The recent explosion of data set size, in number of records as well as of attributes, has triggered the development of a number of big data platforms as well as parallel data analytics algorithms. At the same time though, it has pushed for the usage of data dimensionality reduction procedures. Dealing with a lot of dimensions can be painful for machine learning algorithms. High dimensionality will increase the computational complexity, increase the risk of over fitting (as your algorithm has more degrees of freedom) and the sparsity of the data will grow. Hence, dimensionality reduction will project the data in a space with fewer dimensions to limit these phenomena.

### What is Dimensionality Reduction?

The problem of unwanted increase in dimension is closely related to fixation of measuring / recording data at a far granular level then it was done in past. This is no way suggesting that this is a recent problem. It has started gaining more importance lately due to surge in data.

In machine learning classification problems, there are often too many factors on the basis of which the final classification is done. These factors are basically variables called features. The higher the number of features, the harder it gets to visualize the training set and then work on it. Sometimes, most of these features are correlated, and hence redundant. This is where **dimensionality reduction** algorithms come into play. *Dimensionality reduction* is the process of reducing the number of random variables under consideration, by obtaining a set of principal variables. It can be divided into feature selection and feature extraction. * Dimensionality reduction* is not only useful to speed up algorithm execution, but actually might help with the final classification/clustering accuracy as well. Too much noisy or even faulty input data often lead to a less than desirable algorithm performance. Removing un-informative or dis-informative data columns might indeed help the algorithm find more general classification regions and rules and overall achieve better performances on new data.

Manifold learning is a significant problem across a wide variety of information processing fields including pattern recognition, data compression, machine learning, and database navigation. In many problems, the measured data vectors are high-dimensional but we may have reason to believe that the data lie near a lower-dimensional manifold. In other words, we may believe that high-dimensional data are multiple, indirect measurements of an underlying source, which typically cannot be directly measured. Learning a suitable low-dimensional manifold from high-dimensional data is essentially the same as learning this underlying source. **Dimensionality reduction** can also be seen as the process of deriving a set of degrees of freedom which can be used to reproduce most of the variability of a data set. Consider a set of images produced by the rotation of a face through different angles.

**Example of dimensionality reduction**

An intuitive example of *dimensionality reduction* can be discussed through a simple e-mail classification problem, where we need to classify whether the e-mail is spam or not. This can involve a large number of features, such as whether or not the e-mail has a generic title, the content of the e-mail, whether the e-mail uses a template, etc. However, some of these features may overlap. In another condition, a classification problem that relies on both humidity and rainfall can be collapsed into just one underlying feature, since both of the aforementioned are correlated to a high degree. Hence, we can reduce the number of features in such problems. A 3-D classification problem can be hard to visualize, whereas a 2-D one can be mapped to a simple 2 dimensional space, and a 1-D problem to a simple line. The below figure 1 illustrates this concept, where a 3-D feature space is split into two 1-D feature spaces, and later, if found to be correlated, the number of features can be reduced even further.

### Common Method for Dimensionality Reduction

There are many methods to perform Dimension reduction. I have listed the most common methods below:

**Principal Component Analysis (PCA): **In this technique, variables are transformed into a new set of variables, which are linear combination of original variables. These new set of variables are known as **principle components.** They are obtained in such a way that first principle component accounts for most of the possible variation of original data after which each succeeding component has the highest possible variance.

**Factor Analysis: **Let’s say some variables are highly correlated. These variables can be grouped by their correlations i.e. all variables in a particular group can be highly correlated among themselves but have low correlation with variables of other group(s). Here each group represents a single underlying construct or factor. These factors are small in number as compared to large number of dimensions. However, these factors are difficult to observe. There are basically two methods of performing factor analysis:

- EFA (Exploratory Factor Analysis)
- CFA (Confirmatory Factor Analysis)

**Decision Trees:** It is one of my favorite techniques. It can be used as a ultimate solution to tackle multiple challenges like missing values, outliers and identifying significant variables. Several data scientists used decision tree and it worked well for them.

**Random Forest: **Similar to decision tree is Random Forest. I would also recommend using the in-built feature importance provided by random forests to select a smaller subset of input features. Just be careful that random forests have a tendency to bias towards variables that have more no. of distinct values i.e. favor numeric variables over binary/categorical values.

**High Correlation:** Dimensions exhibiting higher correlation can lower down the performance of model. Moreover, it is not good to have multiple variables of similar information or variation also known as “Multicollinearity”. You can use Pearson (continuous variables) or Polychoric (discrete variables) correlation matrix to identify the variables with high correlation and select one of them using VIF (Variance Inflation Factor). Variables having higher value (VIF > 5) can be dropped.

### Advantages of Dimensionality Reduction

- It helps in data compression, and hence reduced storage space.
- It reduces computation time.
- It also helps remove redundant features, if any.

### Disadvantages of Dimensionality Reduction

- It may lead to some amount of data loss.
- PCA tends to find linear correlations between variables, which is sometimes undesirable.
- PCA fails in cases where mean and covariance are not enough to define datasets.
- We may not know how many principal components to keep- in practice, some thumb rules are applied.

### References

[1] Ali Ghodsi, “Dimensionality Reduction A Short Tutorial”, Waterloo, Ontario, Canada, 2006

[2] Sunil Ray, “Beginners Guide To Learn Dimension Reduction Techniques”, July 28, 2015, available online at: https://www.analyticsvidhya.com/blog/2015/07/dimension-reduction-methods/

[3] “Introduction to Dimensionality Reduction”, available online at: https://www.geeksforgeeks.org/dimensionality-reduction/

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An Example of Principal Component Analysis

Introduction of Linear Discriminant Analysis (LDA)