# Stability and Dynamics of Complex Order Fractional Difference Equations

Sachin Bhalekar<EMAIL_ADDRESS><EMAIL_ADDRESS>(Corresponding Author) Prashant M. Gade<EMAIL_ADDRESS>Divya Joshi
<EMAIL_ADDRESS>School of Mathematics and Statistics, University of
Hyderabad, Hyderabad, 500046 India  Department of Physics, RTM Nagpur
University, Nagpur

###### Abstract

We extend the definition of $n$-dimensional difference equations to complex
order $\alpha\in\mathbb{C}$. We investigate the stability of linear systems
defined by an $n$-dimensional matrix $A$ and derive conditions for the
stability of equilibrium points for linear systems. For the one-dimensional
case where $A=\lambda\in\mathbb{C}$, we find that the stability region, if any
is enclosed by a boundary curve and we obtain a parametric equation for the
same. Furthermore, we find that there is no stable region if this parametric
curve is self-intersecting. Even for $\lambda\in\mathbb{R}$, the solutions can
be complex and dynamics in one-dimension is richer than the case for
$\alpha\in\mathbb{R}$. These results can be extended to $n$-dimensions. For
nonlinear systems, we observe that the stability of the linearized system
determines the stability of the equilibrium point.

††journal:

##  1 Introduction

The generalization of various mathematical notions such as functions or even
operators has importance that goes beyond mathematical curiosity. The Gamma
function is a generalization of factorial for a real argument that is unique
under certain constraints [1]. It can even be generalized to complex arguments
[2]. It has been used in complex analysis, statistics, number theory, and even
string theory in physics. Similarly, the Riemann zeta function was introduced
by Euler for real argument and was later extended to complex argument [3]. It
has turned out to be an extremely important function in physics and
mathematics [4]. The notion of derivatives has even been extended to
functional derivatives [5]. Another example is the q-deformation of numbers
and functions [6]. It has found applications in quantum groups and statistical
physics [7]. Even q-derivatives and q-integrals have been defined. Of course,
these generalizations need not be unique and different generalizations can be
used in a different contexts.

One of the important generalizations of the concept of derivatives has been
fractional calculus [8]. Several definitions have been proposed for extending
the concept of fractional derivatives for real numbers. Some of them are
extended to complex numbers and even derivatives of fractional complex order
have been introduced. Differential equations of complex fractional order
(which should be distinguished from complex differential equations) have been
studied in the context of viscoelasticity, control systems, etc. Time domain,
frequency domain and stability analysis of linear systems represented by
differential equations with complex order derivative has been carried out [9].
It has found applications such as the design of controller for fractional-
order DC motor system [10], in PID controller and low-pass filter [11]. Such
systems have been found to have large stability regions for certain
parameters. The dynamic response of elastic foundations was modeled using
complex order differential equations and was useful in predicting the response
for various vibration modes over the entire frequency range of interest [12].
Particle swarm optimization is a well-studied optimization technique. It has
been used in several constraints. Complex order derivatives have found
application in this context as well [13]. It has been studied in discrete-time
control of linear and nonlinear systems [14]. In biophysics, atrial
fibrillation is an important research topic and a mathematical model based on
fractional-order complex derivatives has been recently proposed [15].
Fractional-order circuit theory has been popular in recent times. It has been
extended to fractional-order derivatives in circuit elements [16]. Thus
complex order differential equations have found applications in several
situations as mentioned before and the difference equations of complex order
can be useful in those contexts. As Oono and Puri pointed out ”Nature gives
physicists phenomena, not equations.” [17], new mathematical tools have always
found applications in a variety of fields and these difference equations can
be useful in numerous phenomena in nature.

Difference equations can be viewed as an attempt to solve the differential
equation by finite difference method and the notion of fractional order
difference equation has been introduced in this context [18]. We note that in
fields such as economics and biology, difference equations appear naturally in
modeling. Several dynamical phenomena obtained in differential equations are
seen in difference equations as well [19]. Many schemes for control of chaos
have been applicable to both differential equations as well as maps. The
notion of fractional order differential equation has been extended to
fractional order difference equation and few definitions have been proposed
[20]. Dynamics of linear and nonlinear systems have been investigated for
fractional-order difference equations [21] and even spatially extended
dynamical systems have been defined as well as investigated [22].

We are not aware of any attempt to define and study the difference equation of
complex fractional order. In this work, we define the difference equation of
complex fractional order in Caputo-like definition. We study the stability of
linear systems which is an important and useful starting point for
understanding the dynamics. We give stability conditions for linear systems
and results can be extended to higher dimensions without loss of generality.
Finally, we study nonlinear difference equations of complex order and
investigate their dynamics.

##  2 Preliminaries

###### Definition 2.1

(see [23]). The Z-transform of a sequence $\\{y(n)\\}_{n=0}^{\infty}$ is a
complex function given by

$Y(z)=Z[y](z)=\sum_{k=0}^{\infty}y(k)z^{-k}$

where $z\in\mathbb{C}$ is a complex number for which the series converges
absolutely.

###### Definition 2.2

(see [24, 25]). Let $h>0,\;a\in\mathbb{R}$ and
$(h\mathbb{N})_{a}=\\{a,a+h,a+2h,\ldots\\}$. For a function
$x:(h\mathbb{N})_{a}\rightarrow\mathbb{C}$, the forward h-difference operator
if defined as

$(\Delta_{h}x)(t)=\frac{x(t+h)-x(t)}{h},$

where t $\in(h\mathbb{N})_{a}$.

Throughout this article, we take $a=0$ and $h=1$. We write $\Delta$ for
$\Delta_{1}$. Now, we generalize the fractional order operators defined in
[23, 24, 25] to include the complex order $\alpha$.

###### Definition 2.3

For a function $x:(h\mathbb{N})_{a}\rightarrow\mathbb{C}$ the fractional h-sum
of order $\alpha=u+\iota v\in\mathbb{C},u>0$ is given by

$(_{a}\Delta_{h}^{-\alpha}x)(t)=\frac{h^{\alpha}}{\Gamma(\alpha)}\sum_{s=0}^{n}\frac{\Gamma(\alpha+n-s)}{\Gamma(n-s+1)}x(a+sh),\\\
$

where, $t=a+(\alpha+n)h,\;n\in\mathbb{N_{\circ}}$.

For $h=1$ and $a=0$ , we have

$\displaystyle(\Delta^{-\alpha}x)(t)$ $\displaystyle=$
$\displaystyle\frac{1}{\Gamma(\alpha)}\sum_{s=0}^{n}\frac{\Gamma(\alpha+n-s)}{\Gamma(n-s+1)}x(s)$
$\displaystyle=$
$\displaystyle\sum_{s=0}^{n}\left(\begin{array}[]{c}n-s+\alpha-1\\\ n-s\\\
\end{array}\right)x(s).$

Here, we used the generalized binomial coefficient

$\left(\begin{array}[]{c}\mu\\\ \eta\\\
\end{array}\right)=\frac{\Gamma(\mu+1)}{\Gamma(\eta+1)\Gamma(\mu-\eta+1)},\\\
\;\mu,\eta\in\mathbb{C},\;\text{Re}(\mu)>0,\;\text{and Re}(\eta)>0.$

If $n$ $\in$ $\mathbb{N_{\circ}}$ then

$\displaystyle\left(\begin{array}[]{c}\mu\\\
n\end{array}\right)=\frac{(\mu+1)}{n!\Gamma(\mu-n+1)}=\frac{\mu(\mu-1)\ldots(\mu-n-1)}{n!}.$

###### Definition 2.4

For $n\in\mathbb{N_{\circ}}$ and $\alpha=u+\iota v\in\mathbb{C},u>0,$ we
define

$\displaystyle\tilde{\phi}_{\alpha}(n)=\left(\begin{array}[]{c}n+\alpha-1\\\
n\\\ \end{array}\right)=(-1)^{n}\left(\begin{array}[]{c}-\alpha\\\
n\end{array}\right).$

Note: The convolution $\tilde{\phi}_{\alpha}*x$ of the sequences
$\tilde{\phi}_{\alpha}$ and $x$ is defined as

$\left(\tilde{\phi}_{\alpha}*x\right)(n)=\sum_{s=0}^{n}\tilde{\phi}_{\alpha}(n-s)x(s)$
$\therefore(\Delta^{-\alpha}x)(n)=(\tilde{\phi}_{\alpha}*x)(n)\\\ .$
$\therefore Z(\Delta^{-\alpha}x)(n)=Z\left(\tilde{\phi}(n)\right)Z(x(n))\\\
=(1-z^{-1})^{-\alpha}X(z),$

where $X$ is $Z$ transform of $x$.

###### Lemma 2.1

For $\alpha\in\mathbb{C},\;\text{Re}(\alpha)>0$,

$Z(\tilde{\phi}_{\alpha}(t))=\frac{1}{(1-z^{-1})^{\alpha}}.$

Proof: We have,

$\displaystyle Z(\tilde{\phi}_{\alpha}(t))$ $\displaystyle=$
$\displaystyle\sum_{j=0}^{\infty}\tilde{\phi}_{\alpha}(j)z^{-j}$
$\displaystyle=$
$\displaystyle\sum_{j=0}^{\infty}\left(\begin{array}[]{c}j+\alpha-1\\\
j\end{array}\right)z^{-j}$ $\displaystyle=$
$\displaystyle\sum_{j=0}^{\infty}(-1)^{j}\left(\begin{array}[]{c}-\alpha\\\
j\end{array}\right)z^{-j}$ $\displaystyle=$
$\displaystyle(1-z^{-1})^{-\alpha}.$

by using Newton’s generalization of Binomial Theorem [26, 27].

##  3 Stability Analysis

We consider the linear fractional order difference equation

$(\Delta^{\alpha}x)(t)=(A-I)x(t+\alpha-1),\\\ $ (6)

where
$t\in\mathbb{N}_{1-\alpha}=\\{1-\alpha,2-\alpha,3-\alpha,\ldots\\},\;\alpha\in\mathbb{C},\;\text{Re}(\alpha)\in(0,1),$
$x(t)\in\mathbb{C}^{n},\;A$ is $n\times n$ complex matrix and $I$ is $n\times
n$ identity matrix and $x(0)=x_{0}$.
Initial value problem (6) is equivalent to [28]

$x(t)=x_{0}+\frac{1}{\Gamma(\alpha)}\sum_{s=1+\alpha}^{t+\alpha}\frac{\Gamma(t-s)(A-I)x(s+\alpha-1)}{\Gamma(t-s-\alpha+1)}.$

Putting $s+\alpha-1=j$, we get

$\displaystyle x(t)$ $\displaystyle=$ $\displaystyle
x_{0}+\sum_{j=0}^{t-1}\frac{\Gamma(t-j+\alpha-1)}{\Gamma(\alpha)\Gamma(t-j)}(A-I)x(j)$
$\displaystyle=$ $\displaystyle x_{0}+(A-I)(\tilde{\phi}_{\alpha}*x)(t-1).$
$\therefore\;x(t+1)=x_{0}+(A-I)(\tilde{\phi}_{\alpha}*x)(t),\;t=0,1,2\ldots.$
(7)

If $Z(x(t))=X(z),$ then $Z(x(t+1))=zX(z)-zx_{0}$.
Taking Z-transform of (7), we get

$zX(z)-zx_{0}=\frac{x_{0}}{1-z^{-1}}+(A-I)\frac{1}{(1-z^{-1})^{\alpha}}X(z)\\\
.$

provided, $|z|>1$ [23].

