Understanding the Applicability of Linear & Non-Linear Models Using a Case-Based Study

International Journal of Artificial Intelligence & Applications (IJAIA), Vol. 5, No. 6, November 2014
UNDERSTANDING THE APPLICABILITY OF
LINEAR & NON-LINEAR MODELS USING A
CASE-BASED STUDY
Gaurav Singh Thakur1,Anubhav Gupta 2, Ankur Bhardwaj3 and Biju R Mohan4
Department of Information Technology, National Institute of Technology Karnataka,
Surathkal
Abstract
This paper uses a case based study – “product sales estimation” on real-time data to help us understand
the applicability of linear and non-linear models in machine learning and data mining. A systematic
approach has been used here to address the given problem statement of sales estimation for a particular set
of products in multiple categories by applying both linear and non-linear machine learning techniques on
a data set of selected features from the original data set. Feature selection is a process that reduces the
dimensionality of the data set by excluding those features which contribute minimal to the prediction of the
dependent variable. The next step in this process is training the model that is done using multiple
techniques from linear & non-linear domains, one of the best ones in their respective areas. Data Re-modeling
has then been done to extract new features from the data set by changing the structure of the
dataset & the performance of the models is checked again. Data Remodeling often plays a very crucial and
important role in boosting classifier accuracies by changing the properties of the given dataset. We then try
to explore and analyze the various reasons due to which one model performs better than the other & hence
try and develop an understanding about the applicability of linear & non-linear machine learning models.
The target mentioned above being our primary goal, we also aim to find the classifier with the best possible
accuracy for product sales estimation in the given scenario.
Keywords
Machine Learning, Prediction, Linear and Non-linear models, Linear Regression, Random Forest,
Dimensionality Reduction,Feature Selection, Homoscedasticity, Overfitting, Regularization, Occam Razor
Hypothesis, Elastic Net
1. Introduction
According to Arthur Samuel (1959), Machine learning is a field of study that gives computers
the ability to learn without being explicitly programmed. For example, given a purchase history
for a customer and a large inventory of products, a machine learning algorithm can be used to
identify those products in which that customer will be interested and likely to purchase. There are
various types of Machine Learning Algorithms; two of the main types are Supervised Learning
(where we provide the algorithm with a training dataset in which the right answers are given i.e.
for each training example, the right output is given), and Unsupervised Learning (where we do
DOI : 10.5121/ijaia.2014.5603 27

not provide the algorithm with the right answers
belong to Supervised learning[3].
echniques which
Predicting the future sales of a new product in the market has intrigued many scholars and
industry leaders as a difficult and challenging problem. It involves customer sciences and helps
the company by analyzing data and applying insights from a large number of customers across
the globe to predict the sales in the upcoming time in near future. The success or failure of a new
product launch is often evident within the first few weeks of sales. Therefore, i
forecast the sales of the product in the near future by analyzing its sales in the first few weeks.
We propose to predict the success or failure of each of product launches 26 weeks after the
launch, by estimating their sales in the 26
after launch. We intend to do so by combining data analysis with machine learning techniques
and use the results for forecasting.
We have divided the work into following phases:
t possible to
i) Dimensionality reductio
ii) Application of Linear & Non
iii) Data Re-modeling
iv) Re-application of learning
v) Evaluation of the performance of the learning models through comparative study &
Normality tests.
vi) Boosting the accuracy of the
evaluation.
To develop a forecasting system for this problem statement
for nearly 2000 Products belonging to 198 categories
as units_sold_that_week, Stores_selling
different customer groups etc are used
variable- “Sales_in_the_nth_week”
26th week.
In Section 2, we discuss about the methodology and work done
the results & discussion in Section 3
applications, followed by the references
2. Methodology and Work Done
A basic block diagram to explain the entire process of Machine Learning is given below.
Figure
answers). In this paper, we discuss the techniques
alyzing it is
26th week based only on information up to the 13th week
reduction (Feature selection)
Non-Linear Learning Models
models.
ation model that better suits the problem based on their
statement, we gathered 26 weeks information
to train our model. Various attributes such
selling_in_the_nth_week, Cumulative units sold to a number of
as independent variables to train & predict the dependent
week”. However, our task here is only to predict their sales in the
in each of the phases,
3.Finally, we draw a conclusion in Section 4 along with its
references.
1 : Machine Learning Life Cycle
28
eeks , o followed by

