# Balancing Explainability-Accuracy of Complex Models

Poushali Sengupta
Department of Informatics
The University of Oslo
Oslo, Norway
<EMAIL_ADDRESS>
&Yan Zhang
Professor
Institute for Informatics
The University of Oslo
Oslo, Norway
<EMAIL_ADDRESS>
&Sabita Maharjan
Professor
Institute for Informatics
The University of Oslo
Oslo, Norway
<EMAIL_ADDRESS>
&Frank Eliassen
Professor Emeritus
Institute for Informatics
The University of Oslo
Oslo, Norway
<EMAIL_ADDRESS>

###### Abstract

Explainability of AI models is an important topic that can have a significant
impact in all domains and applications from autonomous driving to healthcare.
The existing approaches to explainable AI (XAI) are mainly limited to simple
machine learning algorithms, and the research regarding the explainability-
accuracy tradeoff is still in its infancy especially when we are concerned
about complex machine learning techniques like neural networks and deep
learning (DL). In this work, we introduce a new approach for complex models
based on the co-relation impact which enhances the explainability considerably
while also ensuring the accuracy at a high level. We propose approaches for
both scenarios of independent features and dependent features. In addition, we
study the uncertainty associated with features and output. Furthermore, we
provide an upper bound of the computation complexity of our proposed approach
for the dependent features. The complexity bound depends on the order of
logarithmic of the number of observations which provides a reliable result
considering the higher dimension of dependent feature space with a smaller
number of observations.

##  1 Indroduction

Figure 1: The graph shows the trade-off between explainability-accuracy; More
complex models are hard to explain.

Artificial Intelligence (AI) is envisioned to play an important role in
various technical fields from the energy market to clinical domains[3, 9, 10].
The machine learning algorithm, especially with the recent development in deep
learning, has proven to be an indispensable technology to any system including
critical infrastructure[12, 13, 14, 15]. As a consequence, there is a growing
interest in understanding the execution and performance of these
algorithms[22, 23]. Explainable AI (XAI)[1, 2] can be a promising research
field to understand and interpret the "black box" behaviour of machine
learning (ML) algorithms[24, 25, 26].

SHAP: Shapley Additive Explanations[18]and LIME[19] are the most used
algorithms for the explainability of an ML model. The SHAP-based approach
calculates the weighted marginal contribution called Shapley value for each
feature. The Shapley value of a feature represents its contribution to one or
several sets of features. On the other hand, LIME focuses on the local
faithfulness of a model. Although LIME[19] has the desirable property of
additivity[18], it has weaknesses regarding the lack of consistency[16],
missingness[17], and stability[20, 27]. SHAP fulfils these and hence is
commonly used. The sum of SHAP values associated with all features is equal to
the final contribution. LIME assumes that the local model is linear, whereas
SHAP can be applied to any simple nonlinear model[21]. However, for complex ML
models, the SHAP algorithm simplifies the model first and then calculates the
feature importance[28]. SHAP considers all possible combinations of features,
which becomes hard to apply with high dimensional dependent feature space. To
this end, while more complex ML/DL models yield better accuracy, the
explainability of such models is a challenging and largely unexplored problem
(see figure 1 for better understanding). In particular, enhancing the
explainability of a model while also ensuring a high level of accuracy is not
a trivial task[29]. Also, most of the existing algorithms including SHAP, are
hard to apply when features are dependent [30]. Furthermore, existing XAI
models including SHAP-based approaches interpret the result without measuring
the uncertainty[32, 33] regarding the feature’s contributions to the output.
On this background, the three main challenges addressed by us, 1) The
explainability-Accuracy tradeoff in a complex model, 2) Large number of
dependent input features, and  3) Uncertainty associated with each feature and
output.
Motivated by these considerations, we propose a new approach ExCIR:
Explainability through Correlation Impact Ratio that can maintain an accuracy-
explainability trade-off for complex ML models, considering both dependent and
independent features in higher dimensional feature space. ExCIR also considers
the uncertainty associated with a feature contribution by measuring Shannon
entropy [31] for each feature. ExCIR adjusts the distance between features and
output vectors in hyper-dimensional space for both the original and the
lightweight model environment to ensure the accuracy of the model. Instead of
SHAPLY values[21, 23], ExCIR calculates the correlation impact ratio (CIR) to
explain the relation distance of a target feature on the output variable. The
main contributions of this work are:

  1. 1.

Novel framework for Accuracy-Explainability tradeoff: We create a lightweight
model to explain the feature impacts on the output by creating a suitable
environment such that the lightweight model works with sample data having less
input instead of choosing different sets of features.

  2. 2.

Introducing new metric to calculate the uncertainty of explainability: To
calculate this metric we will consider the notion of uncertainty [6] which
measures the uncertainty between the features and output.

  3. 3.

Novel metrics of Explainability for both feature independencies and
dependencies:  We propose different measures to compute the relation impact
for the cases of independent features and dependent features. Both the metrics
can be directly applied to the nonlinear model and as a result, the
explainability is enhanced while ensuring the same level of accuracy.

##  2 ExCIR: System Model and Design

We consider a data set with $n$ rows and $k$ features, denoted as
${\underset{\sim}{f_{1}}},{\underset{\sim}{f_{2}}},...,{\underset{\sim}{f_{k}}};\
{\underset{\sim}{f_{i}}}\epsilon||F||^{(k\times n)}$ where $||F||^{(k\times
n)}$ is the $(k\times n)$ dimensional feature space having $k$ different
features with the $n$ observations. Here, the $i^{th}$ feature is denoted as
${\underset{\sim}{f_{i}}};\ i=1:k$ and
${\underset{\sim}{f_{i}}}=(f_{i1},f_{i2},.....,f_{in})$. The machine learning
model $M(f)$ learns all data history and predicts the output vector
${\underset{\sim}{Y}}=(y_{1},y_{2},...,y_{2})$. We are interested in finding
how the features are related to each other and how it affects the result. The
correct and detailed explanation can help the model to perform better and
reduce the loss in future predictions. We derive a lightweight explanation
model that works with the sample of the original data. We create a suitable
environment for our XAI model where the accuracy of the lightweight model is
almost the same as the original model. We achieve the model output accuracy by
equating the projection and the embedding distance[37]. This strategy helps to
create the twin environment by securing the same positions of input data
distributions in both high-dimensional spaces. As the environments are similar
for the original and lightweight models, the explainability of the lightweight
model is good enough to explain the original model. The reason for using a
lightweight model is that a higher dimension of input data with bigger feature
space can make ExCIR more complicated to apply in real-life practice. Also, we
calculate mutual information and uncertainty between features and the output
to explain the impact of the dependent feature because lower dimensional input
space provides lesser bias in computing the uncertainty associated with the
features. The main difference between our ExCIR model with the surrogate
model[39] is, the ExCIR model is a copy of the original model which works with
the same data set and chooses multiple numbers of combinations of features to
calculate feature importance. On the other hand, our model is equivalent to
the original model but works with less data and considers all features for the
calculation of the feature impact ratio. In ExCIR, we do not need to consider
different combinations of feature sets, whereas we calculate Shannon entropy
considering the uncertainty of the features, and based on that we introduce
Conditional Multivariate Mutual Information CMMI. The main reason for
achieving the same environment is,

  1. 1.

Both the lightweight model and the original model’s environments are
equivalent, so we are securing the accuracy guarantee at first.

  2. 2.

We apply our new XAI approach to the lightweight model and as the lightweight
model works with a lower-dimension environment, it is more flexible to
calculate complex metrics like CMMI for each feature.

  3. 3.

As we prove that both models have the same environment (see next section), we
can claim the impact of the feature in the lightweight model will be the same
as the original one, i.e., the feature’s contributions will be the same for
both models.

Let $M^{\prime}$ denote the lightweight model, trained over the lower
dimensional data set having the same number of features $k$ as
${\underset{\sim}{f_{1}}},{\underset{\sim}{f_{2}}},...,{\underset{\sim}{f_{k}}};\
{\underset{\sim}{f_{i}}}\epsilon||F||^{k\times n^{\prime}}$ and
${\underset{\sim}{f_{i}}}=(f_{i1},f_{i2},.....,f_{in^{\prime}})$, where
$i=1:k$ and $n^{\prime};n^{\prime}<n$ is the number of rows in the data set.
Let $Y^{\prime}$ denote the output variable.
$||F||^{(k\times n)}$ is an $(k\times n)$ dimensional feature space that
contains each feature’s distributions. A feature distribution refers to a
distribution that is followed by all the data points of specific features. So
we can consider unified $([k+1]\times n)$ dimensional super space
$\mathcal{U}$ such that $||F||^{(k\times n)}\subset\mathcal{U}$. $\mathcal{U}$
contains all feature distributions as well as target output distribution. We
assume that $(k+1)$th distribution is the output distribution. Let,
$\mathcal{D}_{i};\ i=1:[k+1]$ denote the distribution of the $i^{th}$ feature,
and $\mathcal{D}(Y)$ is the output distribution of the original model. Then we
can have,
$\mathcal{U}=[\mathcal{D}_{1}({\underset{\sim}{f_{1}}})\cup\mathcal{D}_{2}({\underset{\sim}{f_{2}}})\cup...\cup\mathcal{D}_{k}({\underset{\sim}{f_{k}}})\cup\mathcal{D}(Y))]$.
On the other hand, $\mathcal{D}(Y^{\prime})$ is the output distribution from
the lightweight explainable model. Without loss of generality,
$\mathcal{U^{\prime}}$ is the superspace for the lightweight model where
$\mathcal{U^{\prime}}\subset\mathcal{U}$ and we can have,
$\mathcal{U^{\prime}}=[\mathcal{D}_{1}({\underset{\sim}{f_{1}}})\cup\mathcal{D}_{2}({\underset{\sim}{f_{2}}})\cup...\cup\mathcal{D}_{k}({\underset{\sim}{f_{k}}})\cup\mathcal{D}(Y^{\prime}))].$
Our main idea is to find the relation among features and how it affects the
output. These relations can be non-linear and dependent. ExCIR will work
directly with the non-linear model for dependent as well as independent
features.

##  3 Accuracy of ExCIR

To maintain the lightweight model’s accuracy, the environment of the
lightweight model must be almost the same as the original model. Because
within the same input-output environment both the models should behave in the
same manner. Here, the environment refers to the features’ distributions and
their positions in the superspace. Every feature has some impact on generating
the output. Keeping this in mind, we equate the distance between each feature
and output distribution for both spaces $\mathcal{U}$ and
$\mathcal{U^{\prime}}$. If the distance of the same feature to the output is
the same in both spaces, we can claim that the positions of features in both
spaces are similar. So, $\mathcal{U^{\prime}}$ will create a twin environment
of $\mathcal{U}$ with a lower dimension. Once we secure the feature and output
distribution position, in the next step, we use projection and embedding
distance [37] through f-divergence so that the lightweight model output
distribution becomes a mirror image of the original model output. More
specifically, we can claim that the lightweight model environment is the same
as the original model when

  1. (i)

The distances between features and output distributions are the same in both
spaces.

  2. (ii)

Output distributions in both spaces are mirror images of each other.

Let
${\underset{\sim}{\mathcal{F}}}=(\mathcal{F}_{1},\mathcal{F}_{2},....,\mathcal{F}_{n})^{\prime}$
denote the input column vector, where $\mathcal{F}_{j};j=1:n$ is the $jth$
input containing all $k$ features. That means,
$\mathcal{F}_{j}=[f_{1j},f_{2j},.....,f_{kj}]$; $j=1:n$. The output vector is
$Y=(y_{1},y_{2},....,y_{n})^{\prime}$. Now, each of $y_{i}$ s is explained by
the $k$ features of the $jth$ input vector. that means, $y_{j}$’s are
explained by $\mathcal{F}_{j}$s. Now if we consider a $k$ dimensional input
space, then by the definition of Euclidean distance [38] we define the local
distance for a single output $y_{i}$ as follows:

$D^{2}_{ij}=\sum_{j=1}^{k}(y_{i}-f_{ji})^{2}$ (1)

The average $k$ dimensional distance between $Y$ and
${\underset{\sim}{\mathcal{F}}}$ can provide the exact position of the
distribution of $Y$ in a super space. It will help us build our lightweight
model environment to produce a highly accurate output space with similar
influences of features as the original data. So to calculate the distance
between $Y$ and ${\underset{\sim}{\mathcal{F}}}$, we have to consider all
local distances. The final average distance for the original and sample space
can be defined as respectively:

$D^{2}_{\text{final}}=\frac{1}{n}\sum_{j=1}^{k}\sum_{i=1}^{n}(y_{i}-f_{ji})^{2},\
\text{and}\ D^{\prime
2}_{\text{final}}=\frac{1}{n^{\prime}}\sum_{j=1}^{k}\sum_{i=1}^{n^{\prime}}(y^{\prime}_{i}-f_{ji})^{2}$
(2)

The idea is to equalize two distances so that the output distribution for two
data sets has the same distance from their feature i.e., exact same positions
in the high dimensional space. We will have to achieve the same output
environment in lightweight space so that we can get the exact accurate
relation impact of each feature on the output vector. To create such a
situation we have to minimize the difference between two distance measurements
which means $|D^{2}_{\text{final}}-D^{\prime 2}_{\text{final}}|\rightarrow 0$.
We define the following condition for achieving an equivalent environment for
the lightweight explainable model:

Definition 1: For an ML model, let ${\underset{\sim}{\mathcal{F}}}$ be the
input vector having $k$ number of features, $Y$ be the output vector and for a
lightweight model, ${\underset{\sim}{\mathcal{F}^{\prime}}}$ be the input
vector having the same $k$ features and $Y^{\prime}$ be the output vector. In
a multidimensional space, the average distances from the feature space to the
output vector for the original ML model and for the lightweight model are
denoted as $D^{2}_{\text{final}}$, and $D^{\prime 2}_{\text{final}}$
respectively. Then we can claim that both the spaces’ features and output
distribution have similar positions with the exact same amount of relation
impact of each feature if and only if :

$\lim_{y\rightarrow y^{\prime}}|D^{2}_{\text{final}}-D^{\prime
2}_{\text{final}}|=0$ (3)

More fundamentally, when two spaces have the equivalent environment and the
output distributions are in similar positions i.e., the average distance from
one local output to the corresponding feature input is approximately the same
for both spaces and given the fact that two distributions are exactly mirror
image of each other (proved later in this section), we can claim that the
impact of the $ith$ feature on the output vector is exactly the same in both
multi-dimensional spaces. So, if we consider the lightweight space to
calculate the relation impact of each feature for generating the report, it
will be able to accurately explain the original model without loss of quality
in the result. So, in the next step, we will minimize the distance between the
original and the lightweight model output distribution in different dimensions
to achieve the mirror effect.
In ExCIR, after passing the sample dataset to the lightweight model, the model
will generate an initial output, defined as $Y^{\prime}$. Then we will
minimize the loss function $\mathcal{L}(Y,Y^{\prime})$ with respect to the
distance from output distribution to feature distributions. Now the
lightweight model super space $\mathcal{U^{\prime}}$ and original model super
space $\mathcal{U}$ can be of different dimensions, as we are working with the
low dimensional sample data set. In this case, we need to consider calculating
the output distribution distance which belongs to a different dimension space
that makes it challenging to compute distances. At first, we will use two
different approaches Projection distance and Embedded distance [37] to measure
the distance of output distributions of the original and lightweight model.

let, $m(\Omega)$ denotes all sets of borel probability measures where $\Omega\
\epsilon\ \mathbb{R}^{n}$, and let $m^{p}(\Omega)\subseteq m(\Omega)$ refer
those finite $pth$ moment , where $p\ \epsilon\ \mathbb{N}$. Then for any
$n^{\prime},n\ \epsilon\ \mathbb{N}$ and $n^{\prime}\ \leq\ n$, we can write

$O(n^{\prime},n)=\\{P\epsilon\mathbb{R}^{n^{\prime}\times
n}:PP^{T}=I_{n^{\prime}}\\}$ (4)

the $n^{\prime}\times n$ , metrics consists of orthonormal rows and we can
write $O(n)=O(n,n)$ for the orthonormal group. $P^{T}=$ Transpose of $P$ where
for any $P\ \epsilon\ O(n^{\prime},n)$ and $b\ \epsilon\
\mathbb{R}^{n^{\prime}}$,

$\Phi_{P,b}:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n^{\prime}},\Phi_{P,b(x)}=Px+b;$
(5)

and for any $\mu\ \epsilon\ m(\mathbb{R}^{n})$,
$\Phi_{P,b(\mu)}=\mu\circ\Phi_{P,b}^{-1}$ is the pushforward measure where
$\Phi_{P}=\Phi_{P,0}$ if $b=0$. More specifically, for any measurable map
$\Phi:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n^{\prime}}$,
$\Phi(\mu)=\mu\circ\Phi$ is the pushforwad measure. Here for our method,
without loss of generality, we assume that $\mu=\mathcal{D}({Y})$ is the
lightweight model output probability measure and $\delta=\mathcal{D}(Y)$ is
the original model output probability measure. We define a distance
$d(\mu,\delta)$ where, $\mu\ \epsilon\ m(\Omega_{1})$ and $\delta\ \epsilon\
m(\Omega_{2})$; $\Omega_{1}\subseteq\mathbb{R}^{n^{\prime}}$, and
$\Omega_{2}\subseteq\mathbb{R}^{n}$. Here, $n,n^{\prime}\ \epsilon\
\mathbb{N}$ and $n^{\prime}\leq n$. We consider here the notion of
f-divergence to measure distance function. According to [37] any method of
f-divergence (kl divergence [42], Jensen- Shanon divergence [41] etc ) will
satisfy the algorithm. Without loss of generality, we can consider,

$\Omega_{1}=\mathbb{R}^{n^{\prime}},\Omega_{2}=\mathbb{R}^{n}$ (6)

Now we are interested in calculating the distance with projection and
embedding measures [37].
Definition 2:  let $n^{\prime},n\ \epsilon\ \mathbb{N}$, $n^{\prime}\leq n$.
For any $\mu\ \epsilon\ m(\mathbb{R}^{n^{\prime}})$ and $\delta\ \epsilon\
m(\mathbb{R}^{n})$, the embedding of $\mu$ into $\mathbb{R}^{n}$ are the set
of $n$-dimensional measures

$d^{+}(\mu,n)=\\{\alpha\ \epsilon m(\mathbb{R}^{n}):\Phi_{P,b}(\alpha)=\mu\
\text{for some}\ P\epsilon O(n^{\prime},n),b\ \epsilon\
\mathbb{R}^{n^{\prime}}\\};$ (7)

and,
Defination 3: let $n^{\prime},n\ \epsilon\ \mathbb{N}$, $n^{\prime}\leq n$.
For any $\mu\epsilon\ m(\mathbb{R}^{n^{\prime}})$ and $\delta\epsilon
m(\mathbb{R}^{n})$. The projection of $\delta$ onto $\mathbb{R^{n^{\prime}}}$
are the $n^{\prime}$\- dimensional measures,

$\vspace{-1mm}d^{-}(\delta,n^{\prime})=\\{\beta\ \epsilon\
m(\mathbb{R}^{n^{\prime}}):\Phi_{P,b}(\beta)=\delta\ \text{for some }\ P\
\epsilon\ O(n^{\prime},n),b\ \epsilon\ \mathbb{R}^{n^{\prime}}\\}$ (8)

Let $d$ be any notion of distance on $m(\mathbb{R}^{n})$ for any $n\ \epsilon\
\mathbb{N}$. Then the projection distance will be :

$\vspace{-2mm}d^{-}(\mu,\delta)=\operatornamewithlimits{inf\vphantom{p}}_{\beta\in
d^{+}(\delta,n^{\prime})}\text{d}(\mu,\beta)$ (9)

and the embedding distance will be,

$d^{+}(\mu,\delta)=\operatornamewithlimits{inf\vphantom{p}}_{\alpha\in
d^{+}(\mu,n)}\text{d}(\delta,\alpha)$ (10)

Both $d^{-}(\mu,\delta)$ and $d^{+}(\mu,\delta)$ calculate the distance
between two probability measures $\mu$ and $\delta$ of different dimensions.
In [37] it is shown that if $d$ is an f-divergence, then
$d^{-}(\mu,\delta)=d^{+}(\mu,\delta)=\hat{d}(\mu,\delta)$. The authors in [37]
also generalized the theorem and proved that
$d^{-}(\mu,\delta)=d^{+}(\mu,\delta)=\hat{d}(\mu,\delta)=0$ if and only if
$\Phi_{P,b}(\delta)=\mu$ for some $P\ \epsilon\ O(n^{\prime},n)$, and $b\
\epsilon\ \mathbb{R}^{n^{\prime}}$. This means if the expected distance
$\hat{d}(\mu,\delta)$ can be minimized to $0$, then the necessary and
sufficient condition for claiming that two probability measures $\mu$ and
$\delta$ are rotated and translated copies of each other, modulo embedding in
a higher-dimensional ambient space where $n^{\prime}\neq n$. Here, our loss
function is the expected distance and the goal is to minimize the expected
distance between the two output distributions $\mu=\mathcal{D}(Y^{\prime})$
and $\delta=\mathcal{D}(Y)$ as close as $0$ through a risk generator. This
will help to boost the accuracy of the model. So, we can define the loss
function as :

$\mathcal{L}(Y,Y^{\prime})=E_{\mu\epsilon m(\mathbb{R}^{n}),\delta\epsilon
m(\mathbb{R}^{n^{\prime}})}(\hat{d}(\mu,\delta))=E_{Y\epsilon\mathcal{U},Y^{\prime}\epsilon\mathcal{U^{\prime}}}(\hat{d}(\mathcal{D}(Y),\mathcal{D}(Y^{\prime})))$
(11)
$\mathcal{L}(Y,Y^{\prime})=E[\hat{d}(\mathcal{D}(Y),\mathcal{D}(Y^{\prime}))|Y\epsilon\mathcal{U},Y^{\prime}\epsilon\mathcal{U^{\prime}}]$
(12)

To minimize the loss function, we introduce a risk generator function, say
$\mathcal{R}(d*)\epsilon\mathcal{H}$ which will give us the desired result for
higher accuracy. Here, $\mathcal{H}$ is a hypothesis class that contains all
possible risk generators. We will choose the final risk generator
$\mathcal{R}(d*)$ when
$\hat{d}(\mathcal{D}(Y),\mathcal{D}(Y^{\prime}))\rightarrow 0$. We can define
$\mathcal{R}(d*)$ as:

$\mathcal{R}(d*)=\operatornamewithlimits{argmin\vphantom{p}}_{\mathcal{R}(d)in\mathcal{H},\hat{d}\rightarrow
0}\int_{\mathcal{D}(Y)\epsilon\mathcal{U}}\int_{\mathcal{D}(Y^{\prime})\epsilon\mathcal{U^{\prime}}}E[\mathcal{L}(Y,Y^{\prime})]+\lambda(.)$
(13)

Here $\lambda$ is the regularized parameter. Clearly from the equation, we can
see when $E[\mathcal{L}(Y,Y^{\prime})]=\mathcal{L}(Y,Y^{\prime})\rightarrow 0$
, then $\hat{d}\rightarrow 0$ implies $Y\rightarrow Y^{\prime}$. So, it will
achieve the necessary and sufficient condition where two probability measures
$\mathcal{D}(Y)$ and $\mathcal{D}(Y^{\prime})$ is rotated and translated
copies of each other, and $n^{\prime}\neq n$. As a result, the accuracy of the
output is the same as the original model.

##  4 Explainability of ExCIR

Figure 2: The graph reflects the behaviour of the model in equation 14
considering the feature’s uncertainty to contribute to output generation and
the table indicates the presence and absence of the feature for generating
output, green represents the presence, and red represents absence.

We use CIR to measure the feature relation impact on the final output and show
how it works in two different environments regarding independent and dependent
features. To check the relation between a feature
${\underset{\sim}{f_{1}}},{\underset{\sim}{f_{2}}},...,{\underset{\sim}{f_{k}}}$
and output (linear or nonlinear) we need to hold one parameter constant and
also hold input features $({\underset{\sim}{f_{1}}})$ constant. Then we vary
the remaining parameters and watch how ${\underset{\sim}{Y^{\prime}}}$
changes. If the change in ${\underset{\sim}{Y^{\prime}}}$ is non-linear with
the change in the parameter which is varied, and if this is true for all the
parameters in the model, then that model is said to be non-linear."Non-Linear"
describes the model, not the graph of
$({\underset{\sim}{f_{1}}}...{\underset{\sim}{f_{k}}})$ vs.
${\underset{\sim}{Y^{\prime}}}$.

An example of non-linear regression would be something like this:

$M(f_{(f,\beta)})=\frac{\beta_{1}f_{1}+\beta_{2}f_{2}}{\beta_{3}f_{3}+\beta_{4}f_{4}+.......+\beta_{n}f_{n}}$
(14)

The equation 14, may behave like a linear model depending on parameter
choosing in the general case. In our work, as we consider the uncertainty
associated with the feature’s contribution i.e., a feature may or may not
contribute to generating an output, equation 14 should behave like a nonlinear
model. For a set of empirical outputs, figure 2 demonstrates the curve of how
the model in equation 14 would behave considering the feature’s uncertainty.
For our model, we also assume that at least two features (one is conversely
and another is directly related to the output) are present.

###  4.1 Partial correlation Impact ratio for independent features

We have a data set with $n^{\prime}$ rows and $k$ features, denoted as
${\underset{\sim}{f_{1}}},{\underset{\sim}{f_{2}}},...,{\underset{\sim}{f_{k}}};{\underset{\sim}{f_{i}}}\epsilon||F||^{k\times
n^{\prime}}$. The machine learning model $M^{\prime}(f)$ learns all data
history and predicts the output vector
${\underset{\sim}{Y}}^{\prime}=(y^{\prime}_{1},y^{\prime}_{2},...,y^{\prime}_{n^{\prime}})$.
Changing the notation for more easy understanding and without loss of
generality, let $f_{ij}$ denote the $jth$ observation of the $ith$ feature;
where $i=1:k$ and $j=1:n^{\prime}$ and let $y^{\prime}_{j}$ denote the jth
observation of the output vector ${\underset{\sim}{Y^{\prime}}}$;
$j=1:n^{\prime}$. then the mean of the feature $f_{i};i=1:k$, the mean of the
output vector $Y$ will be, and the joint weighted mean of feature $f_{i}$ ;
$i=1:k$ and $Y$ will be respectively,

$\hat{f_{i}}=\frac{\sum_{j}f_{ij}}{n^{\prime}},\
\hat{y^{\prime}}=\frac{\sum_{j}y_{j}}{n^{\prime}},\ \text{and}\
\hat{f_{i}y^{\prime}}=\frac{\hat{f_{i}}+\hat{y^{\prime}}}{2}$ (15)

To measure the impact of each feature on the output, we calculate the relation
impact for each feature separately. Then we define the PCIR of feature
${\underset{\sim}{f_{i}}}$ on the output vector
${\underset{\sim}{Y^{\prime}}}$ by a ratio $\eta_{f_{i}}={\beta_{f_{i}}}^{2}$
which is formulated as:

$\eta_{f_{i}}=\frac{n^{\prime}[(\hat{f_{i}}-\hat{f_{i}y^{\prime}})^{2}+(\hat{y^{\prime}}-\hat{f_{i}y^{\prime}})^{2}]}{\sum_{j}(f_{ij}-\hat{f_{i}y^{\prime}})^{2}+\sum_{j}(y^{\prime}_{j}-\hat{f_{i}y^{\prime}})^{2}}$
(16)

i.e., the weighted variance of the $ith$ feature and output variable with
respect to joint mean $\hat{f_{i}y^{\prime}}$ divided by the variance of all
values with respect to joint mean $\hat{f_{i}y^{\prime}}$. This correlation
impact ratio $\eta_{f_{i}}$; $i=1:k$ takes values between $0$ and $1$.

For $\eta_{f_{i}}=1$, the dispersion is the same for both the output and
features. That means a small change in feature observation leads to changes in
output observation. So, we can say the $ith$ feature has a great impact on
output. Whereas, $\eta_{f_{i}}=0$ refers to the case when $ith$ features have
almost no impact on the output. We divide the features into two groups, the
first group is directly related to output; i.e., changes in the same
direction. On the other hand, another group of features is conversely related
to the output; i.e., (changes in a different direction). For the sake of
simplicity, we take
$({\underset{\sim}{f_{1}}},{\underset{\sim}{f_{2}}},....,{\underset{\sim}{f_{m}}});0<m\leq
k$ directly related to output and belongs to numerator and
${\underset{\sim}{(f_{(m+1)}}},{\underset{\sim}{f_{(m+2)}}},....,{\underset{\sim}{f_{k})}}$
conversely related to output and belongs to denominator. Then we can write
ExCIR model as:

$M^{\prime}(f_{(f,\beta^{2})})=\frac{\beta_{f_{1}}^{2}{\underset{\sim}{f_{1}}}+\beta_{f_{2}}^{2}{\underset{\sim}{f_{2}}}+......+\beta_{f_{m}}^{2}{\underset{\sim}{f_{m}}}}{\beta_{f_{m+1}}^{2}{\underset{\sim}{f_{m+1}}}+\beta_{f_{m+2}}^{2}{\underset{\sim}{f_{m+2}}}+.......+\beta_{f_{k}}^{2}{\underset{\sim}{f_{k}}}}$
(17)

where $0<m\leq k$ and we can modify it to the following form :

$M^{\prime}(f_{(f,\eta)})=\frac{\eta_{f_{1}}{\underset{\sim}{f_{1}}}+\eta_{f_{2}}{\underset{\sim}{f_{2}}}+......+\eta_{f_{m}}{\underset{\sim}{f_{m}}}}{\eta_{f_{m+1}}{\underset{\sim}{f_{m+1}}}+\eta_{f_{m+2}}{\underset{\sim}{f_{m+2}}}+.......+\eta_{f_{k}}{\underset{\sim}{f_{k}}}}$
(18)

For the $jth;j=1:n$ observations of features $f_{1j},f_{2j},.....,f_{kj}$ ,
the local output $y_{j}$ can be expressed as :

$y^{\prime}_{j}=M^{\prime}(f_{j(f,\eta)})=\frac{\eta_{f_{1}}f_{1j}+\eta_{f_{2}}f_{2j}+......+\eta_{f_{m}}f_{mj}}{\eta_{f_{m+1}}f_{m+1,j}+\eta_{f_{m+2}}f_{m+2,j}+.......+\eta_{f_{k}}f_{kj}}$
(19)

Here , $f_{ij};i=1:k,j=1:n$ is the binary variable. $f_{ij}=1$, refers to the
presence of the $ith$ feature in the $jth$ entry. If we can prove that the
changes in the output by changing a single input in the $ith$ feature are
actually dependent on its corresponding correlation ratio, i.e.,
$\eta_{f_{i}}$, we can claim that our proposed correlation ratio can reassure
the feature importance.

Theorem 1: The changes in a particular feature input can affect the changes in
local output. So the changes must either depend on the corresponding
correlation ratio of that feature or is constant, given that all the features
are independent of each other. that is:

$\displaystyle\frac{dy}{d{\underset{\sim}{f_{j}}}}=c_{1}.\eta_{f_{j}};\text{
when $j$ < $m$},\text{, and \
}\frac{dy}{d{\underset{\sim}{f_{j}}}}=\frac{c_{2}}{(2K_{2}-\eta_{f_{j}}^{2})};\text{when
$j$ $\geq$ $m$}$ (20)

Here, $c_{1},c_{2},$ and $k_{2}$ are constants. proof is in supplementary
subsection 6.2.

Corollary 1:  When the particular feature is positively related to the output,
the expected change in the output due to the change in the input feature is
directly proportional to its correlation ratio impact. On the other hand, if
the particular feature is negatively related to the output, the expected
change in output due to the change in input of that feature is inversely
proportional to its correlation ratio impact. We can express it mathematically
as follows:
$E(\frac{dy^{\prime}}{d{\underset{\sim}{f_{j}}}})\varpropto\eta_{f_{j}};$if
$j\leq m$ , and
$E(\frac{dy^{\prime}}{d{\underset{\sim}{f_{j}}}})\varpropto\frac{1}{\eta_{f_{j}}}$
if $j>m$

###  4.2 Mutual Correlation Impact Ratio for dependent features

In section 4.1, we introduced the correlation impact ratio for the nonlinear
model assuming that the features are independent of each other. Though the
proposed PCIR is easy to compute and able to keep the balance between
interpretability and accuracy, PCIR is only suitable for the environment where
the features are independent. So, the current section is only dedicated to the
interpretability theorem when the features are dependent. For the multivariate
dependent feature space, it is challenging to compute the relation impact of a
targeted feature on output due to the influence of other features. To address
the issue, we propose a new metric called Mutual Correlation Impact Ratio
(MCIR). The state-of-the Conditional Mutual Information CMI [45] considers the
situation where more than one feature depends on the same feature. But, in our
case, every feature is dependent on multiple features and for this reason, we
can not use the existing CMI concept. We also propose a new concept called
Conditional Multivariate Mutual Information (CMMI) considering the case when
every feature is dependent on other features. MCIR is calculated based on
(CMMI) assuming that the given features
$({\underset{\sim}{f_{1}}},{\underset{\sim}{f_{2}}},...,{\underset{\sim}{f_{k}}}\epsilon||F||^{(k\times
n^{\prime})})$ follow either multivariate probability density function (pdf)
or multivariate probability mass function(pmf); features can be both discrete
and continuous. Our proposed metric CMMI represents the mutual dependency
among the targeted feature and the output variable considering the targeted
feature is dependent on the rest of the features belonging to a
multidimensional feature space. When two non-identical conditional probability
distributions are taken, there should be a divergence between their cross-
entropy and their individual entropies [43]. In a multivariate environment,
this divergence can be called Joint Mutual Information (JMI). MCIR is
calculated based on (JMI), and CMMI. MCIR measures how the changes in the
targeted feature’s input affect the output of the model. It also considers the
measurement of uncertainty associated with the feature’s contributions. At
first, for the sake of simplicity, we consider ${\underset{\sim}{f_{i}}}$
depends on ${\underset{\sim}{f_{j}}}$, while ${\underset{\sim}{f_{i}}}$ is
independent of the rest of the features; $i=1(1)k,j=1(1)k$, and $i\neq j$.
Then, we find the impact of ${\underset{\sim}{f_{i}}}$ on
${\underset{\sim}{Y^{\prime}}}$, given the fact that
${\underset{\sim}{f_{i}}}$ depends on ${\underset{\sim}{f_{j}}}$. This impact
can be explained by the information theory, if and only if we can compute
$I({\underset{\sim}{Y^{\prime}}};{\underset{\sim}{f_{i}}}|{\underset{\sim}{f_{j}}});\forall
i,j=1(1)k,i\neq j$. The previously described MI can not provide the desired
result. To achieve our goal we have to first calculate the Conditional Mutual
Impact [44] [45].

