Denoising time series e.g. historical price data using Least Squares Method in python using NumPy.

The denoise function in lsm-denoise.py simply could be used by passing a 1-dim numpy array of time series data to the function.

example:

denoised_prices = denoise(
    price_vec,
    LAMBDA=100,
    plot=True
    )

suppose y is the time series vector, x is the denoised vector and v is an unknown noise vector

to find x we need to solve the following minimization problem using least squares method:

this problem minimizes the noise factor v with the penalty of the difference between two consequetive x values become minimized.

the penalty term could be written as:

where D is an (n-1)x(n) matrix like:

and finally the main minimization problem could be re-written as:

an example of denoised time series by different LAMBDA valuse plotted below: