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Файл:Ординатура / Хирургия / Библиотека им академика М.И. Перельмана / Книга_5486_Библиотеки_им_академика_М_И_Перельмана
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also facilitates the identification of critical descriptors that contribute most to the var-
iance in biological activity. PCA also serves to mitigate multicollinearity issues and di-
minish noise, which often arises in QSAR datasets due to high intercorrelation
between descriptors. By retaining a subset of PCs that explain the majority of the vari-
ance, PCA streamlines subsequent modeling efforts, resulting in more robust and ac-
curate predictive models. PCA within MVDA for QSAR in drug discovery acts as a
transformative process, translating intricate molecular information into a concise re-
presentation while preserving the essential characteristics driving biological activity.
This technique empowers researchers to uncover patterns, prioritize descriptors, and
expedite the identification of promising drug candidates within the vast chemical
space [29, 30].
With the evolution of QSAR techniques in the past few decades, the number of
descriptors has also increased enormously. Creating MLR from this large pool of inde-
pendent variables (descriptors) has now become a tedious task due to two main rea-
sons, that is, complexity in the MLR equation generation with a large number of
descriptors and colinearity among the descriptors. PCA in simple words is a data re-
duction statistical approach in which the number of descriptors is reduced rationally
without losing the important information among them [31].
To understand this, suppose initially we have a pool of 1,000 descriptors of the
molecules. Creating an MLR equation correlating the biological activity of the mole-
cules with these descriptors will be a very tedious task. Also, there will be high co-
linearity among some of these descriptors. To overcome all this PCA is applied. Each of
the 1,000 descriptors will be converted to 1,000 PCs numbered PC
1
,PC
2
,PC
3
,PC
4
,...,
PC
1,000
. Interestingly, these PCs are derived in such a way that the highest information
of the data is contained by the PC
1,
and as we move from PC
1
to other components in-
formation (variation) enclosed by them decreases. Suppose in this case PC
1
contains
80% of the information (variation), PC
2
will have lesser information (say 14%), PC
3
will
have even lesser information (say 5%), and moving further the information carried by
the PCs will reduce subsequently. Therefore, if we select only the first three PCs from
the 1,000 derived for our data analysis, that is, PC
1
,PC
2
, and PC
3
, we will be using 99%
of the information of these descriptors. In other words, at the cost of losing 1% accu-
racy, we can simplify our QSAR model generation by reducing the 1,000 descriptors to
3 useful variables. The choice of the number of PCs to be incorporated solely depends
upon how much accuracy is desired.
The exact methodology for the PCA is complex and is done through machine
learning. The basic steps involved in the PCA are depicted below:
– Define the data: This is the very first step in which original data of the biological
activity of all the molecules along with their descriptor values is generated.
– Creating mean- centerin g/normalization of data: The data obtained in the first
step is then made mean-centered or normalization of the same is done. Normalization
is done to make all the data at the same scale. The purpose of the mean-centered data
6.3 Multivariate data analysis (MVDA) 119
is to ensure that the PC will be driven toward maximum variance. The general for-
mula for obtaining mean-centered data is given by the following equation:
x
i
= X
i
− X
mean
(6:16)
where x
i
is the mean-centered value of the descriptor X, X
i
is the original data value,
and X
mean
is the mean of all the values of descriptor X. For normalization of the data
we can use the following equation:
z =
x
i
SD
(6:17)
where z is the normalized data of the descriptor, x
i
represents its mean-centered
value, and SD is the standard deviation value calculated by the following equation:
SD =
p
Σðx
i
Þ
2
p
N
(6:18)
where N is the total number of observations.
The calculation of mean-centered data and normalization is illustrated in Tab. 6.6
with a hypothetical example.