The conditional mutual information between the output variable
${\underset{\sim}{Y^{\prime}}}$ and the target feature
$({\underset{\sim}{f_{i}}}|{\underset{\sim}{f_{j}}})$, $\forall
i,j=1(1)k;i\neq j$ is defined as [44]:

$\displaystyle
I({\underset{\sim}{Y^{\prime}}};{\underset{\sim}{f_{i}}}|{\underset{\sim}{f_{j}}})=I({\underset{\sim}{f_{i}}},{\underset{\sim}{f_{j}}})-I({\underset{\sim}{Y^{\prime}}},{\underset{\sim}{f_{i}}}|{\underset{\sim}{f_{j}}})$
(21) $\displaystyle=\sum_{[f_{j}=\mathrm{f};f_{j}\epsilon||F||^{k\times
n^{\prime}}]}\sum_{[f_{i}=\mathrm{f^{*}};f_{j}\epsilon||F||^{k\times
n^{\prime}}]}\sum_{Y^{\prime}=y;Y^{\prime}\epsilon||Y^{\prime}||^{n^{\prime}}}P(\mathrm{f^{*}},\mathrm{f},y)\log_{2}[\frac{P(y,\mathrm{f^{*}}|\mathrm{f})}{P(y|\mathrm{f})P(\mathrm{f^{*}}|\mathrm{f})}]$

If any of the features $f_{i},f_{j}$, or $Y^{\prime}$ is continuous, the
summation operator can be replaced by the integral operator.

####  4.2.1  MCIR; with two dependent features

When any two of the $k$ features are dependent on each other and other
features are independent, the state-of-the-art CMI is sufficient to explain
the mutual dependency of $Y^{\prime}$ and $(f_{i}|f_{j})$. But the value of
CMI varies from $0$ to $\infty$ which is an open bound, thus making
scalability a major challenge. So, to scale it down in between $[0,1]$, we
derive MCIR and as,

$\displaystyle
C({\underset{\sim}{Y^{\prime}}};{\underset{\sim}{f_{i}}}|{\underset{\sim}{f_{j}}})$
$\displaystyle=\frac{I({\underset{\sim}{Y^{\prime}}};{\underset{\sim}{f_{i}}}|{\underset{\sim}{f_{j}}})}{I({\underset{\sim}{Y^{\prime}}};{\underset{\sim}{f_{i}}}|{\underset{\sim}{f_{j}}})+I({\underset{\sim}{Y^{\prime}}},{\underset{\sim}{f_{i}}},{\underset{\sim}{f_{j}}})}$
(22)

where, $0\leq
I({\underset{\sim}{Y^{\prime}}};{\underset{\sim}{f_{i}}}|{\underset{\sim}{f_{j}}})\leq\infty$
and $0\leq
I({\underset{\sim}{Y^{\prime}}},{\underset{\sim}{f_{i}}},{\underset{\sim}{f_{j}}})\leq\infty$.
Then,
$0\leq\frac{I({\underset{\sim}{Y^{\prime}}};{\underset{\sim}{f_{i}}}|{\underset{\sim}{f_{j}}})}{I({\underset{\sim}{Y^{\prime}}};{\underset{\sim}{f_{i}}}|{\underset{\sim}{f_{j}}})+I({\underset{\sim}{Y^{\prime}}},{\underset{\sim}{f_{i}}},{\underset{\sim}{f_{j}}})}\leq
1$. So we can claim, $0\leq
C({\underset{\sim}{Y^{\prime}}};{\underset{\sim}{f_{i}}}|{\underset{\sim}{f_{j}}})\leq
1$.
$C({\underset{\sim}{Y^{\prime}}};{\underset{\sim}{f_{i}}}|{\underset{\sim}{f_{j}}})$
is considered as CMI of the target feature ${\underset{\sim}{f_{i}}}$ on
output variable ${\underset{\sim}{Y^{\prime}}}$ when
${\underset{\sim}{f_{i}}}$ depends on ${\underset{\sim}{f_{j}}}$.
$C({\underset{\sim}{Y^{\prime}}};{\underset{\sim}{f_{i}}}|{\underset{\sim}{f_{j}}})$
can capture how the changes in inputs of target variable
${\underset{\sim}{f_{i}}}$ can affect the output variable
${\underset{\sim}{Y^{\prime}}}$ considering the fact that the input of
${\underset{\sim}{f_{j}}}$ may or may not be changed.
$I({\underset{\sim}{Y^{\prime}}},{\underset{\sim}{f_{i}}},{\underset{\sim}{f_{j}}})$
is called Joint Mutual Information $(\mbox{JMI})$. For simplifying the
notations, we consider a simple environment where ${\underset{\sim}{f_{1}}}$
and ${\underset{\sim}{f_{2}}}$ are dependent on each other and
${\underset{\sim}{f_{3}}},{\underset{\sim}{f_{4}}},....,{\underset{\sim}{f_{k}}}$
are independent features. We also assume that,
$({\underset{\sim}{f_{1}}},{\underset{\sim}{f_{3}}},....,{\underset{\sim}{f_{m}}})$
are directly related to output variable ${\underset{\sim}{Y}}$ and
$({\underset{\sim}{f_{2}}},{\underset{\sim}{f_{p}}},....{\underset{\sim}{f_{k}}})$
are conversely related to ${\underset{\sim}{Y^{\prime}}}$; $m,p\leq k$. Then
using 22 on equation (14), our proposed ExCIR model takes the form:

$E({\underset{\sim}{Y^{\prime}}})=M^{\prime}(f_{j(f,C)}))=\frac{C({\underset{\sim}{Y^{\prime}}};{\underset{\sim}{f_{1}}}|{\underset{\sim}{f_{2}}})f_{1}+\eta_{f_{3}}f_{3}+......+\eta_{f_{m}}f_{m}}{C({\underset{\sim}{Y^{\prime}}};{\underset{\sim}{f_{2}}}|{\underset{\sim}{f_{1}}})f_{2}+\eta_{f_{p}}f_{p}+.......+\eta_{f_{k}}f_{k}}$
(23)

where $\eta_{f_{i}};i=3,...,m,....,p,...,k;m,p\leq k$ is the correlation
impact for the independent feature
${\underset{\sim}{f_{i}}},i=3,...,m,...,p,...,k;m,p\leq k$.
$I({\underset{\sim}{Y^{\prime}}};{\underset{\sim}{f_{1}}}|{\underset{\sim}{f_{2}}})$
and
$I({\underset{\sim}{Y^{\prime}}};{\underset{\sim}{f_{2}}}|{\underset{\sim}{f_{1}}})$
can be called MCIRs for the dependent features $f_{1}$ and $f_{2}$
respectively. But, this theory is only based on the fact that any two of the
features are dependent on each other. However, in real-life cases, a feature
can be dependent on multiple other features. In that case, one has to consider
multivariate distribution considering multiple cases of dependency. Keeping
this in mind, we propose a new concept for CMMI in the next section.

####  4.2.2 Mutual Correlation Impact Ratio; when all features are dependent

In ExCIR, it is necessary to calculate the mutual impact of a feature on the
output variable while assuming the target feature is dependent on other
features. But before we define MCIR for multivariate cases, we have to derive
CMMI. In [45] the authors derived CMI for a multivariate environment where the
features are dependent on another variable. i.e., they considered the case
when all variables are dependent on one common variable. But in our work, we
address the case when all the features are dependent on each other. So, if we
want to calculate the mutual dependence between a targeted feature and the
output variable we have to calculate the CMMI given that target feature is
dependent on multiple features. More specifically, in existing works
[44][45][46], the notion of
$I({\underset{\sim}{Y^{\prime}}};{\underset{\sim}{f_{1}}},{\underset{\sim}{f_{2}}},...,{\underset{\sim}{f_{k-1}}}|{\underset{\sim}{f_{k})}}$
is derived and used in many real-life cases. However, this approach cannot be
directly applied in our environment. We therefore introduce a new matrix
$I({\underset{\sim}{Y^{\prime}}};{\underset{\sim}{f_{i}}}|\\{{\underset{\sim}{f_{j}}}\\}\subseteq\\{||F||^{k\times
n^{\prime}}-{\underset{\sim}{f_{i}}}\\};i\neq j)$].
Definition 4: CMMI:  If an environment has dataset including $k$ number of
dependent features
${\underset{\sim}{f_{1}}},{\underset{\sim}{f_{2}}},...,{\underset{\sim}{f_{k}}}\epsilon||F||^{k\times
n^{\prime}}$; $f_{i}=(f_{i1},f_{i2},...,f_{in})$, $i=1(1)k$ with the output
variable
${\underset{\sim}{Y^{\prime}}}=(y^{\prime}_{1},y^{\prime}_{2},....y^{\prime}_{n^{\prime}})$,
then the CMMI between output variable $Y$ and any targeted feature $f_{i}$
will be:

$\displaystyle
I({\underset{\sim}{Y^{\prime}}};{\underset{\sim}{f_{i}}}|\\{{\underset{\sim}{f_{j}}}\\}\subseteq\\{||F||^{k\times
n^{\prime}}-f_{i}\\};i\neq j))$ (24)
$\displaystyle=\sum_{\phi}\int_{\psi}\int_{f_{i}}\sum_{Y^{\prime}}P(Y^{\prime},f_{1},..,f_{i},..f_{k})\log_{2}[\frac{P(Y^{\prime}|\\{f_{j}\\}\subseteq||F||^{k\times
n^{\prime}})}{P(Y^{\prime}|\\{{\underset{\sim}{f_{j}}}\\}\subseteq\\{||F||^{k\times
n^{\prime}}-f_{i}\\};i\neq j)}]$

where $\phi=f_{j}\subseteq\\{||F||^{k\times n^{\prime}}-f_{i}\\}_{d}$ is the
set of discrete features, and $\psi=f_{j}\subseteq\\{||F||^{k\times
n^{\prime}}-f_{i}\\}_{c}$ \- the set of continuous features not including
$f_{i}$. The derivation of the equation 24 is based on the concept of MI[46]
and it can be found in the supplementary subsection 6.3. CMMI could take the
values between $0$ to $\infty$, unlike the strict bound of $[0,1]$ that the
normal correlation coefficient has.

