Influence over the Dimensionality Reduction and Clustering for Air Quality Measurements using PCA and SOM

International Journal of Advanced Engineering, Management and Science (IJAEMS) [Vol-3, Issue-11, Nov- 2017]
https://dx.doi.org/10.24001/ijaems.3.11.4 ISSN: 2454-1311
www.ijaems.com Page | 1044
Influence over the Dimensionality Reduction and
Clustering for Air Quality Measurements using
PCA and SOM
Navya H.N
Bangalore, 560072, India
Abstract—The current trend in the industry is to analyze
large data sets and apply data mining, machine learning
techniques to identify a pattern. But the challenges with
huge data sets are the high dimensions associated with it.
Sometimes in data analytics applications, large amounts
of data produce worse performance. Also, most of the
data mining algorithms are implemented column wise and
too many columns restrict the performance and make it
slower. Therefore, dimensionality reduction is an
important step in data analysis. Dimensionality reduction
is a technique that converts high dimensional data into
much lower dimension, such that maximum variance is
explained within the first few dimensions.
This paper focuses on multivariate statistical and
artificial neural networks techniques for data reduction.
Each method has a different rationale to preserve the
relationship between input parameters during analysis.
Principal Component Analysis which is a multivariate
technique and Self Organising Map a neural network
technique is presented in this paper. Also, a hierarchical
clustering approach has been applied to the reduced data
set. A case study of Air quality measurement has been
considered to evaluate the performance of the proposed
techniques.
Keywords — Air Quality Dimensionality reduction,
Hierarchical Clustering,Principal Component Analysis,
Self Organising Maps.
I. INTRODUCTION
Multivariate Statistical Analysis is useful in the case
where data is of high dimension. Since human vision is
limited to 3 dimensions, all application above 2 or 3
dimension is an ideal case for data to be analyzed through
Multivariate Statistical Analysis (MVA). This analysis
provides joint analysis and easy visualization of the
relationship between involved variables. As a result,
knowledge that was unrevealed and hidden among vast
amounts of data can be obtained. PCA is a powerful
multivariate technique in reducing the dimension of data
set and revealing the hidden relationship among the
different variables without losing much of information
[1].
Artificial Neural Network (ANN) is an information
processing computational model based on biological
neural networks and is composed of several
interconnected processing elements (neurons) that work
in parallel to solve a generic problem. In general ANN are
used to identify complex relations between input and
output or to find patterns in a given data set. ANN is an
adaptive system that can change its structure based on the
flow of information through the network during the
training phase.
The two important learning paradigms in ANNs are
supervised learning and unsupervised learning. In
supervised learning, there exists a training data that helps
in the construction of the model by specifying classes
and by providing positive and negative objects that
belong to those classes. In unsupervised learning, there is
no preexisting taxonomy and the algorithm classifies the
output into different classes. Kohonen’s Self Organising
Map is an unsupervised learning technique that reduces a
high dimensional data into a 2 dimensional space. This
reduction in dimensionality helps in understanding the
relationship quickly and also SOM provides better
visualization of components [2].
The rest of the paper is organized as follows. Section 2
discusses related work and their findings. Section 3
briefly explains the existing Principal Component
Analysis and Self Organising Map techniques used in our
work. Section 4 consists of a case study on air quality to
evaluate the performance of the proposed techniques,
along with expected results. Finally the concluding points
are given in section 5.
II. RELATED WORK
Some of the related work pertaining to Principal
Component Analysis and Self Organising Maps are
discussed in this section. In [3] multivariate analysis used
to analyze and interpret data from a large chemical
process. PCA was used to identify correct correlations
between the variables to reduce the dimensionality of

process data. In [4] Multivariate Statistical Analysis is
applied for Dermatological Disease Diagnosis. There are
few diseases like psoriasis, seborrhoeic dermatitis, lichen
planus, pityriasis, chronic dermatitis and pityriasis rubra
piaris in dermatology that share the same clinical features.
