Fault detection and diagnosis in the pumping unit is a challenging industrial problem for the system that exhibits nonlinearity, coupled parameters and time-varying noise. This paper proposes a novel unscented Kalman filter - radical basis function (UKF-RBF) method based on an adaptive noise factor (NAUKF-RBF) for fault diagnosis in the pumping unit. First, to avoid the curse of dimensionality, the Fourier descriptor method based on an approximate polygon is introduced to extract the features of the indicator diagram curve. Then, an RBF neural network is applied to establish the fault diagnosis model, and UKF is used to optimize the weighting, center, and width of the RBF neural network due to its strong nonlinear tracking performance. In particular, the adaptive noise factor method is presented to address the unknown and time-varying noise statistics in the actual production process. The proposed method is applied to the pumping unit system, and experimental results show the effectiveness and favorable recognition rate in classifying multiple faults.