– Creation of the covariance matrix: The covariance matrix is a square symmetrical
matrix of order n (n is the number of rows and columns). The purpose of creating the
covariance matrix is to identify the correlation among the descriptors. A negative co-
variance between two descriptors means a negati ve correlation whereas a positive
covariance between them demonstrates a positive correlation between them. As in
the hypothetical example given in Tab. 6.6, a covariance matrix of order 3 will be ob-
tained from the normalized values of the descriptors as illustrated below:
Tab. 6.6: Example of calculation of mean-centered data and normalization.
S. no. Descriptor
(X)
Descriptor
(X)
Descriptor
(X)
x mean-
centered
x mean-
centered
x mean-
centered
z z z
− − − −. −. −.
. .
− −. −. −.
− −. .
. . .
Mean = Mean = Mean =
120 Chapter 6 Statistical approaches used in the QSAR
Cov
z1z1
Cov
z1z2
Cov
z1z3
Cov
z2z1
Cov
z2z2
Cov
z2z3
Cov
z3z1
Cov
z3z2
Cov
z3z3
2
6
4
3
7
5
Individual covariance between the two variables (suppose x and y) can be calculated
by the following equation:
Covarience
xyðÞ
=
Σ x
i
− x
mean
ðÞx
i
− x
mean
ðÞ
N
(6:19)
where x
i
represents the individual value of the x variable, y
i
is the value of the indi-
vidual y variable, x
mean,
and y
mean
represent their respective mean values, and N rep-
resents the total number of molecules in the study.
– Calculations of eigenvalues and eigenvectors: Eigenvalues and eigenvectors are
calculatedfromthecovariancematrixgiven above. These two terms are obtained
from linear algebra and are used in computing the PCs. Eigenvectors tell us about the
direction toward which maximum information of the variables (variance) is available
that constitutes our PC and eigenvalues tell us about the magnitude of that PC. It must
be noted that both these terms exist in pairs and are equal to the number of variables
in our study.
Consider an example where three variables are in the initial study. We will get
three eigenvalues denoted by λ1, λ2, and λ3, where λ1 will be the largest value and λ3
is the smallest value. Eigenvectors associated with them will be denoted by V1, V2,
and V3, respectively. Now suppose if the value of λ1 is 10, λ2 is 3, and λ3 is 1, then the
total of these three values is 14 and it implies that PC1 corresponding to λ1 and V1 will
have 71.42% [(10/14) × 100] of the total information (variance). Similarly, PC2 will have
21.42% of the information, and the remaining 7.14% of the information will be pos-
sessed by the PC3.
– Formation of the feature vector: As of now, we already know that the number of
PCs identified is equal to the number of initial variables in the study. The next step in
the PCA is to keep the N number of the PC from all the computed ones. The choice of
the number of PCs to be retained and to discard the rest of them totally depends upon
the user and can either be done statistically or manually. Generally, PCs that explain
95% of the variance are retained in the study. The next step is the fo rmation of the
feature vector, which is simply a matrix containing the eigenvectors in columns. Each
column of the vector matrix is made of an eigenvector and the number of columns is
equal to the number of PCs we want to retain in the study.
– Reducing the dimensionality of the dataset: This is the final step of the PCA in
which the dataset is rearranged as per the new PCs computed. The values of PCs are
calculated simply by multiplying the transpose of the abovementioned feature vector
matrix and the transpose of the original dataset.
6.3 Multivariate data analysis (MVDA) 121
Here one must understand that PCA is just a data mining technique in which an initial
large pool of descriptors is reduced to a few numbers that contain the maximum infor-
mationofthevariables.ThePCs,denotedasPC1,PC2,PC3,...,PCn, are then employed
for the generation of regression equations for the development of the correlation be-
tween the biological activity and these newly calculated variables, that is, PCs. This
method is known as PC regression (PCR). The general equation obtained from PCR is
given as in the following equation:
Biological activity = m
1
PC
1
+ m
2
PC
2
+ m
3
PC
3
+ + m
n
PC
n
+ C (6:20)
where PC1, PC2, . . . , PCn represent the PCs obtained from PCA, m
1
, m
2
,. . ., m
n
repre-
sent the respective coefficients of PCs, and C is the intercept.
