运动意象中的因果关系分析：格兰杰因果与新因果的对比
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"这篇研究论文对比分析了格兰杰因果关系(Granger Causality, GC)和新因果关系(New Causality, NC)，并探讨它们在运动意象(Motor Imagery, MI)中的应用。文章指出GC在估计自回归模型时存在的问题，即其采用的逆向递归操作在许多不可逆化学反应模型中无法实现，从而引出NC作为更敏感揭示因果关系的理论。通过使用GC和NC分析2008年脑机接口竞赛IV中九名受试者的头皮脑电图数据，研究MI过程中的因果流动力学，关注大脑中央皮层的Cz区和左侧C3区等关键区域。"
在IT领域，尤其是神经网络和学习系统的研究中，因果关系分析是一个重要的工具，用于理解复杂系统的动态行为。格兰杰因果关系是一种统计学概念，用于判断一个时间序列是否对另一个时间序列有预测能力，从而推断因果效应。然而，GC方法依赖于对时间序列的自回归模型估计，这在处理某些非线性、非平稳或者物理上不可逆的过程时可能存在局限。
新因果关系（New Causality）是针对GC的局限性提出的一种改进方法。如论文中所述，NC理论上更敏感，能更好地识别真实存在的因果关系，特别是在处理那些不能用简单逆向操作描述的动态系统时。这在理解不可逆化学反应或其他复杂生物过程，如大脑功能活动时，具有显著优势。
运动意象是指人们在没有实际执行动作的情况下，在大脑中模拟运动的能力。这种心理过程在认知神经科学和心理学中有重要研究价值，尤其在脑机接口(Brain-Computer Interface, BCI)领域，MI被广泛用于开发非侵入式通信和控制通道。通过分析MI过程中的因果流，可以深入了解大脑如何编码和处理运动指令，从而优化BCI的设计和性能。
在论文中，研究人员使用2008年脑机接口竞赛IV的数据，该竞赛旨在推动BCI技术的发展。通过对九个受试者的头皮脑电图（EEG）信号进行分析，他们特别关注Cz和C3这两个大脑区域，因为这些区域通常与运动想象相关联。通过GC和NC的对比，可能揭示大脑不同区域在MI过程中如何相互作用，以及哪些区域起主导作用，这对于理解大脑的工作机制以及改进BCI技术至关重要。
这篇研究论文提供了关于如何利用新的因果分析方法改进对大脑活动理解的见解，特别是在运动意象研究中的应用。这不仅有助于深化我们对大脑功能的认识，也有助于推动脑机接口技术的进步，为未来开发更高效、更精确的BCI系统奠定基础。
1432 IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS, VOL. 27, NO. 7, JULY 2016
very small and can be neglected. On the other hand, in terms
of |a
22,1
| < 1 we can have lim
n→+∞
a
12,1
a
n−1
22,1
X
2,t−n
= 0.
Therefore, when m
is large enough and n m
, (11) can be
approximately written as
X
1,t
= a
11,1
X
1,t−1
+ a
12,1
a
21,1
×
X
1,t−2
+ a
22,1
X
1,t−3
+···+a
m
−2
22,1
X
1,t−m
+a
12,1
η
2,t−1
+ a
22,1
η
2,t−2
+···+a
m
−1
22,1
η
2,t−m
+η
1,t
. (13)
Moreover, we can easily observe the fact that the larger m
is chosen, the more close to the real X
1,t
the right side
of (13) is. Comparing (13) with X
1,t
in the autoregressive
model (1), one can have a
1,1
= a
11,1
, a
1,2
= a
12,1
a
21,1
,
a
1,3
= a
12,1
a
21,1
a
22,1
,...,a
1,m
= a
12,1
a
21,1
a
m
−2
22,1
,
1,t
=
a
12,1
η
2,t−1
+ a
22,1
η
2,t−2
+···+a
m
−1
22,1
η
2,t−m
+η
1,t
.
