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A unified perspective on some autocorrelation measures in different fields: A note

  • Hiroshi Yamada EMAIL logo
Published/Copyright: April 1, 2023
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Open Mathematics
From the journal Open Mathematics Volume 21 Issue 1

Abstract

Using notions from linear algebraic graph theory, this article provides a unified perspective on some autocorrelation measures in different fields. They are as follows: (a) Orcutt’s first serial correlation coefficient, (b) Anderson’s first circular serial correlation coefficient, (c) Moran’s r 11 , and (d) Moran’s I . The first two are autocorrelation measures for one-dimensional data equally spaced, such as time series data, and the last two are for spatial data. We prove that (a)–(c) are a kind of (d). For example, we show that (d) such that its spatial weight matrix equals the adjacency matrix of a path graph is the same as (a). The perspective is beneficial because studying the properties of (d) leads to studying the properties of (a)–(c) at the same time. For example, the bounds of (a)–(c) can be found from the bounds of (d).

Keywords: Orcutt’s first serial correlation coefficient; Anderson’s first circular serial correlation coefficient; Moran’s r 11 ; Moran’s I ; path graph; cycle graph; two-dimensional lattice graph; serial correlation; spatial autocorrelation
MSC 2010: 62M10; 62H11; 05C50

1 Introduction

In this article, using notions from linear algebraic graph theory, we provide a unified perspective on some autocorrelation measures in different fields. They are as follows:

  1. Orcutt’s first serial correlation coefficient [1],

  2. Anderson’s first circular serial correlation coefficient [2,3],

  3. Moran’s r 11 [4], and

  4. Moran’s I [5–7].

The first two are autocorrelation measures for one-dimensional data equally spaced, such as time series data, and the last two are for spatial data.[1] In this article, we prove that (a)–(c) are a kind of (d). For example, we show that (d) such that its spatial weight matrix equals the adjacency matrix of a path graph is the same as (a). The perspective is beneficial because studying the properties of (d) leads to studying the properties of (a)–(c) at the same time. For example, the bounds of (a)–(c) can be found from the bounds of (d).

This article is organized as follows. In Section 2, we introduce some notions for documenting our main results. In Section 3, we present the main results of the article. Section 4 concludes the article. Appendix A provides the proof of the main results.

2 Preliminaries

In this section, for later exposition,

  1. we explicitly present the four autocorrelation measures, (a)–(d),

  2. we review three undirected graphs: a path graph, a cycle graph, and a two-dimensional lattice graph, and

  3. we represent (d) in matrix form.

Let y 1 , … , y n be observations such that ∑ i = 1 n ( y i − y ¯ ) 2 > 0 , where y ¯ = 1 n ∑ i = 1 n y i . In addition, when n = p q for p , q ∈ N , denote y q ( i − 1 ) + j by x i , j for i = 1 , … , p and j = 1 , … , q . Accordingly, x ¯ = 1 p q ∑ i = 1 p ∑ j = 1 q x i , j equals y ¯ .

2.1 Four autocorrelation measures

In this subsection, we explicitly present the four autocorrelation measures, (a)–(d).

(a) and (b) Orcutt’s first serial correlation coefficient and Anderson’s first circular serial correlation coefficient are, respectively, defined by

(1) ϕ = n n − 1 ∑ i = 2 n ( y i − y ¯ ) ( y i − 1 − y ¯ ) ∑ i = 1 n ( y i − y ¯ ) 2

and

(2) ψ = ∑ i = 1 n ( y i − y ¯ ) ( y i − 1 − y ¯ ) ∑ i = 1 n ( y i − y ¯ ) 2 ,

where y 0 = y n .

(c) and (d) Moran’s r 11 is defined by

(3) r 11 = p q 2 p q − p − q ∑ i = 1 p ∑ j = 1 q − 1 z i , j z i , j + 1 + ∑ i = 1 p − 1 ∑ j = 1 q z i , j z i + 1 , j ∑ i = 1 p ∑ j = 1 q z i , j 2 ,

where z i , j = x i , j − x ¯ for i = 1 , … , p , and j = 1 , … , q . Moran’s I is defined by

(4) I = n ∑ i = 1 n ∑ j = 1 n w i , j ∑ i = 1 n ∑ j = 1 n w i , j ( y i − y ¯ ) ( y j − y ¯ ) ∑ i = 1 n ( y i − y ¯ ) 2 ,

where w i , j for i , j = 1 , … , n denote nonnegative spatial weights.