2.1) Feature selection
We use Greedy Stepwise[2]mechanism for feature
gives us a list of important feature
stores_selling_in_the_nth_weekand weeks_since_launch
features with maximum sales predicting power
selection[2]. The process of feature selection
features from the original feature set. Here
and have been the two most important
in the original data set.
The results from this procedure can be backed up using the scatter plots. The scatter plots are
used for the feature “Total Units sold
in nth week” plotted ed against other features. Their
variations
are then studied and can be used d as a reference
to justify the results from feature selection
scatter plots with random nature can be easily identified and discarded.
Figure 2: Cumulative Units sold to very price sensitive customers
selection. Those
: vs. Total units Sold
The above scatter plot is between:
y-axis Total Units sold
x-axis Units sold to Very
Price sensitive
customers
We e can clearly see, the scatter plot between the two features does not show any trend as it is
completely random in nature. A
those obtained from feature selection
processthese scatter plots showed
are the necessary features for regression.
similar scatter plot was seen for most of the features, except for
f
selection. For the ones obtained from the feature selection
hese some relation between them which confirms the fact that they
he 29

Figure 3 : Total Units Sold Vs Stores Selling
y-axis Total Units sold
x-axis Stores Selling
Hence, this allows us to reduce the number of features that must be used to train our model.
2.2) Linear Model
Regression[1] for our linear learning model. The equation for Multiple
We used Multiple Linear Regression
Linear Regression is given below.
F(X) = + i X
Where, X is the set of input vector with coefficients/weights
bias. F(X) is the approximated Linear function to be used for regression.
i and constant value of
This model needs to be optimized by minimizing the Mean Square Error produced by the model.
The cost function in this case is:
J()=(1/2m) (F (xi)- yi)^2
Where, F (xi) is the predicted value, y
for training. This is the cost function which has been optimized
Algorithm [4].
) yi is the actual value, and ‘m’ is the number of tup
using Gradient Descent
We have applied this linear learning model on the data set of selected features. The results
obtained have been mentioned in the next section.
2.3) Non-Linear Model
We use Random Forest, a bagging based ensemble learning technique for non
Random Forest[9] consists of a collection or ensemble of
non-linear training
basedecision tree predictors
30
(Eq.. 1)
called the
(Eq.. 2)
e, tuples used
training. A
predictors/classifiers,

International Journal of Artificial Intelligence Applications (IJAIA), Vol. 5, No. 6, November 2014
each capable of producing a response when presented with a set of predictor input values. For
classification problems, this response takes the form of a class membership, which associates, or
classifies, a set of independent predictor values with one of the categories present in the
dependent variable. Alternatively, for regression problems, the tree response is an estimate of the
dependent variable given the predictors, taking the average from each tree as the net output of the
model.
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Each tree is grown as follows[3]:
1. Firstly, create n_tree bootstrap samples allowing selection with replacement, where each
sample will be used to create a tree.
2. If there are M input variables, a number mM is specified such that at every node, m variables
are selected at random out of the M and the best splitting attribute off these m is selected. The
value of m is kept constant during the process.
3. Each tree is grown completely without pruning.
This technique was implemented in R tool, with the parameter values as n_trees = 500 and m
(variables tried as each split) = sqrt(Number of features).
The accuracy of Random forests is calculated from the out-of-bag MSE which provides an
unbiased result and eliminates the need for cross-validation.
MSE = (1/n) (yi – F_oob)^2 (Eq.. 3)
2.4) Data Remodeling
Data Remodeling is a phase that requires some domain specific knowledge and use of problem
specific information to restructure the data. Another approach could be the Brute Force technique
which is not a good practice. We have made use of certain basic assumptions related to the
market activities to make changes in this stage. We progressively make changes to the data set
and analyze their results with the aim to improve them further.
2.4.1) Stage 1
The Data set provided for the problem statement originally had the following structure.
• Independent Variables – product id, product category, weeks since launch, stores selling
in that week and various sales data to categorical customers.
• Dependent Variables – The total sales in the nth week.
The current approach is basically dividing the dataset based on the “product_category” and
training the model for each one of them separately. This goes by the intuition that the market
sales patterns and demands-supply varies differently for different categories. And hence we
regress separately for each category and use that model to predict the sales of a product for the
26th week.

