In this section, we give a detailed description of the SSLV that we proposed and the problems which are brought by local vote. Section 3.1 presents the scan statistics with local vote decision algorithm. The correlation of local sensors is presented in Section 3.2. According to our analysis, it turns out that the results provided by sensors are not independent and identically distributed (i.i.d) anymore after the local vote. It makes an expression (4) cannot be used in the SSLV, but a new expression for the SSLV is deduced in Section 3.3. Section 3.4 introduces a variable-step-parameter into the SSLV in order to take full advantage of the local vote.

### 3.1 Scan statistics with local vote decision

Precisely, in this part, we will give an introduction about the SSLV in two-dimensional region. The underlying assumptions in the last part are still suitable here. The difference is that we let sensors make a local vote decision before scan statistics. According to [6], we can take various neighborhood algorithms, such as fixed distance *r* or fixed size. Any one of those algorithms can be selected, then all corresponding parameters can be confirmed. For better description, we will redefine some variables. Let *X*
_{
i,j
} be the event that has been observed in the rectangular sub-regions [*i*
*h*
_{1},(*i* + 1)*h*
_{1}]×[*j*
*h*
_{2},(*j* + 1)*h*
_{2}] after local vote where 1 ≤ *i* ≤ *N*
_{1} and 1 ≤ *j* ≤ *N*
_{2}, *N*
_{
i
} = *N*
^{′}
_{
i
}−2, and for simplicity, we exclude the rectangular sub-regions on the edge of the field in this part. For 1 ≤ *i* ≤ *N*
_{1} and 1 ≤ *j* ≤ *N*
_{2}, let *m*
_{1} and *m*
_{2} be the positive integers, 1 ≤ *m*
_{1} ≤ *N*
_{1} and 1 ≤ *m*
_{2} ≤ *N*
_{2}.

$$ {{v}_{{{i}_{1}},{{i}_{2}}}}~=~\sum\limits_{j~=~{{i}_{2}}}^{{{i}_{2}}~+~{{m}_{2}}~-~1}{\sum\limits_{i~=~{{i}_{1}}}^{{{i}_{1~}}+~{{m}_{1}}~-~1} {{{X}_{i,j}}}} $$

(5)

Similarly, if \({v_{^{{i_{1}}, {i_{2}}}}}\) exceeds a pre-set value of *k*, the MA makes the final decision that a target is present. The largest number of events in an agent region can be expressed as

$$\begin{array}{@{}rcl@{}} {{S}_{{{m}_{1}}~\times~ {{m}_{2}};~ {{N}_{1}}z~\times~ {{N}_{2}}}}&=&\max~\{{{v}_{{{i}_{1}}, {{i}_{2}}}}; \text{}1~\le~ {{i}_{1}}~\le~ {{N}_{1}}~-~{{m}_{1}}\notag\\ &&+~1, 1~\le~ {{N}_{2}}~-~{{m}_{2}}~+~1\} \end{array} $$

(6)

For simplicity, we abbreviate \({{S}_{{{m}_{1~}}\times ~ {{m}_{2}}; {{~N}_{1}}~\times ~ {{N}_{2}}}}\phantom {\dot {i}\!}\) to *S*. The next step is to obtain the expression of \(P\left (S_{m_{1}~\times ~ m_{2}~N~\times ~ N} ~\geq ~ k\right)\) to make the SSLV useful.

### 3.2 Correlation of sensors

Our algorithm introduces local vote decision into the traditional scan statistics. Therefore, we should figure out what has changed after the combination of two algorithms. The dependence among sensors should be examined first. For any sensor detection event *Z*
_{
i
}, we start by calculating the expected value *μ*
_{
i
} and variance *σ*
_{
i
}
^{2} of the updated decision.

$$ {{\mu }_{i}}~=~P~\left({{Z}_{i}}~=~1 \right)=\sum\limits_{n~={{~M}_{x}}}^{{{M}_{i}}}{\left(\begin{array}{c} {{M}_{i}} \\ n \\ \end{array} \right){{\alpha }^{n}}{{(1~-~\alpha)}^{{{M}_{i~}}-~n}}} $$

(7)

where *M*
_{
i
} is the number of neighbors which depends on local vote decision algorithm. *M*
_{
x
} is a variable that has a significant influence on the performance. *σ*
_{
i
}
^{2} = *μ*
_{
i
}(1 − *μ*
_{
i
}).

