# Inaccuracy measures for concomitants of generalized order statistics in
Morgenstern family

Safieh Daneshi This work is partially supported by the Clinical Research
Development Center (CRDC) of the Bushehr University of Medical
SciencesDepartment of Statistics, Faculty of Mathematical Sciences, Shahrood
University of Technology, Shahrood, Iran.
11email<EMAIL_ADDRESS>of Statistics, Faculty of
Mathematical Sciences, Shahrood University of Technology, Shahrood, Iran.
11email<EMAIL_ADDRESS>Department of Statistics, Persian Gulf
University, Bushehr, Iran.
11email<EMAIL_ADDRESS>di Matematica e Applicazioni
Universit$\acute{a}$ di Napoli Federico II Via Cintia, I-80126 Napoli, Italy.
11email<EMAIL_ADDRESS>Ahmad Nezakati Department of Statistics,
Faculty of Mathematical Sciences, Shahrood University of Technology, Shahrood,
Iran.
11email<EMAIL_ADDRESS>of Statistics, Faculty of
Mathematical Sciences, Shahrood University of Technology, Shahrood, Iran.
11email<EMAIL_ADDRESS>Department of Statistics, Persian Gulf
University, Bushehr, Iran.
11email<EMAIL_ADDRESS>di Matematica e Applicazioni
Universit$\acute{a}$ di Napoli Federico II Via Cintia, I-80126 Napoli, Italy.
11email<EMAIL_ADDRESS>Saeid Tahmasebi Department of Statistics,
Faculty of Mathematical Sciences, Shahrood University of Technology, Shahrood,
Iran.
11email<EMAIL_ADDRESS>of Statistics, Faculty of
Mathematical Sciences, Shahrood University of Technology, Shahrood, Iran.
11email<EMAIL_ADDRESS>Department of Statistics, Persian Gulf
University, Bushehr, Iran.
11email<EMAIL_ADDRESS>di Matematica e Applicazioni
Universit$\acute{a}$ di Napoli Federico II Via Cintia, I-80126 Napoli, Italy.
11email<EMAIL_ADDRESS>Maria Longobardi This work is partially
supported by the GNAMPA research group of INDAM (Istituto Nazionale di Alta
Matematica) and MIUR-PRIN 2017Department of Statistics, Faculty of
Mathematical Sciences, Shahrood University of Technology, Shahrood, Iran.
11email<EMAIL_ADDRESS>of Statistics, Faculty of
Mathematical Sciences, Shahrood University of Technology, Shahrood, Iran.
11email<EMAIL_ADDRESS>Department of Statistics, Persian Gulf
University, Bushehr, Iran.
11email<EMAIL_ADDRESS>di Matematica e Applicazioni
Universit$\acute{a}$ di Napoli Federico II Via Cintia, I-80126 Napoli, Italy.
11email<EMAIL_ADDRESS>
###### Abstract

In this paper, we obtain a measure of inaccuracy between rth concomitant of
generalized order statistic and the parent random variable in Morgenstern
family. Applications of this result are given for concomitants of order
statistics and record values. We also study some results of cumulative past
inaccuracy (CPI) between the distribution function of rth concomitant of order
statistic (record value) and the distribution function of parent random
variable. Finally, we discuss on a problem of estimating the CPI by means of
the empirical CPI in concomitants of generalized order statistics.

###### keywords:

Measure of inaccuracy, Cumulative inaccuracy, Concomitants, Generalized order
statistics

###### :

primary 62B10, secondary 62G30

##  1 Introduction

Let $(X_{i},Y_{i})$, $i=1,2,\cdots,n$ be independent and identically
distributed random variables from a continuous bivariate distribution
$F_{X,Y}(x,y)$. If $X_{(r:n)}$ denotes the $r$th order statistic, then the
$Y$’s associated with $X_{(r:n)}$ denoted by $Y_{[r:n]}$ is called the
concomitant of $r$th order statistic. The concomitants are of interest in
selection and prediction problems. The concept of generalized order statistics
(GOS) was introduced by [7] as a unified approach to a variety of models of
ordered random variables such as ordinary order statistics, sequential order
statistics, progressive type-II censoring, record values and Pfeifers records.
The random variables $X(1,n,m,k),X(2,n,m,k),\cdots,X(n,n,m,k)$ are called
generalized order statistics based on the absolutely continuous distribution
function(cdf) $F$ with density function $f$, if their joint density function
is given by

$\displaystylef^{X(1,n,m,k),...,X(n,n,m,k)}(x_{1},...,x_{n})$
$\displaystyle=$ $\displaystyle
k\left(\prod_{j=1}^{n-1}\gamma_{j}\right)\left(\prod_{i=1}^{n-1}(1-F(x_{i}))^{m}f(x_{i})\right)$
$\displaystyle\times$ $\displaystyle(1-F(x_{n}))^{k-1}f(x_{n}),$
$\displaystyle$ $\displaystyle F^{-1}(0)\leq x_{1}\leq x_{2}\leq...\leq
x_{n}\leq F^{-1}(1),$

with parameters $n\in\mathbb{N},k>0,m\in\mathbb{R}$, such that
$\gamma_{r}=k+(n-r)(m+1)>0$, for all $1\leq r\leq n$. Similarly, concomitants
can also be defined in the case of GOS.

[9] defined a class of bivariate distributions with the probability density
function (pdf) given by

${}f_{X,Y}(x,y)=f_{X}(x)f_{Y}(y)\left[1+\alpha(2F_{X}(x)-1)(2F_{Y}(y)-1)\right],\;\;\;\;|\alpha|\leq
1,$ (1)

where $\alpha$ is the association parameter (see [5], and references therein
for more details). For the Morgenstern family with pdf given by (1), the
density function and distribution function of the concomitant of rth GOS
(denoted by $Y_{[r,n,m,k]},1\leq r\leq n$), are given by [2] as follows:

${}g_{[r,n,m,k]}(y)=f_{Y}(y)\left[1+\alpha
C^{*}(r,n,m,k)(1-2F_{Y}(y))\right],$ (2)

${}G_{[r,n,m,k]}(y)=F_{Y}(y)\left[1+\alpha C^{*}(r,n,m,k)(1-F_{Y}(y))\right],$
(3)

where
$C^{*}(r,n,m,k)=\frac{2\prod_{j=1}^{r}\gamma_{j}}{\prod_{i=1}^{r}(\gamma_{i}+1)}-1$.
In the special case of GOS in Morgenstern family, if $Y_{[r:n]}$ denotes the
concomitant of rth order statistic $X_{(r:n)}$, then the pdf and cdf of
$Y_{[r:n]}$ in Morgenstern family are given by

$\displaystyle
f_{Y_{[r:n]}}(y)=f_{Y}(y)\left[1+\alpha\left(\frac{n-2r+1}{n+1}\right)(1-2F_{Y}(y))\right],$

and

$\displaystyle
F_{Y_{[r:n]}}(y)=F_{Y}(y)\left[1+\alpha\left(\frac{n-2r+1}{n+1}\right)(1-F_{Y}(y))\right],$

respectively. We refer the reader to [1] for more details.

Let $(X_{1},Y_{1}),(X_{2},Y_{2}),\cdots$ be a sequence of bivariate random
variables from a continuous distribution. If $\\{R_{n},n\geq 1\\}$ is the
sequence of upper record values in the sequence of $X$’s, then the $Y$ which
corresponds with the nth-record will be called the concomitant of the nth-
record, denoted by $R_{[n]}$. The concomitants of record values arise in a
wide variety of practical experiments such as industrial stress testing, life
time experiments, meteorological analysis, sporting matches and some other
experimental fields. For other important applications of record values and
their concomitants see [1]. The pdf and cdf for $R_{[n]}$ has been obtained as
follows:

$f_{R_{[n]}}(y)=f_{Y}(y)[1+\alpha_{n}(1-2F_{Y}(y))],\;\;n\geq 1,$ (4)

and

$F_{R_{[n]}}(y)=F_{Y}(y)[1+\alpha_{n}(1-F_{Y}(y))],$ (5)

where $\alpha_{n}=\alpha(2^{1-n}-1)$ .

Let $X$ and $Y$ be two non-negative random variables with distribution
functions $F(x)$ and $G(x)$, respectively. If $f(x)$ is the actual probability
density function (pdf) corresponding to the observations and $g(x)$ is the
density assigned by the experimenter, then the inaccuracy measure of $X$ and
$Y$ is defined by [8] as follows:

$I(f,g)=-\int_{0}^{+\infty}f(x)\log g(x)dx.$

Analogous to this measure of inaccuracy, [12] proposed a cumulative past
inaccuracy (CPI) measure as

$I(F,G)=-\int_{0}^{+\infty}F(x)\log G(x)dx.$

Several authors have worked on measures of inaccuracy for ordered random
variables. [11] proposed the measure of inaccuracy between the ith order
statistic and the parent random variable. [12] developed measures of dynamic
cumulative residual and past inaccuracy. They studied characterization results
of these dynamic measures under proportional hazard model and proportional
reversed hazard model. Recently [13] have introduced the measure of residual
inaccuracy of order statistics and proved a characterization result for it.
Motivated by some of the articles mentioned above, in this paper we aim to
present some results on inaccuracy for concomitants of GOS in Morgenstern
family. The paper is organized as follows: In Section 2, we obtain a measure
of inaccuracy between $g_{[r,n,m,k]}(y)$ and $f_{Y}(y)$ in Morgenstern family.
Another applications of this result are given for concomitants of order
statistics and record values. We also study some results of CPI between
$G_{[r,n,m,k]}(y)$ and $F_{Y}(y)$. Applications of CPI are given for
concomitants of order statistics and record values. Finally, in Section 3, we
discuss on a problem of estimating the CPI by means of the empirical CPI for
concomitants of GOS.

##  2 Inaccuracy measures for concomitants of GOS

If $Y_{[r,n,m,k]}$ is the concomitant of rth GOS from (1), then the inaccuracy
measure between $g_{[r,n,m,k]}(y)$ and $f_{Y}(y)$ for $1\leq r\leq n$,
$\alpha\neq 0$ is given by

$\displaystyle I(g_{[r,n,m,k]},f_{Y})$ $\displaystyle=$
$\displaystyle-\int_{0}^{\infty}g_{[r,n,m,k]}(y)\log{f_{Y}(y)}dy$ (6)
$\displaystyle=$ $\displaystyle\left[1+\alpha C^{*}(r,n,m,k)\right]H(Y)$
$\displaystyle+$ $\displaystyle 2\alpha
C^{*}(r,n,m,k)\int_{0}^{\infty}f_{Y}(y)F_{Y}(y)\log{f_{Y}(y)}dy$
$\displaystyle=$ $\displaystyle\left[1+\alpha C^{*}(r,n,m,k)\right]H(Y)$
$\displaystyle+$ $\displaystyle 2\alpha C^{*}(r,n,m,k)\int_{0}^{1}u\log
f_{Y}(F^{-1}_{Y}(u))du$ $\displaystyle=$ $\displaystyle\left[1+\alpha
C^{*}(r,n,m,k)\right]H(Y)+2\alpha C^{*}(r,n,m,k)\phi_{f}(u),$

where $\phi_{f}(u)=\int_{0}^{1}u\log f_{Y}(F_{Y}^{-1}(u))du$ and

$H(Y)=-\int_{0}^{\infty}f_{Y}(y)\log{f_{Y}(y)}dy$

is the Shannon entropy of the random variable $Y$.

As an application of the representation (6), we consider the following special
cases.
Case 1: According to (6), if we put $m=0$ and $k=1$, then an inaccuracy
measure between $f_{Y_{[r:n]}}$ (density function of rth concomitant of order
statistic) and $f_{Y}$ in Morgenstern family is obtained as follows:

$\displaystyle I(f_{Y_{[r:n]}},f_{Y})$ $\displaystyle=$
$\displaystyle-\int_{0}^{\infty}f_{Y_{[r:n]}}(y)\log{f_{Y}(y)}dy$ (7)
$\displaystyle=$
$\displaystyle\left[1+\alpha\left(\frac{n-2r+1}{n+1}\right)\right]H(Y)+2\alpha\left(\frac{n-2r+1}{n+1}\right)\phi_{f}(u)$
$\displaystyle=$ $\displaystyle
H(Y)+\frac{3(n-2r+1)}{2(n+1)}\left[I(f_{Y_{[1:2]}},f_{Y})-I(f_{Y_{[2:2]}},f_{Y})\right].$

In the following, we present some examples and properties of
$I(f_{Y_{[r:n]}},f_{Y})$.