PCA is applied to identify the relationship between 12
clinical attributes and the circle of correlations depicts
patterns of variable associations. Monitoring abnormal
changes in the concentration of ozone in the troposphere
is of great interest because of its negative influence on
human health, vegetation and materials. Modeling ozone
is very difficult because formation mechanisms in
troposphere are very complex and adding to it is the
uncertainty regarding the meteorological condition in
urban areas. PCA is used as a data detection method for
highly correlated variables in [5]. PCA was applied for
the yeast sporulation data for simplification of analysis
and visualization of gene expression data. Data was
collected for 7 different gene expressions over time and it
was observed that much of variability was explained
within first two principal components in [6].
In [7] both principal component analysis and self-
organizing has been applied to classify and visualize fire
risks in forest regions. On application of PCA first two
principal component has explained most of the variance
but SOM was successful in effective visualization and
clustering of nodes to depict fire risks. In [8] PCA, cluster
analysis and SOM was applied to a large environmental
data set to assess the quality of river water. The results
indicated the power of classification of SOM when
compared to other traditional methods. In [9] Forest
Inventory (FI) contains useful information on forest
conditions. To interpret such large data sets, SOM
technique is applied that helps in fast and easy
visualization, analysis of multidimensional data sets. In
[10] PCA and SOM was applied in cellular manufacturing
system for visual clustering of machine-part cell
formation. PCA was used for reducing the dimensionality
of data set and was projected on to 2 dimensional space.
The unsupervised SOM technique was applied for data
visualization and also to solve the problem of cell
formation. In [11] SOM was applied to complex
geospatial datasets for knowledge discovery and
information visualization. The dataset consisted of socio
economic indicators mapped to municipalities in
Netherlands. By the application of SOM, structure and
patterns of dataset was uncovered and graphical
representation helped in discovery of knowledge and
better understanding.
III. EXISTING TECHNIQUES
Multivariate Statistical Analysis is useful in the case
where data is of high dimension. Since human vision is
limited to 3 dimensions, all application above 2 or 3
dimensions is an ideal case for data to be analyzed
through Multivariate Statistical Analysis (MVA). This
analysis provides joint analysis and easy visualization of
the relationship between involved variables. As a result,
knowledge that was unrevealed and hidden among vast
amounts of data can be obtained.
A. Principal Component Analysis (PCA)
PCA is a powerful multivariate technique in reducing the
dimension of data set and revealing the hidden
relationship among the different variables without losing
much of information [12]. The steps involved are:
1. Calculate the covariance matrix
Covariance is the measure of how one variable varies
with respect to other variable. If the variables are more
than two, then the covariance matrix needs to be
calculated. The covariance matrix can be obtained by
calculating the covariance values for different
dimensions. The formula for obtaining the covariance
matrix is
𝐶 𝑛 𝑋 𝑛
= (𝑐𝑖 ,𝑗 , 𝑐𝑖 ,𝑗 = 𝑐𝑜𝑣(𝐷𝑖𝑚𝑖 , 𝐷𝑖𝑚𝑗) )
where Cn x n
is a matrix with n rows and n columns
i, j is the row and column indices and covariance is
calculated using below formula
𝑐𝑜𝑣(𝑋, 𝑌) = ∑ [(𝑋𝑖
𝑛
𝑖=1
− 𝑋̅)(𝑌𝑖 − 𝑌̅)]/(𝑛 − 1)
where Xi is the value of X at ith
position
Yi is the value of Y at ith
position
𝑋̅, 𝑌̅ indicates the mean values of X and Y
respectively.
2. Find Eigen Vectors for covariance matrix
Eigen vectors can be calculated only for square matrix
and not all square matrix has eigen vectors. In case n x n
matrix contains eigen vectors, then total number of eigen
vectors is equal to n. Another important property is that
the eigen vectors are always perpendicular to each other.
Eigen values are associated with Eigen vector and
describe the strength of the transformation.
3. Sort Eigen vectors in decreasing order of Eigen values
The highest eigen value corresponds to be the principal
component which explains the maximum variance among
the data points with minimum error. The first few
principal components explain most of the variance and
eigen values with lesser values can be neglected with very
little loss of information.
4. Derive the new data set
The original data will be retained back but will be in
terms of the selected eigen vectors without loss of data.
B. Self Organising Map (SOM)
Self Organising Map is a popular technique in Artificial
Neural Network (ANN) under the category of
unsupervised learning. In conventional ANN approach,

the input vector is presented to a multilayer feed forward
network and the generated output is compared with the
target vector. When there exists a difference, the weights
are altered to minimize the error in output. This phase is
repeated many times with different sets, until the desired
output is achieved. However, SOM does not require a
target vector during the training phase. A SOM without
any external supervision, learns to classify the training
data.