$\therefore[z(1-z^{-1})^{\alpha}I-(A-I)]X(z)\\\ =z(1-z^{-1})^{\alpha-1}x_{0},$

where $|z|>1$.
We can solve this equation for $X(z)$ if the matrix
$(z(1-z^{-1})^{\alpha}I-(A-I))$ is invertible matrix, i.e. if
$det(z(1-z^{-1})^{\alpha}I-(A-I))\neq 0$ $\forall$ $z$ with $|z|>1$ [29, 30].
Therefore, we have following theorem.

###### Theorem 3.1

The zero solution of (6) or (7) is asymptotically stable if and only if all
the roots of $det(z(1-z^{-1})^{\alpha}I-(A-I))=0$ satisfy $|z|<1$.

###  3.1 Sketching the boundary of stable region

Without loss, we can assume that the matrix $(A-I)$ is diagonal. Suppose that
($\lambda-1$) is an arbitrary entry on the diagonal. For stability, all the
roots of characteristic equation

$z(1-z^{-1})^{\alpha}-(\lambda-1)=0$ (8)

should satisfy $|z|<1$. On the boundary of stable region, we must have
$z=e^{\iota t}$, $0\leq t\leq 2\pi$. Therefore, the characteristic equation
(8) becomes $e^{\iota t}(1-e^{-\iota t})^{\alpha}=(\lambda-1)$. Therefore, we
have

$\lambda=2^{\alpha}\left(\sin\frac{t}{2}\right)^{\alpha}e^{\iota\left[\frac{\alpha\pi}{2}+t\left(1-\frac{\alpha}{2}\right)\right]}+1.$

Therefore, the parametric representation of boundary curve is

$\displaystyle\gamma(t)=\left(Re[2^{\alpha}\left(\sin\frac{t}{2}\right)^{\alpha}e^{\iota\left[\frac{\alpha\pi}{2}+t\left(1-\frac{\alpha}{2}\right)\right]}]+1,Im[2^{\alpha}\left(\sin\frac{t}{2}\right)^{\alpha}e^{\iota\left[\frac{\alpha\pi}{2}+t\left(1-\frac{\alpha}{2}\right)\right]}]\right),\;t\in[0,2\pi].$
(9)

If all the eigenvalues of matrix A lie inside this simple closed curve
$\gamma(t)$ then the system will be asymptotically stable.

###  3.2 Condition for simple curve

###### Theorem 3.2

The curve $\gamma(t)$ defined by (8) is simple curve for $\alpha=u+\iota v$,
$u\in(0,1)$ if and only if $0<v<\sqrt{2u-u^{2}}$.

Proof: We have

$\beta(t)=e^{\iota t}(1-e^{-\iota t})^{\alpha},$

where $\alpha=u+\iota v$. There is self intersection or cusp in this
parametric curve if and only if $\exists$ $t_{1}\neq t_{2}$ such that

$\displaystyle\beta(t_{1})$ $\displaystyle=$
$\displaystyle\beta(t_{2}),\;t_{1},t_{2}\in[0,2\pi].$ $\displaystyle\iff
e^{\iota t_{1}}(1-e^{\iota t_{1}})^{u+\iota v}$ $\displaystyle=$
$\displaystyle e^{it_{2}}(1-e^{\iota t_{2}})^{u+\iota v}$
$\displaystyle\iff\left(\sin\frac{t_{1}}{2}\right)^{u}e^{\frac{vt_{1}}{2}}e^{\iota[v\log\sin\frac{t_{1}}{2}+t_{1}(1-\frac{u}{2})]}$
$\displaystyle=$
$\displaystyle\left(\sin\frac{t_{2}}{2}\right)^{u}e^{\frac{vt_{2}}{2}}e^{\iota[v\log\sin\frac{t_{2}}{2}+t_{2}(1-\frac{u}{2})]}$
and $\displaystyle
v\log\left(\left(\sin\frac{t_{1}}{2}\right)+t_{1}\left(1-\frac{u}{2}\right)\right)$
$\displaystyle=$ $\displaystyle
v\log\left(\left(\sin\frac{t_{2}}{2}\right)+t_{2}\left(1-\frac{u}{2}\right)\right)+2k\pi,\;k\in\mathbb{Z}$
$\displaystyle\iff\frac{v^{2}}{2}$ $\displaystyle=$
$\displaystyle\left(1-\frac{u}{2}\right)u-2k\pi u$ $\displaystyle\therefore
v^{2}$ $\displaystyle=$ $\displaystyle(2-u)u-4k\pi u$ $\displaystyle\iff v$
$\displaystyle=$ $\displaystyle\sqrt{(2+4k\pi)u+u^{2}}$

This is non real if $k=-1,-2,-3,\ldots$. Further $v$ is minimum if $k=0$ and
we have $v=\sqrt{2u-u^{2}}$ or $(u-1)^{2}+v^{2}=1$. Therefore, if
$0<v<\sqrt{2u-u^{2}}\;$ then $\gamma(t)$ is a simple curve. This completes the
proof of theorem.
Observation:

Figure 1: Region for simple curve

Observation: If there exists multiple points i.e. if $v>\sqrt{2u-u^{2}}$ then
the system is unstable for all eigenvalues.

##  4 Illustrative Examples

In this section we verify the stability results described in the previous
section.

###### Example 4.1

Figure 2: Stable region for $\alpha=e^{\frac{\iota\pi}{4}}$

(a)

Figure 3: Example 1 stable solution for $f(x)=(0.2+0.5\iota)x$

(a)

Figure 4: Example 1 unstable solution for $f(x)=(0.1-2\iota)x$

(a)

Figure 5: Example 1 unstable solution for $f(x)=(0.1-2\iota)x$

We take $\alpha=e^{\frac{\iota\pi}{4}}=0.7071(1+\iota)$. The stable region for
this $\alpha$ is sketched in Figure 2. Consider the $1$-dimensional system (7)
with $f(x)=(0.2+0.5\iota)x$. In this case, $\lambda=0.2+0.5\iota$ is inside
the stable region of this system as shown in Figure 5. On the other hand, the
eigenvalue of $\lambda=0.1-2\iota)x$ lies outside the stable region. The
unstable trajectory of this system is traced in Figures 5 and 5.

###### Example 4.2

Figure 6: Multiple curve for $\alpha=0.4+0.9\iota$

(a)

Figure 7: Example 2 unstable trajectory for $\lambda=1.1-0.1\iota$

(a)

Figure 8: Example 2 unstable solution for $\lambda=1.1-0.1\iota$

(a)

Figure 9: Example 2 unstable solution for $\lambda=0.5+0.1\iota$

Now we take $\alpha=0.4+0.9\iota$. Here $v>\sqrt{2u-u^{2}}$. Therefore, the
boundary curve $\gamma(t)$ will have multiple points as shown in Figure 6. In
this case, we observe the unstable solutions for all eigenvalues. We take
$f(x)=\lambda x$ with $\lambda=1.1-0.1\iota,-0.1-0.1\iota$ and $0.5+0.1\iota$
inside and outside the curve $\gamma(t)$. The unstable solutions for these
values are sketched in Figures 9, 9 and 9 respectively.

##  5 Nonlinear System

We consider,

$x(t)=x_{0}+\frac{1}{\Gamma(\alpha)}\sum_{j=0}^{t-1}\frac{\Gamma(t-j+\alpha-1)}{\Gamma(t-j)}[f(x(j))-x(j)],\;t=1,2,\ldots.$
(10)

where $x(t)\in\mathbb{C}^{n}$ and $f:\mathbb{C}^{n}\rightarrow\mathbb{C}^{n}$
is continuously differentiable. A steady state solution $x_{*}$ of (10) is a
complex number satisfying $f(x_{*})=x_{*}$.

Following definitions are generalizations of those given in [31, 32].

###### Definition 5.1

We say that $x_{*}$ is stable if for each $\epsilon>0,\;\exists\;\delta>0$
such that $\|x_{0}-x_{*}\|<\delta$ $\implies\;\|x(t)-x_{*}\|<\epsilon$,
$t=1,2,\ldots$

###### Definition 5.2

Equilibrium point $x_{*}$ is asymptotically stable if it is stable and
$\exists\;\delta>0$ such that
$\|x_{0}-x_{*}\|<\delta\;\implies\lim_{t\rightarrow\infty}x(t)=x_{*}$.

Note: If $x_{*}$ is not stable then it is unstable.
The linearization of nonlinear system (10) in the neighborhood of $x_{*}$ is
given by

$x(t)=x_{0}+\sum_{j=0}^{t-1}\frac{\Gamma(t-j+\alpha-1)}{\Gamma(\alpha)\Gamma(t-j)}(A-I)x(j)$
(11)

where $A=f^{\prime}(x_{*})$ is the Jacobian matrix.

The local stability properties of equilibrium point $x_{*}$ of (10) are same
as those of linearization (11) i.e. the equilibrium point $x_{*}$ is
asymptotically stable if all the eigenvalues of Jacobian $A$ lie inside the
stable region.

###  5.1 Complex order logistic map

We consider

$f(x)=\lambda x(1-x),$

where $\lambda>0$.
The logistic map of complex order $\alpha$, with Re$(\alpha)>0$ is given by

$x(t)=x_{0}+\sum_{j=0}^{t-1}\frac{\Gamma(t-j-\alpha-1)}{\Gamma(\alpha)\Gamma(t-j)}[f(x(j))-x(j)],\;t=1,2,\ldots.$
(12)

Equilibrium points of (12) are $x_{1*}=0$ and
$x_{2*}=(\frac{\lambda-1}{\lambda})$. Linearization of (12) in the
neighborhood of equilibrium $x_{*}$ is given by

$x(t)=x_{0}+\sum_{j=0}^{t-1}\frac{\Gamma(t-j-\alpha-1)}{\Gamma(\alpha)\Gamma(t-j)}[(f^{\prime}(x_{*})-1)x(j)].$
(13)

Stability of $x_{1*}=0$:
Here $f^{\prime}(x_{1*})=f^{\prime}(0)=\lambda$. Therefore, the equilibrium
point $x_{1*}$ is asymptotically stable if $\lambda$ is inside the stable
region bounded by (9).
Stability of $x_{2*}=0$:
In this case,

$\displaystyle f^{\prime}(x_{2*})$ $\displaystyle=$
$\displaystyle\lambda-2\lambda\left(\frac{\lambda-1}{\lambda}\right)$
$\displaystyle=$ $\displaystyle 2-\lambda.$

Therefore, $x_{2*}$ is asymptotically stable if ($2-\lambda$) lies inside the
stable region bounded by (9).