######  Example 1.

Let $(X_{i},Y_{i})$, $i=1,2,...,n$ be a random sample from Gumbel bivariate
exponential distribution (GBED) with cdf

$F(x,y)=\left(1-\exp\left(\frac{-x}{\theta_{1}}\right)\right)\left(1-\exp\left(\frac{-y}{\theta_{2}}\right)\right)\left[1+\alpha\exp\left(\frac{-x}{\theta_{1}}-\frac{y}{\theta_{2}}\right)\right].$
(8)

From (7), we find

$\displaystyle
I(f_{Y_{[r:n]}},f_{Y})=[1+\log\theta_{2}]-\frac{\alpha}{2}\left(\frac{n-2r+1}{n+1}\right).$
(9)

By using (9), we get

$\displaystyle
A_{\alpha}(n)=I(f_{Y_{[n:n]}},f_{Y})-I(f_{Y_{[1:n]}},f_{Y})=\alpha\left(\frac{n-1}{n+1}\right),$

which is positive, negative or zero whenever $0<\alpha\leq 1,n>1$;
$-1\leq\alpha<0,n>1$ or $n=1\;or\;\alpha=0$, respectively. Also, the
difference between $I(f_{Y_{[r:n]}},f_{Y})$ and $H(Y)$ is

$\displaystyle
B_{\alpha,n}(r)=I(f_{Y_{[r:n]}},f_{Y})-H(Y)=-\frac{\alpha}{2}\left(\frac{n-2r+1}{n+1}\right).$

$B_{\alpha,n}(r)$ is positive for $-1\leq\alpha<0$ , $1\leq r<\frac{n+1}{2}$
(or $0<\alpha\leq 1$, $\frac{n+1}{2}<r\leq n)$. Also, it is negative for
$-1\leq\alpha<0$ , $\frac{n+1}{2}<r\leq n$( or $0<\alpha\leq 1$ ,$1\leq
r<\frac{n+1}{2}$).

Now, if $n$ is odd, then numerical computations indicate that
$I(f_{Y_{[r:n]}},f_{Y})$ is increasing (decreasing) in $r$ for $1\leq
r<\frac{n+1}{2}$, $0<\alpha\leq 1$ ($\frac{n+1}{2}<r\leq n$,
$-1\leq\alpha<0$).

######  Example 2.

Let $(X_{i},Y_{i})$, $i=1,2,\cdots,n$ be a random sample from Morgenstern type
bivariate Logistic distribution with cdf

$F(x,y)=\left(1+\exp(-x)\right)^{-1}\left(1+\exp(-y)\right)^{-1}\left(1+\frac{\alpha
e^{-x-y}}{(1+e^{-x})(1+e^{-y})}\right).$

Computation shows that

$I(f_{Y_{[r:n]}},f_{Y})=1-0.6\alpha\left(\frac{n-2r+1}{n+1}\right).$ (10)

By using (10), we get

$\displaystyle
D_{\alpha}(n)=I(f_{Y_{[n:n]}},f_{Y})-I(f_{Y_{[1:n]}},f_{Y})=1.2\alpha\left(\frac{n-1}{n+1}\right),$

which is positive, negative or zero whenever $0<\alpha\leq 1,n>1$;
$-1\leq\alpha<0,n>1$ or $n=1\;or\;\alpha=0$, respectively.

######  Example 3.

Let $(X_{i},Y_{i})$, $i=1,2,...,n$ be a random sample from Morgenstern type
bivariate Rayleigh distribution with cdf

$F(x,y)=\left(1-\exp(-\frac{x^{2}}{2\sigma_{1}^{2}})\right)\left(1-\exp(-\frac{y^{2}}{2\sigma_{2}^{2}})\right)\left(1+\alpha\exp\left(-\frac{x^{2}}{2\sigma_{1}^{2}}-\frac{y^{2}}{2\sigma_{2}^{2}}\right)\right).$

From (7), we find

$I(f_{Y_{[r:n]}},f_{Y})=\frac{\alpha(n-2r+1)}{n+1}(\log\sqrt{2}-\frac{1}{2})+1-\frac{1}{2}\psi(1)+\log(\frac{\sigma_{2}}{\sqrt{2}}).$
(11)

Using (11), we have

$\displaystyle
W_{\alpha}(n)=I(f_{Y_{[n:n]}},f_{Y})-I(f_{Y_{[1:n]}},f_{Y})=2\alpha\left(0.5-\log\sqrt{2}\right)\left(\frac{n-1}{n+1}\right),$

which is positive, negative or zero whenever $0<\alpha\leq 1,n>1$;
$-1\leq\alpha<0,n>1$ or $n=1\;or\;\alpha=0$, respectively.

######  Example 4.

Let $(X_{i},Y_{i})$, $i=1,2,...,n$ be a random sample from Morgenstern type
bivariate generalized exponential distribution (MTBGED) with cdf

$\displaystyle
F_{X,Y}(x,y)=\\{(1-e^{-\theta_{1}x})(1-e^{-\theta_{2}y})\\}^{\lambda}[1+\alpha(1-(1-e^{-\theta_{1}x})^{\lambda})(1-(1-e^{-\theta_{2}y})^{\lambda})].$

By using (7), we get

$\displaystyle
I(f_{Y_{[r:n]}},f_{Y})=-\log(\lambda\theta_{2})+B(\lambda)-\frac{\alpha(n-2r+1)}{n+1}D(\lambda)+\frac{\lambda-1}{\lambda}[1+\frac{\frac{\alpha(n-2r+1)}{n+1}}{2}],$
(12)

where $B(\lambda)=\psi\left(\lambda+1\right)-\psi\left(1\right)$ and
$D(\lambda)=B(2\lambda)-B(\lambda)$. Using (12), we have

$\displaystyle
Q_{\alpha,\lambda}(n)=I(f_{Y_{[n:n]}},f_{Y})-I(f_{Y_{[1:n]}},f_{Y})=\frac{\alpha(n-1)}{n+1}\left[2D(\lambda)-\frac{\lambda-1}{\lambda}\right],$

which is positive, negative or zero whenever $0<\alpha\leq 1,n>1$;
$-1\leq\alpha<0,n>1$ or $n=1\;or\;\alpha=0$, respectively.

######  Remark 1.

Let $(X_{i},Y_{i})$, $i=1,2,\cdots,n$ be a random sample of size $n$ with pdf
(1). Then, from (7) we have

$H(Y)=\frac{I(f_{Y_{[n:n]}},f_{Y})+I(f_{Y_{[1:n]}},f_{Y})}{2}.$

######  Remark 2.

Let $(X_{i},Y_{i})$, $i=1,2,\cdots,n$ be a random sample of size $n$ with pdf
(1). If $\lambda\geq 1$ is an integer number and we change $r$ to $r\lambda$
and $n$ to $(n+1)\lambda-1$. Then, from (7) we have

$I(f_{Y_{[r:n]}},f_{Y})=I(f_{Y_{[r\lambda:(n+1)\lambda-1]}},f_{Y}).$

We consider the concomitants of order statistics whenever
$(X_{1},Y_{1}),(X_{2},Y_{2}),\ldots,(X_{n},Y_{n})$ are independent but
otherwise arbitrarily distributed. Let us consider the Morgenstern family with
cdf

${}F_{X_{i},Y_{i}}(x,y)=F_{X_{i}}(x)F_{Y_{i}}(y)\left[1+\alpha_{i}(1-F_{X_{i}}(x))(1-F_{Y_{i}}(y))\right].$
(13)

Now, suppose that $F_{X_{i}}(x)=F_{X}(x)$ , $F_{Y_{i}}(y)=F_{Y}(y)$ and
$|\alpha_{i}|\leq 1$. Then in this particular case, the pdf’s of $Y_{[1:n]}$
and $Y_{[n:n]}$ are given by [6] as follows:

$f_{[1:n]}(y)=f_{Y}(y)\left[1+\frac{n-1}{(n+1)n}\sum_{j=1}^{n}\alpha_{j}(1-2F_{Y}(y))\right],$
(14)
$f_{[n:n]}(y)=f_{Y}(y)\left[1-\frac{n-1}{(n+1)n}\sum_{j=1}^{n}\alpha_{j}(1-2F_{Y}(y))\right].$
(15)

Now, in the following, the measures of inaccuracy for concomitants of extremes
of order statistics is represented.

######  Example 5.

Let $(X_{i},Y_{i}),\,i=1,2,\ldots,n$ be independent random vectors from (13).
If $Y_{[1:n]}$ and $Y_{[n:n]}$ are concomitants of extremes of order
statistics, then

${}I(f_{[1:n]},f_{Y})=\left(1+\frac{n-1}{(n+1)n}\sum_{j=1}^{n}\alpha_{j}\right)H(Y)+2\frac{n-1}{(n+1)n}\sum_{j=1}^{n}\alpha_{j}\phi_{f}(u),$
(16)
${}I(f_{[n:n]},f_{Y})=\left(1-\frac{n-1}{(n+1)n}\sum_{j=1}^{n}\alpha_{j}\right)H(Y)-2\frac{n-1}{(n+1)n}\sum_{j=1}^{n}\alpha_{j}\phi_{f}(u).$
(17)

By using (16) and (17) we have

$A_{n}=I(f_{[n:n]},f_{Y})-I(f_{[1:n]},f_{Y})=-\frac{2(n-1)}{n(n+1)}\Delta,$

where we have set
$\Delta=H(Y)\sum_{j=1}^{n}\alpha_{j}+2\sum_{j=1}^{n}\alpha_{j}\phi_{f}(u)$. If
$\Delta>0$ ($\Delta<0$), then $A_{n}<0$ ($A_{n}>0$). Finally, we get

$I(f_{[n:n]},f_{Y})+I(f_{[1:n]},f_{Y})=2H(Y).$

Case 2: According to (6), if we put $m=-1$ and $k=1$, then an inaccuracy
measure between $f_{R_{[r]}}$ (density function of the concomitant of rth-
record value) and $f_{Y}$ in Morgenstern family is obtained as follows:

$\displaystyle
I(f_{R_{[r]}},f_{Y})=\left(1+\alpha(2^{1-r}-1)\right)H(Y)+2\alpha(2^{1-r}-1)\phi_{f}(u).$
(18)

######  Example 6.

Let $(X_{i},Y_{i})$, $i=1,2,...,n$ be a random sample of GBED with cdf

$F(x,y)=\left(1-\exp\left(\frac{-x}{\theta_{1}}\right)\right)\left(1-\exp\left(\frac{-y}{\theta_{2}}\right)\right)\left[1+\alpha\exp\left(\frac{-x}{\theta_{1}}-\frac{y}{\theta_{2}}\right)\right].$
(19)

From (18), we find

$\displaystyle
I(f_{R_{[r]}},f_{Y})=[1+\log\theta_{2}]+\frac{\alpha}{2}\left(2^{1-r}-1\right).$
(20)

By using (20), we get

$\displaystyle
A_{\alpha}(r)=I(f_{R_{[r]}},f_{Y})-I(f_{R_{[r-1]}},f_{Y})=-\alpha 2^{-r},$

which is positive, negative or zero whenever $-1\leq\alpha<0,r>1$;
$0<\alpha\leq 1,r>1$ or $\alpha=0$, respectively. Also, the difference between
$I(f_{R_{[r]}},f_{Y})$ and $H(Y)$ is

$\displaystyle
B_{\alpha,n}(r)=I(f_{R_{[r]}},f_{Y})-H(Y)=\frac{\alpha}{2}\left(2^{1-r}-1\right).$

$B_{\alpha,n}(r)$ is positive, negative or zero whenever $-1\leq\alpha<0,r>1$;
$0<\alpha\leq 1,r>1$ or $r=1\;or\;\alpha=0$, respectively.

######  Remark 3.

In analogy with (6), a measure of inaccuracy associated with $f_{Y}(y)$ and
$g_{[r,n,m,k]}(y)$ is given by

$\displaystyle I(f_{Y},g_{[r,n,m,k]})=H(Y)-E\left[\log\left(1+\alpha
C^{*}(r,n,m,k)\left(1-2U\right)\right)\right],$

where U is uniformly distributed in (0,1).