Definition 5:  Mutual Correlation Impact Ratio:  Suppose an environment with a
dataset has $k$ number of features
${\underset{\sim}{f_{1}}},{\underset{\sim}{f_{2}}},...,{\underset{\sim}{f_{k}}}\epsilon||F||^{k\times
n^{\prime}}$ ; ${\underset{\sim}{f_{i}}}=(f_{i1},f_{i2},...,f_{in^{\prime}})$,
$i=1(1)k$ and the features are dependent on each other. The output variable is
${\underset{\sim}{Y^{\prime}}}=(y^{\prime}_{1},y^{\prime}_{2},....y^{\prime}_{n^{\prime}})$.
Then, MCIR can be defined as:

$\displaystyle
C({\underset{\sim}{Y^{\prime}}};{\underset{\sim}{f_{i}}}|\\{{\underset{\sim}{f_{j}}}\\}\subseteq\\{||F||^{k\times
n^{\prime}}-{\underset{\sim}{f_{i}}}\\};i\neq j))$ (25) $\displaystyle=$
$\displaystyle\frac{I({\underset{\sim}{Y^{\prime}}};{\underset{\sim}{f_{i}}}|\\{{\underset{\sim}{f_{j}}}\\}\subseteq\\{||F||^{k\times
n^{\prime}}-{\underset{\sim}{f_{i}}}\\};i\neq
j)}{I({\underset{\sim}{Y^{\prime}}};{\underset{\sim}{f_{i}}}|\\{{\underset{\sim}{f_{j}}}\\}\subseteq\\{||F||^{k\times
n^{\prime}}-{\underset{\sim}{f_{i}}}\\};i\neq
j)+I({\underset{\sim}{Y^{\prime}}},{\underset{\sim}{f_{1}}},{\underset{\sim}{f_{2}}},...{\underset{\sim}{f_{j-1}}},{\underset{\sim}{f_{j}}},{\underset{\sim}{f_{j+1}}},..,{\underset{\sim}{f_{k})}}}$

Result 1:
$C({\underset{\sim}{Y^{\prime}}};{\underset{\sim}{f_{i}}}|\\{{\underset{\sim}{f_{j}}}\\}\subseteq\\{||F||^{k\times
n^{\prime}}-{\underset{\sim}{f_{i}}}\\};i\neq j))$ lies between $0$ and $1$.
(See supplementary subsection 6.4 for the proof). Then the ExCIR model with
$k$ dependent features can be written as :

$E({\underset{\sim}{Y^{\prime}}})=M^{\prime}(f_{j(f,\mathfrak{C})})=\mathfrak{J}+\frac{\mathfrak{C}_{f_{1}}f_{1}+......+\mathfrak{C}_{f_{m}}f_{m}}{\mathfrak{C}_{f_{p}}f_{p}+.......+\mathfrak{C}_{f_{k}}f_{k}}$
(26)

where,$\mathfrak{C}_{f_{i}}=C(Y^{\prime};f_{i}|\\{f_{j}\\}\subseteq\\{||F||^{n}-f_{i}\\};i\neq
j))$ and
$\mathfrak{J}=C({\underset{\sim}{Y^{\prime}}},{\underset{\sim}{f_{1}}},{\underset{\sim}{f_{2}}},...,{\underset{\sim}{f_{k}}})$.
$C({\underset{\sim}{Y^{\prime}}},{\underset{\sim}{f_{1}}},{\underset{\sim}{f_{2}}},...,{\underset{\sim}{f_{k}}})=\frac{I({\underset{\sim}{Y^{\prime}}},{\underset{\sim}{f_{1}}},{\underset{\sim}{f_{2}}},....,{\underset{\sim}{f_{k}}})}{I({\underset{\sim}{Y^{\prime}}};{\underset{\sim}{f_{i}}}|\\{{\underset{\sim}{f_{j}}}\\}\subseteq\\{||F||^{k\times
n^{\prime}}-{\underset{\sim}{f_{i}}}\\};i\neq
j)+I({\underset{\sim}{Y^{\prime}}},{\underset{\sim}{f_{1}}},{\underset{\sim}{f_{2}}},...{\underset{\sim}{f_{j-1}}},{\underset{\sim}{f_{j}}},{\underset{\sim}{f_{j+1}}},..,{\underset{\sim}{f_{k}}})}$
is called Joint Mutual Impact.

Feature dependencies create a complex environment where computational
complexity can not be ignored. By the theorem of Kolmogorov Complexity
[48][49], we calculate the computational complexity of ExCIR to generate
output $Y^{\prime}$ when features are dependent and we found that the
complexity is upper bounded by
$K_{\mathbb{A}}({\underset{\sim}{Y^{\prime}}}|I({\underset{\sim}{Y^{\prime}}}))+\frac{k}{2}O(\log{n^{\prime}})+Const_{\mathbb{A}}$,
where $\mathbb{A}$ is any other computer and there exist a constant
$Const_{\mathbb{A}}$ for all strings
${\underset{\sim}{Y^{\prime}}}=(y^{\prime}_{1},y^{\prime}_{2},...,y^{\prime}_{n^{\prime}})\epsilon\\{0,1\\}$,
$I({\underset{\sim}{Y^{\prime}}})=$ length of the string of $Y^{\prime}$.
$Const_{\mathbb{A}}$ does not depends on the output
${\underset{\sim}{Y^{\prime}}}$ [47]. We also found that ExCIR would take no
more than $\frac{1}{2}O(\log{n^{\prime}})+\zeta_{\mathbb{A}}$ time complexity
to calculate each MCIR associated with features. Here, $\zeta_{\mathbb{A}}$ is
a constant associated with the $ith$ MCIR for any other computer $\mathbb{A}$;
$i=1:k$ (See supplementary subsection 6.5 for detailed calculation). Both the
complexity bounds depend on the number of observations, not the number of
features. As our algorithm is based on the lightweight model and the less
number of data observations, both the upper bounds are very promising.

##  5 Conclusion

ExCIR can balance the tradeoff between explainability and accuracy
irrespective of a large number of dependent and independent features. This
approach also considers the uncertainty associated with features and output.
We provide a time complexity upper bound when features are dependent. The
upper bound only depends on input observations and as we used a lightweight
model with a sample dataset for our whole approach, the time complexity can be
considered as a reliable result.

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##  6 Supplementary

###  6.1 Table for Notation Referances

Notations | Descriptions
---|---
$||F||^{(k\times n)}$ |  $(k\times n)$ dimensional feature space.
$k$ | Number of features in both original and sample datasets.
$n$ | Number of observations in each feature in the original database.
$n^{\prime}$ | Number of observations in each feature in the sample database.
${\underset{\sim}{Y}}=(y_{1},..,y_{n})$ | Output vector in original space.
${\underset{\sim}{Y^{\prime}}}=(y^{\prime}_{1},..,y^{\prime}_{n^{\prime}})$ | Output vector in sample space.
$\mathcal{U}$ | Unified superspace for original model.
$\mathcal{U}^{\prime}$ | Unified superspace for lightweight model.
$\mathcal{D}_{i}({\underset{\sim}{f_{i}}});\ i=1:k$ |  Distribution of the $i^{th}$ feature.
$\mathcal{D}({\underset{\sim}{Y}})$ | Distribution of output in original space.
$\mathcal{D}({\underset{\sim}{Y^{\prime}}})$ | Distribution of output in lightweight space.
${\underset{\sim}{\mathcal{F}}}=(\mathcal{F}_{1},\mathcal{F}_{2},....,\mathcal{F}_{n})^{\prime}$ | Input column vector.
$D^{2}_{\text{final}}$ | The final average distance from the features to output in original space.
$D^{{}^{\prime}2}_{\text{final}}$ | The final average distance from the features to output in lightweight space.
$\mu$ | Lightweight model output probability measure.
$\delta$ | Original model probability measure.
$d^{+}$ | Embedding distance.
$d^{-}$ | Projection distance.
$\mathcal{L}(Y,Y^{\prime})$ | Loss between original and lightweight model outputs.
$\mathcal{R}(d*)$ | Risk generator.
$\lambda$ | Regularized parameter.
$\hat{f_{i}}$ |  The mean of the feature $f_{i};i=1:k$.
$\hat{y^{\prime}}$ |  the mean of the output vector $Y$.
$\hat{f_{i}y^{\prime}}$ |  and the joint weighted mean of feature $f_{i}$ ; $i=1:k$ and $Y$.
$\eta_{f_{i}}$ |  PCIR for feature $f_{i}$.
$\mathfrak{C}_{f_{i}}$ |  MCIR\- Mutual Correlation Impact Ratio for feature $f_{i}$.
$\mathfrak{J}$ |  JMI\- Join Mutual Impact.

###  6.2 Proof of theorem 1

Proof: In the following, we show that each local output can be expressed as a
combination of the feature input.
case 1 : When, $j\leq m$;

$\displaystyle\frac{dy^{\prime}}{d{\underset{\sim}{f_{j}}}}=$
$\displaystyle\frac{d}{d{\underset{\sim}{f_{j}}}}(\frac{\eta_{f_{1}}{\underset{\sim}{f_{1}}}+\eta_{f_{2}}{\underset{\sim}{f_{2}}}+......+\eta_{f_{m}}{\underset{\sim}{f_{m}}}}{\eta_{f_{m+1}}{\underset{\sim}{f_{m+1}}}+\eta_{f_{m+2}}{\underset{\sim}{f_{m+2}}}+.......+\eta_{f_{k}}{\underset{\sim}{f_{k}}}})$
(27) $\displaystyle=$
$\displaystyle[\frac{d}{d{\underset{\sim}{f_{j}}}}(\eta_{f_{1}}{\underset{\sim}{f_{1}}})+\frac{d}{df_{j}}(\eta_{f_{2}}{\underset{\sim}{f_{2}}})+....+\frac{d}{df_{j}}(\eta_{f_{j}}{\underset{\sim}{f_{j}}})+....+\frac{d}{df_{j}}(\eta_{f_{m}}{\underset{\sim}{f_{m}}})]\times$
$\displaystyle\frac{d}{df_{j}}[\frac{1}{\eta_{f_{m+1}}{\underset{\sim}{f_{m+1,j}}}+\eta_{f_{m+2}}{\underset{\sim}{f_{m+2,j}}}+.......+\eta_{f_{k}}{\underset{\sim}{f_{kj}}}}]$
$\displaystyle=$
$\displaystyle[\eta_{f_{1}}\frac{d{\underset{\sim}{f_{1}}}}{d{\underset{\sim}{f_{j}}}}+\eta_{f_{2}}\frac{d{\underset{\sim}{f_{2}}}}{d{\underset{\sim}{f_{j}}}}+....+\eta_{f_{j}}\frac{d{\underset{\sim}{f_{j}}}}{d{\underset{\sim}{f_{j}}}}+....+\eta_{f_{m}}\frac{d{\underset{\sim}{f_{m}}}}{d{\underset{\sim}{f_{j}}}}]\times$
$\displaystyle\frac{d}{d{\underset{\sim}{f_{j}}}}[\frac{1}{\eta_{f_{m+1}}{\underset{\sim}{f_{m+1,j}}}+\eta_{f_{m+2}}{\underset{\sim}{f_{m+2,j}}}+.......+\eta_{f_{k}}{\underset{\sim}{f_{kj}}}}]$

Since the features are independent of each other when we consider the impact
of the feature ${\underset{\sim}{f_{j}}}$ on the output $Y^{\prime}$, we fix
the values of features. Then we have;

$\displaystyle\frac{dy^{\prime}}{d{\underset{\sim}{f_{j}}}}=\eta_{f_{j}}\frac{d{\underset{\sim}{f_{j}}}}{d{\underset{\sim}{f_{j}}}}\times\frac{1}{\mathbb{K}}=c_{1}*\eta_{f_{j}}$
(28)

Here,
$c_{1}=\frac{1}{\mathbb{K}}=\frac{1}{\eta_{f_{m+1}}{\underset{\sim}{f_{m+1,j}}}+\eta_{f_{m+2}}{\underset{\sim}{f_{m+2,j}}}+.......+\eta_{f_{k}}{\underset{\sim}{f_{kj}}}}$
is a constant that can be expressed as the combination of the rest of the
features and their correlation ratio values which are fixed in our
environment.
case 2: when $j>m$;