Figure 10: Stability region for logistic map

(a)

Figure 11: $x_{1*}$ is asymptotically stable for $\lambda=-0.1$

(a)

Figure 12: $x_{2*}$ is unstable for $\lambda=-0.1$

(a)

Figure 13: $x_{1*}$ is unstable for $\lambda=1.5$

(a)

Figure 14: $x_{2*}$ is asymptotically stable for $\lambda=1.5$

Let us take $\alpha=0.8+0.7\iota$. The stable region is given in Figure 10.
Therefore, $x_{1*}$ is asymptotically stable if
$\lambda\in(-0.2774,0.6432)\cup(1,1.0754)$ and the $x_{2*}$ is asymptotically
stable if $\lambda\in(0.9246,1)\cup(1.3568,2.2774)$. For $\lambda=-0.1$,
$x_{1*}$ is asymptotically stable whereas $x_{2*}=11$ is unstable. In Figure
14, we take $x_{0}=0.3$ where the trajectory converges to $x_{1*}$. We take
$x_{0}=10.2$ in Figure 14 and show that the trajectory is diverging. The
equilibrium point $x_{1*}$ is unstable for $\lambda=1.5$ whereas
$x_{2*}=0.3333$ is asymptotically stable. This result is verified in Figures
14 and 14 by selecting appropriate initial conditions viz. $x_{0}=0.3$ and
$x_{0}=-0.1$ respectively.

###  5.2 Two-dimensional system

Now, we consider two-dimensional system

$\displaystyle
x(t)=x_{0}+\frac{1}{\Gamma(\alpha)}\sum_{j=0}^{t-1}\frac{\Gamma(t-j+\alpha-1)}{\Gamma(t-j)}[f_{1}(x(j),y(j))-x(j)],$
$\displaystyle
y(t)=y_{0}+\frac{1}{\Gamma(\alpha)}\sum_{j=0}^{t-1}\frac{\Gamma(t-j+\alpha-1)}{\Gamma(t-j)}[f_{2}(x(j),y(j))-y(j)],$
(14) $\displaystyle t=1,2,\ldots$

where, $f_{1}(x,y)=\lambda x(y+1)+\mu(x^{2}+1)y$ and $f_{2}(x,y)=\lambda
y(x+1)-\mu(y+1)^{2}x.$ Clearly, origin $(0,0)$ is an equilibrium point. The
Jacobian matrix at origin

$\displaystyle J=\left[\begin{array}[]{cc}\lambda&\mu\\\
\\-\mu&\lambda\end{array}\right]$

Figure 15: Stable region for $2$-dimensional system (14)

(a)

Figure 16: Stable orbits for $(\lambda,\mu)=(-0.2,0.1)$

(a)

Figure 17: Unstable orbits for $(\lambda,\mu)=(-0.2,0.5)$

has eigenvalues $\lambda\pm\iota\mu$. For $\alpha=0.7+0.4\iota$, the stable
region is shown in Figure 15. For the value $(\lambda,\mu)=(-0.2,0.1)$, both
the eigenvalues $-0.2\pm 0.1\iota$ of $J$ lie inside the stable region. The
stable orbits $|x(j)|$ and $|y(j)|$ are shown in Figure 17. If we set
$(\lambda,\mu)=(-0.2,-0.5)$ then the eigenvalue $-0.2-0.5\iota$ lies inside
the stable region whereas the eigenvalue $-0.2+0.5\iota$ lies outside the
stable region. Therefore, the equilibrium is unstable. The unstable orbits are
shown in Figure 17.

##  6 Results and Conclusion

We have used Z-transform for carrying out stability analysis of equilibrium
points. This technique is fairly general. Stability for the linear system is
achieved if the zero solutions of the system are asymptotically stable i.e.,
the roots of equation (8) satisfy $|z|<1$. If all the roots of the system lie
within the boundary curve defined by (9) then it is considered to be
asymptotically stable otherwise, unstable. We have demonstrated that this
result can be extended to non-linear systems using the logistic map. For a
non-linear system, the first step is to linearize it around its equilibrium
points. These points are asymptotically stable if the eigenvalues of the
Jacobian matrix lie inside the stable region. For higher dimensions, we
considered the example of a two-dimensional system and found similar results
for stability analysis. All the eigenvalues of the Jacobian matrix must lie
inside the curve for the equilibrium point to be stable as seen in the
examples. This criterion is very similar to one obtained for integer-order
difference equations though the stability region is very different.

However, we note that the qualitative dynamics is dissimilar in many contexts.
For example, for $x_{n+1}=\lambda x_{n}$, we observe monotonic decrease or
increase for $\lambda\in\mathbb{R}$. On the other hand, trajectories can
spiral in or out even for $\lambda\in\mathbb{R}$ for complex order difference
equations even for real initial conditions. It is possible that these
difference equations can lead to stable limit cycles or quasi-periodic cycles
in nonlinear systems even in $1$-dimension and it is of interest in the future
if it is possible in one or higher dimensions. It has been shown that periodic
orbits do not exist for fractional-order differential equations for real order
[33]. Complex order differential has been studied in several contexts as
mentioned in the introduction. In viscoelastic systems, some researchers have
demanded that the output should be real for real input, and hence the dynamics
is governed by the sum of two complex order differential equations of complex
conjugate order [34]. Our work can be easily extended in this direction and
the stable region, in this case, would be the intersection set of the two
difference equations.

This work is mainly motivated by mathematical curiosity and we have
generalized the notion of difference equations to complex order. The limited
studies in nonlinear systems do not show chaos or quasi-periodic cycles. It
would be interesting to further examine whether the system shows chaos under
certain conditions.

##  7 Acknowledgment

P. M. Gade thanks DST-SERB for financial assistance (Ref. EMR/2016/006686 and
CRG/2020/003993). S. Bhalekar acknowledges the Science and Engineering
Research Board (SERB), New Delhi, India for the Research Grant (Ref.
MTR/2017/000068) under Mathematical Research Impact Centric Support (MATRICS)
Scheme and the University of Hyderabad for Institute of Eminence-Professional
Development Fund (IoE-PDF) by MHRD (F11/9/2019-U3(A)).

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# Depth of factors of square free monomial ideals

Dorin Popescu  Dorin Popescu, ”Simion Stoilow” Institute of Mathematics of
Romanian Academy, Research unit 5, P.O.Box 1-764, Bucharest 014700, Romania
<EMAIL_ADDRESS>
###### Abstract.

Let $I$ be an ideal of a polynomial algebra over a field, generated by $r$
square free monomials of degree $d$. If $r$ is bigger (or equal, if $I$ is not
principal) than the number of square free monomials of $I$ of degree $d+1$,
then $\operatorname{depth}_{S}I=d$. Let $J\subsetneq I$, $J\not=0$ be
generated by square free monomials of degree $\geq d+1$. If $r$ is bigger than
the number of square free monomials of $I\setminus J$ of degree $d+1$, or more
generally the Stanley depth of $I/J$ is $d$, then
$\operatorname{depth}_{S}I/J=d$. In particular, Stanley’s Conjecture holds in
theses cases.
Key words  : Monomial Ideals, Depth, Stanley depth.
2000 Mathematics Subject Classification: Primary 13C15, Secondary 13F20,
13F55, 13P10.

The support from grant ID-PCE-2011-1023 of Romanian Ministry of Education,
Research and Innovation is gratefully acknowledged.

## Introduction

Let $S=K[x_{1},\ldots,x_{n}]$ be the polynomial algebra in $n$ variables over
a field $K$, $d$ a positive integer and $I\supsetneq J$, be two square free
monomial ideals of $S$ such that $I$ is generated in degrees $\geq d$,
respectively $J$ in degrees $d+1$. Let $\rho_{d}(I)$ be the number of all
square free monomials of degree $d$ of $I$. It is easy to note (see Lemma 1.1)
that $\operatorname{depth}_{S}I/J\geq d$. Our Theorem 2.2 gives a sufficient
condition which implies $\operatorname{depth}_{S}I/J=d$, namely this happens
when

$\rho_{d}(I)>\rho_{d+1}(I)-\rho_{d+1}(J).$

Suppose that this condition holds. Then the Stanley depth of $I/J$ (see [11],
[2], or here Remark 2.6) is $d$ and if Stanley’s Conjecture holds then
$\operatorname{depth}_{S}I/J\leq d$, that is the missing inequality. Thus to
test Stanley’s Conjecture means to test the equality
$\operatorname{depth}_{S}I/J=d$, which is much easier since there exist very
good algorithms to compute $\operatorname{depth}_{S}I/J$ but not so good to
compute the Stanley depth of $I/J$. After a lot of examples computed with the
computer algebra system SINGULAR we understood that a result as Theorem 2.2 is
believable. The above condition is not necessary as Example 2.4 shows.
Necessary and sufficient conditions could be possible found classifying some
posets (see Remark 2.5) but this is not the subject of this paper.

The proof of Theorem 2.2 uses the Koszul homology and can be read without
other preparation. Our first section gives easy proofs in special cases, but
they are mainly an introduction in the subject. Remarks 1.7, 1.9 show that the
Koszul homology seems to be the best tool in our problems. Section $2$ starts
with one example, where we give the idea of the proof of Theorem 2.2.

If $I$ is generated by more (or equal, if $I$ is not principal) square free
monomials of degree $d$ than ${n\choose d+1}$, or more general than
$\rho_{d+1}(I)$, then $\operatorname{depth}_{S}I=d$ as shows our Corollary
3.4. This extends [9, Corollary 3], which was the starting point of our
research, the proof there being easier. Remark 3.5 says that the condition of
Corollary 3.4 is tight.

The conditions above are consequences of the fact that
$\operatorname{sdepth}I/J=d$, as we explained in Remark 2.6, and we saw that
they imply $\operatorname{depth}I/J=d$. But what happens if we just suppose
that $\operatorname{sdepth}I/J=d$? Then there exists a monomial square free
ideal $I^{\prime}\subset I$ such that
$\rho_{d}(I^{\prime})>\rho_{d+1}(I^{\prime})-\rho_{d+1}(I^{\prime}\cap J)$
using our Theorem 4.1 (somehow an extension of [10, Lemma 3.3]) and it follows
also $\operatorname{depth}_{S}I/J=d$ by our Theorem 4.3.

We owe thanks to a Referee who found gaps in a preliminary version of our
paper.

