Alzheimer's disease (AD) is characterized by deficits in cognition, behavior, and intellectual functioning, and Mild Cognitive Impairment (MCI) refers to individuals whose cognitive impairment deviates from what is expected for their age but does not significantly interfere with daily activities. Because there is no treatment for AD, early prediction of AD can be helpful to reducing the progression of this disease. This study examines the Electroencephalography (EEG) signal of 3 distinct groups including AD, MCI, and healthy individuals. Recognizing the non-stationary nature of EEG signals, two nonlinear approaches, Poincare and Entropy, are employed for meaningful feature extraction. To extract features from EEG signal, data should segmented into epochs and for each one, feature extraction approaches are implemented. The obtained features are given to machine learning algorithms to classify the subjects. Extensive experiments were conducted to analyze the features comprehensively The results demonstrate that, our proposed method surpasses previous studies in terms of accuracy, sensitivity, and specificity, indicating its effectiveness in classifying individuals with AD, MCI, and those without cognitive impairment.