# Noise Enhancement for Weighted Sum of Type I and II Error Probabilities with Constraints

^{*}

## Abstract

**:**

## 1. Introduction

- Formulation of the optimization problem for minimizing the noise modified weighted sum of type I and II error probabilities under the constraints on the two error probabilities is presented.
- Derivations of the optimal noise that minimizes the weighted sum and sufficient conditions for improvability and nonimprovability for a general composite hypothesis testing problem are provided.
- Analysis of the characteristics of the optimal additive noise that minimizes the weighted sum for a simple hypothesis testing problem is studied and the corresponding algorithm to solve the optimization problem is developed.
- Numerical results are presented to verify the theoretical results and to demonstrate the superior performance of the proposed detector.

## 2. Noise Enhanced Composite Hypothesis Testing

#### 2.1. Problem Formulation

#### 2.2. Sufficient Conditions for Improvability and Non-improvability

**Theorem**

**1.**

**Theorem**

**2.**

- (1)
- $e{r}^{(1)}(\mathbf{x},\mathbf{g}){|}_{\mathbf{x}=0}<0$, ${a}_{{\theta}_{0}}^{(1)}(\mathbf{x},\mathbf{g}){|}_{\mathbf{x}=0}<0$, ${b}_{{\theta}_{1}}^{(1)}(\mathbf{x},\mathbf{g}){|}_{\mathbf{x}=0}<0$;
- (2)
- $e{r}^{(1)}(\mathbf{x},\mathbf{g}){|}_{\mathbf{x}=0}>0$, ${a}_{{\theta}_{0}}^{(1)}(\mathbf{x},\mathbf{g}){|}_{\mathbf{x}=0}>0$, ${b}_{{\theta}_{1}}^{(1)}(\mathbf{x},\mathbf{g}){|}_{\mathbf{x}=0}>0$;
- (3)
- $e{r}^{(2)}(\mathbf{x},\mathbf{g}){|}_{\mathbf{x}=0}<0$, ${a}_{{\theta}_{0}}^{(2)}(\mathbf{x},\mathbf{g}){|}_{\mathbf{x}=0}<0$, ${b}_{{\theta}_{1}}^{(2)}(\mathbf{x},\mathbf{g}){|}_{\mathbf{x}=0}<0$.

**Theorem**

**3.**

**Corollary**

**1.**

#### 2.3. Optimal Additive Noise

**Theorem**

**4.**

## 3. Noise Enhanced Simple Hypothesis Testing

#### 3.1. Problem Formulation

#### 3.2. Algorithm for the Optimal Additive Noise

**Theorem**

**5.**

- (1)
- If ${Q}_{e}\cap {Q}_{1}\ne \varnothing $, then $\eta =1$ and ${\mathbf{n}}_{1}\in {Q}_{e}\cap {Q}_{1}$ such that $E{r}_{opt}^{\mathbf{y}}=Er({\mathbf{n}}_{1})=\underset{\mathbf{n}}{\mathrm{min}}Er(\mathbf{n})$.
- (2)
- If ${Q}_{e}\cap {Q}_{2}\ne \varnothing $ and ${Q}_{e}\cap {Q}_{3}\ne \varnothing $ are true, then we have ${\mathbf{n}}_{1}\in {Q}_{e}\cap {Q}_{2}$, ${\mathbf{n}}_{2}\in {Q}_{e}\cap {Q}_{3}$, ${\eta}_{1}\le \eta \le {\eta}_{2}$, and $E{r}_{opt}^{\mathbf{y}}=\underset{\mathbf{n}}{\mathrm{min}}Er(\mathbf{n})$.
- (3)
- If ${Q}_{e}\subset {Q}_{2}$, then $E{r}_{opt}^{\mathbf{y}}$ is obtained when $\eta ={\eta}_{2}$, and the corresponding $E\{{A}_{0}(\mathbf{n})\}$ achieves the minimum and $E\{{B}_{1}(\mathbf{n})\}={\beta}_{o}$.
- (4)
- If ${Q}_{e}\subset {Q}_{3}$, then $E{r}_{opt}^{\mathbf{y}}$ is achieved when $\eta ={\eta}_{1}$, and the corresponding $E\{{A}_{0}(\mathbf{n})\}={\alpha}_{o}$ and $E\{{B}_{1}(\mathbf{n})\}$ reaches the minimum.