Quantile functions are efficient alternatives to the distribution function in
modelling and analysis of statistical data. The quantile function is defined
by,

$Q(u)=F^{-1}(u)=\inf\\{y:F(y)\geq u\\},\qquad 0<u<1.$

Noting that $F(Q(u))=u$ and differentiating it with respect to $u$ yields
$q(u)f(Q(u))=1$. Let $Y$ be a nonnegative random variable with pdf $f(\cdot)$
and quantile function $Q(\cdot)$, then $f(Q(u))$ is called the density
quantile function and $q(u)=Q^{\prime}(u)$ is known as the quantile density
function of $Y$. Now using (6), the corresponding quantile based
$I(g_{[r,n,m,k]},f_{Y})$ is defined as

$\displaystyle I(g_{[r,n,m,k]},f_{Y})=E(\log q(U))+\alpha
C^{*}(r,n,m,k)E\left[(1-2U)\log q(U)\right].$ (21)

###  2.1 CPI between $Y_{[r,n,m,k]}$ and $Y$

If $Y_{[r,n,m,k]}$ is the concomitant of rth GOS from (1), then the CPI
measure between $G_{[r,n,m,k]}(y)$ and $F_{Y}(y)$ for $1\leq r\leq n$,
$\alpha\neq 0$ is given by

$\displaystyle{}I(G_{Y_{[r,n,m,k]}},F_{Y})$ $\displaystyle=$
$\displaystyle-\int_{0}^{\infty}G_{[r,n,m,k]}(y)\log{F_{Y}(y)}dy$ (22)
$\displaystyle=$ $\displaystyle\left[1+\alpha
C^{*}(r,n,m,k)\right]{\mathcal{CE}}(Y)$ $\displaystyle+$ $\displaystyle\alpha
C^{*}(r,n,m,k)\int_{0}^{\infty}F^{2}_{Y}(y)\log{F_{Y}(y)}dy$ $\displaystyle=$
$\displaystyle\left[1+\alpha C^{*}(r,n,m,k)\right]{\mathcal{CE}}(Y)$
$\displaystyle-$
$\displaystyle\frac{\alpha}{2}C^{*}(r,n,m,k){\mathcal{CE}}(Y_{(2:2)}),$

where ${\mathcal{CE}}(Y)$ and ${\mathcal{CE}}(Y_{(2:2)})$ are the cumulative
entropy of the random variables $Y$ and $Y_{(2:2)}$, respectively (see [4]).

######  Remark 4.

In analogy with (22), a measure of inaccuracy associated with $F_{Y}$ and
$G_{[r,n,m,k]}$ is given by

$\displaystyle
I(F_{Y},G_{[r,n,m,k]})=\mathcal{CE}(Y)-E\left[\frac{U\log{\left(1+\alpha
C^{*}(r,n,m,k)\left(1-U\right)\right)}}{f(F^{-1}(U))}\right].$

Case 1: If we put $m=0$ and $k=1$, then a measure of inaccuracy between
$F_{Y_{[r:n]}}$ (distribution function of rth concomitant of order statistic)
and $F_{Y}$ is presented as

$\displaystyle I(F_{Y_{[r:n]}},F_{Y})$ $\displaystyle=$
$\displaystyle-\int_{0}^{\infty}F_{Y_{[r:n]}}(y)\log{F_{Y}(y)}dy$ (23)
$\displaystyle=$
$\displaystyle\left[1+\alpha\left(\frac{n-2r+1}{n+1}\right)\right]{\mathcal{CE}}(Y)$
$\displaystyle+$
$\displaystyle\alpha\left(\frac{n-2r+1}{n+1}\right)\int_{0}^{\infty}F^{2}_{Y}(y)\log{F_{Y}(y)}dy$
$\displaystyle=$
$\displaystyle\left[1+\alpha\left(\frac{n-2r+1}{n+1}\right)\right]{\mathcal{CE}}(Y)$
$\displaystyle-$
$\displaystyle\alpha\left(\frac{n-2r+1}{2(n+1)}\right){\mathcal{CE}}(Y_{(2:2)}).$

######  Example 7.

Let $(X_{i},Y_{i})$, $i=1,2,\cdots,n$ be a random sample from Morgenstern type
bivariate uniform distribution (MTBUD) with cdf

$F(x,y)=\frac{xy}{\theta_{1}\theta_{2}}\left[1+\alpha(1-\frac{x}{\theta_{1}})(1-\frac{y}{\theta_{2}})\right],\;\;0<x<\theta_{1},\;\;0<y<\theta_{2}.$

Computation shows that

$\displaystyle I(F_{Y_{[r:n]}},F_{Y})$ $\displaystyle=$
$\displaystyle\left[1+\alpha\left(\frac{n-2r+1}{n+1}\right)\right]\frac{\theta_{2}}{4}-\alpha\left(\frac{n-2r+1}{n+1}\right)\frac{\theta_{2}}{9}$
(24) $\displaystyle=$
$\displaystyle\frac{\theta_{2}}{4}+\alpha\left(\frac{n-2r+1}{n+1}\right)\frac{5\theta_{2}}{36}.$

Using (24), we have

$\displaystyle
D_{\alpha,\theta_{2}}(n)=I(F_{Y_{[n:n]}},F_{Y})-I(F_{Y_{[1:n]}},F_{Y})=\frac{5\alpha\theta_{2}(-n+1)}{18(n+1)}.$

which is positive, negative or zero whenever $-1\leq\alpha<0$, $0<\alpha\leq
1$ or $\alpha=0$, respectively.

######  Example 8.

Let $(X_{i},Y_{i})$, $i=1,2,\cdots,n$ be a random sample from GBED. Then,
computation shows that

$\displaystyle I(F_{Y_{[r:n]}},F_{Y})$ $\displaystyle=$
$\displaystyle\left[1+\alpha\left(\frac{n-2r+1}{n+1}\right)\right]\left[\frac{\pi^{2}}{6}-1\right]\theta_{2}$
(25) $\displaystyle-$
$\displaystyle\alpha\left(\frac{n-2r+1}{n+1}\right)\left[\frac{\pi^{2}}{6}-\frac{5}{4}\right]\theta_{2}$
$\displaystyle=$
$\displaystyle\left[\frac{\pi^{2}}{6}-1\right]\theta_{2}+\frac{\alpha\theta_{2}}{4}\left(\frac{n-2r+1}{n+1}\right).$

Using (25), we have

$\displaystyle
Q_{\alpha,\theta_{2}}(n)=I(F_{Y_{[n:n]}},F_{Y})-I(F_{Y_{[1:n]}},F_{Y})=\frac{\alpha\theta_{2}(-n+1)}{2(n+1)},$

which is positive, negative or zero whenever $-1\leq\alpha<0$, $0<\alpha\leq
1$ or $\alpha=0$, respectively.

######  Example 9.

Let $(X_{i},Y_{i})$, $i=1,2,\cdots,n$ be a random sample from Morgenstern type
bivariate inverse Weibull distribution with cdf

$\displaystyle F(x,y)$ $\displaystyle=$
$\displaystyle\exp\left[-\left(\frac{\theta_{1}}{x}\right)^{\beta_{1}}-\left(\frac{\theta_{2}}{y}\right)^{\beta_{2}}\right]$
$\displaystyle\times$
$\displaystyle\left[1+\alpha\left(1-\exp\left[-\left(\frac{\theta_{1}}{x}\right)^{\beta_{1}}\right]\right)\left(1-\exp\left[-\left(\frac{\theta_{2}}{y}\right)^{\beta_{2}}\right]\right)\right].$

Computation shows that

$\displaystyle I(F_{Y_{[r:n]}},F_{Y})$ $\displaystyle=$
$\displaystyle\left[1+\alpha\left(\frac{n-2r+1}{n+1}\right)\right]\frac{\theta_{2}}{\beta_{2}}\Gamma\left(\frac{\beta_{2}-1}{\beta_{2}}\right)$
$\displaystyle-$
$\displaystyle\alpha\left(\frac{n-2r+1}{n+1}\right)\frac{2^{\frac{1}{\beta_{2}}-1}\theta_{2}}{\beta_{2}}\Gamma\left(\frac{\beta_{2}-1}{\beta_{2}}\right)$
$\displaystyle=$
$\displaystyle\frac{\theta_{2}}{\beta_{2}}\Gamma\left(\frac{\beta_{2}-1}{\beta_{2}}\right)+\alpha\left(\frac{n-2r+1}{n+1}\right)\frac{\theta_{2}}{\beta_{2}}\Gamma\left(\frac{\beta_{2}-1}{\beta_{2}}\right)\left(1-2^{\frac{1}{\beta_{2}}-1}\right).$

Using (9), we have

$\displaystyle
D_{\alpha,\theta_{2}}(n)=I(F_{Y_{[n:n]}},F_{Y})-I(F_{Y_{[1:n]}},F_{Y})=\alpha\frac{\theta_{2}(1-n)}{\beta_{2}(n+1)}\Gamma\left(\frac{\beta_{2}-1}{\beta_{2}}\right)\left(1-2^{\frac{1}{\beta_{2}}-1}\right).$

which is positive, negative or zero whenever $-1\leq\alpha<0$, $0<\alpha\leq
1$ or $\alpha=0$, respectively.

######  Proposition 1.

Let $(X_{i},Y_{i})$, $i=1,2,\cdots,n$ be a random sample from Morgenstern
family. Then for $1\leq r\leq\frac{n+1}{2}$, we have

$I(F_{Y_{[r:n]}},F_{Y})\leq(\geq){\mathcal{CE}}(Y),\;\;-1\leq\alpha<0\;(0<\alpha\leq
1).$ (27)

Proof. The proof follows by recalling Proposition 4.8 of [4] .

Case 2: If we put $m=-1$ and $k=1$, then a measure of inaccuracy between
$F_{R_{[r]}}$ (distribution function of nth concomitant of upper record value)
and $F_{Y}$ is presented as

$\displaystyle I(F_{R_{[r]}},F_{Y})$ $\displaystyle=$
$\displaystyle-\int_{0}^{\infty}F_{R_{[r]}}(y)\log{F_{Y}(y)}dy$ (28)
$\displaystyle=$
$\displaystyle[1+\alpha(2^{1-r}-1)]{\mathcal{CE}}(Y)+\alpha(2^{1-r}-1)\int_{0}^{\infty}F^{2}_{Y}(y)\log
F_{Y}(y)dy$ $\displaystyle=$
$\displaystyle[1+\alpha(2^{1-r}-1)]{\mathcal{CE}}(Y)-\frac{\alpha}{2}(2^{1-r}-1){\mathcal{CE}}(Y_{(2:2)}).$

######  Proposition 2.

Let $(X_{i},Y_{i})$, $i=1,2,\cdots,n$ be a random sample from Morgenstern
family. Then, we have

$I(F_{R_{[r]}},F_{Y})\leq(\geq){\mathcal{CE}}(Y),\;\;0<\alpha\leq
1(-1\leq\alpha<0).$ (29)

Proof. The proof follows by recalling Proposition 4.8 of [4] .

##  3 Empirical CPI for concomitants of GOS

In this section we address the problem of estimating the CPI for concomitants
by means of the empirical CPI. Let $(X_{i},Y_{i})$, $i=1,2,\cdots,n$ be a
random sample of size $n$ from Morgenstern family. Then according to (22), the
empirical CPI between $G_{Y_{[r,n,m,k]}}$ and $F_{Y}$ can be obtained as
follows:

$\displaystyle{}\widehat{I}(G_{Y_{[r,n,m,k]}},F_{Y})$ $\displaystyle=$
$\displaystyle\left[1+\alpha
C^{*}(r,n,m,k)\right]\sum_{j=1}^{n-1}U_{j}\left(\frac{j}{n}\right)\left(-\log\frac{j}{n}\right)$
$\displaystyle-$ $\displaystyle\alpha
C^{*}(r,n,m,k)\sum_{j=1}^{n-1}U_{j}\left(\frac{j}{n}\right)^{2}\left(-\log\frac{j}{n}\right)$
$\displaystyle=$
$\displaystyle\sum_{j=1}^{n-1}U_{j}\left(\frac{j}{n}\right)\left(-\log\frac{j}{n}\right)\left[1+\alpha
C^{*}(r,n,m,k)\left(1-\frac{j}{n}\right)\right],$

where $U_{j}=Z_{(j+1)}-Z_{(j)},j=1,2,...,n-1$ are the sample spacings based on
ordered random samples.