The dependence between *Z*
_{
i
} and *Z*
_{
j
} has relations with the intersection of their respective neighborhoods *U*(*i*) and *U*(*j*). The number of sensors in the intersection *U*(*i*)∩*U*(*j*) can be denoted by *n*
_{
i,j
}. According to the expression of covariance, we first compute *E*(*Z*
_{
i
}
*Z*
_{
j
}) = *P*(*Z*
_{
i
} = *Z*
_{
j
} = 1) and then calculate the covariance between *Z*
_{
i
} and *Z*
_{
j
}. We divide the neighborhoods into three parts. Suppose that A is the number of positive decisions in *U*(*i*)∩*U*(*j*) and B is the number of positive decisions in *U*(*i*), but not in *U*(*j*), while C is the number of positive decisions in *U*(*j*), but not in *U*(*i*). Noting that A, B, and C are independent, we can have

$$ \begin{aligned} E\left({{Z}_{i}}{{Z}_{j}} \right)~&=~\sum\limits_{k~=~0}^{{{n}_{i,j}}}P(A~=~k)P(B~>~{{M}_{x}}_{i}~-~k)\\ &\quad \times P(C~>~{{M}_{xj}}~-~k) \end{aligned} $$

(8)

$$ P\left(A~=~k \right)=\left(\begin{array}{c} {{n}_{i, j}} \\ k \\ \end{array} \right){{\alpha }^{_{k}}}{{\left(1~-~\alpha \right)}^{{{n}_{i, j~}}-~k}} $$

(9)

$$\begin{array}{@{}rcl@{}} P\left(B>{{M}_{\mathrm{x}i}}~-~k \right) ~&=~\sum\limits_{q~=~{{M}_{\text{xi}}}~-~k~+~1}^{{{M}_{i}}~-~{{n}_{i, j}}}\left(\begin{array}{c} {{M}_{i}}~-~{{n}_{i, j}} \\ q \\ \end{array} \right)\notag\\ &{{\alpha }^{q}}{{(1~-~\alpha)}^{{{M}_{i}}~-~{{n}_{i,j}}~-~q}} \end{array} $$

(9)

$$\begin{array}{@{}rcl@{}} P\left(C~>~{{M}_{{{x}_{j}}}}~-~k \right) =\sum\limits_{q~=~{{M}_{{{x}_{j}}}}~-~k~+~1}^{{{M}_{j}}~-~{{n}_{_{i,j}}}}{\left(\begin{array}{c} {{M}_{j}}~-~{{n}_{^{i, j}}} \\ q \\ \end{array} \right)}\notag\\ {{\alpha }^{q}}{{(1~-~\alpha)}^{{{M}_{j}}~-~{{n}_{i,j}}~-~q}} \end{array} $$

(10)

The covariance is then given by

$$ Cov({Z_{i}},{Z_{j}}) ~=~ \left[ {E({Z_{i}},{Z_{j}}) ~-~ {\mu_{i}}{\mu_{j}}} \right]I({n_{i, j}} > 0) $$

(11)

According to the deductions above, we can find out that decision *X*
_{
i,j
} is not i.i.d anymore after the local vote.

### 3.3 Approximation for *P*(*S* ≥ *k*)

In [16], the authors give the proof of approximation when the *X*
_{
i,j
} is i.i.d with the Markov Chain imbeddable systems [17]. Obviously, it is not applicable here. Luckily, there are different ways to give the accurate approximation for *P*(*S* ≥ *k*) and one of them is using the Haiman theorem [18–20].