##  1\. Factors of square free monomial ideals

Let $J\subsetneq I$, be two nonzero square free monomial ideals of $S$ and $d$
a positive integer. Let $\rho_{d}(I)$ be the number of all square free
monomials of degree $d$ of $I$. Suppose that $I$ is generated by square free
monomials $f_{1},\ldots,f_{r}$, $r>0$, of degrees $\geq d$ and $J$ is
generated by square free monomials of degree $\geq d+1$. Set
$s:=\rho_{d+1}(I)-\rho_{d+1}(J)$ and let $b_{1},\ldots,b_{s}$ be the square
free monomials of $I\setminus J$ of degree $d+1$.

######  Lemma 1.1.

$\operatorname{depth}_{S}I,\ \operatorname{depth}_{S}I/J\geq d$.

###### Proof.

By [2, Proposition 3.1] we have $\operatorname{depth}_{S}I\geq d$,
$\operatorname{depth}_{S}J\geq d+1$. The conclusion follows from applying the
Depth Lemma in the exact sequence $0\rightarrow J\rightarrow I\rightarrow
I/J\rightarrow 0$.

######  Lemma 1.2.

Suppose that $J=E+F$, $F\not\subset E$, where $E,F$ are ideals generated by
square free monomials of degree $d+1$, respectively $>d+1$. Then
$\operatorname{depth}_{S}I/J=d$ if and only if
$\operatorname{depth}_{S}I/E=d$.

###### Proof.

We may suppose that in $E$ there exist no monomial generator of $F$. In the
exact sequence

$0\rightarrow J/E\rightarrow I/E\rightarrow I/J\rightarrow 0$

we see that the first end is isomorphic with $F/(F\cap E)$ and has depth $\geq
d+2$ by Lemma 1.1. Apply the Depth Lemma and we are done.

Before trying to extend the above lemma it is useful to see the next example.

######  Example 1.3.

Let $n=4$, $d=1$, $I=(x_{2})$, $E=(x_{2}x_{4})$, $F=(x_{1}x_{2}x_{3})$. Then
$\operatorname{depth}_{S}I/E=3$ and $\operatorname{depth}_{S}I/(E+F)=2$.

######  Lemma 1.4.

Let $H$ be an ideal generated by square free monomials of degrees $d+1$. Then
$\operatorname{depth}_{S}I/J=d$ if and only if
$\operatorname{depth}_{S}(I+H)/J=d$.

###### Proof.

By induction on the number of the generators of $H$ it is enough to consider
the case $H=(u)$ for some square free monomial $u\not\in I$ of degrees $d+1$.
In the exact sequence

$0\rightarrow I/J\rightarrow(I+(u))/J\rightarrow(I+(u))/I\rightarrow 0$

we see that the last term is isomorphic with $(u)/I\cap(u)$ and has depth
$\geq d+1$ by Lemma 1.1, since $I\cap(u)$ has only monomials of degrees
$>d+1$. Using the Depth Lemma the first term has depth $d$ if and only if the
middle has depth $d$, which is enough.

Using Lemmas 1.2, 1.4 we may suppose always in our consideration that $I$, $J$
are generated in degree $d$, respectively $d+1$, in particular $f_{i}$ have
degree $d$.

######  Lemma 1.5.

Let $e\leq r$ be a positive integer and $I^{\prime}=(f_{1},\ldots,f_{e})$,
$J^{\prime}=J\cap I^{\prime}$. If
$\operatorname{depth}_{S}I^{\prime}/J^{\prime}=d$ then
$\operatorname{depth}_{S}I/J=d$ .

###### Proof.

In the exact sequence

$0\rightarrow I^{\prime}/J^{\prime}\rightarrow I/J\rightarrow
I/(I^{\prime}+J)\rightarrow 0$

the right end has depth $\geq d$ by Lemma 1.1 because

$I/(I^{\prime}+J)\cong(f_{e+1},\ldots,f_{r})/((J+I^{\prime})\cap(f_{e+1},\ldots,f_{r})))$

and $(J+I^{\prime})\cap(f_{e+1},\ldots,f_{r})$ is generated by monomials of
degree $>d$. If the left end has depth $d$ then the middle has the same depth
by the Depth Lemma.

######  Lemma 1.6.

Suppose that there exists $i\in[r]$ such that $f_{i}$ has in $J$ all square
free multiples of degree $d+1$. Then $\operatorname{depth}_{S}I/J=d$.

###### Proof.

We may suppose $i=1$. By our hypothesis $J:f_{1}$ is generated by $(n-d)$
variables. If $r=1$ then the depth of $I/J\cong S/(J:f_{1})$ is $d$. If $r>1$
apply the above lemma for $e=1$.

######  Remark 1.7.

Suppose in the proof of the above lemma that $f_{1}=x_{1}\cdots x_{d}$. Then
the hypothesis says that $(x_{d+1},\ldots,x_{n})f_{1}\subset J$. It follows
that $z=f_{1}e_{\sigma_{1}}$, $e_{\sigma_{1}}=e_{d+1}\wedge\ldots\wedge e_{n}$
induces a nonzero element in the Koszul homology module $H_{n-d}(x;I/J)$ of
$I/J$ (some details from Koszul homology theory are given in Example 2.1).
Thus $\operatorname{depth}_{S}I/J\leq d$ by [1, Theorem 1.6.17], the other
inequality follows from Lemma 1.1. This gives a different proof of the above
lemma using stronger tools, which will be very useful in the next section. We
also remind that
$H_{n-d}(x;I/J)\cong\operatorname{Tor}_{n-d}^{S}(K,I/J)\not=(0)$ gives
$\operatorname{pd}_{S}I/J\geq n-2$, which means
$\operatorname{depth}_{S}I/J\leq 2$ by Auslander-Buchsbaum Theorem [1, Theorem
1.3.3].

######  Lemma 1.8.

Suppose that $r\geq 2$ and the least common multiple $b=[f_{1},f_{2}]$ has
degree $d+1$ and it is the only monomial of degree $d+1$ which is in
$(f_{1},f_{2})\setminus J$. Then $\operatorname{depth}_{S}I/J=d$.

###### Proof.

Apply induction on $r\geq 2$. Suppose that $r=2$. By hypothesis the greatest
common divisor $u=(f_{1},f_{2})$ have degree $d-1$ and after renumbering the
variables we may suppose that $f_{i}=x_{i}u$ for $i=1,2$. By hypothesis the
square free multiples of $f_{1},f_{2}$ by variables $x_{i}$, $i>2$ belongs to
$J$. Thus we see that $I/J$ is a finite module over a polynomial ring in
$(d+1)$-variables and we get $\operatorname{depth}_{S}I/J\leq d$ since $I/J$
it is not free. Now it is enough to apply Lemma 1.1. If $r>2$ then apply Lemma
1.5 for $e=2$.

######  Remark 1.9.

We see in the proof of the above lemma (similarly as in Remark 1.7) that if
$u=x_{n-d+2}\cdots x_{n}$ then $z=f_{1}e_{\sigma_{1}}-f_{2}e_{\sigma_{2}}$,
$e_{\sigma_{1}}=e_{2}\wedge\ldots\wedge e_{n-d+1}$,
$e_{\sigma_{2}}=e_{1}\wedge e_{3}\wedge\ldots\wedge e_{n-d+1}$ induces a
nonzero element in $H_{n-d}(x;I/J)$. Thus $\operatorname{depth}_{S}I/J\leq d$
again by [1, Theorem 1.6.17].

######  Proposition 1.10.

Let $b_{1},\ldots,b_{s}$ be the monomials of degree $d+1$ from $I\setminus J$.
Suppose that $r>s$ and for each $i\in[r]$ there exists at most one $j\in[s]$
with $f_{i}|b_{j}$. Then $\operatorname{depth}_{S}I/J=d$.

###### Proof.

If there exists $i\in[r]$ such that $f_{i}$ has in $J$ all square free
multiples of degree $d+1$, then we apply Lemma 1.6. Otherwise, each $f_{i}$
has a square free multiple of degree $d+1$ which is not in $J$. By hypothesis,
there exist $i,j\in[r]$, $i\not=j$ such that $f_{i},f_{j}$ have the same
multiple $b$ of degree $d+1$ in $I\setminus J$. Now apply the above lemma.

######  Corollary 1.11.

Suppose that $r>s\leq 1$. Then $\operatorname{depth}_{S}I/J=d$.

######  Proposition 1.12.

Suppose that $r>s=2$. Then $\operatorname{depth}_{S}I/J=d$.

###### Proof.

Using Lemma 1.5 for $e=3$ we reduce to the case $r=3$. By Lemma 1.6 we may
suppose that each $f_{i}$ divides $b_{1}$, or $b_{2}$. By Proposition 1.10 we
may suppose that $f_{1}|b_{1}$, $f_{1}|b_{2}$, that is $f_{1}$ is the greatest
common divisor $(b_{1},b_{2})$. Assume that $f_{2}|b_{1}$. If $f_{2}|b_{2}$
then we get $f_{2}=(b_{1},b_{2})=f_{1}$, which is false. Similarly, if
$f_{3}|b_{1}$ then $f_{3}\not|b_{2}$ and we may apply Lemma 1.8 to
$f_{2},f_{3}$. Thus we reduce to the case when $f_{3}|b_{2}$ and
$f_{3}\not|b_{1}$. We may suppose that $b_{1}=x_{1}f_{1}$, $b_{2}=x_{2}f_{1}$
and $x_{1},x_{2}$ do not divide $f_{1}$ because $b_{i}$ are square free. It
follows that $b_{1}=x_{i}f_{2}$, $b_{2}=x_{j}f_{3}$ for some $i,j>2$ with
$x_{i},x_{j}|f_{1}$.

Case $i=j$

Then we may suppose $i=j=3$ and $f_{1}=x_{3}u$ for a square free monomial $u$
of degree $d-1$. It follows that $f_{2}=x_{2}u$, $f_{3}=x_{1}u$. Let
$S^{\prime}$ be the polynomial subring of $S$ in the variables
$x_{1},x_{2},x_{3}$ and those dividing $u$. Then for each variable
$x_{k}\not\in S^{\prime}$ we have $f_{i}x_{k}\in J$ and so $I/J\cong
I^{\prime}/J^{\prime}$, where $I^{\prime}=I\cap S^{\prime}$, $J^{\prime}=J\cap
S^{\prime}$. Changing from $I,J,S$ to $I^{\prime},J^{\prime},S^{\prime}$ we
may suppose that $n=d+2$ and $u=\Pi_{i>3}^{n}x_{i}$. Then
$I/J\cong(I:u)/(J:u)\cong(x_{1},x_{2},x_{3})S/(x_{1}x_{2})S$. Then
$\operatorname{depth}_{S}I/J=d-1+\operatorname{depth}_{T}(x_{1},x_{2},x_{3})T/(x_{1}x_{2})$.
By Lemma 1.2 it is enough to see that
$\operatorname{depth}_{T}(x_{1},x_{2},x_{3})T/(x_{1}x_{2})T=1$.