2.2 Three undirected graphs

In this subsection, we review three undirected graphs: a path graph, a cycle graph, and a two-dimensional lattice graph. These undirected graphs are related to (a)–(c), respectively. Let V = { 1 , … , n } .

Path graph. Consider an undirected graph G p = ( V , E p ) , where E p = { { 1 , 2 } , … , { n − 1 , n } } . Then, G p = ( V , E p ) is a path graph with n vertices. Figure 1 depicts G p = ( V , E p ) for n = 6 . Denote the adjacency matrix of G p = ( V , E p ) by A p . Then, A p is a tridiagonal symmetric matrix as follows:

A p = 0 1 0 ⋯ 0 0 1 0 1 ⋱ 0 0 ⋱ ⋱ ⋱ ⋱ ⋮ ⋮ ⋱ ⋱ ⋱ ⋱ 0 0 ⋱ 1 0 1 0 0 ⋯ 0 1 0 ∈ R n × n .

We remark that, e.g., ( 1 , 2 ) and ( 2 , 1 ) entries of A p are equal to 1 as vertex 1 is adjacent to vertex 2, as shown in Figure 1. Likewise, ( 1 , 3 ) and ( 3 , 1 ) entries of A p are equal to 0 as vertex 1 is not adjacent to vertex 3. For more details of linear algebraic graph theory, see, e.g., [9–11].

Figure 1 A path graph with six vertices.
Figure 1

A path graph with six vertices.

Cycle graph. Consider an undirected graph G c = ( V , E c ) , where E c = { { 1 , 2 } , … , { n − 1 , n } , { n , 1 } } . Then, G c = ( V , E c ) is a cycle graph with n vertices. Figure 2 depicts G c = ( V , E c ) for n = 6 . Denote the adjacency matrix of G c = ( V , E c ) by A c . Explicitly, it is

A c = 0 1 0 ⋯ 0 1 1 0 1 ⋱ 0 0 ⋱ ⋱ ⋱ ⋱ ⋮ ⋮ ⋱ ⋱ ⋱ ⋱ 0 0 ⋱ 1 0 1 1 0 ⋯ 0 1 0 ∈ R n × n .

Figure 2 A cycle graph with six vertices.
Figure 2

A cycle graph with six vertices.

Two-dimensional lattice graph. Consider an undirected graph G g = ( V , E g ) , where E g = E g ∗ ⋃ E g ∗ ∗ . Here, E g ∗ = ⋃ i = 1 p { { q ( i − 1 ) + 1 , q ( i − 1 ) + 2 } , … , { q ( i − 1 ) + ( q − 1 ) , q ( i − 1 ) + q } } = ⋃ i = 1 p ⋃ j = 1 q − 1 { q ( i − 1 ) + j , q ( i − 1 ) + j + 1 } and E g ∗ ∗ = ⋃ i = 1 p − 1 { { q ( i − 1 ) + 1 , q i + 1 } , … , { q ( i − 1 ) + q , q i + q } } = ⋃ i = 1 p − 1 ⋃ j = 1 q { q ( i − 1 ) + j , q i + j } . Figure 3 depicts G g = ( V , E g ) for p = 2 and q = 4 . Denote the adjacency matrix of G g = ( V , E g ) by A g . For example, when p = 2 and q = 4 , given E g ∗ = { { 1 , 2 } , { 2 , 3 } , { 3 , 4 } , { 5 , 6 } , { 6 , 7 } , { 7 , 8 } } and E g ∗ ∗ = { { 1 , 5 } , { 2 , 6 } , { 3 , 7 } , { 4 , 8 } } , it follows that

A g = 0 1 0 0 1 0 0 0 1 0 1 0 0 1 0 0 0 1 0 1 0 0 1 0 0 0 1 0 0 0 0 1 1 0 0 0 0 1 0 0 0 1 0 0 1 0 1 0 0 0 1 0 0 1 0 1 0 0 0 1 0 0 1 0 ∈ R 8 × 8 .