###
**Theorem 1**

Let {*X*
_{
i
}} be a stationary 1-dependent sequence of r.v’s and for *x* < *w*, *w* = sup{*u*;*P*(*X*
_{1}≤*u*)<1}, let *q*
_{
n
} = *q*
_{
n
}(*x*) = *P*{max(*X*
_{1},…*X*
_{
n
}) ≤ *x*}. For any x such that *P*(*X*
_{1}>*x*) = 1 − *q*
_{1} ≤ 0.025 and any integer *n* > 3 such that \(3.3n{{(1~-~{{q}_{^{1}}})}^{2}}~\le ~ 1\), we have

$$ \frac{\left| {{q}_{n}}-\frac{(2{{q}_{1}}~-~{{q}_{2}})}{{{(1~+~{{q}_{1}}~-~{{q}_{2}}~+~2{{({{q}_{1}}~-~{{q}_{2}})}^{2}})}^{n}}} \right|}{{{q}_{n}}}\le 3.3n{{(1~-~{{q}_{1}})}^{2}} $$

(12)

According to the Haiman theorem, we need to construct a stationary 1-dependent sequence. Supposing that *N*
_{1} = *K*
*m*
_{1} and *N*
_{2} = *L*
*m*
_{2}, where *K* and *L* are positive integers, we have

$$ {{Z}_{k}}=\underset{\begin{array}{c} (k~-~1){{m}_{1}}~<~ t~\le~ k{{m}_{1}} \\ 0~<~s~\le~(L-1){{m}_{2}} \end{array}}{\mathop{\max }}\,{{v}_{ts}},k~=~1, 2,\ldots K~-~1 $$

(13)

{*Z*
_{
k
}}_{
k = 1,…,k − 1} is a 1-dependent stationary sequence and \(P\left (S\le n \right)=P(\underset {k~=~1\ldots K~-~1}{\mathop {\max }}\,\{{{Z}_{k}}\}~\le ~n)\). Let *Q*
_{2} = *P*(*Z*
_{1} ≤ *n*) and *Q*
_{3} = *P*(*Z*
_{1} ≤ *n*,*Z*
_{2} ≤ *n*). Then, if 1-*Q*
_{2} ≤ 0.025, we can get approximation from Haiman theorem

$$ \begin{aligned} P\left(S~\le~ n \right)&\approx (2{{Q}_{2}}~-~{{Q}_{3}})[1~+~{{Q}_{2}}~-~{{Q}_{3}}\\ &\quad+~2{{({{Q}_{2}}-{{Q}_{3}})}^{2}}]^{-(K~-~1)} \end{aligned} $$

(14)

with an error of about 3.3(*K* − 1)(1 − *Q*
_{2})^{2}. To evaluate (15), one needs approximations for *Q*
_{2} and *Q*
_{3}. Hence, the question is transformed into evaluating *Q*
_{2} and *Q*
_{3}. We may apply Theorem 1 again considering the two sequences of random variables defined by

$${{Y}_{l}}=\underset{\begin{array}{c} 0 < t~\le~ {{m}_{1}} \\ (l~-~1){{m}_{2}}~<~ s~\le~ l{{m}_{2}} \\ \end{array}}{\mathop{\max }}\,{{v}_{ts}} $$

and

$${{Z}_{l}}=\underset{\begin{array}{c} 0 ~<~t\le 2{{m}_{1}} \\ (l~-~1){{m}_{2}}~<~ s\le l{{m}_{2}} \\ \end{array}}{\mathop{\max }}\,{{v}_{ts}},l=1,2,\ldots L-1 $$

which are also stationary and 1-dependent. Put *Q*
_{22}=*P*(*Y*
_{1}≤*n*), *Q*
_{23}=*P*(*Y*
_{1}≤*n*,*Y*
_{2}≤*n*), *Q*
_{32}=*P*(*Z*
_{1} ≤ *n*) and *Q*
_{33} = *P*(*Z*
_{1} ≤ *n*,*Z*
_{2} ≤ *n*). We have

$${{Q}_{22}}~=~P\left(S({{m}_{1}},{{m}_{2}},2{{m}_{1}},2{{m}_{2}})~\le~ n \right), $$

$${{Q}_{23}}~=~P\left(S({{m}_{1}},{{m}_{2}},2{{m}_{2}},3{{m}_{2}})~\le~ n \right), $$

$${{Q}_{32}}~=~P(S({{m}_{1}},{{m}_{2}},3{{m}_{1}},2{{m}_{2}})~\le~ n), $$

$${{Q}_{33}}~=~P(S({{m}_{1}},{{m}_{2}},3{{m}_{1}},3{{m}_{2}})~\le~ n). $$

Then, if 1-*Q*
_{22} ≤ 0.025 and 1-*Q*
_{32} ≤ 0.025, we can still get the approximations from Theorem 1.

