Typical shape detection algorithms

Question

Here is my target, given a set of consecutive discrete 1-d data points within a window. I want to know whether there are good algorithms to detect certain typical shapes of the signal in the window. Including:

1. A positive step
2. A negative step
3. An upward line
4. A downward line
5. A hill shape
6. A bowl shape

The size of the given window could be of any length. And the data is from financial markets so it could be highly random. I wonder if there are good methods to detect whether the above typical shapes exist in the window, which could of course be given by a probability for each shape. I don't need very concrete descriptions of each algorithm, just need to get a direction for my further research.

Answer 1

Do you know Calculus?

All those shapes correspond to bounded values over an interval of the function or its first and second derivatives.

There are many ways to approximate derivatives in discrete sequences.

You can design a metric for each of your conditions based on the derivatives, then evaluate those metrics on a sliding basis.

1) A step occurs when the first derivative is relatively close to zero, then has a spike, then returns to zero.

2) Same as 1, opposite sign.

3) An upward line exists when the first derivative is bound near a positive value and the second derivative is bound near zero.

4) Same as 3, opposite sign

5) A hill (approximated by a parabola) occurs when the second derivative is bound near a negative value.

6) Same as 5, opposite sign

Answer 2

• A first comment is that the situation you describe can be mixed. A bowl shape can happen "during" a wide upward line. This questions the scale of the shape with respect to the size of the window.

• A second comment is that your situations don't describe all possible situations. There might be at least a last "trash" or "underdetermined" situation (every other possibility).

• A third comment is that each pair sounds relatively symmetrical, so with careful examination, it is possible to reduce, or nest, the sets of possiblities.

I would evoke two far-away types of techniques.

1. Context-based: you define contexts, a set of mutually exclusive behaviors that covers all possibilities. For instance, you can define $S_-/S_+$, $L_-/L_+$, $C_-/C_+$ and $O$ for steps (positive/negative), line (upward, downward), circle (up and down) and other. Then define a metric (based on gradient, laplacian, possibly relative to scale) that provides the best match in a window. From the set of metric values, you can get a probability of looking like a context. We somehow used a similar approach (but not going to scales) in CHOPtrey: contextual online polynomial extrapolation for enhanced multi-core co-simulation of complex systems. This approach was chosen because of real-time constraints and a lack of knowledge in the generating cybersystems

2. Learning-based: the current time is concentrated to deep-learning. So if you can build a database of trained labels (by experts) you can hope to achieve state of the art performance with neural networks and their avatars. Some recent advances seem to be scale-invariant and noise robust.

Between the above options, many intermediates can be found.