After some study, we had already identified in the initial phase that, for a particular category of
product, only the weeks since launch and number of stores selling have a major effect in
predicting the total sales for that week. But this model was not exactly suitable as:
• Firstly, the sales in the 26th week apart from stores selling are also dependent on the sales
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in the previous weeks which were not being considered in the previous data model.
• Secondly, since in the test cases, the data provided is only for 13 weeks, the training must
also not include any consumer specific sales data from beyond 13 weeks.
• The independent variable to be predicted must be the “Total Sales in the 26th week” and
not the “Total Sales in the nth week”.
Hence, we have modified the data set such that we use the sales in every week upto 13 weeks
along with the stores selling in week 26 as a feature set to estimate the sales in week 26. This way
we can also measure the predictive power of sales in each week and how do they affect the sales
in the later stages. This has been analyzed using feature selection on the new set and also through
performing an Autocorrelation analysis on the ‘sales_in_nth_week’ to find the correlation
between the sales series with itself for a given lag. The acf value for lag 1 was 0.8 and for lag 2
was 0.6 showing that with so much persistence, there is a lot of predictive power in the Total
sales in a given week that can help us predict the sales for atleast two more weeks.Finally, the
new structure of the dataset for this problem statement is as follows:
• Independent Variables – Sales in week1, Sales in week2, Sales in week3… Sales in
week13, stores_selling_in_the_13th_week
• Dependent variable - Total Sales in the 26th week.
This dataset was then subjected to both Linear Non-linear learning models. The one which
performs better would then be used to train on the next phase of Data Remodeling.
2.4.2) Stage 2
Weneeded to further modify the data structure to improve results and also find a method by which
we could regress the data set for all the categories together. This meant trying to find a model that
allowed us to train a single model that could work on all the categories together. To do this, we
used the following strategy:
1. Let ‘usn’ represent Units_sold_in_week_n and ‘ssn’ represent Stores_selling_in_week_n.
2. Now, as we had previously obtained the hypothesis from Linear Regression (Eq.. 1), in the
form of:
Sales_n = () x Stores_n (Eq.. 4)
Where is the co-efficient of stores_n. Note that is the only factor which would vary from
category to category.
3. Therefore, from Eq.. 4, we get
Sales_26 = () x Stores_26 (Eq.. 5)