$\displaystyle\frac{dy^{\prime}}{d{\underset{\sim}{f_{j}}}}=$
$\displaystyle\frac{d}{d{\underset{\sim}{f_{j}}}}(\frac{\eta_{f_{1}}{\underset{\sim}{f_{1}}}+\eta_{f_{2}}{\underset{\sim}{f_{2}}}+......+\eta_{f_{m}}{\underset{\sim}{f_{m}}}}{\eta_{f_{m+1}}{\underset{\sim}{f_{m+1}}}+\eta_{f_{m+2}}{\underset{\sim}{f_{m+2}}}+.......+\eta_{f_{k}}{\underset{\sim}{f_{k}}}})$
(29) $\displaystyle=$
$\displaystyle[\frac{d}{d{\underset{\sim}{f_{j}}}}(\eta_{f_{1}}{\underset{\sim}{f_{1}}})+\frac{d}{d{\underset{\sim}{f_{j}}}}(\eta_{f_{2}}{\underset{\sim}{f_{2}}})+........+\frac{d}{d{\underset{\sim}{f_{j}}}}(\eta_{f_{m}}{\underset{\sim}{f_{m}}})]\times$
$\displaystyle\frac{d}{d{\underset{\sim}{f_{j}}}}[\frac{1}{\eta_{f_{m+1}}{\underset{\sim}{f_{m+1,j}}}+\eta_{f_{m+2}}{\underset{\sim}{f_{m+2,j}}}+....+\eta_{f_{j}}{\underset{\sim}{f_{j}}}+...+\eta_{f_{k}}{\underset{\sim}{f_{kj}}}}]$
$\displaystyle[{\eta_{f_{m+1}}{\underset{\sim}{f_{m+1,j}}}+\eta_{f_{m+2}}{\underset{\sim}{f_{m+2,j}}}+....+\eta_{f_{j}}{\underset{\sim}{f_{j}}}+...+\eta_{f_{k}}{\underset{\sim}{f_{kj}}}}]\frac{d}{df_{j}}(\eta_{f_{1}}{\underset{\sim}{f_{1}}}+\eta_{f_{2}}{\underset{\sim}{f_{2}}}+......+\eta_{f_{m}}{\underset{\sim}{f_{m}}})$
$\displaystyle=$
$\displaystyle-(\eta_{f_{1}}{\underset{\sim}{f_{1}}}+\eta_{f_{2}}{\underset{\sim}{f_{2}}}+......+\eta_{f_{m}}{\underset{\sim}{f_{m}}})\frac{d}{df_{j}}[{\eta_{f_{m+1}}{\underset{\sim}{f_{m+1,j}}}+\eta_{f_{m+2}}{\underset{\sim}{f_{m+2,j}}}+....+\eta_{f_{j}}{\underset{\sim}{f_{j}}}+...+\eta_{f_{k}}{\underset{\sim}{f_{kj}}}}]\times$
$\displaystyle\frac{1}{[\frac{d}{df_{j}}[\eta_{f_{m+1}}{\underset{\sim}{f_{m+1,j}}}+\eta_{f_{m+2}}{\underset{\sim}{f_{m+2,j}}}+....+\eta_{f_{j}}{\underset{\sim}{f_{j}}}+...+\eta_{f_{k}}{\underset{\sim}{f_{kj}}}]^{2}]}$
$\displaystyle=$
$\displaystyle\frac{-k-1.\eta_{f_{j}}}{2.[\eta_{f_{m+1}}{\underset{\sim}{f_{m+1,j}}}+\eta_{f_{m+2}}{\underset{\sim}{f_{m+2,j}}}+....+\eta_{f_{j}}{\underset{\sim}{f_{j}}}+...+\eta_{f_{k}}{\underset{\sim}{f_{kj}}}]\frac{d}{d{\underset{\sim}{f_{j}}}}[\eta_{f_{j}}^{2}{\underset{\sim}{f_{j}}}^{2}-2\eta_{f_{j}}(\eta_{f_{1}}+\eta_{f_{2}}+...\eta_{f_{k}})]}$
$\displaystyle=$
$\displaystyle\frac{-k_{1}.\eta_{f_{j}}}{2.\eta_{f_{j}}(\eta_{f_{j}}^{2}-2K_{2})}$
$\displaystyle=$ $\displaystyle\frac{k_{1}}{2(2K_{2}-\eta_{f_{j}}^{2})}$
$\displaystyle=$ $\displaystyle\frac{c_{2}}{(2K_{2}-\eta_{f_{j}}^{2})}$

Equations (2)-(3) imply that the correlation ratio is able to capture and
reflect the impact of a certain feature on the generated output.
Mathematically when the feature has a direct positive relation with the
output, it should belong to the numerator of the complex model. On the other
hand, the feature belonging to the denominator of the model shows a constant
inverse relation with the output. In the second case, we can say that the
features are conversely correlated to the output.

###  6.3 Derivation of equation 24

For the sake of simplicity, Let consider we have only three features
${\underset{\sim}{f_{1}}},{\underset{\sim}{f_{2}}},{\underset{\sim}{f_{3}}}$
dependent on each other. If we want to find the CMMI between
${\underset{\sim}{Y}}$ and ${\underset{\sim}{f_{1}}}$ where
${\underset{\sim}{f_{1}}}$ is dependent on ${\underset{\sim}{f_{2}}}$ and
${\underset{\sim}{f_{3}}}$. We also assume that ${\underset{\sim}{f_{1}}}$ is
continuous. On the other hand,
${\underset{\sim}{f_{2}}},{\underset{\sim}{f_{3}}}$, and
${\underset{\sim}{Y}}$ are discrete in nature. Then using the concept of MI
[46], the CMMI will be :

$\displaystyle
I({\underset{\sim}{Y^{\prime}}};{\underset{\sim}{f_{1}}}|{\underset{\sim}{f_{2}}},{\underset{\sim}{f_{3}}})$
$\displaystyle=\sum_{f_{3}}\sum_{f_{2}}\int_{f_{1}}\sum_{Y}P(Y,f_{1},f_{2},f_{3})\log_{2}[\frac{P(Y,f_{1}|f_{2},f_{3})}{P(Y|f_{2},f_{3})P(f_{1}|f_{2},f_{3})}]$
(30)
$\displaystyle=\sum_{f_{3}}\sum_{f_{2}}\int_{f_{1}}\sum_{Y}P(Y,f_{1},f_{2},f_{3})\log_{2}[\frac{\frac{P(Y,f_{1},f_{2},f_{3})}{P(f_{2},f_{3})}}{P(Y|f_{2},f_{3})P(f_{1}|f_{2},f_{3})}]$

By using Bayes’ theorem it can be expressed as :

$\displaystyle
I({\underset{\sim}{Y^{\prime}}};{\underset{\sim}{f_{1}}}|{\underset{\sim}{f_{2}}},{\underset{\sim}{f_{3}}})$
$\displaystyle=\sum_{f_{3}}\sum_{f_{2}}\int_{f_{1}}\sum_{Y^{\prime}}P(Y^{\prime},f_{1},f_{2},f_{3})\log_{2}[\frac{\frac{P(Y^{\prime}|f_{1},f_{2},f_{3})P(f_{1}|f_{2},f_{3})P(f_{2}|f_{3})P(f_{3})}{P(f_{2}|f_{3})P(f_{3})}}{P(Y|f_{2},f_{3})P(f_{1}|f_{2},f_{3})}]$
(31)
$\displaystyle=\sum_{f_{3}}\sum_{f_{2}}\int_{f_{1}}\sum_{Y^{\prime}}P(Y,f_{1},f_{2},f_{3})\log_{2}[\frac{P(Y^{\prime}|f_{1},f_{2},f_{3})}{P(Y|f_{2},f_{3})}]$

Similarly, using Bayes’ theorem, we get:

$\displaystyle
I({\underset{\sim}{Y^{\prime}}};{\underset{\sim}{f_{2}}}|{\underset{\sim}{f_{1}}},{\underset{\sim}{f_{3}}})$
$\displaystyle=\sum_{f_{3}}\sum_{f_{2}}\int_{f_{1}}\sum_{Y^{\prime}}P(Y^{\prime},f_{1},f_{2},f_{3})\log_{2}[\frac{P(Y^{\prime}|f_{1},f_{2},f_{3})}{P(Y^{\prime}|f_{1},f_{3})}]$
(32) $\displaystyle
I({\underset{\sim}{Y^{\prime}}};{\underset{\sim}{f_{3}}}|{\underset{\sim}{f_{1}}},{\underset{\sim}{f_{2}}})$
$\displaystyle=\sum_{f_{3}}\sum_{f_{2}}\int_{f_{1}}\sum_{Y^{\prime}}P(Y^{\prime},f_{1},f_{2},f_{3})\log_{2}[\frac{P(Y^{\prime}|f_{1},f_{2},f_{3})}{P(Y^{\prime}|f_{1},f_{2})}]$

we can also write in the following way:

$\displaystyle
I({\underset{\sim}{Y^{\prime}}};{\underset{\sim}{f_{1}}}|{\underset{\sim}{f_{2}}},{\underset{\sim}{f_{3}}})$
$\displaystyle=\sum_{f_{3}}\sum_{f_{2}}\int_{f_{1}}\sum_{Y^{\prime}}P(Y^{\prime},f_{1},f_{2},f_{3})\log_{2}[\frac{P(Y^{\prime}|\\{f_{j}\\}\epsilon||F||^{k\times
n^{\prime}})}{P(Y|\\{f_{j}\\}\epsilon\\{||F||^{k\times
n^{\prime}}-f_{1}\\})}]$ (33) $\displaystyle
I({\underset{\sim}{Y^{\prime}}};{\underset{\sim}{f_{2}}}|{\underset{\sim}{f_{1}}},{\underset{\sim}{f_{3}}})$
$\displaystyle=\sum_{f_{3}}\sum_{f_{2}}\int_{f_{1}}\sum_{Y^{\prime}}P(Y^{\prime},f_{1},f_{2},f_{3})\log_{2}[\frac{P(Y^{\prime}|\\{f_{j}\\}\epsilon||F||^{k\times
n^{\prime}})}{P(Y^{\prime}|\\{f_{j}\\}\epsilon\\{||F||^{k\times
n^{\prime}}-f_{2}\\})}]$ $\displaystyle
I({\underset{\sim}{Y^{\prime}}};{\underset{\sim}{f_{3}}}|{\underset{\sim}{f_{1}}},{\underset{\sim}{f_{2}}})$
$\displaystyle=\sum_{f_{3}}\sum_{f_{2}}\int_{f_{1}}\sum_{Y^{\prime}}P(Y^{\prime},f_{1},f_{2},f_{3})\log_{2}[\frac{P(Y^{\prime}|\\{f_{j}\\}\epsilon||F||^{k\times
n^{\prime}})}{P(Y^{\prime}|\\{f_{j}\\}\epsilon\\{||F||^{k\times
n^{\prime}}-f_{3}\\})}]$

In the second step, we take four features
${\underset{\sim}{f_{1}}},{\underset{\sim}{f_{2}}},{\underset{\sim}{f_{3}}},{\underset{\sim}{f_{4}}}$
depending on each other. Again, we assume, ${\underset{\sim}{f_{1}}}$ is
continuous and
${\underset{\sim}{f_{4}}},{\underset{\sim}{f_{3}}},{\underset{\sim}{f_{2}}}$ ,
and ${\underset{\sim}{Y^{\prime}}}$ are discrete in nature. Then CMMI of
${\underset{\sim}{Y^{\prime}}}$ and
$({\underset{\sim}{f_{1}}}|{\underset{\sim}{f_{2}}},{\underset{\sim}{f_{3}}},{\underset{\sim}{f_{4})}}$
will be :