The basic principle of SOM is when the weight of nodes
matches the input vector, then that portion of lattice is
selectively optimized to resemble the input vector. Each
node receives every element of input or training data in
vector format one at a time. A calculation is done between
the element and weight of the node to determine the fit
between them. The calculation performed is to determine
the distance between the two and usually it is Euclidian
distance or any other distance measure can be used. A
winning node that best describes the training element can
be obtained and it has the smallest distance between the
input element and node’s weight. The neighbors of
winning node should be identified. Then, these neighbors
and winning node is updated to represent the new training
element. By this procedure, the map learns through
individual elements [13].
SOM Algorithm
1. Initialize the weights for each node either randomly or
by pre-computed values.
2. For every input element
a) Get the input element and convert to a vector
b) For every node in the map
i) Compare the input vector with the node
c) Every node weight is examined to see which one
matches and is closest to the input vector. The
node with the smallest distance is declared as the
winning unit and is commonly referred as Best
Matching Unit (BMU).
Distance between the between input vector and
node’s weight is calculated using Euclidean
Distance formula.
where V is the current input vector
W is the node’s weight vector
d) The radius of neighborhood of the BMU is
calculated. All nodes within this radius range
will be updated in the next iterations. Initially,
the radius will be set to the radius of lattice and it
decreases in each step.
e) The weights of each neighboring nodes are
adjusted according to the below equation
𝑥(𝑡 + 1) = 𝑥(𝑡) + 𝑁(𝑥, 𝑡) 𝛼 (𝑡) (𝜕(𝑡) − 𝑥(𝑡))
where x(t+1) is the next value of weight vector
x(t) is current value of weight vector
N(x, t) is the neighbourhood function, which
decreases as a function of time
α(t) is the learning rate, which decreases as a
function of time
∂(t) is the vector representing the input document
IV. CASE STUDY
A case study is considered to demonstrate the concept of
dimensionality reduction on a data set using Principal
Component Analysis and Self Organising Map. Later,
hierarchical clustering is applied to the obtained SOM
results. The case study considered determines the quality
of air in regions of New York city. The data set consists
of variables like Ozone, Solar radiation, Wind,
Temperature for different regions. Ozone variable
consists of numeric values in parts per billion, Solar
radiation depicts the radiation in Langleys with a
frequency band 4000-7700 Angstroms, Wind variable has
numeric values in miles per hour, Temperature indicates
in degrees the maximum daily temperature.
A. PCA Application
PCA reduces the dimension and explores the hidden
information. The first few principal components explain
most of the variance. Table 1 indicates the summary of
principal component analysis which explains the standard
deviation between the principal components, proportion
of variance and cumulative proportion.
Table.1: indicates the standard deviation, variance and
cumulative proportion associated with principal
components
PC1 PC2 PC3 PC4
Standard
Deviation
1.5343 0.9529 0.6890 0.51441
Proportion of
Variance
0.5886 0.2266 0.1187 0.06615
Cummulative
Proportion
0.5886 0.8152 0.9338 1.00000
The important task in PCA is deciding the number of
principal components to be considered for further
analysis. Fig. 1. Indicates a scree plot for determining the
number of PCs. The elbow point shows the number of
components to be considered and beyond this the eigen
values are small and indicates negligible data loss.

Fig. 1. indicates a scree plot for determining the number
of PCs.
The relationship between variables can be identified using
biplot. Fig. 2. represents variables and observation of
multi-dimensional data. Fig.2 indicates the variables
Temperature and Ozone are closely related to each other
and Wind is negatively correlated with other variables.
The angle between vectors represent an approximation of
covariance. A small angle between vectors indicates the
variables are highly correlated, an angle of 90 degrees
indicates variables are not correlated and 180 degrees
represents that the variables are negatively correlated.
Fig. 2. Represents variables and observation of multi-
dimensional data.
B. SOM Application
Before the application of SOM, the data was prepared for
analysis by identifying the duplicates, removal of outliers,
data conversion, removal of null values and the variables
are scaled to provide equal importance during training
phase. A SOM grid is created with a size of 10 x 10 and
there is no explicit rule for selecting the number of nodes,
except that it should allow easy visualization of the SOM.
