using the Sklearn library without paying much attention to the number of learners and tree depth can lead to over-fitting; so care must be taken. One disadvantage common to all ensemble methods is that the simplicity of understanding model that one gets using a single tree is lost. A speedier and more accurate and scalable version of gradient boosted trees is the extreme gradient boosted trees (XGBoost). The performance of XGBoost algorithm is at par with deep learning models and this model is very popular. 

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Information Retrieval Vs. Data Retrieval

Often there is a confusion about data retrieval and information retrieval. This short post will attempt to clarify the difference between the two. 

Information retrieval (IR) is a field of study dealing with the representation, storage, organization of, and access to documents. The documents may be books, reports, pictures, videos, web pages or multimedia files. The whole point of an IR system is to provide a user easy access to documents containing the desired information. One of the best known example of an IR system is Google search engine. A data retrieval system is a database management system (DBMS).

The difference between an information retrieval system and a data retrieval system is that

– IR deals with unstructured/semi-structured data while a data retrieval (a database management system or DBMS) deals with structured data with well-defined semantics

– Querying a DBMS system produces exact/precise results or no results if no exact match is found

– Querying an IR system produces multiple results with ranking. Partial match is allowed.

Document Characterization

Three kinds of characteristics are associated with a document.

    Metadata characterization

This kind of characterization refers to ownership, authorship and other items of information about a document. The Library of Congress subject coding is also an example of metadata. Another example of metadata is the category headings at Internet search engine Yahoo. To standardize category headings, many areas use specific ontologies, which are hierarchical taxonomies of terms describing certain knowledge topics.

    Presentation Characterization

This refers to attributes that control the formatting or presentation of a document.

Content Characterization

This refers to attributes that denote the semantic content of a document. Content characterization is of primary interest in IR. The common practice in IR is to represent a textual document by a set of keywords called index terms or simply terms. An index term is a word or a phrase in a document whose semantics give an indication of the document’s theme. The index terms, in general, are mainly nouns because nouns have meaning by themselves.

The same concept can be applied to images/multimedia documents to characterize them in terms of words using the representation known aptly as the Bag of Word (BoW) representation.

However, what are the words for images is a tricky question. One way to define such words for images is through the use of vector quantization which yields a set of code words (A code word is simply an image patch of certain size) that can be used to assemble given images. Once we are able to characterize images in terms of code words, various attributes such as frequencies and joint frequencies of code words occurrences can be computed to group images into meaningful groups. An example of one such grouping is shown in the figure below where the grouping of pictures is shown in the form of a graph.