33
Sales_13 = () x Stores_13 (Eq.. 6)
=Sales_26/Sales_13 = Stores_26/Stores_13 (Eq.. 7)
=Sales_26= Stores_26/ Stores_13 * Sales_13 (Eq.. 8)
4.In this way, we remove the need for finding the need of the Coefficient of Stores_n for each
category and simply keep an additional attribute ‘Stores_26/ Stores_13 * Sales_13’.
5. Also, instead of keeping only the “Stores_selling” in week 26, we decided to keep
“Stores_selling” from week 14 to 26 to further incorporate the trend (if any) of the Stores_selling
against Units_sold_in_week_26. This was the only useful feature provided beyond 13th week. The
number of stores could help us identify the trends in the sales of the product hence further
improving the accuracy of our predictions in the 26th week.
Hence the structure of our new dataset was as follows:
1. For each of weeks 1 to 13, the ratio of stores in week 26 to stores in week 13, multiplied
by the sales in that week.
2. The raw sales in weeks 1 through 13
3. The number of stores in weeks 14 through 26.
The results obtained from these changes are explained later in the next section.
2.5) Understanding the Applicability of Models
Any Linear model can only be applied on a given dataset assuming that it encompasses the
following properties, else it performs poorly.
1. Linearity of the relationship between dependent and independent variables.
2. Independence of the errors (no serial correlation).
3. Homoscedasticity[11]means that the residuals are not related to the variable plotted on X-axis.
4. Normality of the error distribution.
In this case we test these properties to understand and justify the performance of Linear Models
against a Non-Linear Models in this domain. These tests are conducted by:
1. Linear relationship among the features is a domain based question. For example does the
“sales to price sensitive customer” affect its “stores selling in nth week”. Such errors can
be fixed only by applying transformations that take into account the interactions between
the features.
2. Independence of errors is tested by plotting the Autocorrelation graph for the residuals.
Serial correlation in the residuals implies scope for improvement and extreme serial
correlation is a symptom of a bad model.
3. If the variance of the errors increases with time, confidence intervals for out of-sample
predictions tend to be unrealistically narrow. To test this we look at plots of residuals vs.

time and residuals vs. predicted value, and look for residuals thatincrease (i.e., more
spread-out) either as a function of time or the predicted value.
4. The best test for normally distributed errors is a normal probability plot of the residuals.
This is a plot of the fractiles of error distribution versus the fractiles of a normal
distribution having the same mean and variance. If the distribution is normal, the points
on this plot fall close to the diagonal line.
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The results obtained in these tests are given in the next section.
2.6) Regularized Linear Regression
As we have already performed the normality tests on the data, and it is clear that the Linear
Models are most likely to perform better than any other model, our next aim here is to further
improve upon the accuracy obtained from the application of Linear Regression Model on this
problem. To achieve this we use a concept of Regularization.
In supervised learning problems, it has been found that when we have a large number of input
features as compared to the training examples, algorithm can suffer from the problem of
overfitting(i.e., if we have too many input features, the learned hypothesis may fit the training set
very well( J() is almost equal to 0), but fail to generalize to new examples).
Regularization is one of the most effective mechanisms to avoid overfitting problem. It basically
introduces an additional term in the cost function which helps penalize modelwith extreme
parameter values. A theoretical justification for regularization mechanism is that it strives to
impose Occam’s razor hypothesis(which states that a simple hypothesis generalizes better) on
the solution.
We apply the two standard regularization mechanisms in machine learning i.e., L1
regularizationand L2 regularizationon the data. L1 regularization uses a penalty term that makes
the sum of the absolute values of the parameters small. L2 regularization makes the sum of the
squares of the parameters small. Linear least-squares regression with L1 regularization is called
the Lassoalgorithm[14] and linear regression with L2 regularization is called the Ridge regression.
Modified cost function is as follows:
J() = (1/2m) (F (xi)- yi)^2 + |w| (Eq.. 9)
Where, F (xi) is the predicted value, yi is the actual value, ‘m’ is the number of tuples used for
training, is regularization parameter that has to be tuned empirically and w is model's weight
vector, ||.|| is either the L1 norm or the squared L2 norm.
The regularization parameter, , controls a trade-offbetween two different goals. The first goal is
captured by the first term((1/2m) (F (xi)- yi)^2) in the cost function(Eq..9)which takes care of
the training set to fit well. The second goal, captured by the second term(|w|), is that parameters
need to be kept small and therefore keeping the hypothesis relatively simple to avoid overfitting.
Cost function for Lasso Regression is as follows:
J() = (1/2m) i=1 to m (F (xi)- yi)^2 + j=1 to n |j | (Eq.. 10)