Case 1: If we put $m=0$ and $k=1$, then the empirical CPI between
$F_{Y_{[r:n]}}$ and $F_{Y}$ is given by

$\displaystyle\widehat{I}(F_{Y_{[r:n]}},F_{Y})=\sum_{j=1}^{n-1}U_{j}\frac{j}{n}\left(-\log\frac{j}{n}\right)\left[1+\alpha\left(\frac{n-2r+1}{n+1}\right)\left(1-\frac{j}{n}\right)\right].$
(31)

Case 2: If we put $m=-1$ and $k=1$, then the empirical CPI between
$F_{R_{[r]}}$ and $F_{Y}$ can be written as

$\displaystyle{}\widehat{I}(F_{R_{[r]}},F_{Y})=\sum_{j=1}^{n-1}U_{j}\frac{j}{n}\left(-\log\frac{j}{n}\right)\left[1+\alpha\left(2^{1-r}-1\right)\left(1-\frac{j}{n}\right)\right].$
(32)

######  Example 10.

Let $(X_{i},Y_{i})$, $i=1,2,...,n$ be a random sample from MTBGED with
$\lambda=1$, then the sample spacings $U_{j}$ are independent and
exponentially distributed with mean $\frac{1}{\theta_{2}(n-j)}$ (for more
details see [10]). Now from (32) we obtain

$\displaystyle
E[\widehat{I}(F_{R_{[r]}},F_{Y})]=\frac{1}{\theta_{2}}\sum_{j=1}^{n-1}\frac{j}{n(n-j)}\left(-\log\frac{j}{n}\right)\left[1+\alpha\left(2^{1-r}-1\right)\left(1-\frac{j}{n}\right)\right],$
(33)

and

$\displaystyle
Var[\widehat{I}(F_{R_{[r]}},F_{Y})]=\frac{1}{\theta_{2}^{2}}\sum_{j=1}^{n-1}\left(\frac{j}{n(n-j)}(-\log\frac{j}{n})\left[1+\alpha\left(2^{1-r}-1\right)\left(1-\frac{j}{n}\right)\right]\right)^{2}.$

We have computed the values of $E[\widehat{I}(F_{R_{[r]}},F_{Y})]$ and
$Var[\widehat{I}(F_{R_{[r]}},F_{Y})]$ for sample sizes $n=10,15,20$,
$\theta_{2}=0.5,1,2$, $\alpha=-1,-0.5,0.5,1$ and $r=2$ in Table 1. We can
easily see that $E[\widehat{I}(F_{R_{[r]}},F_{Y})]$ and
$Var[\widehat{I}(F_{R_{[r]}},F_{Y})]$ are decreasing in $\alpha$ and
$\theta_{2}$. Also, we consider that
$\lim_{n\rightarrow\infty}Var[\widehat{I}(F_{R_{[r]}},F_{Y})]=0$.

######  Example 11.

Let $(X_{i},Y_{i})$, $i=1,2,\cdots,n$ be a random sample from
MTBUD with $\theta_{1}=\theta_{2}=1$. Then the sample spacings $U_{j}$ are
independent of beta distribution with parameters 1 and $n$ (for more details
see [10]). Now from (32) we obtain

$\displaystyle
E[\widehat{I}(F_{R_{[r]}},F_{Y})]=\frac{1}{n+1}\sum_{j=1}^{n-1}\frac{j}{n}\left(-\log\frac{j}{n}\right)\left[1+\alpha\left(2^{1-r}-1\right)\left(1-\frac{j}{n}\right)\right],$
(35)

and

$\displaystyle
Var[\widehat{I}(F_{R_{[r]}},F_{Y})]=\frac{n}{(n+1)^{2}(n+2)}\sum_{j=1}^{n-1}\left(\frac{j}{n}(-\log\frac{j}{n})\left[1+\alpha\left(2^{1-r}-1\right)\left(1-\frac{j}{n}\right)\right]\right)^{2}.$

We have computed the values of $E[\widehat{I}(F_{R_{[r]}},F_{Y})]$ and
$Var[\widehat{I}(F_{R_{[r]}},F_{Y})]$ for sample sizes $n=10,15,20$,
$\alpha=-1,-0.5,0.5,1$ and $r=2$ in Table 2. We can easily see that
$E[\hat{I}(F_{R_{[r]}},F_{Y})]$ and $Var[\widehat{I}(F_{R_{[r]}},F_{Y})]$ are
decreasing in $\alpha$. Also, we consider that
$\lim_{n\rightarrow\infty}Var[\widehat{I}(F_{R_{[r]}},F_{Y})]=0$.

Table 1: Numerical values of $\mathbb{E}[\widehat{I}(F_{R_{[r]}},F_{Y})]$ and $Var[\widehat{I}(F_{R_{[r]}},F_{Y})]$ for MTBGED with $\lambda=1$. | $\mathbb{E}[\widehat{I}(F_{R_{[r]}},F_{Y})]$
---|---
$\theta_{2}$ | 0.5 | 1 | 2 | 0.5 | 1 | 2 | 0.5 | 1 | 2 | 0.5 | 1 | 2
$n$ |  | $\alpha=-1$ |  |  | $\alpha=-0.5$ |  |  | $\alpha=0.5$ |  |  | $\alpha=1$ |
10 | 1.429 | 0.714 | 0.357 | 1.306 | 0.653 | 0.326 | 1.061 | 0.530 | 0.265 | 0.938 | 0.469 | 0.234
15 | 1.468 | 0.734 | 0.367 | 1.344 | 0.672 | 0.336 | 1.096 | 0.548 | 0.274 | 0.972 | 0.486 | 0.243
20 | 1.487 | 0.743 | 0.372 | 1.362 | 0.681 | 0.340 | 1.114 | 0.557 | 0.278 | 0.989 | 0.494 | 0.247
| $Var[\widehat{I}(F_{R_{[r]}},F_{Y})]$
$\theta_{2}$ | 0.5 | 1 | 2 | 0.5 | 1 | 2 | 0.5 | 1 | 2 | 0.5 | 1 | 2
$n$ |  | $\alpha=-1$ |  |  | $\alpha=-0.5$ |  |  | $\alpha=0.5$ |  |  | $\alpha=1$ |
10 | 0.241 | 0.060 | 0.015 | 0.205 | 0.051 | 0.013 | 0.144 | 0.036 | 0.009 | 0.119 | 0.030 | 0.007
15 | 0.165 | 0.041 | 0.010 | 0.141 | 0.035 | 0.009 | 0.100 | 0.025 | 0.006 | 0.083 | 0.021 | 0.005
20 | 0.126 | 0.031 | 0.008 | 0.108 | 0.027 | 0.007 | 0.077 | 0.019 | 0.005 | 0.064 | 0.016 | 0.004
Table 2: Numerical values of $\mathbb{E}[\widehat{I}(F_{R_{[r]}},F_{Y})]$ and $Var[\widehat{I}(F_{R_{[r]}},F_{Y})]$ for MTBUD with $\theta_{1}=\theta_{2}=1$. | $\mathbb{E}[\widehat{I}(F_{R_{[r]}},F_{Y})]$ | $Var[\widehat{I}(F_{R_{[r]}},F_{Y})]$
---|---|---
$n$ | $\alpha=-1$ | $\alpha=-0.5$ | $\alpha=0.5$ | $\alpha=1$ | $\alpha=-1$ | $\alpha=-0.5$ | $\alpha=0.5$ | $\alpha=1$
10 | 0.285 | 0.254 | 0.192 | 0.162 | 0.008 | 0.007 | 0.004 | 0.003
15 | 0.297 | 0.264 | 0.200 | 0.168 | 0.006 | 0.005 | 0.003 | 0.002
20 | 0.302 | 0.270 | 0.204 | 0.171 | 0.005 | 0.004 | 0.002 | 0.001

######  Theorem 1.

Let $(X_{i},Y_{i})$, $i=1,2,\cdots,n$ be a random sample of size $n$ from
Morgenstern family. Then we have

$\displaystyle\widehat{I}(F_{R_{[r]}},F_{Y})\,\longrightarrow
I(F_{R_{[r]}},F_{Y})\;\;\;a.s\;as\;n\rightarrow\infty.$

Proof. From relation (32), we obtain

$\displaystyle\widehat{I}(F_{R_{[r]}},F_{Y})=\left[1+\alpha\left(2^{1-r}-1\right)\right]\widehat{\mathcal{CE}}(Y)-\frac{\alpha}{2}\left(2^{1-r}-1\right)\widehat{\mathcal{CE}}(Y_{(2:2)}).$

Since $\widehat{\mathcal{CE}}(Y)\longrightarrow\mathcal{CE}(Y)$ and
$\widehat{\mathcal{CE}}(Y_{(2:2)})\longrightarrow\mathcal{CE}(Y_{(2:2)})$,
then proof follows by [4].

######  Theorem 2.

Let $(X_{i},Y_{i})$, $i=1,2,...,n$ be a random sample from MTBGED with
$\lambda=1$, then

$\displaystyle
Z_{n}:=\frac{\widehat{I}(F_{R_{[r]}},F_{Y})-E\left[\widehat{I}(F_{R_{[r]}},F_{Y})\right]}{\sqrt{Var\left[\widehat{I}(F_{R_{[r]}},F_{Y})\right]}}$

converges in distribution to a standard normal variable as
$n\rightarrow\infty$.

Proof. First the empirical measure $\hat{I}(F_{R_{[r]}},F_{Y})$ can be
expressed as the following sum of independent random variables as

$\displaystyle\widehat{I}(F_{R_{[r]}},F_{Y})=\sum_{j=1}^{n-1}W_{j},$

where
$W_{j}=U_{j}\frac{j}{n}\left(-\log\frac{j}{n}\right)\left[1+\alpha\left(2^{1-r}-1\right)\left(1-\frac{j}{n}\right)\right]$
are independent random variables with the mean and variance given by

$E[W_{j}]=\frac{1}{n\theta_{2}(1-\frac{1}{j/n})}\left(\log\frac{j}{n}\right)\left[1+\alpha\left(2^{1-r}-1\right)\left(1-\frac{j}{n}\right)\right],$
$Var[W_{j}]=\frac{1}{n^{2}\theta^{2}_{2}(1-\frac{1}{j/n})^{2}}\left(\log\frac{j}{n}\right)^{2}\left[1+\alpha\left(2^{1-r}-1\right)\left(1-\frac{j}{n}\right)\right]^{2}.$

Since $E[|W_{j}-E(W_{j})|^{3}]=2e^{-1}(6-e)[E(W_{j})]^{3}$ for any
exponentially distributed random variable $W_{j}$, by setting
$\alpha_{j,k}=E[|W_{j}-E(W_{j})|^{k}]$ the following approximations hold for
large n:

$\displaystyle\sum_{j=1}^{n}\alpha_{j,2}$ $\displaystyle=$
$\displaystyle\frac{1}{n^{2}\theta_{2}^{2}}\sum_{j=1}^{n}\frac{1}{\left(1-\frac{1}{j/n}\right)^{2}}\left(\log\frac{j}{n}\right)^{2}\left[1+\alpha\left(2^{1-r}-1\right)\left(1-\frac{j}{n}\right)\right]^{2}$
$\displaystyle\approx$ $\displaystyle\frac{c_{2}}{n\theta_{2}^{2}},$
$\displaystyle\sum_{j=1}^{n}\alpha_{j,3}$ $\displaystyle=$
$\displaystyle\frac{2(6-e)}{en^{3}\theta_{2}^{3}}\sum_{j=1}^{n}\frac{1}{\left(1-\frac{1}{j/n}\right)^{3}}\left(\log\frac{j}{n}\right)^{3}\left[1+\alpha\left(2^{1-r}-1\right)\left(1-\frac{j}{n}\right)\right]^{3}$
$\displaystyle\approx$
$\displaystyle\frac{2(6-e)c_{3}}{en^{2}\theta_{2}^{3}},$

where

$\displaystyle c_{k}:=\int_{0}^{1}\left(\frac{\log
x}{1-1/x}\right)^{k}\left[1+\alpha\left(2^{1-r}-1\right)\left(1-x\right)\right]^{k}.$

Hence, Lyapunov’s condition of the central limit theorem is satisfied (see
[3]):

$\displaystyle\frac{(\alpha_{1,3}+\cdots+\alpha_{n,3})^{1/3}}{(\alpha_{1,2}+\cdots+\alpha_{n,2})^{1/2}}\approx\frac{[2(6-e)c_{3}]^{1/3}}{e^{1/3}c_{2}^{1/2}}n^{-1/6}\rightarrow
0\;\;\;\;\;as\;\;n\rightarrow\infty,$

which completes the proof.