Case $i\not=j$

Then we may suppose $i=3$, $j=4$ and $f_{1}=x_{3}x_{4}v$ for a square free
monomial $v$ of degree $d-2$. It follows that
$f_{2}=x_{1}f_{1}/x_{3}=x_{1}x_{4}v$, $f_{3}=x_{2}f_{1}/x_{4}=x_{2}x_{3}v$.
Let $S^{\prime\prime}$ be the polynomial subring of $S$ in the variables
$x_{1},x_{2},x_{3},x_{4}$ and those dividing $v$. As above $I/J\cong
I^{\prime\prime}/J^{\prime\prime}$, where $I^{\prime\prime}=I\cap
S^{\prime\prime}$, $J^{\prime\prime}=J\cap S^{\prime\prime}$. Changing from
$I,J,S$ to $I^{\prime\prime},J^{\prime\prime},S^{\prime\prime}$ we may suppose
that $n=d+2$ and $v=\Pi_{i>4}^{n}x_{i}$. Then

$I/J\cong(I:v)/(J:v)\cong(x_{1}x_{4},x_{2}x_{3},x_{3}x_{4})S/(x_{1}x_{2}x_{3},x_{1}x_{2}x_{4})S.$

Then

$\operatorname{depth}_{S}I/J=d-2+\operatorname{depth}_{T^{\prime}}(x_{1}x_{4},x_{2}x_{3},x_{3}x_{4})T^{\prime}/(x_{1}x_{2}x_{3},x_{1}x_{2}x_{4})T^{\prime}.$

By Lemma 1.2 it is enough to see that

$\operatorname{depth}_{T^{\prime}}(x_{1}x_{4},x_{2}x_{3},x_{3}x_{4})T^{\prime}/(x_{1}x_{2}x_{3},x_{1}x_{2}x_{4})T^{\prime}=2.$

######  Lemma 1.13.

Suppose that $d=1$, $f_{i}=x_{i}$, $i\in[r]$ and $b_{j}\in
S^{\prime}=K[x_{1},\ldots,x_{r}]$ for all $j\in[s]$. Then
$\operatorname{depth}_{S}I/J=1$ independently of $r,s$ ($s$ may be greater
than $r$).

###### Proof.

Set $I^{\prime}=I\cap S^{\prime}$ and $J^{\prime}=J\cap S^{\prime}$. Then
$\operatorname{depth}_{S^{\prime}}S^{\prime}/I^{\prime}=0$ and
$\operatorname{depth}_{S^{\prime}}S^{\prime}/J^{\prime}>0$ by Lemma 1.1. From
the following exact sequence

$0\rightarrow I^{\prime}/J^{\prime}\rightarrow
S^{\prime}/J^{\prime}\rightarrow S^{\prime}/I^{\prime}\rightarrow 0$

it follows that $\operatorname{depth}_{S^{\prime}}I^{\prime}/J^{\prime}=1$ by
the Depth Lemma. If $r<n$ then note that $(x_{r+1},\ldots,x_{n})I\subset J$
and so
$\operatorname{depth}_{S}I/J=\operatorname{depth}_{S}(I^{\prime}S/J^{\prime}S)-(n-r)=\operatorname{depth}_{S^{\prime}}I^{\prime}/J^{\prime}=1$.

######  Proposition 1.14.

Suppose that $d=1$ and $r>s$. Then $\operatorname{depth}_{S}I/J=1$.

###### Proof.

By Lemma 1.13 we may suppose that $I=(x_{1},\ldots,x_{r})$ with $r<n$. Using
Lemma 1.6 we may suppose that each $x_{i}$, $i\in[r]$ divides a certain
$b_{k}$. Apply induction on $s$, the case $s\leq 2$ being done in Proposition
1.12. Assume that $s>2$. We may suppose that each $b_{k}$ is a product of two
different $x_{i}$, $i\in[r]$ because if let us say $b_{s}$ is just a multiple
of one $x_{i}$, $i\in[r]$, for example $x_{r}$, then we may take
$I^{\prime}=(x_{1},\ldots,x_{r-1})$, $J^{\prime}=J\cap I^{\prime}$ and we get
$\operatorname{depth}_{S}I^{\prime}/J^{\prime}=1$ by induction hypothesis on
$s$ since $r-1>s-1$, that is $\operatorname{depth}_{S}I/J=1$ by Lemma 1.5. But
if each $b_{k}$ is a product of two different $x_{i}$, $i\in[r]$ we see that
$b_{j}\in S^{\prime}=K[x_{1},\ldots,x_{r}]$ for all $j\in[s]$ and we may apply
again Lemma 1.13.

##  2\. Main result

We want to extend Proposition 1.14 for the case $d>1$. Next example is an
illustration of our method.

######  Example 2.1.

Let $n=6$, $d=2$, $f_{1}=x_{1}x_{6}$, $f_{2}=x_{1}x_{5}$, $f_{3}=x_{1}x_{3}$,
$f_{4}=x_{3}x_{4}$,
$f_{5}=x_{2}x_{4}$,

$J=(x_{1}x_{2}x_{4},x_{1}x_{2}x_{5},x_{1}x_{2}x_{3},x_{1}x_{2}x_{6},x_{1}x_{3}x_{6},x_{1}x_{4}x_{5},x_{1}x_{4}x_{6},$
$x_{2}x_{4}x_{5},x_{2}x_{4}x_{6},x_{3}x_{4}x_{5},x_{3}x_{4}x_{6})$

and $I=(f_{1},f_{2},f_{3},f_{4},f_{5})$. We have $s=4$,
$b_{1}=x_{5}f_{1}=x_{6}f_{2}$, $b_{2}=x_{3}f_{2}=x_{5}f_{3}$,
$b_{3}=x_{4}f_{3}=x_{1}f_{4}$, $b_{4}=x_{2}f_{4}=x_{3}f_{5}$. Let
$\partial_{i}:K_{i}(x;I/J)\rightarrow K_{i-1}(x;I/J)$, $K_{i}(x;I/J)\cong
S^{{6\choose i}}$, $i\in[6]$ be the Koszul derivation given by

$\partial_{i}(e_{j_{1}}\wedge\ldots\wedge
e_{j_{i}})=\sum_{k=1}^{i}(-1)^{k+1}x_{j_{k}}e_{j_{1}}\wedge\ldots\wedge
e_{j_{k-1}}\wedge e_{j_{k+1}}\wedge\ldots\wedge e_{j_{i}}.$

We consider the following elements of $K_{4}(x;I/J)$

$e_{\sigma_{1}}=e_{2}\wedge\ldots\wedge e_{5},\
e_{\sigma_{2}}=e_{2}\wedge\ldots\wedge e_{4}\wedge e_{6},\
e_{\sigma_{3}}=e_{2}\wedge e_{4}\wedge\ldots\wedge e_{6},$
$e_{\sigma_{4}}=e_{1}\wedge e_{2}\wedge e_{5}\wedge e_{6},\
e_{\sigma_{5}}=e_{1}\wedge e_{3}\wedge e_{5}\wedge e_{6}.$

Then the element

$z=f_{1}e_{\sigma_{1}}-f_{2}e_{\sigma_{2}}-f_{3}e_{\sigma_{3}}-f_{4}e_{\sigma_{4}}+f_{5}e_{\sigma_{5}}$

satisfies

$\partial_{4}(z)=(-b_{1}+b_{1}))e_{2}\wedge e_{3}\wedge
e_{4}+(b_{2}-b_{2})e_{2}\wedge e_{4}\wedge e_{6}+$ $(b_{3}-b_{3})e_{2}\wedge
e_{5}\wedge e_{6}+(b_{4}-b_{4})e_{1}\wedge e_{5}\wedge e_{6}=0,$

since $(b_{k})$ are the only monomials of degree $3$ which are not in $J$.
Note that in a term $ue_{\sigma}$ of an element from
$\operatorname{Im}\partial_{5}$ we have $u$ of degree $\geq 3$ because $I$ is
generated in degree $2$. Thus $z\not\in\operatorname{Im}\partial_{5}$ induces
a nonzero element in $H_{4}(x;I/J)$. By [1, Theorem 1.6.17] we get
$\operatorname{depth}_{S}I/J\leq 2$, which is enough.

######  Theorem 2.2.

If $r>s$ then $\operatorname{depth}_{S}I/J=d$, independently of the
characteristic of $K$.

###### Proof.

Let $\operatorname{supp}f_{i}=\\{j\in[n]:x_{j}|f_{i}\\}$,
$e_{\sigma_{i}}=\wedge_{j\in([n]\setminus\operatorname{supp}f_{i})}\ e_{j}$
and
$e_{\tau_{k}}=\wedge_{j\in([n]\setminus\operatorname{supp}b_{k})}\ e_{j}$. By
[1, Theorem 1.6.17] it is enough to show, as in the above example, that there
exist $y_{i}\in K$, $i\in[r]$ such that
$z=\sum_{i=1}^{r}y_{i}f_{i}e_{\sigma_{i}}$ induces a nonzero element of
$H_{n-d}(x;I/J)$. Let $\partial_{i}$ be the Koszul derivation as above. Then

$\partial_{n-d}(z)=\sum_{k=1}^{s}(\sum_{i\in[r],f_{i}|b_{k}}\varepsilon_{ki}y_{i})b_{k}$

for some $\varepsilon_{ki}\in\\{1,-1\\}$. Thus $\partial_{n-d}(z)=0$ if and
only if $\sum_{i\in[r],f_{i}|b_{k}}\varepsilon_{ki}y_{i}=0$ for all $k\in[s]$.
This is a system of $s$ homogeneous linear equations in $r$ variables $Y$,
which must have a nonzero solution in $K$ because $r>s$. As in the above
example $z\not\in\operatorname{Im}\partial_{n-d+1}$ if $I$ is generated in
degree $d$ (this may be supposed by Lemmas 1.2, 1.4).

The condition given in Theorem 2.2 is tight as shows the following two
examples.

######  Example 2.3.

Let $n=4$, $d=2$, $f_{1}=x_{1}x_{3}$, $f_{2}=x_{2}x_{4}$, $f_{3}=x_{1}x_{4}$
and $I=(f_{1},\ldots,f_{3})$, $J=(x_{2}x_{3}x_{4})$ be ideals of $S$. We have
$r=s=3$, $b_{1}=x_{1}x_{2}x_{3}$, $b_{2}=x_{1}x_{2}x_{4}$,
$b_{3}=x_{1}x_{3}x_{4}$, and $\operatorname{depth}_{S}I/J=d+1$.