6.3.2 Partial least square (PLS) analysis
PLS analysis is a pivotal technique within MVDA that serves as a potent tool to unravel
intricate relationships between molecular descriptors and biological activities, facilitat-
ing the predictive modeling of a compound’s pharmacological behavior [32]. In the con-
text of QSAR, where datasets entail numerous variables, PLS ef ficiently tackles the
challenge of multicollinearity and high dimensionality [33]. This method constructs la-
tent variables or components that encompass both descriptor and activity information,
emphasizing the most influential variables driving compound-activity correlations. By
simultaneously considering all variables, PLS captures shared variance and retains pre-
dictive power, even in scenarios with noisy or incomplete data [34].
PLS operates by iteratively extracting components that maximize the covariance
between descriptors and activities, consequently constructing a model that not only
describes the data’s structure but also enables accurate activity predictions for new
compounds [35]. PLS’s ability to handle complex relationships and its capacity to man-
age “small sample, high dimension” scenarios make it a staple in QSAR applications.
PLS analysis within the framework of MVDA in QSAR offers an efficient means to un-
ravel intricate molecular-activity associations, thereby revolutionizing the process of
drug development through data-driven insights [36].
PLS is considered as the extension of the PCA and is an MVDA approach used for
the reduction of variables when we have a large pool of variables and ultimately to
develop a regression model that correlates the biological activity with the new varia-
bles. It basically satisfies two conditions: first variables chosen are highly correlated
with biological activity and secondly, these variables contain the maximum amount
of variance among them [37, 38].
In PCA, PCs are calculated based on covariance among the variables . However,
one major drawback of this approach is that small PCs that are obtained sometimes
hold a very high correlation with the output variable, that is, with the biological activ-
ity, and therefore, should not be neglected (Fig. 6.4). To overcome this, the PLS ap-
122 Chapter 6 Statistical approaches used in the QSAR
proach is applied in which, instead of PCs, latent variables are computed based on the
covariance between the independent variables and the dependent variables.
Let us understand this with a hypothetical example given in Tab. 6.7. Assume we have
six molecules along with their biological activity and descriptor values. In this exam-
ple, if we go for traditional MLR for the development of the correlation between the
biological activity and two variables (X1 and X2), we will find that there is very strong
colinearity between these two variables. Therefore, it will hinder the quality of the
generated QSAR model. By applying PCA, we can get PC1 but how it is correlated with
the BA is not taken into consideration. Hence a better approach PLS is applied to cal-
culate the latent variable. This is a combination of X1 and X2 in such a way that it is
correlated with the biological activity rationally.
Fig. 6.4: The principal component sometimes has a small weightage but it has a high contribution to the
biological activity, and hence must be included in the model generation.
6.3 Multivariate data analysis (MVDA) 123
A new variable, termed the latent variable (LV), will be obtained by the combination
of the variables X1 and X2 and can be represented as follows:
LV = α
1
X1 + α
2
X2(6:21)
where α
1
and α
2
are the weights corresponding to the concerned variables and the sum
of their square should always be equal to one as shown in the following equation:
α
2
1
α
2
2
= 1orα
2
=
ffiffi
1
p
− α
2
1
(6:22)
We can have numerous combinations of α
1
and α
2
that will give the sum of their
squares as one. The role of PLS comes into play in this situation for deciding the ap-
propriate combination. Suppose, α
1
is 0.5, then from the above equation α
2
will be
0.87. The new variable, LV, can be calculated as LV = 0.5X1 + 0.87X2 and the covariance
between the biological activity and this LV can be calculated. These covariance values
obtained from the different combinations of α
1
and α
2
are plotted against the α
1
(Fig. 6.5)
.
The value of α
1
at which covariance is highest is selected. The α
2
is calculated
for the α
1
by employing equation (6.21). In this example, α
1
came out as 0.52 and the
corresponding α
2
value comes as 0.85. The LV1 obtained can be depicted as shown in
the following equation:
LV1 = 0.52X1 + 0.85X2(6:23)
The next step will be the calculation of LV for the molecules from the above equation
given in Tab. 6.8.