$\displaystyle
I({\underset{\sim}{Y^{\prime}}};{\underset{\sim}{f_{1}}}|{\underset{\sim}{f_{2}}},{\underset{\sim}{f_{3}}},{\underset{\sim}{f_{4}}})$
$\displaystyle=\sum_{f_{4}}\sum_{f_{3}}\sum_{f_{2}}\int_{f_{1}}\sum_{Y^{\prime}}P(Y^{\prime},f_{1},f_{2},f_{3},f_{4})\log_{2}[\frac{P(Y^{\prime},f_{1}|f_{2},f_{3},f_{4})}{P(Y^{\prime}|f_{2},f_{3},f_{4})P(f_{1}|f_{2},f_{3},f_{4})}]$
(34)
$\displaystyle=\sum_{f_{4}}\sum_{f_{3}}\sum_{f_{2}}\int_{f_{1}}\sum_{Y^{\prime}}P(Y^{\prime},f_{1},f_{2},f_{3},f_{4})\log_{2}[\frac{\frac{P(Y^{\prime},f_{1},f_{2},f_{3},f_{4})}{P(f_{2},f_{3},f_{4})}}{P(Y^{\prime}|f_{2},f_{3},f_{4})P(f_{1}|f_{2},f_{3},f_{4})}]$
$\displaystyle=\sum_{f_{4}}\sum_{f_{3}}\sum_{f_{2}}\int_{f_{1}}\sum_{Y^{\prime}}P(Y^{\prime},f_{1},f_{2},f_{3},f_{4})\log_{2}[\frac{\frac{P(Y^{\prime}|f_{1},f_{2},f_{3},f_{4})P(f_{1}|f_{2},f_{3},f_{4})P(f_{2}|f_{3},f_{4})P(f_{3}|f_{4})P(f_{4})}{P(f_{2}|f_{3},f_{4})P(f_{3}|f_{4})P(f_{4})}}{P(Y^{\prime}|f_{2},f_{3},f_{4})P(f_{1}|f_{2},f_{3},f_{4})}]$
$\displaystyle=\sum_{f_{4}}\sum_{f_{3}}\sum_{f_{2}}\int_{f_{1}}\sum_{Y^{\prime}}P(Y^{\prime},f_{1},f_{2},f_{3},f_{4})\log_{2}[\frac{P(Y^{\prime}|f_{1},f_{2},f_{3},f_{4})}{P(Y^{\prime}|f_{2},f_{3},f_{4})}]$
$\displaystyle=\sum_{f_{4}}\sum_{f_{3}}\sum_{f_{2}}\int_{f_{1}}\sum_{Y^{\prime}}P(Y^{\prime},f_{1},f_{2},f_{3},f_{4})\log_{2}[\frac{P(Y^{\prime}|\\{f_{j}\\}\epsilon||F||^{k\times
n^{\prime}})}{P(Y^{\prime}|\\{f_{j}\\}\epsilon\\{||F||^{k\times
n^{\prime}}-f_{1}\\})}]$

Similarly, it can be written as:

$\displaystyle
I({\underset{\sim}{Y^{\prime}}};{\underset{\sim}{f_{2}}}|{\underset{\sim}{f_{1}}},{\underset{\sim}{f_{3}}},{\underset{\sim}{f_{4}}})$
$\displaystyle=\sum_{f_{4}}\sum_{f_{3}}\sum_{f_{2}}\int_{f_{1}}\sum_{Y^{\prime}}P(Y^{\prime},f_{1},f_{2},f_{3},f_{4})\log_{2}[\frac{P(Y^{\prime}|\\{f_{j}\\}\epsilon||F||^{k\times
n^{\prime}})}{P(Y^{\prime}|\\{f_{j}\\}\epsilon\\{||F||^{k\times
n^{\prime}}-f_{2}\\})}]$ (35) $\displaystyle
I({\underset{\sim}{Y^{\prime}}};{\underset{\sim}{f_{3}}}|{\underset{\sim}{f_{2}}},{\underset{\sim}{f_{1}}},{\underset{\sim}{f_{4}}})$
$\displaystyle=\sum_{f_{4}}\sum_{f_{3}}\sum_{f_{2}}\int_{f_{1}}\sum_{Y^{\prime}}P(Y^{\prime},f_{1},f_{2},f_{3},f_{4})\log_{2}[\frac{P(Y^{\prime}|\\{f_{j}\\}\epsilon||F||^{k\times
n^{\prime}})}{P(Y^{\prime}|\\{f_{j}\\}\epsilon\\{||F||^{k\times
n^{\prime}}-f_{3}\\})}]$ $\displaystyle
I({\underset{\sim}{Y^{\prime}}};{\underset{\sim}{f_{4}}}|{\underset{\sim}{f_{2}}},{\underset{\sim}{f_{3}}},{\underset{\sim}{f_{1}}})$
$\displaystyle=\sum_{f_{4}}\sum_{f_{3}}\sum_{f_{2}}\int_{f_{1}}\sum_{Y^{\prime}}P(Y^{\prime},f_{1},f_{2},f_{3},f_{4})\log_{2}[\frac{P(Y^{\prime}|\\{f_{j}\\}\epsilon||F||^{k\times
n^{\prime}})}{P(Y^{\prime}|\\{f_{j}\\}\epsilon\\{||F||^{k\times
n^{\prime}}-f_{4}\\})}]$

In this way we can claim that:

$\displaystyle
I({\underset{\sim}{Y^{\prime}}};{\underset{\sim}{f_{i-1}}}|\\{{\underset{\sim}{f_{j}}}\\}\subseteq\\{||F||^{k\times
n^{\prime}}-{\underset{\sim}{f_{i-1}}}\\};i\neq j))=$ (36)
$\displaystyle\sum_{f_{j}\subseteq\\{||F||^{k\times
n^{\prime}}-f_{i-1}\\}_{d}}\int_{f_{j}\subseteq\\{||F||^{k\times
n^{\prime}}-f_{i-1}\\}_{c}}\int_{f_{i-1}}\sum_{Y^{\prime}}P(Y^{\prime},f_{1},f_{2},..,f_{i},f_{j},..f_{k})\log_{2}[\frac{P(Y^{\prime}|\\{f_{j}\\}\subseteq||F||^{k\times
n^{\prime}})}{P(Y^{\prime}|\\{f_{j}\\}\subseteq\\{||F||^{k\times
n^{\prime}}-f_{i-1}\\};i\neq j)}]$

So, if we consider the environment for $k$ number of features, where we want
to find the CMMI between ${\underset{\sim}{Y^{\prime}}}$ and
${\underset{\sim}{f_{i}}}$; $i=1(1)k$. ${\underset{\sim}{f_{i}}}$ is dependent
on the set of the features
$\\{{\underset{\sim}{f_{j}}}\\}\epsilon\\{||F||^{k\times n^{\prime}}-f_{i}\\}$
,

$\displaystyle
I({\underset{\sim}{Y^{\prime}}};{\underset{\sim}{f_{i}}}|\\{{\underset{\sim}{f_{j}}}\\}\subseteq\\{||F||^{k\times
n^{\prime}}-{\underset{\sim}{f_{i}}}\\};i\neq j))=$ (37)
$\displaystyle\sum_{f_{j}\subseteq\mathcal{A}}\int_{f_{j}\subseteq\mathcal{B}}\int_{f_{i}}\sum_{Y^{\prime}}\Phi\log_{2}[\frac{P(Y^{\prime},f_{i}|\\{f_{j}\\}\subseteq\\{||F||^{k\times
n^{\prime}}-f_{i}\\};i\neq
j)}{P(Y^{\prime}|\\{f_{j}\\}\subseteq\\{||F||^{k\times
n^{\prime}}-f_{i}\\})P(f_{i}|\\{f_{j}\\}\subseteq\\{||F||^{k\times
n^{\prime}}-f_{i}\\})}]$
$\displaystyle\sum_{f_{j}\subseteq\mathcal{A}}\int_{f_{j}\subseteq\mathcal{B}}\int_{f_{i}}\sum_{Y^{\prime}}\Phi\log_{2}[\frac{\frac{P(Y^{\prime},f_{1},f_{2},f_{3},..,f_{i-1},f_{i},f_{i+1},..,f_{k})}{P(f_{1},f_{2},..f_{i-1},f_{i+1},..,f_{k})}}{P(Y^{\prime}|\\{f_{j}\\}\subseteq\\{||F||^{k\times
n^{\prime}}-f_{i}\\})P(f_{i}|\\{f_{j}\\}\subseteq\\{||F||^{k\times
n^{\prime}}-f_{i}\\})}]$
$\displaystyle=\sum_{f_{j}\subseteq\mathcal{A}}\int_{f_{j}\subseteq\mathcal{B}}\int_{f_{i}}\sum_{Y^{\prime}}$
$\displaystyle\Phi\log_{2}[\frac{\frac{P(Y^{\prime}|||F||^{k\times
n^{\prime}})P(f_{i}|\\{f_{j}\\}\epsilon\\{||F||^{k\times
n^{\prime}}-f_{i}\\})P(f_{1}|\\{f_{j}\\}\epsilon\\{||F||^{k\times
n^{\prime}}-f_{i}-f_{1}\\})P(f_{2}|\\{f_{j}\\}\epsilon\\{||F||^{k\times
n^{\prime}}-f_{i}-f_{1}-f_{2}\\})..P(f_{k})}{P(f_{1}|\\{f_{j}\\}\epsilon\\{||F||^{k\times
n^{\prime}}-f_{i}-f_{1}\\})P(f_{2}|\\{f_{j}\\}\epsilon\\{||F||^{k\times
n^{\prime}}-f_{i}-f_{1}-f_{2}\\})..P(f_{k})}}{P(Y^{\prime}|\\{f_{j}\\}\subseteq\\{||F||^{k\times
n^{\prime}}-f_{i}\\})P(f_{i}|\\{f_{j}\\}\subseteq\\{||F||^{k\times
n^{\prime}}-f_{i}\\})}]$
$\displaystyle=\sum_{f_{j}\subseteq\mathcal{A}}\int_{f_{j}\subseteq\mathcal{B}}\int_{f_{i}}\sum_{Y^{\prime}}\Phi\log_{2}[\frac{P(Y^{\prime}|||F||^{k\times
n^{\prime}})P(f_{i}|\\{f_{j}\\}\epsilon\\{||F||^{k\times
n^{\prime}}-f_{i}\\})}{P(Y^{\prime}|\\{f_{j}\\}\subseteq\\{||F||^{k\times
n^{\prime}}-f_{i}\\})P(f_{i}|\\{f_{j}\\}\subseteq\\{||F||^{k\times
n^{\prime}}-f_{i}\\})}]$
$\displaystyle=\sum_{f_{j}\subseteq\mathcal{A}}\int_{f_{j}\subseteq\mathcal{B}}\int_{f_{i}}\sum_{Y^{\prime}}\Phi\log_{2}[\frac{P(Y^{\prime}|||F||^{k\times
n^{\prime}})}{P(Y^{\prime}|\\{f_{j}\\}\subseteq\\{||F||^{k\times
n^{\prime}}-f_{i}\\})}]$

Here, $\mathcal{A}=\\{||F||^{k\times n^{\prime}}-f_{i}\\}_{d}$ is the set that
contains the discrete features and $\mathcal{B}=\\{||F||^{k\times
n^{\prime}}-f_{i}\\}_{c}$ is the set that contains continuous features.
$\Phi=P(Y^{\prime},f_{1},f_{2},..,f_{i},f_{j},..f_{k})$.
Therefore we can write,

$\displaystyle
I({\underset{\sim}{Y^{\prime}}};{\underset{\sim}{f_{i}}}|\\{{\underset{\sim}{f_{j}}}\\}\subseteq\\{||F||^{k\times
n^{\prime}}-{\underset{\sim}{f_{i}}}\\};i\neq j))=$ (38)
$\displaystyle\sum_{f_{j}\subseteq\\{||F||^{k\times
n^{\prime}}-f_{i}\\}_{d}}\int_{f_{j}\subseteq\\{||F||^{k\times
n^{\prime}}-f_{i}\\}_{c}}\int_{f_{i}}\sum_{Y^{\prime}}P(Y^{\prime},f_{1},f_{2},..,f_{i},f_{j},..f_{k})\log_{2}[\frac{P(Y^{\prime}|\\{f_{j}\\}\subseteq||F||^{k\times
n^{\prime}})}{P(Y^{\prime}|\\{f_{j}\\}\subseteq\\{||F||^{k\times
n^{\prime}}-f_{i}\\};i\neq j)}]$

CMMI could take the values between $0$ to $\infty$, unlike the strict bound of
$(0,1)$ that the normal correlation coefficient has. The infinite range can be
the cause of the problem regarding scalability. So, it will be better to scale
down the dependency. This problem can be solved by our proposed metrics Mutual
Correlation Impact Ratio (MCIR) ratio. MCIR is the ratio of two mutual
information which is defined below. MCIR has a strict bound between $0$ to
$1$.