## Acknowledgement

The authors are thankful to the referee and editor for their valuable
suggestions towards the improvement of the paper.

## References

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# In-filter Computing For Designing Ultra-light Acoustic Pattern Recognizers

Abhishek Ramdas Nair, Shantanu Chakrabartty, and Chetan Singh Thakur Abhishek
Ramdas Nair and Chetan Singh Thakur are with the Department of Electronic
Systems Engineering, Indian Institute of Science, Bangalore, KA, 560012 INDIA
e-mail<EMAIL_ADDRESS>csthakur@iisc.ac.in).Shantanu Chakrabartty
is with Department of Electrical and Systems Engineering, Washington
University in St. Louis,USA, 63130 e-mail: (shantanu@wustl.edu).missing

###### Abstract

We present a novel in-filter computing framework that can be used for
designing ultra-light acoustic classifiers for use in smart internet-of-things
(IoTs). Unlike a conventional acoustic pattern recognizer, where the feature
extraction and classification are designed independently, the proposed
architecture integrates the convolution and nonlinear filtering operations
directly into the kernels of a Support Vector Machine (SVM). The result of
this integration is a template-based SVM whose memory and computational
footprint (training and inference) is light enough to be implemented on an
FPGA-based IoT platform. While the proposed in-filter computing framework is
general enough, in this paper, we demonstrate this concept using a Cascade of
Asymmetric Resonator with Inner Hair Cells (CAR-IHC) based acoustic feature
extraction algorithm. The complete system has been optimized using time-
multiplexing and parallel-pipeline techniques for a Xilinx Spartan 7 series
Field Programmable Gate Array (FPGA). We show that the system can achieve
robust classification performance on benchmark sound recognition tasks using
only ~1.5k Look-Up Tables (LUTs) and ~2.8k Flip-Flops (FFs), a significant
improvement over other approaches.

##  I Introduction

Internet-of-Things like unattended ground sensors[1] (UGS), intruder detection
systems[2] [3], wild-life tracking [4] or, structural health monitoring
systems [5] generally operate in remote locations. They have to be active at
all times to ensure that it can detect events of interest. In most cases, the
events of interest are infrequent or rare. As a result, most of these IoT
systems use an embedded pattern classifier to relax data storage and wireless
transmission requirements [6]. An example of such a system is illustrated in
Fig. 1. Here the system wirelessly transmits alerts only when it detects
signatures (acoustic or visual) about a target (for example a wild life
species).

Figure 1: Smart IoT architecture using a classifier to reduce wireless
transmission bandwidth

Based on this selective transmission, the IoT platform can conserve a
significant battery power and hence prolong its operational life. However, the
key challenge in designing such IoTs is that the integrated classifier needs
to be robust and highly energy-efficient. While deep neural network (DNN)
based classification systems can achieve very high accuracy[7][8], there exist
certain limitations when applying them to IoTs for rare-event detection.
First, by the nature of the problem, the training data corresponding to the
rare event is sparse and might not be suitable for DNNs. Even if it were
possible to train a DNN for deployment, a compressed or a quantized variant of
DNN, like the Binary Neural Networks (BNNs)[9], has to be used to optimize
computational resources. Retraining BNNs on the IoT platform to account for
data and hardware drifts is challenging due to quantization effects. Full-
precision training is not possible due to limited computational resources.
Also, if the parameters of the DNNs can be quantized, the input features
cannot be significantly quantized without affecting the classification
accuracy. K-Nearest Neighbour (KNN) [10][11] and Support vector machines
(SVMs) [12][13], on the other hand, have been shown to generalize well with
sparse training data [14] . However, SVM is more robust to outliers, and the
convexity of SVM training ensures any recalibration is interpretable and
stable. There have been many instances where SVMs have performed well as an
acoustic classifier [15] [16]. In literature, several approaches have been
proposed to reduce the computational and memory footprint of SVMs
[17][18][19][20]. However, in these platforms, computing features and
classification are generally treated independently, both during training and
inference.

In this paper, we present an in-filter computing framework that exploits the
computing and nonlinear primitives in the feature extraction process to design
ultra-light IoT acoustic classifiers. The approach is motivated by the fact
that acoustic front-ends like the neuromorphic cochlea [21] can be designed to
be highly computationally efficient using different degrees of linear and
nonlinear transformations. Our goal is to systematically exploit and map these
nonlinear transformations into the kernel functions used in SVMs, such that
both classification and feature extraction are co-optimized for training and
inference. This results in a template-based SVM [22][23] architecture that has
an ultra-low computational footprint for inference and training. This feature
not only relaxes communication bandwidth requirements on the IoT system but
also allows recalibration (retraining) to account for statistical drifts. The
main advantage of using template-based SVM is the ability of the framework to
use arbitrary functions without any restriction on its properties, like
positive-definite kernels for traditional SVM. This allows us to use hardware-
friendly mapping or functions that need not be specified in a closed-form,
such as using an ordinary differential equation (ODE). This property is
beneficial especially for hardware implementation, where the inherent
nonlinearity of the device can be used as a kernel rather than engineering a
specific nonlinearity. As a proof-of-concept, we have applied the in-filter
computing framework using an acoustic feature extractor based on Cascade of
Asymmetric Resonators with Inner Hair Cells (CAR-IHC) [21] [24] [25]. The CAR-
IHC model exhibits inherent nonlinearity and hence performs well as a kernel
for classification. We believe that our proposed framework has the following
key advantages:

  * •

A template-based SVM architecture that allows an arbitrary function to be used
as a kernel, unlike a conventional SVM that requires a positive-definite
kernel.

  * •

Combining the feature extraction and SVM kernel into one function makes the
system ultra-light and computationally efficient.

  * •

The memory footprint of the proposed system is user-defined and can be
specified based on the IoT hardware constraints.

  * •

A novel fast training algorithm with reduced training complexity in terms of
memory and computational complexity.

  * •

A system that can scale without affecting significant hardware changes due to
the time-multiplexing approach allows the framework to deploy for more complex
tasks.

As proof of concept IoT implementation, we have implemented this inference
framework on Xilinx Spartan 7 series FPGA [26], a low-cost and low-power FPGA.
We have validated our architecture on various auditory datasets such as the
environmental sound dataset [27] and speech-based dataset[28].

The rest of this paper is organized as follows. In Section II, a brief
discussion of related work is provided, followed by section III, where we
present the modified template-based SVM algorithm and explain the uniqueness
of the formulation. In Section IV, we explain the novel training algorithm
used for our framework. Section V provides the FPGA implementation details.
Section VI provides results obtained with an audio based dataset for detection
and surveillance applications. Section VII concludes this paper and provides
some useful applications, and discusses possible future work using this
framework.

Figure 2: Architecture of : (a) traditional SVM based acoustic classifier (b)
proposed in-filter SVM framework

##  II Related Work

Hardware implementations of SVM using FPGAs have been successfully achieved
over the years with high accuracy and the least possible area and power.
Binary classifications or even multi-class classifications using a Modified
One-against-all (M-OAA) approach for SVMs have been implemented on FPGAs [29]
[30]. Since the kernel is one of the most important parts of the SVM
algorithm, the kernel function consumes maximum resources in implementation.
This is demonstrated in [17] with linear and nonlinear SVM implementations on
FPGA. The authors show that nonlinear kernel implementations use more
resources than linear kernels. However, at the same time, there is a drop in
accuracy by more than 10% when using a linear kernel compared to a nonlinear
kernel. The authors implement a kernel with parallel inputs enabling high
operating frequency but at the cost of high resource utilization in terms of
LUTs and DSPs. This shows that in order to get good classification, we require
a nonlinear kernel, but at the same time, we need to achieve hardware
efficiency for an ultra-light implementation.

Regarding acoustic feature extraction, acoustic signals require a certain
amount of pre-processing to extract the salient features before it is used for
classification. One such FPGA-based approach is detailed in [18]. The authors
use (Discrete Wavelet Transforms) DWTs for feature extraction from a given
audio signal. This DWT feature extraction forms the input to a standard SVM
having a Radial Basis Function (RBF) kernel, which is nonlinear. This
classification system is used for phoneme recognition using data from the
TIMIT dataset. Due to hardware constraints and the complexity of the DWT
algorithm, the authors chose to implement only the SVM classifier on the FPGA.
The acoustic signals are pre-processed using a software implementation of the
DWT algorithm and are provided as inputs to the SVM hardware. This
implementation has the disadvantage of offline software feature extraction,
making the hardware incapable of using unprocessed acoustic signals as inputs.
At the same time, the SVM hardware implementation consumes a high number of
FPGA resources in terms of LUTs and DSPs. Also, the weights and support
vectors from the SVM training are stored in external ROMs. This makes the
implementation impractical for a small IoT-based edge device.

Furthermore, time-series data need not always be a speech signal, and there
may be cases where we may need to classify non-auditory time-series signals.
In [19], authors use Mel-frequency cepstral coefficients (MFCC) technique to
extract salient features from pulmonary sounds to detect wheezing using
standard SVM classification. Here, MFCC, as well as SVM, was implemented on
FPGA. This implementation provided an end-to-end solution on hardware that
could classify between a normal and an abnormal pulmonary sound. In this
implementation, MFCC itself is a resource-heavy algorithm, and additional
hardware is required for the SVM classifier to be implemented. Also, ROMs
store support vectors and weights along with additional registers to store
MFCC coefficients. The MFCC coefficient calculations, which are being done on
hardware, also contribute to high DSP usage. The authors have demonstrated
their hardware capability using only a 6 kHz input sampling frequency, making
the hardware limited in terms of the flexibility of signals that it can
process. Hence, such a system cannot be used in an IoT edge device due to the
high resource utilization and rigidity.

Another representative example of an IoT for acoustic classification is a
speaker identification system used in security systems. One such system was
realized on FPGAs in [20]. Similar to the implementation in [19], the authors
implemented an SVM classifier with MFCC as the feature extractor on hardware.
The input data was sampled at 8 kHz, making it resource-efficient, but at the
same time, it was less flexible in terms of processing signals of higher
sampling frequency. External SRAM was used to store the MFCC coefficients and
training parameters. Despite having a slight improvement in terms of hardware
efficiency compared to the previous implementation, this implementation lacked
flexibility and still had a significant amount of resource usage, given the
hardware constraints applicable for an IoT device.

Our framework addresses all the shortcomings of prior works by having a
neuromorphic cochlea-based CAR-IHC kernel integrated inside a template-based
SVM system. This kernel exhibits nonlinearity for better classification and,
at the same time, inherently provides a robust feature extracting capability
in order to get a good classification. This kernel has multiple tunable
parameters which can be adjusted to get the best feature extraction depending
on the application. The template-based SVM provides the flexibility of
choosing the right number of templates as support vectors, which can be tuned
as per the application. This avoids the additional requirement for the storage
of support vectors. Flexibility, scalability, low resource usage, and low
power make this framework ultra-light and ideal for IoT deployment for many
applications.