Figure 4 depicts two subgraphs of G g = ( V , E g ) for p = 2 and q = 4 , i.e., G g ∗ = ( V , E g ∗ ) (left) and G g ∗ ∗ = ( V , E g ∗ ∗ ) (right).

Figure 3 A two-dimensional lattice graph: G g = ( V , E g ) {G}_{g}=\left(V,{E}_{g}) for p = 2 p=2 and q = 4 q=4 .
Figure 3

A two-dimensional lattice graph: G g = ( V , E g ) for p = 2 and q = 4 .

Figure 4 Subgraphs of G g = ( V , E g ) {G}_{g}=\left(V,{E}_{g}) for p = 2 p=2 and q = 4 q=4 .
Figure 4

Subgraphs of G g = ( V , E g ) for p = 2 and q = 4 .

2.3 Moran’s I in matrix form

In this subsection, we represent Moran’s I in (4) in matrix form. It can be represented as follows:

I = n ι ⊤ W ι y ⊤ Q ι W Q ι y y ⊤ Q ι y ,

where y = [ y 1 , … , y n ] ⊤ , ι = [ 1 , … , 1 ] ⊤ ∈ R n , W = [ w i , j ] ∈ R n × n , which denotes the spatial weight matrix, and Q ι = I n − ι ( ι ⊤ ι ) − 1 ι ⊤ . Here, I n denotes the identity matrix of order n . See, e.g., [12, p. 129].

3 Main results

In this section, we present the main results of the article.

Proposition 1

Denote Moran’s I for the case in which W equals A i by I i for i = p , c , g , i.e.,

I i = n ι ⊤ A i ι y ⊤ Q ι A i Q ι y y ⊤ Q ι y , i = p , c , g .

Then, it follows that

  1. Anderson’s first circular serial correlation coefficient, ψ in (2) equals I c ,

  2. Orcutt’s first serial correlation coefficient, ϕ in (1) equals I p , and

  3. Moran’s r 11 in (3) equals I g .

Proof

See Appendix A.□

We make three remarks on Proposition 1.

  1. The perspective on the four autocorrelation measures in different fields provided by Proposition 1 is beneficial because studying the properties of Moran’s I leads to studying the properties of the other three autocorrelation measures. We give one such example. De Jong et al. [13] derived the bounds of Moran’s I . Then, given the results in Proposition 1, we can find the bounds of the other autocorrelation measures.[2]

  2. Proposition 1(ii) is of particular interest. This is because it is an example showing that time series analysis can be seen as a kind of spatial analysis. See also [14,15] for more such examples.

  3. Let I i ∗ = n ι ⊤ ( w A i ) ι y ⊤ Q ι ( w A i ) Q ι y y ⊤ Q ι y for i = p , c , g , where w is a positive real number. Then it follows that I i ∗ = I i for i = p , c , g . Therefore, I i in Proposition 1 can be replaced by I i ∗ .

4 Concluding remarks

In this article, using notions from linear algebraic graph theory, we have provided a unified perspective on some autocorrelation measures in different fields, namely, Orcutt’s first serial correlation coefficient, ϕ in (1), Anderson’s first circular serial correlation coefficient, ψ in (2), Moran’s r 11 in (3), and Moran’s I in (4). The main results of this article are demonstrated in Proposition 1.

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Acknowledgments

The author would like to thank Kazuhiko Hayakawa, Ryo Okui, and four anonymous referees for their valuable comments on an earlier version of this article. The usual caveat applies.

  1. Funding information: The Japan Society for the Promotion of Science supported this work through KAKENHI Grant No. 20K20759.

  2. Ethical approval: The conducted research is not related to either human or animal use.

  3. Conflict of interest: The author states no conflict of interest.

Appendix A Proof of Proposition 1

In this section, we provide the proof of Proposition 1. Let e i be the i th column of I n , i.e., I n = [ e 1 , … , e n ] and P ι = ι ( ι ⊤ ι ) − 1 ι ⊤ .