Now we can apply SLR on the biological activity and the LV1 values. The correla-
tion equation we will get will be expressed as follows:
Biological activity = m LV1 + C (6:24)
where m is the slope coefficient and C is the intercept of the line. In our example, this
equation can be written as follows:
Biological activity = 1.96 LV1 − 72.8(6:25)
Tab. 6.7: Example of six hypothetical compounds along with their biological
activity (Y) and descriptor values (X1 and X2).
Compound Biological activity (Y) X X
124 Chapter 6 Statistical approaches used in the QSAR
Replacing its value in equation (6.23), we will get the correlation equation as follows:
Biological activity = 1.96 0.52X1 + 0.85X1ðÞ− 72.8(6:26)
6.3.2.1 Advantages of PLS analysis
– Handling multicollinearity: PLS effectively addresses multicollinearity, where
predictor variables (molecular descriptors) are highly correlated. It creates latent
variables (components) that are linear combinations of the or iginal descriptors,
reducing collinearity issues and enhancing model stability.
Fig. 6.5: The plot between the covariance and one of the weight components (α
1
) in the generation of the
partial least square (PLS)-based model.
Tab. 6.8: Calculation of the latent variable from the data obtained from Tab. 6.7.
Compound Biological activity (Y) X X LV
.
.
.
.
.
6.3 Multivariate data analysis (MVDA) 125
– Optimal feature selection: PLS identifies the most informative descriptors con-
tributing to the observed biological activities. This feature selection process sim-
plifies the model by focusing on the most relevant information, leading to more
interpretable and accurate models.
– Predictive power: PLS models have demonstrated strong predictive capabilities,
allowing researchers to make reliable predictions about the activity of new com-
pounds based on their structural characteristics. This is crucial for guiding exper-
imental efforts and prioritizing compounds for further evaluation.
– Handling small sample sizes: In drug discovery, datasets with limited compounds
but numerous descriptors are common. PLS copes well with small sample sizes,
making it suitable for scenarios where traditional regression techniques might fail.
– Robustness to noisy data: PLS is robust in the presence of noise and outliers,
which are common in real-world datasets. It identifies patterns despite the pres-
ence of measurement errors or unusual data points.
– Multivariate relationships: PLS considers both the descriptor-response relationship
and interrelationships among descriptors simultaneously. This ability to capture
complex multivariate relationships improves the understanding of structure–activity
relationships.
– Overfitting prevention: PLS includes a regularization parameter that helps pre-
vent overfitting, maintaining the model’s generalization capabilities even when
dealing with high-dimensional data.
– Visualization and interpretation: PLS generates score plots and loading plots
that enable visualization of compound clusters, trends, and influential descrip-
tors. This aids in data exploration and hypothesis generation.
– Iterative improvement: PLS allows for iterative model refinement by adding or
removing descriptors, enhancing the model’s performance progressively.
– Guiding compound design: PLS-based QSAR models offer insights into the struc-
tural features that impact compound activity, guiding the rational design of new
molecules with desired properties.
6.3.2.2 Limitations of PLS analysis
– Model complexity: PLS models can become quite complex, especially when deal-
ing with a large number of descriptors. This complexity can make it difficult to
interpret the relationships between descriptors and biological activities.
– Sensitivity to outliers: PLS is sensitive to outliers in the data. Outliers can dispro-
portionately influence the model, leading to potentially inaccurate predictions.
– Multicollinearity: PLS assumes that the predictor variables (descriptors) are not
highly correlated with each other. If there is significant multicollinearity among
the descriptors, it can lead to unstable and unreliable model estimates.
– Interpretability: While PLS can predict activity, it might not provide direct insight
into the underlying mechanisms of the molecular interactions. The relationships
126 Chapter 6 Statistical approaches used in the QSAR
between descriptors and biological activities can be challenging to interpret, espe-
cially when dealing with a large number of descriptors.
