# The Generalized Bayes Method for High-Dimensional Data Recognition with Applications to Audio Signal Recognition

## Abstract

**:**

## 1. Introduction

## 2. Methods

**Step 1.**Adopt the k-means approach to classify the observations ${\mathbf{y}}_{1l},l=1,\dots ,n$ to m groups. For an $i\in \{1,\dots ,m\}$, the sample mean ${\mathbf{u}}_{i1}^{\left(0\right)}$ and the sample covariance ${\Sigma}_{i1}^{\left(0\right)}$ of the data which are clustered to the ith group are used as the initial values for ${\mathbf{u}}_{i1}$ and ${\Sigma}_{i1}$, respectively. We can set an initial value for the ${\varphi}_{i1}$ to be the equal weight ${\varphi}_{i1}^{\left(0\right)}=1/m$ for $i=1,\dots ,m$.

**Step 2.**For each $l,l=1,\dots ,n$, obtain probabilities $({\eta}_{l1},\cdots ,{\eta}_{lm})$ in the multinomial distribution ${\mathcal{M}}_{m}(1;{\eta}_{l1},\cdots ,{\eta}_{lm})$ for the missing variable ${z}_{l}$ based on $({\mathbf{u}}_{i1}^{\left(0\right)},{\Sigma}_{i1}^{\left(0\right)},{\varphi}_{i1}^{\left(0\right)})$ values as follows. For $w=1,\dots ,m$, let

**Step 3.**Let ${n}_{i}$ denote the number of ${z}_{l}^{(count-1)}$ equal to i for $i=1,\dots ,m$. If there exists an i such that ${n}_{i}=0$ for some i, go to Step 1 and redo the steps.

**Step 4.**Let

**Step 5.**Repeat Steps 3 and 4 for $count=c$. Let

## 3. Results and Discussion

#### 3.1. Simulation

#### 3.2. A Real Data Example

## 4. Conclusions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

GMMs | Gaussian Mixture Models |

EM | expectation-maximization |

MLE | maximum likelihood estimator |

AIC | Akaike information criterion |

CPU | central processing unit |

MFCCs | Mel-frequency cepstral coefficients |

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**Table 1.**The recognition rates for the cases of $p=3,5$, and 6 when the component number of the Gaussian mixture models (GMMs) is known.