Cost function for Ridge Regression is as follows:
yi)^2 + j=1 to n2
j
J() = (1/2m) i=1 to m (F (xi)- y
where n is number of input features
features.
Regularized regression can also be depicted as a constrained regression problem
between both the mechanisms can
problem.The
be more described with the following figure:
(Eq.. 11)
difference
From the figure, we can say that regularized regression is nothing but
optimizing a data term and
regularization term which balances MSE with the regularization cost. We can visualize this as the
balance of two different losses. T
The square (in graph (a))and the circle (in graph (b))
represent the
regularization costs of Lasso and Ridge regressions ,respectively,
which are minimum
at the
center i.e. origin where all the parameters are zero
(which ensures hypothesis is relatively simple
and thus avoids overfitting).The contours in the plots represent different
values/data
loss which are minimum at those dots in the figurewhich determine how well we can fit the
model through the data(for the unconstrained regression model
model). So, the combination of the data
term and the regularization term will be minimized at some point that touches both surfaces
As mentioned earlier, L1 regularization results in sparse
are zero. A sparse vector is one that lies exactly on some coordinate axis.
cost function
s parameters i.e., most of the parameters
L2 optimum in graph (b) will be sparse onl
only when the minimum MSE point also lies
axis which occurs with zero probability. But, in Lasso regression, L1
optimum can be on the axis
even when the minimum MSE point is not on axis.
Because the contour of L1 regularization cost
is sharp at that point, it is possible for it to intersect with one of the data loss contours even when
the minimum MSE point is not lo
incurring some level of sparseness
computations. Thus, data analysts tend to choose L1 regularization
located on the axis. So, L1 regularized solutions result in
enesswhich makes the model relatively simple and saves a lot
over L2 regularization
35
surfaces.
exactly on
of
regularization.

Andrew Ng has empirically compared L1 and L2 regularization. As per Andrew Ng's Feature
selection, L1 vs. L2 regularization, and rotational invariance paper [15], L1 regularization is
anticipated to perform better than L2 regularization if you have a less number of training
examples as compared to the number of input features.Conversely, if your features are generated
from algorithms like Principal Component Analysis (PCA), Singular Value Decomposition
(SVD), or any other algorithm that assumes rotational invariance, or you have ampletraining
examples, L2 regularization is anticipated to perform better because it is directly related to
minimizing the VC dimension of the learned hypothesis, while L1 regularization does not have
this property.
We have used R(a free software programming language and software environment for statistical
computing and graphics) to implement Lasso and Ridge regressions on our data. The packages
used to implement Lasso and Ridge are lasso2 and ridge respectively and the functions used are
l1ce() and linearRidge() respectively.
One of the important parameters of l1ce() is bound (a constraint(s) that is/are put onto the L1
norm of the parameters). We manually choose the optimal value of bound (i.e. 0.24) to obtain the
minimum RMSE. There is no algorithm implemented in the package - lasso2 for automatically
choosing an optimal value of the bound. However, there exists a number of algorithms to
automate this selection process and may be implemented in the subsequent libraries in the time to
come. One of those algorithms is found in the paper (On Tuning Parameter Selection of Lasso-
TypeMethods - A Monte Carlo Study)[16]
One of the important parameters of linearRidge() is lambda (a ridge regression parameter). We
kept its default value ,i.e. automatic, while implementing the function on the data which causes
ridge regression parameter to be chosen automatically using the method (a semi-automatic
method to guide the choice of ridge parameter in ridge regression)[17]
The results obtained after applying these methods, are given in the next section.
Every algorithm has its own demerits. Lasso fails to utilize the correlation between the input
features. If there is a group of highly correlated features, then the Lasso has tendency to select
one feature from the group and discard the others.
The results can be further improved upon by using another regularization method called Elastic
Net[18] regularization (basically overcomes the limitations of Lasso regression)which linearly
combines the L1 and L2 penalties of the Lasso and Ridge regressions.By incorporating ridge
regression as well, it allows the L2-tendency of shrinking coefficients for correlated predictors
towards each other, while keeping the feature selection property provided by the Lasso
regression.
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The cost function for Elastic Net is as follows:
J() = (1/2m) i=1 to m (F (xi)- yi)^2 + 1 j=1 to n |j | + 2 j=1 to n2
j (Eq.. 12)
As we see, elastic net overcomes the limitations of Lasso by adding the quadratic part to the
penalty (2 j=1 to n2
j) which is used alone in ridge regression.Although this algorithm has not
been used yet in our project, it can later be used to further optimize the results and adapt to any
given system according to the requirements and specifications.