##  III Template-based SVM and In-filter Computing Formulation

Rooted in statistical learning theory, an SVM minimizes the structural risk by
maximizing a classification margin over a set of training samples[31]. In the
case of acoustic classification where the input is a time-series signal, one
can define a data vector (for training and inference) at a time-instant $n$ as
$X_{n}\in\mathbb{R}^{W}$ constructed using a window
$X_{n}=\\{x_{n},x_{n-1},...,x_{n-W+1}\\}$ of previous $W$ samples of the
signal $x_{n}$. An SVM based binary classifier produces a decision label
$y_{n}\in\\{+1,-1\\}$ corresponding to the data vector $X_{n}$ according to

$y_{n}={\rm sgn}(f(X_{n})).$ (1)

where $f:\mathbb{R}^{W}\rightarrow\mathbb{R}$ is given by

$f(X_{n})=\sum\limits_{s=1}^{S}\alpha_{s}y_{s}K(X_{s},X_{n})+b.$ (2)

Here $X_{s}\in\mathbb{R}^{W}$ is a subset of the training vector called
support vectors with their a-priori known decision labels
$y_{s}\in\\{+1,-1\\}$.
$K:\mathbb{R}^{W}\times\mathbb{R}^{W}\rightarrow\mathbb{R}$ is a positive-
definite kernel function that is also chosen a-priori and plays an important
role in implementing nonlinear decision functions. $\alpha_{s}\in\mathbb{R}$
and $b$ are training parameters, corresponding to the support vector $x_{s}$
and is determined by solving a standard quadratic program based training
procedure[31]. Note that the memory requirements to implement an SVM inference
engine in hardware is proportional to the number of support vectors $S$, and
hence in literature numerous techniques exist to reduce $S$ using heuristic
methods [32] [33].

For conventional SVM-based acoustic classifiers[34], as shown in Fig.2, the
raw input signal is pre-processed by a feature extraction module or function
$\Psi:\mathbb{R}^{W}\rightarrow\mathbb{R}^{D}$ before providing as an input to
the SVM kernel. $D$ is the feature dimension. The eq.(2) can be re-expressed
as

$f(X_{n})=\sum\limits_{s=1}^{S}\alpha_{s}y_{s}K(\Psi(X_{s}),\Psi(X_{n}))+b.$
(3)

In literature, the kernel function $K(.,.)$ and the feature extraction
function $\Psi(.)$ are typically chosen independently. As a result, the memory
footprint $S$ of the SVM is determined by the complexity of the problem and
the discriminative power of feature extraction. Note that a typical acoustic
feature extraction function $\Psi(.)$, itself comprises several nonlinear
transformations that could directly be used as SVM kernels. However, for the
SVM formulation to be valid, the nonlinear transformations must be mapped to a
positive-definite kernel. In [22] we reported a mechanism to design SVMs using
arbitrary template functions within a fixed memory footprint. The approach
expressed the kernel in eq.(2), as an outer-product over $P$ template
functions $\Phi_{p}:\mathbb{R}^{W}\rightarrow\mathbb{R}$ as
$K(X_{s},X_{n})=\sum\limits_{p=1}^{P}\Phi_{p}(X_{s})\Phi_{p}(X_{n}),$. The
template functions $\Phi_{p}(\cdot),p=1,..,P$ then could represent $P$ feature
extraction modules. Following the derivations in [22], the SVM function $f(.)$
can be rewritten as:

$\displaystyle f(X_{n})=\sum\limits_{p=1}^{P}Q_{p}\Phi_{p}(X_{n})+b.$ (4)

Here, $Q_{p}=\sum\limits_{s=1}^{S}\alpha_{s}y_{s}\Phi_{p}(X_{s})$ can be
viewed as a consolidated training parameter that can be estimated using a
reduced-complexity training procedure described in section IV. Note that the
memory footprint of the reformulated template-SVM is determined by the number
of template functions $P$, and each of the template functions
$\Phi_{p}(\cdot)$ could be chosen arbitrarily. Here, $\Phi_{p}(\cdot)$ can be
any function that can be used to express the input features in order to make a
classification. This makes this framework flexible to implement various
functions for classification. For example, in Fig.2, we illustrate how a
cascade of filters a CAR-IHC feature extraction module could be used to
implement $\Phi_{p}(\cdot)$, and described in the following section.

Figure 3: Neuromorphic cochlea-based CAR-IHC model

###  III-A CAR-IHC model as SVM Kernel

The biological cochlea is a nonlinear and causal system. This nonlinearity
makes it ideal to use a cochlea model as an SVM kernel since it would give
robust classification in higher dimensional space [35]. One such auditory
filter model is the Cascade of Asymmetric Resonators with Fast-Acting
Compression (CARFAC) model, which is a digital version of the cascade of pole-
zero filter [21][25] [36]. It consists of CAR block, which mimics Basilar
Membrane (BM) functionality, IHC, Ganglion cells, and Outer Hair Cell (OHC).
We use the CAR and IHC modules of this model for our kernel.

Given $X_{n}$ with $W$ samples and each sampled input data as $x_{n}$
described previously. The system receives an audio sample at each sampling
clock which is fed to the first CAR block $H_{1}$. There are $P$ CAR blocks
arranged in a cascaded manner as shown in Fig.3. The eq.(5) denotes the two
pole two zero filter which mimics BM implemented as a CAR filter.

$H_{p}=H(z)=g_{p}\left[\frac{z^{2}+(-2a_{0,p}+k_{p}c_{0,p})r_{p}z+r_{p}^{2}}{z^{2}-2a_{0,p}r_{p}z+r_{p}^{2}}\right].$
(5)

$a_{0,p},c_{0,p},k_{p}$ are the resonator filter coefficients for each filter,
$r_{p}$ is the pole-zero radius in the z-plane, and $g_{p}$ is the DC gain
factor. The CAR block transfer function $H(z)$ is derived in detail in
Appendix.

For the first CAR filter, the input is $x_{n}$, and due to the cascade nature
of the CAR filters, the output of one filter becomes the input of the next
stage filter. The output of each CAR filter is denoted by $b_{p,n}$ as shown
in Fig.3, which forms the input to the IHC blocks in parallel. We use a
simplified model of the IHC implemented using half wave rectifier (HWR),
$HWR(\cdot)\in\mathbb{R}$, as per eq.(6).

$HWR(q)=max(0,q).$ (6)

From Fig.3, $q=b_{p,n}$ in eq.(6), which gives,

$d_{p,n}=HWR(b_{p,n}).$ (7)

The IHC generates output as per eq.(7). The IHC output is summed over $W$
samples, and this forms the input for the standardization (std) blocks in
parallel.

$s_{p}=\sum\limits_{n=1}^{W}d_{p,n}.$ (8)

Here, $s_{p}\in\mathbb{R}$.

$std(S_{p,i})=\frac{S_{p,i}-\mu_{p}}{\sigma_{p}}.$ (9)

where $\\{s_{p}\in S_{p,i}|1\leq i\leq N\\}$ with $N$ as the training samples,
$\mu_{p}=mean(S_{p,1},S_{p,2},..,S_{p,N})$ and
$\sigma_{p}$ =$\sqrt{\frac{1}{N-1}\sum\limits_{i=1}^{N}(S_{p,i}-\mu_{p})^{2}}$

$\Phi_{p}=std(S_{p,i}).$ (10)

Here, $\Phi_{p}\in\mathbb{R}$. The summation over $W$ samples of the output of
IHC is taken as per eq.(8) for each filter. Then standardization technique,
commonly used in neural network optimizations [37], is applied across $N$
training input samples as per eq.(9). Note that $\mu_{p}$ and $\sigma_{p}$ are
calculated only during training, and these vectors are passed as learned
parameters to the inference engine. Therefore, an input signal vector $X_{n}$
sampled at a sampling frequency $f_{s}$ generates $W$ samples with each sample
denoted as $x_{n}$. It is then processed by a cascade parallel arrangement of
neuromorphic cochlea-based CAR-IHC filters to estimate the kernel function
$\Phi_{p}$ with ${p}$ as the filter stage out of $P$ filters as per (10). The
output is a $P\times 1$ kernel vector, as shown in Fig.3. The classification
output is produced using eq.(4) employing this kernel vector, the output
weight vector $Q\in\mathbb{R}^{P}$, and the bias $b\in\mathbb{R}$ obtained
after the training process.

##  IV Template-SVM Training

A conventional SVM training involves solving a quadratic optimization problem
over a set of training data $(X_{m},y_{m}),m=1,..,M$, of size $M$ [31]. The
optimization can be expressed as:

$\displaystyle\underset{\alpha_{m}}{min}\;\frac{1}{2}\sum\limits_{m=1}^{M}\sum\limits_{n=1}^{M}\alpha_{m}\alpha_{n}y_{m}y_{n}K(X_{n},X_{m})-\sum\limits_{m=1}^{M}\alpha_{m}.$
(11)
$\displaystyle\sum\limits_{m=1}^{M}\alpha_{m}y_{m}=0.\qquad\qquad\qquad\quad$
(12) $\displaystyle 0\leq\alpha_{m}\leq C.\qquad\qquad\;\;\qquad\quad$ (13)

Here, $C$ is a hyper-parameter that is chosen through cross-validation and
$n=1,..,M$. Due to the quadratic nature of the optimization problem, the
worst-case complexity of SVM training scales as $\mathcal{O}(M^{2})$. In
practice, the training complexity scales as $\mathcal{O}(MS)$, where $S$ is
the number of support vectors. However, the number of support vectors is
unknown, so any SVM formulation has to accommodate the worst-case scenario.

In the template-based SVM, the kernel is expressed as an outer-product over a
set of $P$ templates as
$K(X_{n},X_{m})=\sum\limits_{p=1}^{P}\Phi_{p}(X_{n})\Phi_{p}(X_{m})$.
Substituting in equation (11), the template-SVM training reduces to a lower
complexity quadratic optimization problem as:

$\displaystyle\underset{Q_{p},\alpha_{m}}{min}\;\frac{1}{2}\sum\limits_{p=1}^{P}Q_{p}^{2}-\sum\limits_{m=1}^{M}\alpha_{m}.$
(14) $\displaystyle
s.t\;\;Q_{p}=\sum\limits_{j=1}^{M}\alpha_{j}y_{j}\Phi_{p}(X_{j}).\;$ (15)
$\displaystyle\sum\limits_{m=1}^{M}\alpha_{m}y_{m}=0.\qquad\quad$ (16)
$\displaystyle 0\leq\alpha_{m}\leq C.\qquad\quad\;\;$ (17)

Here, $j$ is iterated over $M$ training samples. Equations eq.(15), (16), (17)
are the constraints imposed on eq.(14). The equation eq.(14) shows that the
optimization complexity has been reduced to $\mathcal{O}(P+M)$ with $P$
additional constraints that can be controlled based on the number of templates
required for the application. This reduced complexity enables us to use this
training algorithm to implement IoT devices, making them adaptive and
deployable in dynamic environments. Thus, our framework is capable of online
training. For the in-filter computing, the features or templates are computed
as an input stream. Training the template SVM entails solving a simplified
constrained quadratic problem. Thus, the architecture can be trained in an
online manner. However, in a traditional SVM, the features would first need to
be accumulated and fed to a kernel module. Note that there are several ways to
efficiently solve the constrained optimization in eq.(14), including both
batch and online variants. In the accompanying software for template-based
SVM[23], we have used a growth-transform-based approach [38] to solve eq.(14).

Figure 4: High level hardware execution flow diagram of the framework.

##  V FPGA Implementation of our SVM classifier

We demonstrate the efficiency of our in-filter computing architecture by
implementing it on FPGA. This system is configurable as the in-filter
parameters, and weight vectors are tunable based on the application. The
weight vector and biases are trained offline, as mentioned in the previous
sections. Initially, we simulated this framework in floating-point in MATLAB
software tool. In order to estimate the appropriate FPGA implementation, we
simulated the model in fixed-point code. The CAR-IHC kernel is implemented in
a 12-bit fixed-point code. The trained weights ($Q_{p})$ and bias ($b$) are
stored as 8-bit, and the mean ($\mu_{p}$) and standard deviation
($\sigma_{p})$ are stored as 12-bits. In our experiments, we analyzed that
using 12-bits for inputs, filter coefficients and standardization parameters
(mean and standard deviation) with 8-bits for weights and bias resulted in
minimal accuracy degradation and reduced hardware resource utilization. We use
pipelining to speed up the kernel execution in FPGA. The CAR-IHC kernel
filters are executed using the time-multiplexed technique where each filter
uses the same hardware for generating output which makes the design small in
area.