######  Example 2.4.

Let $n=6$, $d=2$, $f_{1}=x_{1}x_{5}$, $f_{2}=x_{2}x_{3}$, $f_{3}=x_{3}x_{4}$,
$f_{4}=x_{1}x_{6}$, $f_{5}=x_{1}x_{4}$, $f_{6}=x_{1}x_{2}$, and
$I=(f_{1},\ldots,f_{6})$,

$J=(x_{1}x_{2}x_{4},x_{1}x_{2}x_{5},x_{1}x_{3}x_{5},x_{1}x_{3}x_{6},x_{1}x_{4}x_{6},x_{2}x_{3}x_{5},x_{2}x_{3}x_{6},x_{3}x_{4}x_{5},x_{3}x_{4}x_{6}).$

We have $r=s=6$ and $b_{1}=x_{1}x_{4}x_{5}$, $b_{2}=x_{2}x_{3}x_{4}$,
$b_{3}=x_{1}x_{2}x_{3}$, $b_{4}=x_{1}x_{5}x_{6}$, $b_{5}=x_{1}x_{3}x_{4}$,
$b_{6}=x_{1}x_{2}x_{6}$ but $\operatorname{depth}_{S}I/J=2$ (although $d=2$).

######  Remark 2.5.

The above example 2.4 shows that one could find a nice class of factors of
square free monomial ideals with $r=s$ but $\operatorname{depth}_{S}I/J=d$
similarly as in [9, Lemma 6]. An important tool seems to be a classification
of the possible posets given on $f_{1},\ldots,f_{r},b_{1},\ldots,b_{s}$ by the
divisibility.

######  Remark 2.6.

Given $J\subsetneq I$ two square free monomial ideals of $S$ as above one can
consider the poset $P_{I\setminus J}$ of all square free monomials of
$I\setminus J$ (a finite set) with the order given by the divisibility. Let
${\mathcal{P}}$ be a partition of ${\mathcal{P}}\ \ P_{I\setminus J}$ in
intervals $[u,v]=\\{w\in P_{I\setminus J}:u|w,w|v\\}$, let us say
$P_{I\setminus J}=\cup_{i}[u_{i},v_{i}]$, the union being disjoint. Define
$\operatorname{sdepth}{\mathcal{P}}=\operatorname{min}_{i}\operatorname{deg}v_{i}$
and
$\operatorname{sdepth}_{S}I/J=\operatorname{max}_{\mathcal{P}}\operatorname{sdepth}{\mathcal{P}}$,
where ${\mathcal{P}}$ runs in the set of all partitions of $P_{I\setminus J}$.
This is the Stanley depth of $I/J$, in fact this is an equivalent definition
(see [11], [2]). If $r>s$ then it is obvious that
$\operatorname{sdepth}_{S}I/J=d$ and so Theorem 2.2 says that Stanley’s
Conjecture holds, that is
$\operatorname{sdepth}_{S}I/J\geq\operatorname{depth}_{S}I/J$. In general the
Stanley depth of a monomial ideal $I$ is greater than or equal with the
Lyubeznik’ size of $I$ increased by one (see [3]). Stanley’s Conjecture holds
for intersections of four monomial prime ideals of $S$ by [5] and [7] and for
square free monomial ideals of $K[x_{1},\ldots,x_{5}]$ by [6] (a short
exposition on this subject is given in [8]). Also Stanley’s Conjecture holds
for intersections of three monomial primary ideals by [13]. In the case of a
non square free monomial ideal $I$ a useful inequality is
$\operatorname{sdepth}I\leq\operatorname{sdepth}\sqrt{I}$ (see [4, Theorem
2.1]).

##  3\. Around Theorem 2.2

Let $S^{\prime}=K[x_{1},\ldots,x_{n-1}]$ be a polynomial ring in $n-1$
variables over a field $K$, $S=S^{\prime}[x_{n}]$ and $U,V\subset S^{\prime}$,
$V\subset U$ be two square free monomial ideals. Set $W=(V+x_{n}U)S$.
Actually, every monomial square free ideal $T$ of $S$ has this form because
then $(T:x_{n})$ is generated by an ideal $U\subset S^{\prime}$ and
$T=(V+x_{n}U)S$ for $V=T\cap S^{\prime}$.

######  Lemma 3.1.

([6]) Suppose that $U\not=V$ and
$\operatorname{depth}_{S^{\prime}}S^{\prime}/U=\operatorname{depth}_{S^{\prime}}S^{\prime}/V=\operatorname{depth}_{S^{\prime}}U/V$.
Then
$\operatorname{depth}_{S}S/W=\operatorname{depth}_{S^{\prime}}S^{\prime}/U$.

######  Lemma 3.2.

Suppose that $U\not=V$ and
$d:=\operatorname{depth}_{S^{\prime}}S^{\prime}/U=\operatorname{depth}_{S^{\prime}}S^{\prime}/V$.
Then $d=\operatorname{depth}_{S^{\prime}}U/V$ if and only if
$d=\operatorname{depth}_{S}S/W$.

###### Proof.

The necessity follows from the above lemma. For sufficiency note that in the
exact sequence

$0\rightarrow VS\rightarrow W\rightarrow US/VS\rightarrow 0$

the depth of the left end is $d+2$ and the middle term has depth $d+1$. It
follows that $\operatorname{depth}_{S}US/VS=d+1$ by the Depth Lemma, which is
enough.

Let $I$ be an ideal of $S$ generated by square free monomials of degree $\geq
d$ and $x_{n}f_{1},\ldots,x_{n}f_{r}$, $r>0$ be the square free monomials of
$I\cap(x_{n})$ of degree $d$. Set $U=(f_{1},\ldots,f_{r})$, $V=I\cap
S^{\prime}$.

######  Theorem 3.3.

If $r>\rho_{d}(U)-\rho_{d}(U\cap V)$ then
$\operatorname{depth}_{S}S/I=\operatorname{depth}_{S^{\prime}}(U+V)/V=d-1$.

###### Proof.

By Theorem 2.2 we have
$\operatorname{depth}_{S^{\prime}}(U+V)/V=\operatorname{depth}_{S^{\prime}}U/(U\cap
V)=d-1$. Using Lemmas 1.2, 1.4 we get

$\operatorname{depth}_{S^{\prime}}(U+V)/V=\operatorname{depth}_{S^{\prime}}((I:x_{n})\cap
S^{\prime})/(I\cap S^{\prime})=d-1.$

If $\operatorname{depth}_{S^{\prime}}S^{\prime}/(I\cap
S^{\prime})=\operatorname{depth}_{S^{\prime}}S^{\prime}/((I:x_{n})\cap
S^{\prime})=d-1$ then $\operatorname{depth}_{S}S/I=d-1$ by Lemma 3.2. If
$\operatorname{depth}_{S^{\prime}}S^{\prime}/((I:x_{n})\cap S^{\prime})=d-2$
then in the exact sequence

$0\rightarrow S/(I:x_{n})\xrightarrow{x_{n}}S/I\rightarrow S^{\prime}/(I\cap
S^{\prime})\rightarrow 0$

the first term has depth $d-1$ and the other two have depth $\geq d-1$ by
Lemma 1.1. By the Depth Lemma it follows that
$\operatorname{depth}_{S}S/I=d-1$.

It remains to consider the case when at least one of
$\operatorname{depth}_{S^{\prime}}S^{\prime}/((I:x_{n})\cap S^{\prime})$ and
$\operatorname{depth}_{S^{\prime}}S^{\prime}/(I\cap S^{\prime})$ is $\geq d$.
Using the Depth Lemma in the exact sequence

$0\rightarrow((I:x_{n})\cap S^{\prime})/(I\cap S^{\prime})\rightarrow
S^{\prime}/(I\cap S^{\prime})\rightarrow S^{\prime}/((I:x_{n})\cap
S^{\prime})\rightarrow 0$

we see that necessarily the depth of the last term is $\geq d$ and the depth
of the middle term is $d-1$. But then the Depth Lemma applied to the previous
exact sequence gives $\operatorname{depth}_{S}S/I=d-1$ too.

The following corollary extends [9, Corollary 3].

######  Corollary 3.4.

Let $I$ be an ideal generated by $\mu(I)>1$ square free monomials of degree
$d$. If $\mu(I)\geq\rho_{d+1}(I)$, in particular if $\mu(I)\geq{n\choose
d+1}$, then $\operatorname{depth}_{S}I=d$.

###### Proof.

We have $I=(V+x_{n}(U+V))S$ as above. Renumbering the variables we may suppose
that $U,V\not=0$. Note that $\mu(I)=r+\rho_{d}(V)$ and
$\rho_{d+1}(I)=\rho_{d+1}(V)+\rho_{d}(U+V)>\rho_{d}(V)+\rho_{d}(U)-\rho_{d}(U\cap
V)$. By hypothesis, $\mu(I)\geq\rho_{d+1}(I)$ and so
$r>\rho_{d}(U)-\rho_{d}(U\cap V)$. Applying Theorem 3.3 we get
$\operatorname{depth}_{S}S/I=d-1$, which is enough.

######  Remark 3.5.

Take in Example 2.3 $S^{\prime}=K[x_{1},\ldots,x_{5}]$ and
$L=(J+x_{5}I)S^{\prime}$. We have $\mu(L)=4<{5\choose 3+1}$, that is the
hypothesis of the above corollary are not fulfilled. This is the reason that
$\operatorname{depth}_{S^{\prime}}L=4$ by Lemma 3.2 since
$\operatorname{depth}_{S}I/J=3$. Thus the condition of the above corollary is
tight.

##  4\. Minimal Stanley depth

Let $S=K[x_{1},\ldots,x_{n}]$ be the polynomial algebra in $n$-variables over
a field $K$, $d$ a positive integer and $J\subsetneq I$, be two square free
monomial ideals of $S$. Let $\rho_{d}(I)$ be the number of all square free
monomials of degree $d$ of $I$. Suppose that $\rho_{d}(I)>0$ and $I$ is
generated in degree $\geq d$. It follows that
$\operatorname{sdepth}_{S}I/J\geq d$.

######  Theorem 4.1.

The following statements are equivalent:

  1. (1)

$\operatorname{sdepth}_{S}I/J=d$

  2. (2)

there exist some square free monomials of degree $d$ in $I$, which generate an
ideal $I^{\prime}$ such that
$\rho_{d}(I^{\prime})>\rho_{d+1}(I^{\prime})-\rho_{d+1}(I^{\prime}\cap J)$.

###### Proof.

If $J\not=0$ then $J$ is generated in degree $\geq d$, even we may suppose
that $J$ is generated in degree $\geq d+1$ using an easy isomorphism. Let
${\mathcal{M}}_{d}(I)$ be the set of all square free monomials of $I$ of
degree $d$ and ${\mathcal{B}}={\mathcal{M}}_{d+1}\setminus J$. We consider the
bipartite graph $G$ defined by $V(G)={\mathcal{M}}_{d}(I)\cup{\mathcal{B}}$,
an edge of $G$ can have only endpoints $f\in{\mathcal{M}}_{d}(I)$ and
$b\in{\mathcal{B}}$ with $f|b$. Given $f\in{\mathcal{M}}_{d}(I)$ let
$\Gamma(f)$ be the set of all vertices $b$ adjacent to $f$ and for
$A\subset{\mathcal{M}}_{d}(I)$ set $\Gamma(A)=\cup_{f\in A}\Gamma(f)$. By P.
Hall’s marriage theorem [12] there is a complete matching from
${\mathcal{M}}_{d}(I)$ to ${\mathcal{B}}$ if and only if $|\Gamma(A)|\geq|A|$
for every subset $A\subset{\mathcal{M}}_{d}(I)$. Thus
$\operatorname{sdepth}_{S}I/J=d$ if and only if there exists no complete
matching above and so there exists a subset $A\subset{\mathcal{M}}_{d}(I)$
such that $|\Gamma(A)|<|A|$, that is $I^{\prime}=(A)$ satisfies the second
statement.