A.1 Proof of Proposition 1(i): ψ = I c

Let J c = [ e 2 , … , e n , e 1 ] . Then, given that A c = J c + J c ⊤ and ι ⊤ A c ι = 2 n , I c can be represented as

(A1) I c = n 2 n y ⊤ Q ι ( J c + J c ⊤ ) Q ι y y ⊤ Q ι y = 1 2 y ⊤ Q ι ( J c + J c ⊤ ) Q ι y y ⊤ Q ι y .

Next, given y 0 = y n , it follows that

( J c − P ι ) y = J c y − P ι y = y n y 1 ⋮ y n − 1 − y ¯ y ¯ ⋮ y ¯ = y 0 − y ¯ y 1 − y ¯ ⋮ y n − 1 − y ¯ ,

which leads to ∑ i = 1 n ( y i − y ¯ ) ( y i − 1 − y ¯ ) = ( Q ι y ) ⊤ { ( J c − P ι ) y } = y ⊤ Q ι ( J c − P ι ) y . Thus, ψ in (2) can be represented in matrix form as ψ = y ⊤ Q ι ( J c − P ι ) y y ⊤ Q ι y . Given J c ι = ι , it follows that J c Q ι = J c − J c P ι = ( J c − P ι ) , which yields ψ = y ⊤ Q ι J c Q ι y y ⊤ Q ι y . Moreover, given that y ⊤ Q ι J c Q ι y is a scalar, it follows that y ⊤ Q ι J c Q ι y = y ⊤ Q ι J c ⊤ Q ι y , which yields y ⊤ Q ι J c Q ι y = 1 2 y ⊤ Q ι ( J c + J c ⊤ ) Q ι y . Thus, we have

(A2) ψ = 1 2 y ⊤ Q ι ( J c + J c ⊤ ) Q ι y y ⊤ Q ι y .

Therefore, from (A1) and (A2), we obtain ψ = I c .

A.2 Proof of Proposition 1(ii): ϕ = I p

Let J p = [ e 2 , … , e n , 0 ] ∈ R n × n . Then, given that A p = J p + J p ⊤ and ι ⊤ A p ι = 2 ( n − 1 ) , I p can be represented as

(A3) I p = n 2 ( n − 1 ) y ⊤ Q ι ( J p + J p ⊤ ) Q ι y y ⊤ Q ι y .

Let y 0 = y ¯ . Then, given ( y 1 − y ¯ ) ( y 0 − y ¯ ) = 0 , we have ∑ i = 2 n ( y i − y ¯ ) ( y i − 1 − y ¯ ) = ∑ i = 1 n ( y i − y ¯ ) ( y i − 1 − y ¯ ) . Let J † = J p + 1 n e 1 ι ⊤ . Then, it follows that

( J † − P ι ) y = J † y − P ι y = y ¯ y 1 ⋮ y n − 1 − y ¯ y ¯ ⋮ y ¯ = y 0 − y ¯ y 1 − y ¯ ⋮ y n − 1 − y ¯ ,

which leads to ∑ i = 2 n ( y i − y ¯ ) ( y i − 1 − y ¯ ) = ∑ i = 1 n ( y i − y ¯ ) ( y i − 1 − y ¯ ) = ( Q ι y ) ⊤ { ( J † − P ι ) y } = y ⊤ Q ι ( J † − P ι ) y . Thus, ϕ can be represented in matrix form as ϕ = n n − 1 y ⊤ Q ι ( J † − P ι ) y y ⊤ Q ι y . Given J † Q ι = ( J p + 1 n e 1 ι ⊤ ) Q ι = J p Q ι and J † ι = ι , it follows that J p Q ι = J † Q ι = J † − J † P ι = ( J † − P ι ) , which yields ϕ = n n − 1 y ⊤ Q ι J p Q ι y y ⊤ Q ι y . Moreover, given that y ⊤ Q ι J p Q ι y is a scalar, it follows that y ⊤ Q ι J p Q ι y = y ⊤ Q ι J p ⊤ Q ι y , which leads to y ⊤ Q ι J p Q ι y = 1 2 y ⊤ Q ι ( J p + J p ⊤ ) Q ι y . Thus, we have

(A4) ϕ = n 2 ( n − 1 ) y ⊤ Q ι ( J p + J p ⊤ ) Q ι y y ⊤ Q ι y .