TABLE I: FPGA implementation summary. | FPGA Implementation Summary
---
Spartan-7 xc7s6cpga196
Clock Frequency | 25 MHz
Audio Sampling Frequency | 16 kHz
Number of Filters | 30
Dynamic Power | 8 mW
DSPs | 4 (Total 10)
LUTs | 1517 (Total 3750)
FFs | 2864 (Total 7500)
TABLE II: Comparison of architecture and resource utilization of related work.

[t]  Related Work Mahmoodi, et al. [17] Cutajar, et al. [18] Boujelben, et.
al. [19] Ramos-Lara et al. [20] This work FPGA Virtex4-xc4vsx35 Virtex-II
XC2V3000 Artix-7 XC7A100T Spartan 3 XCS2000 Spartan 7 xc7s6cpga196 Operating
Frequency 151.286 MHz 42.012 MHz 101.74 MHz 50 MHz 25 MHz Input Sampling
Frequency NA1 16 kHz 6 kHz 8 kHz 16 kHz Flip Flop 11589 1576 17074 5351 2864
LUTs 9141 11943 16563 6785 1517 RAM (18 Kb) 99 NA1 4 NA1 0 DSP 81 64 87 21 4
Kernel type Gaussian RBF Linear RBF CAR-IHC Feature extraction NA1 DWT
(Offline) MFCC (On-Chip) MFCC (On-chip) CAR-IHC (On-chip)

  * 1

These works did not report this entity for their designs.

Figure 5:  Block diagram for FPGA implementation

Figure 6:  CAR-IHC kernel hardware micro-architecture implementation. The
filter coefficients ($a_{0},c_{0},r,k,g$) stored in block RAM and the input
sampled at $f_{s}$ are used as inputs to this block. This block is used in
pipelined manner using select lines to enable the FSM for the design.

Fig.4 shows the hardware execution flow, and Fig.5 shows the FPGA block
architecture of the system. The input sample ($x_{n}$) sampled at $f_{s}$ is
provided as input to CAR block and filter coefficients ($a_{0},c_{0},r,k,g$),
stored in the FCMEM memory block. The CAR block performs filtering as per
eq.(5) followed by the IHC block, which performs half wave rectification as
per eq.(6). The detailed implementation of the CAR-IHC block is shown in
Fig.6. The time-multiplexed processing of each cascaded filter is determined
by the $CAR$ $Done$ select signal. If this signal is set, then the next sample
($x_{n+1}$) enters the CAR block. This signal is set only when the processing
of all the filters is done. Hence, the same CAR block is used to process $P$
filters using the stored filter coefficients for each filter ($p$). The $sel1$
and $sel2$ select lines control this filter coefficient flow. So the input to
output delay is directly proportional to the number of cascaded blocks, $P$ in
this case, i.e., the number of filters used. The multipliers have been
designed to operate in a pipelined manner, where multiplication of the
coefficients will take multiple clock cycles for producing the output by
reusing the hardware. The half wave rectification operation in the IHC block
depends on the Most Significant Bit ($MSB$) select signal, determining the
sign bit of CAR output ($b_{p,n}$). This results in discarding the values
below zero to produce the rectified output.

The output of IHC block ($d_{p,n}$) is summed over the entire window of the
input data, i.e., summation of $W$ input samples across $p$ filters.
Standardization of the output ($s_{p}$) of summation is performed based on
eq.(9) in the STD block. The size of $s_{p}$ is 26-bit due to 16000 ($f_{s}$)
additions, and the standardization parameters, i.e., mean ($\mu_{p}$) and
standard deviation ($\sigma_{p}$), are 12-bits, which are fetched from the
SMEM memory block. We quantize $s_{p}$ to 12-bit and further perform another
level of quantization to 8-bit after the standardization operation.
Heuristically, we found that using these multiple quantization levels achieves
the lowest possible hardware resource utilization without impacting the
classification accuracy. The output of standardization block ($\Phi_{p}$) is
now used to perform multiply-accumulate (MAC) operation with the learned
weights ($Q_{p}$), which are also 8-bits, stored in the WMEM memory block.
Finally, the classified output is obtained after the bias ($b$), stored in the
BMEM memory block, is added to the output of the MAC block, as per eq.(4).

Our system uses a 25 MHz system clock, and 16-bit input sampled at a 16 kHz
audio sampling rate. We have used 30 filters in all the reported results here.
However, it is parameterizable and can be changed based on the application
requirements. Every input data sample takes about 300 clock cycles, i.e.,
12$\mu$s, to be executed through the pipeline. We have an audio input sample
being given to the system every 1562 clock cycles, i.e., 62.5$\mu$s. We have a
buffer of around 1200 clock cycles, i.e., about 48$\mu$s, before the arrival
of next sample, where the, system is idle. This shows that we can increase the
sampling frequency to 80 kHz, i.e., sample at every 312 clock cycles without
impacting the hardware architecture. On the other hand, for a sample of 16 kHz
frequency, we can also increase the number of filters to up to 120 to use up
the extra 48$\mu$s. The number of clock cycles required to execute a single
audio sample increases linearly with the number of filters. There is an
increase of about 2.5 % of area in overall hardware and 0.23 mW increase in
power for every addition of filter. The increase in the number of filters may
be required for complex auditory tasks. We can also reduce the operating
frequency to as low as 400 kHz to reduce the power consumption of the system
to below 1 mW. We can reduce it further to a few kHz if we reduce the input
sampling frequency to a few Hz in other time-series data such as EEG/ECG
signals. Hence, this shows that our system is highly flexible and scalable to
suit any application in the time-series domain.

TABLE III: ESC-10 dataset classification accuracy results in percent. The fixed point code consists of 16-bit inputs and 8-bit weights and biases. Number of filters for our work is fixed at 30.  Classes (Train/Test) | Traditional SVM | In-filter SVM (This Work)
---|---|---
Support Vectors | Floating pt. | Floating pt. | Fixed pt.
Train | Test | Train | Test | Train | Test
Dog (129/33) | 51 | 88 | 87 | 89 | 90 | 89 | 87
Rain (119/40) | 60 | 89 | 87 | 89 | 87 | 82 | 82
Sea_Waves (200/50) | 113 | 86 | 82 | 84 | 78 | 80 | 74
Crying Baby (144/49) | 58 | 91 | 83 | 91 | 87 | 93 | 85
Clock Tick (114/50) | 86 | 91 | 86 | 92 | 88 | 92 | 85
Person Sneeze (101/44) | 43 | 83 | 77 | 89 | 82 | 82 | 80
Helicopter (197/50) | 48 | 94 | 88 | 96 | 90 | 95 | 85
Chainsaw (99/34) | 39 | 92 | 85 | 93 | 85 | 93 | 82
Rooster (124/54) | 36 | 92 | 94 | 93 | 96 | 93 | 96
Fire Crackling (152/66) | 46 | 90 | 89 | 90 | 87 | 89 | 86
TABLE IV: FSDD classification accuracy results in percent. The fixed point code consists of 16-bit inputs and 8-bit weights and biases. Number of filters for our work is fixed at 30. Classes (Train/Test) | Traditional SVM | In-filter SVM (This Work)
---|---|---
Support Vectors | Floating pt. | Floating pt. | Fixed pt.
Train | Test | Train | Test | Train | Test
Theo (761/254) | 247 | 92 | 91 | 93 | 91 | 89 | 88
Nicolas (889/297) | 197 | 98 | 98 | 98 | 97 | 97 | 94
Ywewelver (749/250) | 196 | 92 | 90 | 94 | 91 | 89 | 88
Jackson (796/200) | 35 | 99 | 99 | 99 | 99 | 99 | 98

We use Xilinx Spartan series part xc7s6cpga196, a low-power FPGA manufactured
on a 28 nm technology node.The Spartan series FPGA caters to edge computing
and IoT platform systems, as the area footprint and power envelope are low.
The dynamic power consumption for our system on FPGA is 8 mW. The logic design
i.e., LUTs and registers, consumes around 2 mW of power, whereas the control
signals take up 4 mW. DSPs consume 1 mw of power, and the clocks take up the
remaining 1 mw of power. The weights and bias generated from the training
procedure are quantized to 8-bit, and the CAR-IHC model uses 16-bit input
samples to generate 16-bit output. This 16-bit kernel output data on
accumulation over 16k samples increases to 30-bit, which is then reduced to
8-bit after a standardization and quantization operation. For this design, 16
kHz sampling rate with 30 filters uses 1517 LUTs and 2864 FFs summarized in
Table I. This exhibits that the system can be implemented with minimum area
and low power and hence suitable for IoT deployable edge devices.

The resource utilization comparison contrasts related work with our system, as
shown in Table II. Our work has the advantage of being low in resource
utilization compared to other works. Most of these systems use acoustic
signals as input, with MFCC as the feature extractor and SVM as the
classification algorithm. MFCC is a widely used feature extractor for acoustic
signals since it extracts linearly separable features amongst most acoustic
signals. Our framework uses a neuromorphic cochlea-based kernel that acts as a
feature extractor as well as a nonlinear kernel for the SVM algorithm. This
avoids the need for a separate feature extractor compared to other works. Our
framework also does not require separate storage for support vectors, and at
the same time, we have control over the number of weights that have to be
stored based on the required application. Another advantage our system has
over other systems is that it is highly tunable and is scalable for higher or
lower sampling frequency input signals.

##  VI Results and Discussion

We use datasets from two domains, namely speech and environmental sounds.
Speech datasets prove the usability of our framework in security like voice-
based access, where we can identify the speaker and provide biometric access.
The environmental sounds showcase the framework’s versatility which can be
deployed for multiple sounds as the target for robust classification. We use
MATLAB for software simulations and verification of the algorithm. The FPGA
design implements the MATLAB code using fixed-point arithmetic.

Environmental Sounds Classification (ESC-10) dataset [27] consists of sound
clips constructed from recordings publicly available through the Freesound
project. It consists of 400 environmental recordings with 10 classes, i.e., 40
clips per class and 5 seconds per clip. Each class contains 40 wav format
audio files. These clips had a lot of silence, so we trimmed the silence part
and further trimmed the remaining clips into 1 second version belonging to the
same class, thus increasing the dataset’s number of samples. Table III shows
the class labels, which depict the wide variety of data samples used. The
classes include sounds from dog bark, rain, sea waves, crying baby, clock
ticking, person sneezing, helicopter, chainsaw, crawing rooster and fire
crackling. Here, the dataset was used to create balanced classes to identify
one class versus other classes arranged randomly. The train and test accuracy
values are shown with the train-to-test ratio mentioned in the bracket. One
thing to note from these results is that with less amount of data too, our
framework could classify the sounds. We have compared our results with
traditional SVM, which uses inputs after being pre-processed using the same
CAR-IHC filters. For the traditional SVM, we use the in-built MATLAB library
with default command lines. The number of support vectors for traditional SVMs
is significantly higher than the number of filters used in our work,
indicating that we can get comparable accuracy with lower hardware resources
and can be used in low-powered devices. As the number of samples is low, we
see lower accuracy for classes like clock tick and person sneeze. These
classes have a lot of overlapping information with other classes, causing
confusion.

The Free Spoken Digit Dataset (FSDD) [28] is an open dataset consisting of
recordings of spoken digits in wav files at a sampling rate of 8 kHz. The
recordings are trimmed so that they have near minimal silence at the beginning
and ends. It consists of 4 speakers with 500 recordings per speaker, amounting
to an overall of 2000 recordings. These recordings are English pronunciations
of each digit from 0 to 9 by each speaker. We use our framework to identify
the speaker based on the recordings. We create recordings of each speaker
versus a random pool of remaining speakers. We can tune our system to each
speaker and get a classification to identify whether our target speaker is
speaking or not. Similar to ESC dataset results, FSDD results in Table IV also
show that traditional SVM requires many more support vectors than the number
of filters used in this work. As the number of support vectors is
significantly higher for traditional SVMs, we see a slight reduction in
accuracy for few classes in in-filter SVM. For the proposed in-filter SVM, the
number of template vectors is determined by the fixed number of filters. The
training algorithm tries to find the best possible solution within this fixed
constraint. However, adhering to this constraint is one of the reasons for the
reduction in accuracy. The other constraint with the proposed in-filter SVM
approach is that the final solution is linear for the CAR-IHC (filter)
function. Any nonlinear mapping is implemented only by the CAR-IHC function.
Whereas in a standard SVM formulation that uses the CAR-IHC filter output as
features, there is additional nonlinearity in the kernel mapping. Thus, the
traditional SVM may be able to exploit this cross-filter nonlinearity to
achieve better accuracy. FSDD classification showcases the capability of the
framework to identify the right person, which can be used in giving access to
a secure area or facility.