## 4. Numerical Results

#### 4.1. Rayleigh Distribution Background Noise

#### 4.2. Gaussian Mixture Background Noise

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix A. Proof of Theorem 1

**Proof.**

## Appendix B. Proof of Theorem 2

**Proof.**

- (1)
- Inequalities (A15)–(A17) can be satisfied by setting $k$ as a sufficiently large positive number, if $e{r}^{(1)}(\mathbf{x},\mathbf{g}){|}_{\mathbf{x}=0}<0$, ${a}_{{\theta}_{0}}^{(1)}(\mathbf{x},\mathbf{g}){|}_{\mathbf{x}=0}<0$, ${b}_{{\theta}_{1}}^{(1)}(\mathbf{x},\mathbf{g}){|}_{\mathbf{x}=0}<0$ hold.
- (2)
- Inequalities (A15)–(A17) can be satisfied by setting $k$ as a sufficiently large negative number, if $e{r}^{(1)}(\mathbf{x},\mathbf{g}){|}_{\mathbf{x}=0}>0$, $e{a}_{{\theta}_{0}}^{(1)}(\mathbf{x},\mathbf{g}){|}_{\mathbf{x}=0}>0$, ${b}_{{\theta}_{1}}^{(1)}(\mathbf{x},\mathbf{g}){|}_{\mathbf{x}=0}>0$ hold.
- (3)
- Inequalities (A15)–(A17) can be satisfied by setting $k$ as zero, if $e{r}^{(2)}(\mathbf{x},\mathbf{g}){|}_{\mathbf{x}=0}<0$, ${a}_{{\theta}_{0}}^{(2)}(\mathbf{x},\mathbf{g}){|}_{\mathbf{x}=0}<0$, ${b}_{{\theta}_{1}}^{(2)}(\mathbf{x},\mathbf{g}){|}_{\mathbf{x}=0}<0$ hold. □

## Appendix C. Proof of Theorem 3

**Proof.**

## Appendix D. Proof of Theorem 5

**Proof.**

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**Figure 1.**The minimum noise modified weighted sums of the type I and II error probabilities obtained under no constraint and two constraints, and the original weighted sum for different $\sigma $ when s = 3 and γ = s/2.

**Figure 2.**The type I (

**a**) and II (

**b**) error probabilities corresponding to the weighted sum in Figure 1.

**Figure 3.**The minimum noise modified weighted sums of the type I and II error probabilities obtained under no constraint and two constraints, and the original weighted sum for different $s$ when σ = 1 and γ = s/2.

**Figure 4.**The type I and II error probabilities corresponding to the weighted sum in Figure 3 are shown in (

**a**) and (

**b**), respectively.

**Figure 5.**The minimum noise modified weighted sums of the type I and II error probabilities obtained under no constraint and two constraints, and the original weighted sum for different $\gamma $ when σ = 1 and s = 3.

**Figure 6.**The type I and II error probabilities corresponding to the weighted sum in Figure 5 are shown in (

**a**) and (

**b**), respectively.

**Figure 7.**The weighted sums, type I error probabilities, and type II error probabilities for $\mathrm{\Theta}=s$ and −s of the original detector and the noise enhanced detector for different σ where s = 1 and ρ = 0.6 shown in (

**a**), (

**b**), (

**c**) and (

**d**), respectively

**Figure 8.**The weighted sums, type I error probabilities, and type II error probabilities for $\mathrm{\Theta}=s$ and −s of the original detector and the noise enhanced detector for different s where σ = 0.08 and ρ = 0.6 shown in (a), (b), (c) and (d), respectively.

**Table 1.**The optimal additive noises that minimize the weighted sum under two constraints and no constraint for various $\sigma $ where s = 3 and γ = s/2.

$\mathit{\sigma}$ | Two Constraints | No Constraints | ||
---|---|---|---|---|

${\mathit{n}}_{\mathbf{1}}$ | ${\mathit{n}}_{\mathbf{2}}$ | $\eta $ | ${\mathit{n}}_{\mathit{o}}$ | |

0.950 | - | - | - | −1.7089 |

1.250 | −1.9082 | 1.7963 | 0.6950 | −1.9218 |

2.125 | −2.5136 | 3.1896 | 0.7862 | −2.5136/3.1896 |

3.000 | −3.3771 | 4.6942 | 0.3770 | 4.7449 |

**Table 2.**The optimal additive noises that minimize the weighted sum under two constraints and no constraint for various $s$ where σ = 1 and γ = s/2.