Therefore, from (A3) and (A4), we obtain ϕ = I p .

A.3 Proof of Proposition 1(iii): r 11 = I g

Denote the adjacency matrix of G g ∗ = ( V , E g ∗ ) [resp. G g ∗ ∗ = ( V , E g ∗ ∗ ) ] by A g ∗ [resp. A g ∗ ∗ ]. Then, it follows that A g = A g ∗ + A g ∗ ∗ and these adjacency matrices are explicitly expressed as A g ∗ = J ∗ + J ∗ ⊤ and A g ∗ ∗ = J ∗ ∗ + J ∗ ∗ ⊤ , where J ∗ = [ e 2 , … , e q , 0 , … , e q ( p − 1 ) + 2 , … , e p q , 0 ] ∈ R p q × p q and J ∗ ∗ = [ e q + 1 , … , e 2 q , … , e q ( p − 1 ) + 1 , … , e p q , 0 , … , 0 ] ∈ R p q × p q . Accordingly, given n = p q , ι ⊤ A g ∗ ι = 2 p ( q − 1 ) , and ι ⊤ A g ∗ ∗ ι = 2 q ( p − 1 ) , it follows that

I g = p q 2 ( 2 p q − p − q ) y ⊤ Q ι ( J ∗ + J ∗ ⊤ + J ∗ ∗ + J ∗ ∗ ⊤ ) Q ι y y ⊤ Q ι y .

Thus, given that ∑ i = 1 p ∑ j = 1 q ( x i , j − x ¯ ) 2 = ∑ i = 1 n ( y i − y ¯ ) 2 = y ⊤ Q ι y , if the following (A5) and (A6) hold, then I g = r 11 .

(A5) ∑ i = 1 p ∑ j = 1 q − 1 ( x i , j − x ¯ ) ( x i , j + 1 − x ¯ ) = 1 2 y ⊤ Q ι ( J ∗ + J ∗ ⊤ ) Q ι y ,

(A6) ∑ i = 1 p − 1 ∑ j = 1 q ( x i , j − x ¯ ) ( x i + 1 , j − x ¯ ) = 1 2 y ⊤ Q ι ( J ∗ ∗ + J ∗ ∗ ⊤ ) Q ι y .

Accordingly, we prove them in order. First, given that y ⊤ Q ι J ∗ Q ι y = y ⊤ Q ι J ∗ ⊤ Q ι y , (A5) is equivalent to ∑ i = 1 p ∑ j = 1 q − 1 ( x i , j − x ¯ ) ( x i , j + 1 − x ¯ ) = y ⊤ Q ι J ∗ Q ι y . Here, given

y ⊤ Q ι J ∗ Q ι y = { ( y 1 − y ¯ ) ( y 2 − y ¯ ) + ⋯ + ( y q − 1 − y ¯ ) ( y q − y ¯ ) } + ⋯ + { ( y q ( p − 1 ) + 1 − y ¯ ) ( y q ( p − 1 ) + 2 − y ¯ ) + ⋯ + ( y p q − 1 − y ¯ ) ( y p q − y ¯ ) } = ∑ i = 1 p ∑ j = 1 q − 1 ( y q ( i − 1 ) + j − y ¯ ) ( y q ( i − 1 ) + j + 1 − y ¯ ) ,

x i , j = y q ( i − 1 ) + j for i = 1 , … , p and j = 1 , … , q , and x ¯ = y ¯ , we have (A5). Second, given that y ⊤ Q ι J ∗ ∗ Q ι y = y ⊤ Q ι J ∗ ∗ ⊤ Q ι y , (A5) is equivalent to ∑ i = 1 p − 1 ∑ j = 1 q ( x i , j − x ¯ ) ( x i + 1 , j − x ¯ ) = y ⊤ Q ι J ∗ ∗ Q ι y . Here, given

y ⊤ Q ι J ∗ ∗ Q ι y = { ( y 1 − y ¯ ) ( y q + 1 − y ¯ ) + ⋯ + ( y q − y ¯ ) ( y 2 q − y ¯ ) } + ⋯ + { ( y q ( p − 2 ) + 1 − y ¯ ) ( y q ( p − 1 ) + 1 − y ¯ ) + ⋯ + ( y q ( p − 1 ) − y ¯ ) ( y p q − y ¯ ) } = ∑ i = 1 p − 1 ∑ j = 1 q ( y q ( i − 1 ) + j − y ¯ ) ( y q i + j − y ¯ ) ,

x i , j = y q ( i − 1 ) + j for i = 1 , … , p and j = 1 , … , q , and x ¯ = y ¯ , we have (A6).