$${} {{Q}_{2}}\approx (2{{Q}_{22}}-{{Q}_{23}}){{\left[1+{{Q}_{22}}-{{Q}_{23}}+2{{({{Q}_{22}}-{{Q}_{23}})}^{2}}\right]}^{-(L\text{-}1)}} $$

(15)

with an error of about 3.3(*L*−1)(1−*Q*
_{22})^{2} and

$${} {{Q}_{3}}\approx (2{{Q}_{32}}-{{Q}_{33}}){{[\!1+{{Q}_{32}}-{{Q}_{33}}+2{{({{Q}_{32}}-{{Q}_{33}})}^{2}}]}^{-(L-1)}} $$

(16)

with an error of about 3.3(*L* − 1)(1 − *Q*
_{32})^{2}.Assuming that *L* ≤ *K* and substituting (17) and (16) into (15), we can get the final expression we need.

The total error on the resulting approximation of *P*(*S* ≤ *n*) is bounded by about

$$\begin{array}{@{}rcl@{}} {{E}_{app}} &=3.3(L-1)(K~-~1)({{(1~-~{{Q}_{22}})}^{2}}~+~{{(1~-~{{Q}_{32}})}^{2}}\notag\\ &+(L-1){{({{Q}_{22}}-{{Q}_{23}})}^{2}}). \end{array} $$

(17)

The exact formulas for *Q*
_{
uv
},*u*,*v*∈{2,3} is hard to be obtained. Thus, we can use Monte Carlo simulation to evaluate these quantities. The final expression can be given by

$$\begin{array}{@{}rcl@{}} P\left(S\ge k \right)=1-P\left(S<k \right) =1-P(S\le k-1) \end{array} $$

(18)

where *P*(*S*≤*k* − 1) can be approximated by (15).

### 3.4 The SSLV with variable-step-parameter

The traditional scan statistics is a kind of continuous scan. Disjoint-window scan statistics means the MA travels across the ROI and scans the area using no overlapping windows. In [12], the authors investigate the disjoint-window test and compare its performance with the scan statistics. Obviously, the scan statistics overwhelms the disjoint-window, and its performance is more stable. However, the disjoint-window can shorten scan period. In this section, we will introduce a variable-step-parameter for the SSLV. In the process of scan, the MA makes a choice for the next start position according to the result of the current scan. Since the detection probability is based on the distance between the target and sensors, sensors near the target have a higher probability of detecting the target. If the result of detection is small, we can magnify the value of step to avoid the redundant scan especially when the target is absent. The variable step is given by

$$ step=\max \left\{ \left\lfloor \left(1~-~\frac{{{v}_{ts}}}{\text{k}} \right){{f}_{ov}} \right\rfloor,1 \right\} $$

(19)

The scan region can be a rectangular region given by *R*(*i*
_{1},*i*
_{2}) = [*i*
_{1}
*h*
_{1},(*i*
_{1} + *m*)*h*
_{1}] × [*i*
_{2}
*h*
_{2},(*i*
_{2} + *m*)*h*
_{2}]. Assuming *R*(*i*
_{1},*i*
_{2}) is the scan region at the current time, then the next scan region is *R*[(*i*
_{1}+*s*
*t*
*e*
*p*)*h*
_{1},(*i*
_{1}+*s*
*t*
*e*
*p*+*m*)*h*
_{1}]×[*i*
_{2}
*h*
_{2},(*i*
_{2} + *m*)*h*
_{2}], *i*
_{1}+*s*
*t*
*e*
*p*≤*N* − *m* + 1. We only introduce the step at one-dimensional field for better performance. Whereas, the global false alarm rate can still be evaluated by (15).