For $J=0$ we get the following corollary, which is closed to [10, Lemma 3.3]

######  Corollary 4.2.

The following statements are equivalent:

  1. (1)

$\operatorname{sdepth}_{S}I=d$

  2. (2)

there exist some square free monomials of degree $d$ in $I$, which generate an
ideal $I^{\prime}$ such that $\rho_{d}(I^{\prime})>\rho_{d+1}(I^{\prime})$.

######  Theorem 4.3.

If $\operatorname{sdepth}_{S}I/J=d$ then $\operatorname{depth}_{S}I/J=d$, that
is Stanley’s conjecture holds in this case.

###### Proof.

By Theorem 4.1 there exists a monomial square free ideal $I^{\prime}\subset I$
such that
$\rho_{d}(I^{\prime})>\rho_{d+1}(I^{\prime})-\rho_{d+1}(I^{\prime}\cap J)$.
Then $\operatorname{depth}_{S}I^{\prime}/(I^{\prime}\cap J)=2$ by Theorem 2.2
(if $J=0$ we apply Corollary 3.4). Now it is enough to apply Lemma 1.5.

## References

  * [1] W. Bruns and J. Herzog, Cohen-Macaulay rings, Revised edition. Cambridge University Press (1998).
  * [2] J. Herzog, M. Vladoiu, X. Zheng, How to compute the Stanley depth of a monomial ideal, J. Algebra, 322 (2009), 3151-3169.
  * [3] J. Herzog, D. Popescu, M. Vladoiu, Stanley depth and size of a monomial ideal, Proc. Amer. Math. Soc., 140 (2012), 493-504, arXiv:AC/1011.6462v1.
  * [4] M. Ishaq, Upper bounds for the Stanley depth, Comm. Algebra, 40(2012), 87-97.
  * [5] A. Popescu, Special Stanley Decompositions, Bull. Math. Soc. Sc. Math. Roumanie, 53(101), no 4 (2010), arXiv:AC/1008.3680.
  * [6] D. Popescu, An inequality between depth and Stanley depth, Bull. Math. Soc. Sc. Math. Roumanie 52(100), (2009), 377-382, arXiv:AC/0905.4597v2.
  * [7] D. Popescu, Stanley conjecture on intersections of four monomial prime ideals, to appear in Communications in Algebra, arXiv:AC/1009.5646.
  * [8] D. Popescu, Bounds of Stanley depth, An. St. Univ. Ovidius. Constanta, 19(2),(2011), 187-194.
  * [9] D. Popescu, Depth and minimal number of generators of square free monomial ideals, An. St. Univ. Ovidius, Constanta, 19(3), (2011), 163-166, arXiv:AC/1107.2621.
  * [10] Y.H. Shen, When will the Stanley depth increase, arxiv:AC/1110.3182v1.
  * [11] R. P. Stanley, Linear Diophantine equations and local cohomology, Invent. Math. 68 (1982) 175-193.
  * [12] J. H. van Lint, R.M. Wilson, A course in combinatorics, Cambridge Univ. Press, Cambridge, 2001.
  * [13] A. Zarojanu, Stanley Conjecture on three monomial primary ideals, to appear in Bull. Math. Soc. Sc. Math. Roumanie, arXiv:AC/11073211.


:
2023 Machine Learning for Healthcare Editor’s name

# Data augmentation method for modeling health records
with applications to clopidogrel treatment failure detection

Sunwoong Choi<EMAIL_ADDRESS>
Cipherome Korea
Seoul  Korea  Samuel Kim<EMAIL_ADDRESS>
Cipherome Inc.
San Jose  CA  U.S.A

###### Abstract

We present a novel data augmentation method to address the challenge of data
scarcity in modeling longitudinal patterns in Electronic Health Records (EHR)
of patients using natural language processing (NLP) algorithms. The proposed
method generates augmented data by rearranging the orders of medical records
within a visit where the order of elements are not obvious, if any. Applying
the proposed method to the clopidogrel treatment failure detection task
enabled up to 5.3% absolute improvement in terms of ROC-AUC (from 0.908
without augmentation to 0.961 with augmentation) when it was used during the
pre-training procedure. It was also shown that the augmentation helped to
improve performance during fine-tuning procedures, especially when the amount
of labeled training data is limited.

##  1 Introduction

The Electronic Health Record (EHR) includes diagnoses, procedures,
prescriptions, and other items of protected health information. This data is
usually timestamped, allowing for researchers to study patients’ longitudinal
patterns. Various machine learning algorithms have been devised to model the
longitudinal patterns using health records in EHR. For example, [Choi et
al.(2017)Choi, Bahadori, Kulas, Schuetz, Stewart, and Sun] proposed RETAIN to
detect heart failures using a recurrent neural network (RNN) architecture.

Recently, researchers have applied natural language processing (NLP)
algorithms to EHR data by drawing analogies between medical records and
natural language. In particular, Bidirectional Encoder Representations from
Transformers (BERT) algorithm [Devlin et al.(2019)Devlin, Chang, Lee, and
Toutanova] has been widely used. It is a transformer-based algorithm that
utilizes pre-trained models trained on a large unlabeled dataset and fine-
tunes models with a labeled but relatively small dataset. [Shang et
al.(2019)Shang, Ma, Xiao, and Sun] was for medication recommendation using a
BERT-based graph neural network (GNN) model. [Li et al.(2019)Li, Rao, Solares,
Hassaine, Canoy, Zhu, Rahimi, and Salimi-Khorshidi] proposed BEHRT to learn
patterns of diagnostic codes and used the model to predict unseen diagnoses.
Similarly, [Rasmy et al.(2021)Rasmy, Xiang, Xie, Tao, and Zhi] proposed Med-
BERT which also adapted the BERT framework to predict diseases. [Pang et
al.(2021)Pang, Jiang, Kalluri, Spotnitz, Chen, Perotte, and Natarajan]
incorporated temporal information using a hybrid approach by augmenting the
input to BERT using artificial time tokens, incorporating time, age, and
concept embedding, and introduced a new second learning objective for visit
type. The researchers hypothesized that a patient’s medical history could be
structured like a natural language document, where patient’s visit
corresponded to sentences and the diagnoses, procedures, and medications
correspond to words.

However, there are still challenges in applying NLP algorithms to build models
using EHR data. One challenge is the relatively small amount of training data
available for both pre-training and fine-tuning, compared to the amount of
data available for NLP applications. Another challenge is that the analogy
between natural language and medical history is not perfect; for example,
forcing a certain order for medical records to form a sentence-like sequence
to be fed into NLP algorithms may cause ordinal biases since the order of
medical records during a single visit are rarely significant.

To address these challenges, we developed a data augmentation method to
increase the amount of available training data. This method generates
augmented data by rearranging the order of medical records within a visit. We
hypothesize that this method will mitigate the challenges mentioned above by
not only providing with more data but also mitigating ordinal biases. We argue
that this study demonstrates the potential of applying NLP algorithms to EHR
data, while also highlighting the challenges that need to be addressed.

We applied the proposed strategies to model drug responses with given EHR
records. Drug responses, i.e., adverse drug reactions (ADRs) and treatment
failures (TFs), can occur due to inadequate concentrations of medications.
They significantly impact patients’ lives and are serious burdens to the
healthcare system. As stated in [Pitts(2022)], ADRs are estimated to be
between the fourth and sixth most common cause of death worldwide, taking
their place among other prevalent causes of mortality such as heart disease,
cancer, and strok, while TFs may occur when therapeutic doses are not reached.
The variable presentation makes identifying drug responses difficult, and the
impact is likely underrepresented [Hazell and Shakir(2006)]. The multi-
factorial nature of drug responses adds to the complexity; patient’s age,
renal or liver function, comorbidities, comedications, lifestyle, and genetic
predispositions are all important factors affecting drug responses [Handler et
al.(2008)Handler, Hanlon, Perera, et al., Meyer(2000)].

Among the multitude of drugs, we studied clopidogrel, a P2Y12 inhibitor, one
of the most widely prescribed drugs in the US. Its use with aspirin is
commonly referred to as dual antiplatelet therapy (DAPT), and is the standard
of care following stent placement for cardiovascular disease. However, various
responses to this drug have been reported in several studies raising concerns
for its safety [Cattaneo(2007)]. High concentration of active metabolite can
cause bleeding, while low concentrations of active metabolites can cause
recurrent thrombosis, both of which can lead to medical emergencies. There is
a clear need for a screening tool to better predict which patients may be at
risk for such events.

##  2 Generalizable insights

We believed that our research underscores the need for innovative approaches
to tackle the challenges of utilizing language models in healthcare
applications where data availability may be limited. By addressing these
challenges, our research contributes to more accurate and effective prediction
of medical events, leading to improved patient care and outcomes in the
healthcare domain.

##  3 Data

###  3.1 Dataset

We used the UK Biobank which consists of data collected from 502,527
participants [Sudlow et al.(2015)]. Volunteers aged 40 to 70 were recruited
from England, Scotland, and Wales and invited to assessment centers between
2006 and 2010. Data collected during the visits included biosamples, physical
examination measurements, and questionnaire answers followed by interviews.
Genomic data, both sequencing and genotyping, were generated using the
biosamples collected. Also, an extensive and comprehensive medical history
data was made available through hospital in-patient and primary care data from
external data sources.

We extracted prescriptions, diagnoses and procedure records, along with dates,
for all participants from the UK Biobank. All prescription records were from
general practitioner (GP) data coded in Read and British National Formulary
(BNF) depending on the data supplier. Diagnoses and procedure records were
from hospital in-patient records only and coded in ICD-9/10 and OPCS-3/4
respectively.

###  3.2 Annotation

Treatment failure (TF) was defined as having a TF event within one year of the
very first clopidogrel prescription. Clopidogrel prescriptions were identified
using the substance or brand names of clopidogrel or respective read codes,
while TF events include ischemic stroke, myocardial infarct, stent thrombosis
and recurrent thrombosis or stenting. All of these annotations were carefully
done by clinical professions using ICD-9/10 and OPCS-3/4 codes.

Subjects with events occurring within 7 days of the first prescription were
excluded as it was unclear whether those events were associated with
clopidogrel. The visit had to be through the emergency room to be valid in
order to exclude follow up visits from previous events. From the dataset, we
found 9,867 subjects with clopidogrel prescriptions. Among them, we labeled
1,824 patients as TF cases and 6,859 as control cases; 1,184 subjects were
excluded due to data inconsistencies or ambiguities.