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Received: 2022-09-03
Revised: 2023-02-03
Accepted: 2023-03-06
Published Online: 2023-04-01

© 2023 the author(s), published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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  59. Estimates for certain class of rough generalized Marcinkiewicz functions along submanifolds
  60. On semigroups of transformations that preserve a double direction equivalence
  61. Positive solutions for discrete Minkowski curvature systems of the Lane-Emden type
  62. A multigrid discretization scheme based on the shifted inverse iteration for the Steklov eigenvalue problem in inverse scattering
  63. Existence and nonexistence of solutions for elliptic problems with multiple critical exponents
  64. Interpolation inequalities in generalized Orlicz-Sobolev spaces and applications
  65. General Randić indices of a graph and its line graph
  66. On functional reproducing kernels
  67. On the Waring-Goldbach problem for two squares and four cubes
  68. Singular moduli of rth Roots of modular functions
  69. Classification of self-adjoint domains of odd-order differential operators with matrix theory
  70. On the convergence, stability and data dependence results of the JK iteration process in Banach spaces
  71. Hardy spaces associated with some anisotropic mixed-norm Herz spaces and their applications
  72. Remarks on hyponormal Toeplitz operators with nonharmonic symbols
  73. Complete decomposition of the generalized quaternion groups
  74. Injective and coherent endomorphism rings relative to some matrices
  75. Finite spectrum of fourth-order boundary value problems with boundary and transmission conditions dependent on the spectral parameter
  76. Continued fractions related to a group of linear fractional transformations
  77. Multiplicity of solutions for a class of critical Schrödinger-Poisson systems on the Heisenberg group
  78. Approximate controllability for a stochastic elastic system with structural damping and infinite delay
  79. On extremal cacti with respect to the first degree-based entropy
  80. Compression with wildcards: All exact or all minimal hitting sets
  81. Existence and multiplicity of solutions for a class of p-Kirchhoff-type equation RN
  82. Geometric classifications of k-almost Ricci solitons admitting paracontact metrices
  83. Positive periodic solutions for discrete time-delay hematopoiesis model with impulses
  84. On Hermite-Hadamard-type inequalities for systems of partial differential inequalities in the plane
  85. Existence of solutions for semilinear retarded equations with non-instantaneous impulses, non-local conditions, and infinite delay
  86. On the quadratic residues and their distribution properties
  87. On average theta functions of certain quadratic forms as sums of Eisenstein series
  88. Connected component of positive solutions for one-dimensional p-Laplacian problem with a singular weight
  89. Some identities of degenerate harmonic and degenerate hyperharmonic numbers arising from umbral calculus
  90. Mean ergodic theorems for a sequence of nonexpansive mappings in complete CAT(0) spaces and its applications
  91. On some spaces via topological ideals
  92. Linear maps preserving equivalence or asymptotic equivalence on Banach space
  93. Well-posedness and stability analysis for Timoshenko beam system with Coleman-Gurtin's and Gurtin-Pipkin's thermal laws
  94. On a class of stochastic differential equations driven by the generalized stochastic mixed variational inequalities
  95. Entire solutions of two certain Fermat-type ordinary differential equations
  96. Generalized Lie n-derivations on arbitrary triangular algebras
  97. Markov decision processes approximation with coupled dynamics via Markov deterministic control systems
  98. Notes on pseudodifferential operators commutators and Lipschitz functions
  99. On Graham partitions twisted by the Legendre symbol
  100. Strong limit of processes constructed from a renewal process
  101. Construction of analytical solutions to systems of two stochastic differential equations
  102. Two-distance vertex-distinguishing index of sparse graphs
  103. Regularity and abundance on semigroups of partial transformations with invariant set
  104. Liouville theorems for Kirchhoff-type parabolic equations and system on the Heisenberg group
  105. Spin(8,C)-Higgs pairs over a compact Riemann surface
  106. Properties of locally semi-compact Ir-topological groups
  107. Transcendental entire solutions of several complex product-type nonlinear partial differential equations in ℂ2
  108. Ordering stability of Nash equilibria for a class of differential games