– Selection of components: PLS requires the selection of the number of compo-
nents to retain in the model. Selecting an inappropriate number of components
can impact the model’s performance.
– Data quality: The accuracy and reliability of PLS models heavily depend on the qual-
ity of the input data. Inaccurate or noisy data can lead to biased model outcomes.
– Applicability domain: PLS models might not perform well outside the range of
the training data, as they rely on capturing the relationships observed in the
training set. Extrapolation can be problematic.
– Dimensionality reduction: While PLS aims to reduce dimensionality, it might
not eliminate all irrelevant features. The inclusion of irrelevant or redundant de-
scriptors can degrade model performance.
– Human expertise: PLS requires domain knowledge for proper model interpreta-
tion and validation. Lack of expertise can lead to misinterpretation of results or
inadequate model assessment.
6.3.3 Genetic function approximation (GFA)
Genetic function approximation (GFA) represents a significant advancement in QSAR
modeling for drug discovery. GFA, a machine learning technique, enhances this pre-
dictive capacity by identifying complex relationships between molecular features and
activity. At its core, GFA employs a GA to select and refine relevant molecular descrip-
tors or features from a large pool of potential variables. These descriptors encode var-
ious structural, electronic, and physicochemical characteristics of molecules. By
iteratively selecting, combining, and evolving descriptors, GFA uncovers intricate cor-
relations that traditional QSAR methods might overlook. GFA’s strength lies in its abil-
ity to handle high-dimensional data and capture nonlinear relationships between
molecular features and biological activity. This enables it to provide more accurate
predictions, aiding in the prioritization of potentially active compounds during drug
discovery. Moreover, GFA offers insights into the underlying mechanisms of molecu-
lar interactions, facilitating a deeper understanding of structure– activity relation-
ships. In QSAR for drug discovery, GFA stands as a versatile tool that bridges the gap
between complex chemical information and actionable insights. Its integration has
the potential to accelerate the identification of promising drug candidates while mini-
mizing costly and time-consuming experimental efforts [39–41].
GFA is slightly a different approach in the MVDA. The inspiration behind the GFA
is from Holand’s GA developed in 1975 [42] and Friedman’s multivariate adaptive re-
gression splines (MARS) algorithm developed in 1991 [43]. Unlike approaches dis-
cussed earlier (MLR, PCA, and PLS), where only one model is generated, in the GFA
approach many models are generated out of which the best one is selected. In this
6.3 Multivariate data analysis (MVDA) 127
approach, initially, a population of the parent models is generated, and then based on
the “parent models” further models are constructed which are considered the “off-
spring models.” The offspring models generated are equivalent to or better than the
parent models constructed by applying the principles of genetic evolution and the ter-
minology used for it is GA. In the QSAR model building, the GFA or GA approach is
often used in combination with the MLR or PLS, and this combined approach (GA-
MLR or GA-PLS) is always considered superior to other approaches for the generation
of a robust QSAR model [44–46].
6.3.4 Pattern recognition or multivariate data classification procedures
or cluster analysis
The one major advantage of the MVDA is that it pro vides lots of information about
the data but also comes with a hindrance to comprehension of this large data [47].
Sometimes, the approaches discussed above are not sufficient enough to handle such
large multivariate data in statistics. Hence, multivariate data classification procedures
are introduced to classify the large datasets into smaller groups having similar char-
acteristics or patterns (Fig. 6.6). The two main objectives of these classification proce-
dures are the simplification of the data and developm ent of a model having better
predictability. In these approaches, each independent variable (descriptors) of the
molecule is assigned a certain score value; then a plot is plotted for score values
against variables for each molecule (Fig. 6.7). Before plotting, the score value of each
Fig. 6.6: Variables in the multivariate variable models are grouped into a set of variables based on their
similarity in the patterns.
128 Chapter 6 Statistical approaches used in the QSAR
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