##  4 Methodologies

###  4.1 Data Processing

We gathered various codes together per subject and organized them into visits
with the same date. As a result, a subject has multiple visits and a visit has
multiple codes. Table 1 shows the statistics of the codes and the visits.

Table 1: Statistics of structured EHR data in UKBiobank. # of subjects | 502,527
---|---
# of subjects with 1+ visits | 465,506
# of unique visits | 22,063,417
# of unique codes | 31,589
Average # of visits per subject (max/med/min) | 58.3 (2805 / 13 / 1)
Average # of codes per visit (max/med/min) | 2.7 (61 / 2 / 1)

Note that the number of codes per visit is relatively small compared to the
number of words per sentence in natural language processing (NLP)
applications. Also note that the number of visits per patient is relatively
large and their order is evident due to corresponding timestamps. Therefore,
we flattened the codes of the visits and concatenated them across all visits
so that a patient has a sentence-like sequence of codes that can be fed into
NLP models.

The amount of training data, however, for both pre-training and fine-tuning is
small compared to that for natural language applications. This can be
difficult to get the full expressive power of large language model. Therefore,
we propose to use data augmentation process to generate additional data for
training NLP models.

There are traditional NLP data augmentation methods, such as synonym
replacement [Sennrich et al.(2016)Sennrich, Haddow, and Birch] or back
translation [Zhang et al.(2015)Zhang, Zhao, and LeCun], but they are difficult
to be directly applied to EHR codes. Furthermore, some data augmentation
strategies that generate fake data are subject to criticism especially in
healthcare domain because the generated data does not exist in the real world.

Hence, we propose a data augmentation method that generates multiple sequences
by rearranging orders of codes within a visit. We proposed to use multiple
codes including procedures and prescription codes as well as diagnosis codes.
Considering there exists an inherent order between these types of medical
events (diagnosis, procedures, and prescription), we keep the order of the
medical types while we permute the codes within the same medical types.

Figure 1: Proposed data preparation method that augments data. $\bigotimes$
represents a Cartesian product notation. Each box represents medical record,
and different colors denote different types of records. Boxes with bold lines
simply illustrate different elements from the first sequence.

Fig. 1 illustrates the proposed augmentation method. It has two steps; the
first is to permute medical records within visits, and the second is to
combine permuted records to create a sequence. We can formulate out method as
follows. Suppose a patient has $n$ visits, $\\{V_{1},V_{2},\dots,V_{n}\\}$ and
each visit $V_{i}$ where $i\in{1,\dots,n}$ is composed of diagnoses,
procedures, and prescriptions denoted as $D$, $O$, and $P$, i.e.
$V_{i}=\\{D_{i},O_{i},P_{i}\\}$. When we permute the codes that are within the
same types of codes and preserve the inherent order, we can achieve
$p_{i}=|D_{i}|!\cdot|O_{i}|!\cdot|P_{i}|!$ number of codes, where $!$ is the
factorial notation and $|\cdot|$ is the number of elements. After code
permutation within visits, we combine the permutations to create a sequence,
as a result, the total number of possible sequences is
$\prod\limits_{i=1}^{n}p_{i}$.

In our experiments, we only selected a subset of all these possible sequences
from each patients records for practical reasons because using all the
possible sequences is not computationally feasible. We specified the number of
the augmented data as $\alpha$, that is, if we choose $\alpha$ as 4, the
augmentation method will generate four different sequences of medical records
from one patient. In case the number of possible sequences is less than
$\alpha$, it only generates available sequences.

The proposed algorithms can be applied to both of training procedures of BERT,
i.e., pre-training and fine-tuning.

###  4.2 Model preparation

We applied the proposed data augmentation method to BERT. We used the basic
structure of the BERT model architecture consisting of 12 layers of
transformer encoders [Devlin et al.(2019)Devlin, Chang, Lee, and Toutanova];
it takes 512 tokens as an input and generates a 768-dimensional vector as
output.

For pre-training, we used only masked language modeling (MLM) among BERT’s
pre-training objectives. The next sentence prediction (NSP) which is also
widely used in NLP applications is not appropriate in our data preparation
setup because we concatenate all the codes of a patient into a single
sequence. To build a pre-trained model for BERT, we used the data from the
subjects who did not have clopidogrel prescription to learn general
population’s patterns.

During the fine-tuning, we replaced the last layer of the pre-trained model
with a fully connected layer. Since the target task of this study is a binary
classification, the last layer is set to have one output node with one node
with binary cross-entropy as a loss function. The output of the last
transformer encoder layer was fed into a pooling layer, which aggregates the
output embeddings into a fixed-length vector representation. In this case, the
pooling layer generates a 768-dimensional vector representation of the input
sequence.

###  4.3 Experimental Setup

We have designed an experimental task to detect treatment failures within one
year after the first prescription of clopidogrel. The task is to look back
from one year after the first prescription to determine if there have been any
instances of treatment failure. The model’s access to the data is limited to
up until one year after the first prescription date. A padding strategy was
used to make all the sequences are the same length (512 in this particular
case); the most recent codes were used if there were more codes for a patient.
Fig. 2 shows an example of the task. Suppose a patient is first prescribed the
clopidogrel at visit 4 (V4), then the takes all the medical records up to 1
year after the first prescription day, i.e. from V1 to V7, and predicts if
there exist any treatment failures within one year after the first
prescription.

Figure 2: Detection task; to classify if there exist treatment failures within
one year after the first prescription.

We pre-trained the model with various augmentation factors. As illustrated in
Fig. 3, we utilized each of the augmented sequences generated from a
particular patient as input for the model. The generated sequences were
considered as independent subjects. Since we do not use any label information
during the pre-training, we randomly split the data into training and
validating data with 80/20 ratio.

During the fine-tuning procedures, we randomly split the labeled subjects into
training and testing data with 80/20 ratio. This split was used across all
experiments for fair comparisons. The data augmentation was separately applied
in training and testing dataset to avoid augmented data from the same patients
present in both datasets. Since the fine-tuning procedure requires label
information associated with data, it is reasonable to apply the same label of
the patient for the augmented data while treating each sequence as an
independent patient. This process is illustrated in Fig. 4.

We used receiver operating characteristic (ROC) curves and their area under
curve (AUC) to measure the performance of each experiment.

Figure 3: Pre-training procedure using data augmentation strategy. Figure 4:
Fine-tuning procedure using data augmentation strategy..

##  5 Results and Discussion

Figure 5: ROC curves and area under curves based on augmentation factor in
pre-training

###  5.1 Impact of augmentation in pre-training

To examine the performance of augmentation in pre-training, we used the
augmentation method only during pre-training but not during fine-tuning. Our
experimental results, as shown in Fig. 5, demonstrate a clear relationship
between the amount of augmented data used in pre-training and the model’s
ability to detect clopidogrel treatment failures. As the augmentation factor
increases, the model performance in terms of AUC improves; we were able to
achieve 5.3% absolute improvement compared to the case without augmentation
(from 90.8% without augmentation to 96.1% with augmentation factor 8).

We argue that the improvement stems from two folds: more training data by
providing increased variety and mitigating ordinal biases imposed by
flattening process. This validates our initial hypothesis. The proposed
augmentation method allowed the model to learn more robust and generalized
representations of longitudinal medical records and leads to significant
improvement in a downstream task.

In practice, these findings suggest that data augmentation based on
permutation can be a powerful tool for improving the accuracy and reliability
of the BERT based model in the healthcare domain, where large and diverse
datasets can be difficult to obtain. It also indicates the pre-trained model
with augmented data should be delivered as a foundation model [Wornow et
al.()Wornow, Xu, Thapa, Patel, Steinberg, Fleming, Pfeffer, Fries, and Shah]
to be adopted in various downstream tasks for better performance.

###  5.2 Impact of augmentation in fine-tuning

Figure 6: Performance as the augmentation factor varies with data size

Subsequently, we investigated the effectiveness of data augmentation in the
fine-tuning procedure. Specifically, we performed various experiments in
scenarios with limited available data. We used a pre-trained model with an
augmentation factor 8, which exhibited the best performance in the previous
subsection.

The corresponding results are shown in Fig. 6. The results evidently show that
performance improves as the amount of available data increases; the best
performance could be achieved when all the training dataset is available. More
importantly, we also observed that the benefits of augmentation in most of
cases. The trend is more pronounced when data samples are scarce with a
gradual reduction in performance gain as the size of the data set grows. This
is particularly noteworthy because collecting sufficient amount of data with
appropriate labels can be very challenging. Therefore, based on our
experimental results, our approach seems to offer promising outcomes in real-
world scenarios.

We also investigated if the data augmentation during the inference procedure
would help to improve the detection task performance. This is a test-time
augmentation, i.e. we generated augmented test sequences for one test sample
and fed them into the fine-tuned model to average scores to represent the test
sample. However, the improvement using the test-time augmentation was minimal;
we applied the test-time augmentation to all of the models we built, but the
maximum improvement we achieved was 0.2% while there is no performance
degradation found.

##  6 Conclusions

The utilization of language models, such as BERT, has become increasingly
popular in the medical domain for processing EHR data and related downstream
tasks. However, one of the primary challenges associated with leveraging these
models is the availability of sufficient pre-training data, which is critical
for improving their performance.

To address this challenge, we proposed a novel permutation-based data
augmentation approach. Our approach involved generating permutations of input
records, where we randomly permute the codes within the same medical types,
diagnoses, procedures and prescriptions during a single visit. We hypothesized
that this approach would enable the model to learn more robust and
generalizable representations of longitudinal medical records by not only
providing with more data but also mitigating ordinal biases, thereby improving
its overall performance.

Our experimental results demonstrated that the proposed data augmentation
approach was effective in improving the performance of BERT-based models. In
pre-training, we observed a significant improvement in a downstream
application when utilizing the augmented data compared to the case without
augmentation. Furthermore, we also found that augmentations can provide
performance improvements when there is less data available in the fine-tuning
process. We concluded that our method performs well in scenarios where data is
scarce, highlighting its potential in practical applications where large
amounts of data are difficult to obtain.

Notably, our approach maximizes the use of existing data and avoids generating
synthetic data, which is particularly important in the medical domain due to
various ethical and legal concerns surrounding the use of patient data. This
feature of our approach ensures that the augmented data remains faithful to
the original data, minimizing the risk of introducing biases or
inconsistencies into the model.

Overall, our study provides valuable insights into the use of data
augmentation techniques for improving natural language processing algorithms
in medical domains. Our approach has the potential to address the challenges
associated with limited data availability, making it a promising solution for
practical applications in the healthcare domain. We believe that our findings
can inform future research in this area and contribute to the development of
more effective and reliable language models for medical applications.

##  7 Acknowledgement

This research has been conducted using the UK Biobank Resource under
Application Number 52031.

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