  We address the problem of estimating smoothly varying baseline trends in time
series data. This problem arises in a wide range of fields, including
chemistry, macroeconomics, and medicine; however, our study is motivated by the
analysis of data from low cost air quality sensors. Our methods extend the
quantile trend filtering framework to enable the estimation of multiple
quantile trends simultaneously while ensuring that the quantiles do not cross.
To handle the computational challenge posed by very long time series, we
propose a parallelizable alternating direction method of moments (ADMM)
algorithm. The ADMM algorthim enables the estimation of trends in a piecewise
manner, both reducing the computation time and extending the limits of the
method to larger data sizes. We also address smoothing parameter selection and
propose a modified criterion based on the extended Bayesian Information
Criterion. Through simulation studies and our motivating application to low
cost air quality sensor data, we demonstrate that our model provides better
quantile trend estimates than existing methods and improves signal
classification of low-cost air quality sensor output.
