The trend of a time series can reflect actual production conditions to some extent. For example, a stable curve indicates normal production conditions, while a rapid rise or fall curve suggests unstable conditions and potential anomalies. These stable, rising, and falling curves appear as distinct shapes in time series trend plots. Detecting these shapes from historical data and correlating them with production conditions helps analyze issues in production activities and improve efficiency. This chapter presents how to use mathematical methods to detect specifically shaped curve segments.

Time series exhibit a vast diversity of curve shapes, making it impractical to design a dedicated mathematical method for each shape. We therefore use some atomic features in combination to describe these trends—much like assembling countless shapes with a few specific types of building blocks.

We design the following three categories of atomic features, each with corresponding index (or indices) to represent degree:

1. Rise/Fall feature: Characterizes the rising or falling of a curve

(1) Rise/Fall index: Characterizes the rising or falling state of a curve

2. Amplitude feature: Characterizes changes in a curve’s vibration amplitude

(1) Amplitude index: Characterizes a curve’s vibration amplitude

(2) Amplitude Rise/Fall index: Characterizes the rise/fall in vibration amplitude

3. Frequency feature: Characterizes changes in a curve’s vibration frequency

(1) Frequency index: Characterizes a curve’s vibration frequency

(2) Frequency Rise/Fall index: Characterizes the rise/fall in vibration frequency

By combining these indices, we can identify common shapes. The following table lists some common shapes and their corresponding index combinations.