  109. A new reverse half-discrete Hilbert-type inequality with one partial sum involving one derivative function of higher order
  110. About a dubious proof of a correct result about closed Newton Cotes error formulas
  111. Ricci ϕ-invariance on almost cosymplectic three-manifolds
  112. Schur-power convexity of integral mean for convex functions on the coordinates
  113. A characterization of a ∼ admissible congruence on a weakly type B semigroup
  114. On Bohr's inequality for special subclasses of stable starlike harmonic mappings
  115. Properties of meromorphic solutions of first-order differential-difference equations
  116. A double-phase eigenvalue problem with large exponents
  117. On the number of perfect matchings in random polygonal chains
  118. Evolutoids and pedaloids of frontals on timelike surfaces
  119. A series expansion of a logarithmic expression and a decreasing property of the ratio of two logarithmic expressions containing cosine
  120. The 𝔪-WG° inverse in the Minkowski space
  121. Stability result for Lord Shulman swelling porous thermo-elastic soils with distributed delay term
  122. Approximate solvability method for nonlocal impulsive evolution equation
  123. Construction of a functional by a given second-order Ito stochastic equation
  124. Global well-posedness of initial-boundary value problem of fifth-order KdV equation posed on finite interval
  125. On pomonoid of partial transformations of a poset
  126. New fractional integral inequalities via Euler's beta function
  127. An efficient Legendre-Galerkin approximation for the fourth-order equation with singular potential and SSP boundary condition
  128. Eigenfunctions in Finsler Gaussian solitons
  129. On a blow-up criterion for solution of 3D fractional Navier-Stokes-Coriolis equations in Lei-Lin-Gevrey spaces
  130. Some estimates for commutators of sharp maximal function on the p-adic Lebesgue spaces
  131. A preconditioned iterative method for coupled fractional partial differential equation in European option pricing
  132. A digital Jordan surface theorem with respect to a graph connectedness
  133. A quasi-boundary value regularization method for the spherically symmetric backward heat conduction problem
  134. The structure fault tolerance of burnt pancake networks
  135. Average value of the divisor class numbers of real cubic function fields
  136. Uniqueness of exponential polynomials
  137. An application of Hayashi's inequality in numerical integration
https://doi.org/10.1515/math-2022-0574
Keywords for this article
Orcutt’s first serial correlation coefficient; Anderson’s first circular serial correlation coefficient; Moran’s r 11; Moran’s I; path graph; cycle graph; two-dimensional lattice graph; serial correlation; spatial autocorrelation
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  58. Uniqueness theorems of the Hahn difference operator of entire function with a Picard exceptional value
  59. Estimates for certain class of rough generalized Marcinkiewicz functions along submanifolds
  60. On semigroups of transformations that preserve a double direction equivalence
  61. Positive solutions for discrete Minkowski curvature systems of the Lane-Emden type
  62. A multigrid discretization scheme based on the shifted inverse iteration for the Steklov eigenvalue problem in inverse scattering
  63. Existence and nonexistence of solutions for elliptic problems with multiple critical exponents
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  71. Hardy spaces associated with some anisotropic mixed-norm Herz spaces and their applications
  72. Remarks on hyponormal Toeplitz operators with nonharmonic symbols
  73. Complete decomposition of the generalized quaternion groups
  74. Injective and coherent endomorphism rings relative to some matrices
  75. Finite spectrum of fourth-order boundary value problems with boundary and transmission conditions dependent on the spectral parameter
  76. Continued fractions related to a group of linear fractional transformations
  77. Multiplicity of solutions for a class of critical Schrödinger-Poisson systems on the Heisenberg group
  78. Approximate controllability for a stochastic elastic system with structural damping and infinite delay
  79. On extremal cacti with respect to the first degree-based entropy
  80. Compression with wildcards: All exact or all minimal hitting sets
  81. Existence and multiplicity of solutions for a class of p-Kirchhoff-type equation RN
  82. Geometric classifications of k-almost Ricci solitons admitting paracontact metrices
  83. Positive periodic solutions for discrete time-delay hematopoiesis model with impulses