###  6.4 proof of result 1

Proof:  The mutual correlation impact ratio is a ratio that measures the
relation distance between a target feature and the output variable given the
fact that the target feature is dependent on some other features. More
specifically, MCIR measures how much influence other features have in the
dependence of the target feature and the output. The following Venn diagram
will help to understand how the feature-wise MCIR is calculated. (upload
later) Now by the definition of MI [44], it is known that the value of Mutual
information varies between $0$ to $\infty$. So we can say that,

$0\leq
I({\underset{\sim}{Y^{\prime}}};{\underset{\sim}{f_{i}}}|{{\underset{\sim}{f_{j}}}}\subseteq{||F||^{k\times
n^{\prime}}-{\underset{\sim}{f_{i}}}};i\neq j)\leq\infty$ (39)

Now, obviously,

$0\leq
I({\underset{\sim}{Y^{\prime}}};{\underset{\sim}{f_{i}}}|{{\underset{\sim}{f_{j}}}}\subseteq{||F||^{k\times
n^{\prime}}-{\underset{\sim}{f_{i}}}};i\neq j)\leq\leq
I({\underset{\sim}{Y^{\prime}}};{\underset{\sim}{f_{i}}}|{{\underset{\sim}{f_{j}}}}\subseteq{||F||^{k\times
n^{\prime}}-{\underset{\sim}{f_{i}}}};i\neq
j)+I({\underset{\sim}{Y^{\prime}}},{\underset{\sim}{f_{1}}},{\underset{\sim}{f_{2}}},...,{\underset{\sim}{f_{k}}})\leq\infty$
(40)

as,
$I({\underset{\sim}{Y^{\prime}}},{\underset{\sim}{f_{1}}},{\underset{\sim}{f_{2}}},{\underset{\sim}{f_{3}}},....,{\underset{\sim}{f_{k}}})\geq
0$; so it is very obvious that,

$0\leq\frac{I({\underset{\sim}{Y^{\prime}}};{\underset{\sim}{f_{i}}}|\\{{\underset{\sim}{f_{j}}}\\}\subseteq\\{||F||^{k\times
n^{\prime}}-{\underset{\sim}{f_{i}}}\\};i\neq
j)}{I({\underset{\sim}{Y^{\prime}}};{\underset{\sim}{f_{i}}}|\\{{\underset{\sim}{f_{j}}}\\}\subseteq\\{||F||^{k\times
n^{\prime}}-{\underset{\sim}{f_{i}}}\\};i\neq
j)+I({\underset{\sim}{Y^{\prime}}},{\underset{\sim}{f_{1}}},{\underset{\sim}{f_{2}}},...{\underset{\sim}{f_{j-1}}},{\underset{\sim}{f_{j}}},{\underset{\sim}{f_{j+1}}},..,{\underset{\sim}{f_{k})}}}\leq
1$ (41)

###  6.5  Complexity bound on MCIR

Feature dependencies create a complex environment where computational
complexity can not be ignored. We have used the concept of Mutual Information
to calculate our desired metrics MCIR. In this section, we will calculate the
upper bound of the computation complexity when the features are dependent on
each other. We will use the notion of Kolmogorov Complexity [47] to compute
the complexity bound. Let assume our model for feature dependency is
$M(f_{j(f,\mathfrak{C})})=M*$ and the universal computer is $\mathbb{C}$. by
the definition of Kolmogorov Complexity[48] we can write,

$K_{\mathbb{C}}({\underset{\sim}{Y^{\prime}}}|I({\underset{\sim}{Y^{\prime}}}))=\min_{M*:\mathbb{C}(M*,I({\underset{\sim}{Y^{\prime}}}))={\underset{\sim}{Y^{\prime}}}}I(M*)$
(42)

Here, $I(Y^{\prime})$ denotes the length of the output string
${\underset{\sim}{Y^{\prime}}}$, $I(M*)$ denotes the length of the string of
the program of the model $M*$, and $M*$ is the model running on the universal
computer $\mathbb{C}$ to produce the output $y^{\prime}$. Then, by the theorem
of Universality of conditional Kolmogorov complexity [49] knowing
$I({\underset{\sim}{Y^{\prime}}})$, we can find the upper bound:

$K_{\mathbb{C}}({\underset{\sim}{Y^{\prime}}})\leq
K_{\mathbb{A}}({\underset{\sim}{Y^{\prime}}}|I({\underset{\sim}{Y^{\prime}}}))+\log^{*}I({\underset{\sim}{Y^{\prime}}})+Const_{\mathbb{A}}$
(43)

Where, $\log^{*}I({\underset{\sim}{Y^{\prime}}})=\log
I({\underset{\sim}{Y^{\prime}}})+\log\log
I({\underset{\sim}{Y^{\prime}}})+\log\log\log
I({\underset{\sim}{Y^{\prime}}})+......$ as long as the terms are positive.
$k$ is the total number of features in the data. $\mathbb{A}$ is any other
computer and there exist a constant $Const_{\mathbb{A}}$ for all strings
${\underset{\sim}{Y^{\prime}}}=(y^{\prime}_{1},y^{\prime}_{2},...,y^{\prime}_{n^{\prime}})\epsilon\\{0,1\\}$.
$Const_{\mathbb{A}}$ does not depends on the output
${\underset{\sim}{Y^{\prime}}}$. Here,
$I({\underset{\sim}{Y^{\prime}}})=n^{\prime}$, then, it can be written as,

$K_{\mathbb{C}}({\underset{\sim}{Y^{\prime}}})\leq
K_{\mathbb{A}}({\underset{\sim}{Y^{\prime}}}|I({\underset{\sim}{Y^{\prime}}}))+\frac{k}{2}O(\log{n^{\prime}})+Const_{\mathbb{A}}$
(44)

where we take $O(\log{(\log{(n^{\prime})})})=O(\log{n^{\prime}})$ and ignored
the negetive terms. We will also calculate the complexity bound of each
$I({\underset{\sim}{Y^{\prime}}};{\underset{\sim}{f_{i}}}|\\{{\underset{\sim}{f_{j}}}\\}\subseteq\\{||F||^{k\times
n^{\prime}}-{\underset{\sim}{f_{i}}}\\};i\neq j);i=1:k$ and by the theory of
Kolmogorov Complexity bound for entropy [47], it should be as follows:

$\displaystyle
I({\underset{\sim}{Y^{\prime}}};{\underset{\sim}{f_{i}}}|\\{{\underset{\sim}{f_{j}}}\\}\subseteq\\{||F||^{k\times
n^{\prime}}-{\underset{\sim}{f_{i}}}\\};i\neq
j)=H({\underset{\sim}{Y^{\prime}}},{\underset{\sim}{f_{j}}})-H({\underset{\sim}{Y^{\prime}}};{\underset{\sim}{f_{i}}}|\\{{\underset{\sim}{f_{j}}}\\}\subseteq\\{||F||^{k\times
n^{\prime}}-{\underset{\sim}{f_{i}}}\\};i\neq j)$ (45) $\displaystyle\leq
H({\underset{\sim}{Y}})-H({\underset{\sim}{Y^{\prime}}};{\underset{\sim}{f_{i}}}|\\{{\underset{\sim}{f_{j}}}\\}\subseteq\\{||F||^{k\times
n^{\prime}}-{\underset{\sim}{f_{i}}})+\frac{(||Y||^{k*n^{\prime}}\log{n^{\prime}})}{n^{\prime}}-\frac{([||Y||-1]^{k*n^{\prime}}\log{n^{\prime}})}{n^{\prime}}+\frac{Const_{\mathbb{A}}}{n^{\prime}}$

By taking
$\frac{(||Y||^{k*n^{\prime}}\log{n^{\prime}})}{n^{\prime}}-\frac{([||Y||-1]^{k*n^{\prime}}\log{n^{\prime}})}{n^{\prime}}=O(\log{n^{\prime}})$
we can also rewrite the equation 45 following way:

$\displaystyle
I({\underset{\sim}{Y^{\prime}}};{\underset{\sim}{f_{i}}}|\\{{\underset{\sim}{f_{j}}}\\}\subseteq\\{||F||^{k\times
n^{\prime}}-{\underset{\sim}{f_{i}}}\\};i\neq j)$ $\displaystyle\leq
I({\underset{\sim}{Y^{\prime}}};{\underset{\sim}{f_{i}}}|\\{{\underset{\sim}{f_{j}}}\\}\subseteq\\{||F||^{k\times
n^{\prime}}-{\underset{\sim}{f_{i}}}\\};i\neq
j)+O(\log{n^{\prime}})+\frac{Const_{\mathbb{A}}}{n^{\prime}}$ (46)
$\displaystyle
I({\underset{\sim}{Y^{\prime}}};{\underset{\sim}{f_{i}}}|\\{{\underset{\sim}{f_{j}}}\\}\subseteq\\{||F||^{k\times
n^{\prime}}-{\underset{\sim}{f_{i}}}\\};i\neq j)$ $\displaystyle\leq
O(\log{n^{\prime}})+\mathbb{C_{A}}_{i}$

where,
$\mathbb{C_{A}}_{i}=I({\underset{\sim}{Y^{\prime}}};{\underset{\sim}{f_{i}}}|\\{{\underset{\sim}{f_{j}}}\\}\subseteq\\{||F||^{k\times
n^{\prime}}-{\underset{\sim}{f_{i}}}\\};i\neq
j)+\frac{Const_{\mathbb{A}}}{n^{\prime}}$ Similarly, by Kolmogorov’s theorem
[47][49], we also can get,

$I({\underset{\sim}{Y^{\prime}}},{\underset{\sim}{f_{1}}},{\underset{\sim}{f_{2}}},...{\underset{\sim}{f_{j-1}}},{\underset{\sim}{f_{j}}},{\underset{\sim}{f_{j+1}}},..,{\underset{\sim}{f_{k}}})\leq
I({\underset{\sim}{Y^{\prime}}},{\underset{\sim}{f_{1}}},{\underset{\sim}{f_{2}}},...{\underset{\sim}{f_{j-1}}},{\underset{\sim}{f_{j}}},{\underset{\sim}{f_{j+1}}},..,{\underset{\sim}{f_{k}}})+O(\log{n^{\prime}})+\frac{Const_{\mathbb{A}}}{n^{\prime}}$
(47)

Let denote,
$\frac{Const_{\mathbb{A}}}{n^{\prime}}+I({\underset{\sim}{Y^{\prime}}},{\underset{\sim}{f_{1}}},{\underset{\sim}{f_{2}}},...{\underset{\sim}{f_{j-1}}},{\underset{\sim}{f_{j}}},{\underset{\sim}{f_{j+1}}},..,{\underset{\sim}{f_{k}}})=\mathbb{C^{\prime}_{A}}_{i}$
and we will get the following in-equation for $i=1:k$:

$I({\underset{\sim}{Y^{\prime}}},{\underset{\sim}{f_{1}}},{\underset{\sim}{f_{2}}},...{\underset{\sim}{f_{j-1}}},{\underset{\sim}{f_{j}}},{\underset{\sim}{f_{j+1}}},..,{\underset{\sim}{f_{k}}})\leq
O(\log{n^{\prime}})+\mathbb{C^{\prime}_{A}}_{i}$ (48)

Now, from equation 46 and equation 48, we can get the complexity bound for
calculating each MCIR, and it will be as follows for $i=1:k$;

$\mathfrak{C}_{i}\leq\frac{1}{2}O(\log{n^{\prime}})+\zeta_{\mathbb{A}}$ (49)

here, $\zeta_{\mathbb{A}}$ is a constant associated with $ith$ MCIR for any
other computer $\mathbb{A}$; $i=1,k$. So, we can say from the equation 44 and
equation 49 we can say that, our model would take no more than
$\frac{k}{2}O(\log{n})+Const_{\mathbb{A}}$ time complexity to run the whole
program, whereas it would take no more than
$\frac{1}{2}O(\log{n^{\prime}})+\zeta_{\mathbb{A}}$ time complexity to
calculate each MCIR associated with features.