r | 20 | 40 | 60 | 80 |
---|---|---|---|---|

$n=100,K=3,m=2,p=3,w=1$ | ||||

MLE | 0.4884 | 0.5214 | 0.5152 | 0.5350 |

Generalized Bayes | 0.5058 | 0.5621 | 0.5749 | 0.6427 |

$n=300,K=5,m=4,p=5,w=2$ | ||||

MLE | 0.4974 | 0.5925 | 0.6645 | 0.6463 |

Generalized Bayes | 0.6103 | 0.7355 | 0.7781 | 0.8199 |

$n=500,K=3,m=2,p=3,w=1$ | ||||

MLE | 0.5703 | 0.6309 | 0.7000 | 0.7505 |

Generalized Bayes | 0.5868 | 0.6870 | 0.8000 | 0.7921 |

$n=500,K=5,m=4,p=5,w=2$ | ||||

MLE | 0.5696 | 0.6548 | 0.6833 | 0.7280 |

Generalized Bayes | 0.6526 | 0.7808 | 0.8115 | 0.8672 |

$n=1000,K=5,m=4,p=5,w=3$ | ||||

MLE | 0.7830 | 0.8749 | 0.9210 | 0.9620 |

Generalized Bayes | 0.8240 | 0.9245 | 0.9680 | 0.9840 |

$n=1000,K=5,m=4,p=6,w=4$ | ||||

MLE | 0.8918 | 0.9632 | 0.9871 | 0.9913 |

Generalized Bayes | 0.9333 | 0.9797 | 0.9940 | 0.9980 |

$n=2000,K=5,m=4,p=6,w=4$ | ||||

MLE | 0.9200 | 0.9766 | 0.9950 | 0.9944 |

Generalized Bayes | 0.9390 | 0.9804 | 0.9970 | 0.9981 |

$n=2000,K=3,m=2,p=5,w=5$ | ||||

MLE | 0.9900 | 0.9970 | 1 | 1 |

Generalized Bayes | 0.9920 | 0.9970 | 1 | 1 |

**Table 2.**The recognition rates for the cases of $p=15,20,25$, and 40 when the component number of the GMMs is known.

r | 20 | 40 | 60 | 80 |
---|---|---|---|---|

$n=1000,K=3,m=2,p=15,w=1$ | ||||

MLE | 0.5290 | 0.6161 | 0.7031 | 0.7389 |

Generalized Bayes | 0.6160 | 0.7103 | 0.7241 | 0.8222 |

$n=1000,K=5,m=4,p=15,w=2$ | ||||

MLE | 0.5141 | 0.5834 | 0.7059 | 0.8658 |

Generalized Bayes | 0.6396 | 0.6875 | 0.7353 | 0.9146 |

$n=1500,K=3,m=2,p=20,w=1$ | ||||

MLE | 0.5664 | 0.6030 | 0.6982 | 0.7420 |

Generalized Bayes | 0.5929 | 0.6884 | 0.7658 | 0.8310 |

$n=1500,K=5,m=4,p=20,w=2$ | ||||

MLE | 0.5378 | 0.6067 | 0.6394 | 0.6735 |

Generalized Bayes | 0.7198 | 0.7267 | 0.8115 | 0.8367 |

$n=2000,K=5,m=4,p=25,w=3$ | ||||

MLE | 0.7525 | 0.8043 | 0.8366 | 0.8537 |

Generalized Bayes | 0.7921 | 0.8913 | 0.9543 | 0.9431 |

$n=2000,K=5,m=4,p=25,w=4$ | ||||

MLE | 0.8236 | 0.9554 | 0.9362 | 0.9912 |

Generalized Bayes | 0.9510 | 0.9821 | 0.9912 | 1 |

$n=3000,K=5,m=4,p=40,w=4$ | ||||

MLE | 0.8273 | 0.9388 | 0.9386 | 0.9539 |

Generalized Bayes | 0.9727 | 0.9728 | 0.9911 | 0.9923 |

$n=3000,K=3,m=2,p=40,w=5$ | ||||

MLE | 0.9935 | 1 | 1 | 1 |

Generalized Bayes | 1 | 1 | 1 | 1 |

r | 20 | 40 | 60 | 80 |
---|---|---|---|---|

$n=500,K=3,m=2,p=3,w=1,s=4$ | ||||

MLE n | 0.6000 | 0.5860 | 0.6597 | 0.6732 |

Generalized Bayes | 0.5000 | 0.7044 | 0.7571 | 0.7620 |

$n=1000,K=3,m=2,p=3,w=1,s=4$ | ||||

MLE | 0.5653 | 0.6833 | 0.7280 | 0.7540 |

Generalized Bayes | 0.5859 | 0.7033 | 0.7050 | 0.8080 |

$n=1000,K=5,m=4,p=5,w=2,s=2$ | ||||

MLE | 0.6690 | 0.7750 | 0.8260 | 0.8580 |

Generalized Bayes | 0.7120 | 0.8000 | 0.8670 | 0.8730 |

$n=2000,K=5,m=4,p=5,w=2,s=6$ | ||||

MLE | 0.6310 | 0.7350 | 0.7853 | 0.8430 |

Generalized Bayes | 0.6880 | 0.8120 | 0.8863 | 0.9070 |

1 | 2 | 3 | 4 | 5 | 6 | 7 | Average Rate | |
---|---|---|---|---|---|---|---|---|

MLE | 0.708 | 0.792 | 0.708 | 0.792 | 0.75 | 0.708 | 0.667 | 0.732 |

Generalized Bayes | 0.792 | 0.833 | 0.667 | 0.833 | 0.75 | 0.75 | 0.875 | 0.786 |

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Wang, H. The Generalized Bayes Method for High-Dimensional Data Recognition with Applications to Audio Signal Recognition. *Symmetry* **2021**, *13*, 19.
https://doi.org/10.3390/sym13010019

**AMA Style**

Wang H. The Generalized Bayes Method for High-Dimensional Data Recognition with Applications to Audio Signal Recognition. *Symmetry*. 2021; 13(1):19.
https://doi.org/10.3390/sym13010019

**Chicago/Turabian Style**

Wang, Hsiuying. 2021. "The Generalized Bayes Method for High-Dimensional Data Recognition with Applications to Audio Signal Recognition" *Symmetry* 13, no. 1: 19.
https://doi.org/10.3390/sym13010019