37
3. Results And Discussion
3.1 Feature selection
The list of features obtained from Greedy Stepwise feature selection [2] showed that “Stores
selling in nth week” and “weeks since launch” were the most important features contributing to
the prediction of sales. The variance of these features with the dependent variable, together is
greater than 0.94 showing that they contribute the maximum to the prediction of sales.
3.2 Application of Linear Regression and Random Forest
Linear Regression Considering the top 6 features obtained from feature selection procedure based
on their variances:
Correlation Coefficient 0.9339
Mean Absolute Error 28.37
RMSE 69.9397
Random Forests considering all the features:
OOB-RMSE 46.26
As we currently see, the non-linear model is working better than the linear model. This may lead
to a jumpy conclusion that non-linear model is probably better in this scenario. Moreover the
accuracy of the classifiers is also not great due to the high RMSE values of both the models.
3.3 Application of Learning Models after Data Re-modeling Phase-1
Linear Regression Results:
RMSE 0.9012
Random Forest Results:
OOB-RMSE 7.19
As we see here, the performance of both the models have improved drastically, however, we find
that the linear model outperforms random forest. This finding compelled us to inquire about the
properties of the dataset that satisfied the assumptions of the linear model. We found that:
i) The Franke’s Anscombeexperiment[10] to test the normality of data distribution came out
inconclusive leading us to use the Normal Q-Q plot[12].
ii) The Normal Q-Q plot in R[13] concluded that the dataset follows the normal distribution.

iii) The residuals also follow the normal distribution curve under the Normal Q-Q plot just
38
like the actual data conforming the second assumption of linearity.
iv) We check the Homoscedasticity[11] property by plotting the residuals againstfitted values.
The graph was completely random in nature.
v) Lastly, the linear relationship between features is a domain specific question. The data
collected mostly contains the sales data from local stores, from local manufactures of
items of daily consumption types like – bread, milk_packets, airbags, etc. Since these
types of products belong to a class of items where the stochastic component is negligible,
it makes it easy for us to assume that the linear model can be easily applied to this
problem. This is the reason why linear model is working better compared to the non-linear
model due to negligible interaction of the features.
3.4 Application of Linear Models after Data Re-modeling Phase-2
Linear Regression Results considering all the new features:
RMSE 0.4365
Linear Regression Results considering only top 6 new features after applying feature selection on
the new dataset:
RMSE 0.7021
As we see, the results have improved further, with the accuracy of the classifier going up from
RMSE value of approximately 65 to 0.43. With the final model, we were able to predict the Total
sales of any given product in the test set with an error 1 unit for any category, our best RMSE
achieved being 0.43.
3.5 Application of Lassoand Ridge Regression
We have applied these algorithms on a subset of the actual training dataset with approximately
900 products belonging to 52 categories. Hence, the RMSE value for Simple Linear Regression
for this subset is slightly different than that obtained for the entire dataset.
Lasso, Ridge and Simple linear regression results considering all the new features are as follows:
Lasso Regression Ridge Regression Simple Linear
Regression
RMSE 0.44 0.66 0.73
Mean Absolute
0.35 0.52 0.59
Error
As we see clearly, Lasso regression outperforms Ridge and simple linear regressions.