$\mathit{s}$ | Two Constraints | No Constraints | ||
---|---|---|---|---|

${\mathit{n}}_{\mathbf{1}}$ | ${\mathit{n}}_{\mathbf{2}}$ | $\mathit{\eta}$ | ${\mathit{n}}_{\mathit{o}}$ | |

1.25 | −1.3682 | 1.7327 | 0.2918 | 1.7474 |

1.75 | −1.4408 | 1.6563 | 0.7265 | −1.4408/1.6563 |

2.5 | −1.6052 | 1.4690 | 0.6983 | −1.6201 |

3.25 | - | - | - | −0.5866 |

**Table 3.**The optimal additive noises that minimize the weighted sum under two constraints and no constraint for various $\gamma $ where σ = 1 and s = 3.

$\mathit{\gamma}$ | Two Constraints | No Constraints | ||
---|---|---|---|---|

${\mathit{n}}_{\mathbf{1}}$ | ${\mathit{n}}_{\mathbf{2}}$ | $\mathit{\eta}$ | ${\mathit{n}}_{\mathit{o}}$ | |

0.050 | - | - | - | - |

1.100 | −2.1213 | 0.9341 | 0.2878 | 0.9691 |

1.425 | −1.7947 | 1.2585 | 0.5355 | −1.7957 |

2.250 | −0.9693 | 2.0836 | 0.8867 | −1.1763 |

3.375 | - | - | - | −0.5775 |

**Table 4.**The optimal additive noises that minimize the weighted sum under two constraints for various $\sigma $ where s = 1 and ρ = 0.6.

$\mathit{\sigma}$ | ${\mathit{n}}_{\mathbf{1}}$ | ${\mathit{n}}_{\mathbf{2}}$ | ${\mathit{n}}_{\mathbf{3}}$ | ${\mathit{\eta}}_{\mathbf{1}}$ | ${\mathit{\eta}}_{\mathbf{2}}$ | ${\mathit{\eta}}_{\mathbf{3}}$ |
---|---|---|---|---|---|---|

0.0001 | 0.2286 | - | - | 1.0000 | - | - |

0.02 | 0.2286 | −0.2255 | - | 0.8413 | 0.1587 | - |

0.05 | 0.2287 | −0.2208 | 0.2421 | 0.5310 | 0.3446 | 0.1244 |

0.08 | 0.2180 | −0.2185 | −0.2168 | 0.5943 | 0.2449 | 0.1608 |

**Table 5.**The optimal additive noises that minimize the weighted sum under two constraints for various $s$ where σ = 0.08 and ρ = 0.6.

$\mathit{s}$ | ${\mathit{n}}_{1}$ | ${\mathit{n}}_{2}$ | ${\mathit{n}}_{3}$ | ${\mathit{\eta}}_{1}$ | ${\mathit{\eta}}_{2}$ | ${\mathit{\eta}}_{3}$ |
---|---|---|---|---|---|---|

0.65 | 0.1613 | −0.1613 | - | 0.6267 | 0.3733 | - |

0.75 | 0.2026 | −0.2026 | - | 0.7949 | 0.2051 | - |

0.85 | 0.2148 | −0.2149 | -0.2150 | 0.8262 | 0.1300 | 0.0438 |

0.95 | 0.2195 | −0.2196 | -0.2190 | 0.7006 | 0.1916 | 0.1078 |

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**MDPI and ACS Style**

Liu, S.; Yang, T.; Zhang, K.
Noise Enhancement for Weighted Sum of Type I and II Error Probabilities with Constraints. *Entropy* **2017**, *19*, 276.
https://doi.org/10.3390/e19060276

**AMA Style**

Liu S, Yang T, Zhang K.
Noise Enhancement for Weighted Sum of Type I and II Error Probabilities with Constraints. *Entropy*. 2017; 19(6):276.
https://doi.org/10.3390/e19060276

**Chicago/Turabian Style**

Liu, Shujun, Ting Yang, and Kui Zhang.
2017. "Noise Enhancement for Weighted Sum of Type I and II Error Probabilities with Constraints" *Entropy* 19, no. 6: 276.
https://doi.org/10.3390/e19060276