Figure 7: Impact of increasing the number of filters on accuracy for FSDD
(Yweweler) dataset.

We see from Tables III and IV that the number of support vectors ($S$) for the
traditional SVM is always greater than the number of templates, i.e., filters
($P$) used for the proposed SVM ($P<S$). Traditional SVM has a computational
complexity of $\mathcal{O}(S+MS)$, where $\mathcal{O}(MS)$ is the complexity
of a linear kernel. In contrast, the complexity of the proposed work is
$\mathcal{O}(P)$. Thus, the computational complexity of traditional SVM
increases with an increase in support vectors. We see from the results that
the number of filters for in-filter SVM is less than the support vectors used
in traditional SVM. As a case study, we take Yweweler class data from the FSDD
dataset. The number of MAC operations required to classify this class in
traditional SVM is 5096 compared to 30 MAC operations for in-filter SVM. We
know that MAC operations consume maximum resources and, in turn, would
increase the power consumption in any hardware design. Hence, our framework is
efficient in comparison to an equivalent SVM hardware implementation.

We can tune the number of filters based on the application. The number of
filters is determined by the trade-off between the hardware constraints
(memory and speed) versus accuracy. Empirically, we were able to determine the
optimum number of filters required for most datasets. Reducing the number of
filters reduces the discriminatory information encoded by the features, and
hence we observe a reduction in accuracy. We can tune the number of filters
based on the application. As seen in the added Fig.7, increasing the number of
filters beyond a specific value yields a marginal increase in accuracy. Thus,
this marginal increase in accuracy would come at the cost of latency and
increase in hardware resources, as explained in Section V. We chose 30 filters
to satisfy the constraints of our implemented design. Hence, the same number
of filters were used for the datasets. This shows that we can fix the number
of filters based on the constraints and still obtain comparable results.

We performed an experiment to check the classification robustness of our
framework. For this purpose, we added white Gaussian noise to the test input
signals from the existing dataset and observed the accuracies across different
Signal to Noise Ratios (SNRs). We used the MATLAB tool function $awgn$ to add
the white Gaussian noise to the signals. Fig.8 shows the mean and variance
plot of the test accuracy due to the addition of noise over 10 iterations.
Here, we see that our framework is quite robust when we train the data with
the added noise, and with no noise in training data, the test accuracy falls
below 80 % as we reduce the SNR to below 25 dB.

Figure 8: Impact of noise on test accuracy for FSDD (Yweweler) dataset. TABLE V: List of possible domains where our framework can be implemented by tuning the frequency ranges of the CAR filters. Time-series Data |  | Typical Frequency
---
Range (Hz)
Low | High
Speech [39] | 100 | 8k
Music [39] | 40 | 18k
Accelerometers [40] | 0.5 | 1.5k
ECG [41] | 0.1 | 1k
EEG [42] | 1 | 100
EMG [43] | 24 | 400

In general, we can see from the dataset results that our work produces
comparable results when the number of filters is close to the number of
support vectors used in traditional SVM. This shows that we can choose the
number of filters beforehand and arrive at an acceptable accuracy for the
required application without relying on the algorithm to decide this hardware
parameter. For each type of dataset, we need to tune the filter parameters for
efficient classification. We also need to determine the number of filters
used, i.e., the template vectors in SVM formulation based on multiple runs.
This makes our framework highly flexible and tunable as per the application’s
needs. In all our experiments, we have used 30 filters. The fixed point code
consists of 16-bit CAR-IHC kernel output generated using 16-bit input, and the
weight and bias are limited to 8-bit values. From the results across these
datasets, we see that our framework is good at identifying a person using
speech. Also, the ESC-10 dataset results exhibit the framework’s capability
even to classify inanimate sounds that can be used in systems where such
classifications can trigger a more fine-tuned action for corrective measures.
Hence, by tuning the CAR filters to a certain frequency range, we can classify
different time-series data as per Table V. Here, our framework can be
configured for a wide range of frequencies, enabling it to use various sensors
generating time-series data. This gives the flexibility of programming the
framework for a specific application. Also, by determining the number of
filters required for each type of classification, we can optimize the
classification accuracy for any time-series data.

##  VII Conclusion

In this paper, we have demonstrated our novel SVM-based acoustic classifier
using the cochlea module as kernel and feature extraction stage
simultaneously. The neuromorphic cochlea kernel of our unique algorithm does
not require the kernel to be positive definite. This lack of restriction
compared to traditional SVM enabled us to use cochlea-based CAR-IHC function
as a kernel in our framework. Furthermore, the proposed system has the
flexibility of handling different kinds of time-series data, as the kernel
filter parameters can be tuned as per the frequency range of the input signal.
This template-based SVM has a fixed number of templates in contrast to varying
support vectors in traditional SVMs. We can control the operating frequency by
controlling the number of kernel filters, making it power efficient. This can
be fine-tuned by matching the hardware constraints with the required
application speed. Also, since the complexity of this novel SVM is low
compared to traditional SVM, our framework is capable of performing online
training. This flexibility and dynamic behavior make the framework ideal for
implementing in IoT edge devices. In this paper, we have demonstrated the
hardware efficiency of the in-filter computing framework on FPGA. However,
this can be extended to create a custom hardware and used as a battery-powered
edge device. In the future, we plan to deploy this framework in different
environments as an edge classification device. We can use this algorithm on an
embedded system like a microcontroller for greater flexibility in programming
the device. Leveraging the reprogramability of our framework, we can build an
IoT system that can be used to monitor various time-series data using a
network of sensors placed at various locations for different applications. The
proposed system can have several potential applications ranging from
identifying animal behaviour pattern for ecologists using sensors placed in
strategic locations in a forest area to healthcare data analysis using
wearable sensors which provide time-series data like ECG or EEG data. Based on
bird species sounds or any animal sound, we can track the presence of
different rare species of wildlife in a particular environment over a period
of time. In this case, we can remotely reprogram the hardware to detect
different wildlife species as many of these species might not be active in a
specific region for a particular season. Similarly, such systems can also be
deployed for remote health care applications using signals like ECG/EEG or
ultrasound for early disease detection [44] [45] [46] and for automation of
industrial maintenance of machinery using various time-series data produced by
mounted sensors. All these deployments lead to minimizing human intervention
and reducing errors caused by logistics issues. Since our system can classify
rare events with very low power consumption, we can deploy this system as an
always-on system.

TABLE VI: List of Symbols and Acronyms. Symbols and Acronyms | Description
---|---
${\rm sgn}()$ | Sign function
$\sqrt{}$ | Square root function
$mean()$ | Arithmetic mean function for a series
$\underset{x}{min}f(x)$ | Find $x$ that minimizes the function $f(x)$
$s.t.$ | Such that
$\mathcal{O}()$ | Big O notation for complexity
FPGA | Field Programmable Gate Array
SVM | Support Vector Machine
CAR-IHC |  | Cascade of Asymmetric Resonator
---
with Inner Hair Cells
LUT | Look-Up Table
FF | Flip-Flop
UGS | Unattended Ground Sensor
DNN | Deep Neural Network
BNN | Binary Neural Network
KNN | K-Nearest Neighbour
M-OAA | Modified One-Against-All
DWT | Discrete Wavelet Transform
RBF | Radial Basis Function
DSP | Digital Signal Processing
ROM | Read Only Memory
MFCC | Mel-frequency cepstral coefficient
SRAM | Synchronous Random Access Memory
ODE | Ordinary Differential Equation
CARFAC |  | Cascade of Asymmetric Resonators
---
with Fast-Acting Compression
OHC | Outer Hair Cell
BM | Basilar Membrane
HWR | Half Wave Rectifier
MSB | Most Significant Bit
MAC | Multiply-Accumulate
EEG | Electroencephalography
ECG | Electrocardiography
ESC-10 | Environmental Sound Clips-10
FSDD | Free Spoken Digit Dataset
EMG | Electromyography
SNR | Signal to Noise Ratio

[CAR-IHC Filter Formulation]

A two pole two zero filter forms the asymmetric resonator whose transfer
function is as below:

$\displaystyle
H(z)=\frac{Y}{X}=g[\frac{(z-z_{zero})(z-z^{*}_{zero})}{(z-z_{pole})(z-z^{*}_{pole})}].$
$\displaystyle=g[\frac{z^{2}+(-2a_{0}+kc_{0})rz+r^{2}}{z^{2}-2a_{0}rz+r^{2}}].$
(18)

The two pole coupled form has a pair of conjugate poles ($z_{pole}$ and
$z^{*}_{pole}$):

$\displaystyle
z_{pole},z^{*}_{pole}=\frac{2a_{0}\pm\sqrt{(2a_{0}r)^{2}-4r^{2}}}{2}.$ (19)
$\displaystyle=rcos(\theta_{R})\pm irsin(\theta_{R}).$
$a_{0}=cos(\theta_{R}).$ (20)

where $\theta_{R}$ is the pole angle in the z plane. The conjugate zeros
($z_{zero}$ and $z^{*}_{zero}$) are:

$z_{zero},z^{*}_{zero}=\frac{-(-2a_{0}+kc_{0})r}{2}.\\\
\pm\frac{\sqrt{((-2a_{0}+kc_{0})r)^{2}-4r^{2}}}{2}.$ $=rcos(\theta_{Z})\pm
irsin(\theta_{Z}).$ $a_{0}-\frac{kc_{0}}{2}=cos(\theta_{Z}).$ (21)

where $\theta_{Z}$ is the zero angle in the z plane. The zero radius is the
same as the pole radius, $r$. The condition for complex zeros becomes relevant
for high-frequency channels, where $cos(\theta_{R})<0$:

$a_{0}-\frac{kc_{0}}{2}>-1.$ (22) $k<\frac{2+2a_{0}}{c_{0}}.$ (23)

Here, $r$ can be used to move the zeros and the poles simultaneously while $k$
is fixed. $k$ determines the distance between the poles and the zeros. The
frequency of zeros are kept slightly higher than the poles. If we increase the
value of $k$, the poles and zeros grow further apart, giving a slow roll off
at higher frequencies. On the other hand, if the value of $k$ is decreased to
a low value, the poles and zeros grow closer, giving rise to sharp roll off
making it asymmetric. This sharp roll off is similar to the characteristic
exhibited by auditory filtering. This property also enhances selection of
frequencies. In order to keep the pole frequency half octave below zero
frequency, $k$ is kept sames as $c_{0}$. To get unity gain at DC, we can solve
for g as follows:

$g=\frac{1-2a_{0}r+r^{2}}{1-(2a_{0}-kc_{0})r+r^{2}}.$ (24)

The zero-crossing times of the filter’s impulse response does not change with
respect to time, even when we change $r$.

$r=1-damping\times\frac{2\pi f}{f_{s}}.$ (25)

where $damping$ controls the damping factor, $f$ is defined in eq.(26) and
$f_{s}$ is the sampling frequency. $r$ keeps the damping away from zero and
also makes the damping bounded. Changing $r$ means varying the poles and the
zeros of the filter. This satisfies the biologically observed condition where
variation in stimulus level does not vary the impulse response zero crossings
[25]. For each cascade stage, the initial values for zeros and poles are set.
The Greenwood map function [47] is used to choose equidistant poles of the two
pole two zero resonator. These are placed along the normalized length of the
cochlea.

$f=165.4(10^{2.1x}-1).$ (26)

where, $f$ is the frequency of the pole and $x$ is the normalized position
along the cochlea, varying from 0 at the apex of the BM, to 1 at the basal
end.

## Acknowledgment

This research was supported in part by (i) INSPIRE faculty fellowship
(DST/INSPIRE/04/2016/000216), SPARC grant (SPARC/2018-2019/P606/SL) from
Ministry of Human Resource Development and IMPRINT Grant IMP/2018/000550 from
the Department of Science and Technology, India. The authors would like to
acknowledge the joint Memorandum of Understanding (MoU) between Indian
Institute of Science, Bangalore and Washington University in St. Louis for
supporting this research activity.

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