  84. On Hermite-Hadamard-type inequalities for systems of partial differential inequalities in the plane
  85. Existence of solutions for semilinear retarded equations with non-instantaneous impulses, non-local conditions, and infinite delay
  86. On the quadratic residues and their distribution properties
  87. On average theta functions of certain quadratic forms as sums of Eisenstein series
  88. Connected component of positive solutions for one-dimensional p-Laplacian problem with a singular weight
  89. Some identities of degenerate harmonic and degenerate hyperharmonic numbers arising from umbral calculus
  90. Mean ergodic theorems for a sequence of nonexpansive mappings in complete CAT(0) spaces and its applications
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  92. Linear maps preserving equivalence or asymptotic equivalence on Banach space
  93. Well-posedness and stability analysis for Timoshenko beam system with Coleman-Gurtin's and Gurtin-Pipkin's thermal laws
  94. On a class of stochastic differential equations driven by the generalized stochastic mixed variational inequalities
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  96. Generalized Lie n-derivations on arbitrary triangular algebras
  97. Markov decision processes approximation with coupled dynamics via Markov deterministic control systems
  98. Notes on pseudodifferential operators commutators and Lipschitz functions
  99. On Graham partitions twisted by the Legendre symbol
  100. Strong limit of processes constructed from a renewal process
  101. Construction of analytical solutions to systems of two stochastic differential equations
  102. Two-distance vertex-distinguishing index of sparse graphs
  103. Regularity and abundance on semigroups of partial transformations with invariant set
  104. Liouville theorems for Kirchhoff-type parabolic equations and system on the Heisenberg group
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  106. Properties of locally semi-compact Ir-topological groups
  107. Transcendental entire solutions of several complex product-type nonlinear partial differential equations in ℂ2
  108. Ordering stability of Nash equilibria for a class of differential games
  109. A new reverse half-discrete Hilbert-type inequality with one partial sum involving one derivative function of higher order
  110. About a dubious proof of a correct result about closed Newton Cotes error formulas
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  112. Schur-power convexity of integral mean for convex functions on the coordinates
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  114. On Bohr's inequality for special subclasses of stable starlike harmonic mappings
  115. Properties of meromorphic solutions of first-order differential-difference equations
  116. A double-phase eigenvalue problem with large exponents
  117. On the number of perfect matchings in random polygonal chains
  118. Evolutoids and pedaloids of frontals on timelike surfaces
  119. A series expansion of a logarithmic expression and a decreasing property of the ratio of two logarithmic expressions containing cosine
  120. The 𝔪-WG° inverse in the Minkowski space
  121. Stability result for Lord Shulman swelling porous thermo-elastic soils with distributed delay term
  122. Approximate solvability method for nonlocal impulsive evolution equation
  123. Construction of a functional by a given second-order Ito stochastic equation
  124. Global well-posedness of initial-boundary value problem of fifth-order KdV equation posed on finite interval
  125. On pomonoid of partial transformations of a poset
  126. New fractional integral inequalities via Euler's beta function
  127. An efficient Legendre-Galerkin approximation for the fourth-order equation with singular potential and SSP boundary condition
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  129. On a blow-up criterion for solution of 3D fractional Navier-Stokes-Coriolis equations in Lei-Lin-Gevrey spaces
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  131. A preconditioned iterative method for coupled fractional partial differential equation in European option pricing
  132. A digital Jordan surface theorem with respect to a graph connectedness
  133. A quasi-boundary value regularization method for the spherically symmetric backward heat conduction problem
  134. The structure fault tolerance of burnt pancake networks
  135. Average value of the divisor class numbers of real cubic function fields
  136. Uniqueness of exponential polynomials
  137. An application of Hayashi's inequality in numerical integration
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