Alzheimer's disease is a neurodegenerative disease in old age, early diagnosis will help to delay the progression of the disease. Presently, the features of brain functional diseases can be obtained with EEG analysis, but the relationship between characteristics of EEG and Alzheimer's disease has not been clearly clarified. In this work, we hypothesize that there exist default brain variables (latent factors) across subjects in disease processes, decoding latent factor from brain activity contributes to the study of cognitive impairment. To that end, this work proposes to extract characteristics of Alzheimer's disease by combing latent factors of EEG with variational auto-encoder to realize disease identification. Primarily, power spectrum characteristics is investigated and it is found that the dominant frequency of two groups is different. Further analysis reveals that latent factor distribution of Alzheimer's disease exists obvious differences with normal group in the theta frequency band. Moreover, the latent factors are projected onto the three-dimensional state space and the transient rotation of neural state is found, which shows the dynamic characteristics of latent factors. In addition, Takagi-Sugeno-Kang classifier is adopted and multiple latent factors are fed into Takagi-Sugeno-Kang classifier for decoding. Compared with linear classifier, Takagi-Sugeno-Kang fuzzy classifier has better performance in classification of energy feature from sub-frequency bands of latent factors. The accuracy of identification could up to 98.10% when the combination of energy features of four frequency bands is used as model input. Collectively, this work provides a feasible tool for identification of neurological dysfunction from the view of latent factors, especially contributing to the diagnosis of Alzheimer's disease.