It has been found frequently that L1 regularization (or Lasso regression when regularization
applied to linear regression) in many models penalizes many parameters equal to zero resulting in
the parameter vector being sparse. So, this becomes an obvious choice in feature selection
scenarios, where we believe many input features should be eliminated.Theoretically speaking,
Occam Razor hypothesis justifies that sparse or simple solutions (in case of L1 regularization) are
preferable.
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4. Conclusion
The primary target in machine learning is to produce the best learning models which can provide
accurate results that assist in decision making, forecasting, etc. This brings us to the essential
question of finding the best suitable model that can be applied to any given problem statement.
We have performed a case based study here to understand on how to decide whether a linear or a
non-linear model is best suited for a given application.
We initially follow a basic approach by adopting two leading classifiers from each domain and
evaluate their performances. We then try to boost the accuracies of both the learning models
using data re-structuring. The results obtained from this process help us derive an important
empirical proof that the accuracy of a classifier not just depends on its algorithm. There is no such
certainty that a more complex algorithm will perform better than a simple one. As we see in this
case, Random Forests, which belong to the class of ensemble classifiers bagging based is known
to perform well and produce high accuracies. However, here the simple Multiple Linear
Regression model outperforms the previous one. The accuracy of the model largely depends on
the problem domain where it is being applied and the data set, as the domain decides the
properties that the data set would inherit and this greatly determines the applicability of any
machine learning technique. Hence holding a prejudice for/against any algorithm may not provide
optimal results in machine learning.
The framework developed here has been tested on real-time data and has provided accurate
results. This framework can be used for the forecasting of daily use products, items of everyday
consumption, etc. from local manufacturers, as it follows the assumption that the features have
minimum interaction with each other. Branded products from big manufacturers include many
more market variables, like the effect of political and economic factors, business policies,
government policies, etc. which increase the stochastic factor in the product sales also increase
the interaction among the independent features. This feature interaction is very minimal for local
products. Extending this framework to the “branded” scenario will require significant
changes.However, the current model is well suited to small scale local products and can be easily
used with minimal modifications, for accurate predictions.
5. Acknowledgements
We take this opportunity to express and gratitude and deep regards to our mentor – Mr. Biju R
Mohan for his guidance and constant encouragement. Without his support it would not have been
possible to materialize this project. His blessing, help and shall carry us a long way in the journey
of life on which weare about to embark.

We are obliged to staff members of NITK IT department, for the valuable information provided
by them. We are grateful for their cooperation during the project.Lastly, we thank almighty, our
parents, friends and everyone involved directly or indirectly for their constant encouragement
without which this project would not be possible.
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ridge regression.arXiv: 1205.0686v1 [stat.AP] 3 May 2012
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B (2005) 67, Part 2, pp. 301–320
Authors
Gaurav Singh Thakur
Gaurav Singh Thakur has completed his B.Tech in 2014 in Information Technology
from National Institute of Technology Karnataka, Surathkal and is currently working as
a Software Engineer at Cisco Systems, Inc. Bangalore. His technical areas of interest
include Machine learning, Networking Security, Application Development and
Algorithms.

41
Anubhav Gupta
Anubhav Gupta has pursued his Bachelors, at National Institute of Technology
Karnataka, Surathkal in the Field of Information Technology (IT) and graduated in
2014. His technical areas of interest include Machine learning, Information Security,
Web Development and Algorithms.Currently, he is working as Software Developer
Engineer at Commonfloor (MaxHeap Technologies).
Ankur Bhardwaj
Ankur Bhardwaj completed his B.Tech in Information Technology from National
Institute of Technology Karnataka, Surathkal in 2014. His technical areas of interest
include Machine Learning, Statistics and Algorithms. Currently, he is working as
Associate Professional at Computer Sciences Corporation.
Biju R Mohan
Mr. Biju R Mohan is an Assistant Professor at National Institute of Technology
Karnataka, Surathkal at the Dept. of Information Technology. His technical areas of
interest include Software Patterns, Software Aging, Virtualization, Software
Engineering, Software Architecture Requirements Engineering.

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