The SOM is trained with a learning rate of 0.05 that
gradually declines to 0.01. The radius of neighbourhood
can either be a vector or a single number. If it is a single
number, the radius will vary from current value to the
negative value of the current number. Once, the
neighbourhood becomes small and radius is less than 1,
only the winning node will be updated.
Fig.3. shows a plot of SOM during training phase after
100 iterations. In this experiment, 100 times the complete
data set is presented to the network. The distance between
the weight of the node and input sample reduces as the
training iterations progress. This distance should ideally
reach a minimum value and Fig. 3. shows training
progress over time.
Fig.3. shows a plot of SOM during training phase after
100 iterations.
Unified Distance matrix (U –matrix) is a type of SOM
visual representation in a map of 2 dimensional grid as
shown in fig. 4. U-matrix indicates the distance between
adjacent nodes and it is represented by different grey
shades. A dark color indicates a larger distance between
the neighboring nodes and represents a gap in input
values.
A lighter shade indicates the vectors are close to each
other and they indicate cluster themselves, high values
represents borders among the clusters.
Fig. 4. Indicates the unified distance matrix visualization
Component Planes
Component planes represents clearly the visualization of
individual input variables. They are represented in grey
scale or a combination of different colors. Fig. 5.

represents component planes for each variable. By
inspecting component planes in Fig. 5. it is evident that
the variables Ozone, Solar radiation and Temperature are
positively correlated and variable Wind is negatively
correlated with other variables. Fig. 5. represents
component planes for each variable.
The rate of chemical reactions that produce ozone are
affected by solar radiation and temperature. Therefore, an
increase in these values increases the chemical reactions
and leads to an increase in ozone levels. Wind speed is
negatively related to all the variables. The concentration
of ozone is high, when there is low wind speeds. The
covariance matrix is generated for the input variables and
it reflects the same relationship as the component planes.
Table 1 represents the correlation coefficient values
between the variables. It can be observed that maximum
correlation exists between ozone and temperature. Wind
is most negatively related with ozone. Component planes
help in understanding this concept pictorially, therefore
Self Organizing Map helps in easy visualization of
relationship between variables.
Table 1 represents the correlation coefficient values
between the variables
__________________________________
Variables Ozone Solar.R Wind Temp
Ozone 1.0000
Solar.R 0.3411 1.0000
Wind -0.6265 -0.1147 1.0000
Temp 0.6938 0.2850 -0.4947 1.0000
Clustering of Self Organising Map
A U-matrix may be appropriate to identify the cluster
boundaries. But this may not be effective and therefore, a
hierarchical clustering technique is applied after the SOM
is trained. Fig. 6. Shows the formation of clusters. Here,

there is a formation of 3 clusters – High Ozone, Medium
Ozone and Low Ozone.
Fig. 6. Blue nodes indicate High Ozone, Orange colour
nodes indicate Medium Ozone and Green color nodes
indicate Low Ozone.
Fig. 6. shows the formation of clusters into low ozone,
medium and high ozone that is depicted by green, orange
and blue respectively.
In High Ozone class, the effect of solar radiation and
temperature are above average and wind speed has very
low effect. The effect of solar radiation and temperature is
below average in Low Ozone class and wind speed has
significant effect. In Medium Ozone cluster, solar
radiation, temperature and wind have about average
effect. The clustering results obtained are satisfactory and
Self Organising Maps can be used to classify the quality
of air.
V. CONCLUSION
In this paper, principal component analysis (PCA) and
Self Organising Map (SOM) has been applied to
demonstrate the dimensionality reduction of dataset. A
case study was considered to visualize and classify air
quality data. PCA explained most of the variance in data
but it was difficult to interpret the data pattern. However,
SOM appears to be the best fit to represent complex data.
Especially, the component planes of SOM provides
effective visualization and uncovers the correlation
between the input variables.
U-matrix can be used for data classification, but may not
be a good choice in all cases. Therefore, in this paper
hierarchical clustering technique is applied to the SOM
results obtained. The quality of air is partitioned into 3
clusters – low, medium and high ozone content.
Therefore, it can be concluded that SOM has better
resolving power of classification than traditional methods.
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