The rapid growth of Internet of Things (IoT) and sensing technologies has led to an increasing interest in time-series data analysis. In many domains, detecting patterns of IoT data and interpreting these patterns are challenging issues. There are several methods in time-series analysis that deal with issues such as volume and velocity of IoT data streams. However, analysing the content of the data streams and extracting insights from dynamic IoT data is still a challenging task. In this paper, we propose a pattern representation method which represents time-series frames as vectors by first applying Piecewise Aggregate Approximation (PAA) and then applying Lagrangian Multipliers. This method allows representing continuous data as a series of patterns that can be used and processed by various higher-level methods. We introduce a new change point detection method which uses the constructed patterns in its analysis. We evaluate and compare our representation method with Blocks of Eigenvalues Algorithm (BEATS) and Symbolic Aggregate approXimation (SAX) methods to cluster various datasets. We have evaluated our algorithm using UCR time-series datasets and also a healthcare dataset. The evaluation results show significant improvements in analysing time-series data in our proposed method.