Let ξ
k,t
= η
2,t−k
, k = 1,...,m
. Then ξ
1
,...,ξ
m
are m
random variables. Since η
1
and η
2
are uncorrelated
over time and η
1
and η
2
have zero mean, one can easily see
that: 1) ξ
1
,...,ξ
m
are all uncorrelated and have zero mean,
and
1
has zero mean; 2) σ
2
ξ
k
= σ
2
η
2
, k = 1,...,m
;and
3) E[η
1
ξ
k
]=0, k = 1,...,m
. Then, we can derive the
variance of
1
as
σ
2
1
= E
2
1
= E
⎡
⎢
⎣
⎛
⎝
η
1
+
m
k=1
a
12,1
a
k−1
22,1
ξ
k
⎞
⎠
2
⎤
⎥
⎦
= E
⎡
⎢
⎣
η
2
1
+
⎛
⎝
m
k=1
a
12,1
a
k−1
22,1
ξ
k
⎞
⎠
2
+ 2η
1
m
k=1
a
12,1
a
k−1
22,1
ξ
k
⎤
⎥
⎦
= E
η
2
1
+
m
k=1
a
2
12,1
a
2(k−1)
22,1
E
ξ
2
k
+2
m
k=1
a
12,1
a
k−1
22,1
E[η
1
ξ
k
]
= σ
2
η
1
+ σ
2
η
2
a
2
12,1
m
k=1
a
2(k−1)
22,1
= σ
2
η
1
+ σ
2
η
2
a
2
12,1
1 − a
2m
22,1
1 − a
2
22,1
. (14)
Since |a
22,1
| < 1, when m
is large enough, one can see that
a
2m
22,1
is very small and can be neglected. Thus, from (14) one
can approximately obtain
σ
2
1
= σ
2
η
1
+
σ
2
η
2
a
2
12,1
1 − a
2
22,1
. (15)
Then, from (14) it follows that GC value:
F
X
2
→X
1
= ln
σ
2
1
σ
2
η
1
= ln
1 +
σ
2
η
2
a
2
12,1
1 − a
2m
22,1
σ
2
η
1
1 − a
2
22,1
> 0
(16)
or, from (15), it follows that GC value can be approximately
written as:
F
X
2
→X
1
= ln
σ
2
1
σ
2
η
1
≈ ln
1 +
σ
2
η
2
a
2
12,1
σ
2
η
1
1 − a
2
22,1
> 0 (17)
when m
in (16) is large enough.
Therefore, in a mathematical way we give an exact descrip-
tion for GC from X
2
to X
1
in (16) or (17) from which one can
clearly see that GC from X
2
to X
1
involves the coefficients
(a
12,1
and a
22,1
) and variances (σ
2
η
1
and σ
2
η
2
) of the noises, of
the linear regression model (9).
According to (6), one can see that NC from X
2
to X
1
for
model (9) can be written as
n
X
2
→X
1
=
N
t=1
(a
12,1
X
2,t−1
)
2
2
h=1
N
t=1
(a
1h,1
X
h,t−1
)
2
+ Nσ
2
η
1
(18)
from which one can clearly see that NC in (18) also involves
the coefficients (a
11,1
and a
12,1
) and variance (σ
2
η
1
)ofthe
noise, of the linear regression model (9).
Therefore, we have theoretically shown that both
GC and NC involve coefficients of the linear regression
model. Barrett and Barnett [35] said that NC is a causal
mechanism because it includes the coefficients of the linear
regression model. This statement is not correct because GC
in a mathematical way also involves the coefficients of the
linear regression model. Thus, NC like GC and all GC-alike
methods reveals causal effect (causal influence, causal flow, to
name a few). In fact, causal mechanism represents a process
and is a different concept from causal effect (see the diagram
of a simple causal mechanism in [36, Fig. 1]).
C. GC in Frequency Domain
Granger causal influence from X
2
to X
1
in frequency
domain is defined by
I
X
2
→X
1
( f ) =−ln
⎛
⎜
⎜
⎝
1 −
σ
η
2
2
−
σ
η
1
η
2
2
σ
η
1
2
|H
12
( f ) |
2
S
X
1
X
1
⎞
⎟
⎟
⎠
∈[0, +∞) (19)
where the transfer function H ( f ) = A
−1
( f ) whose compo-
nents are
H
11
( f ) =
1
det(A)
¯a
22
( f ), H
12
( f ) =
1
det(A)
¯a
12
( f )
H
21
( f ) =
1
det(A)
¯a
21
( f ), H
22
( f ) =
1
det(A)
¯a
11
( f )
A =[¯a
ij
]
2×2
, ¯a
kk
( f ) = 1−
m
j=1
a
kk, j
e
−i2π fk
k =1, 2
¯a
hl
( f ) =−
m
j=1
a
hl, j
e
−i2π fk
, h, l = 1, 2, h = l.
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