  We introduce the concept of time series motifs for time series analysis. Time
series motifs consider not only the spatial information of mutual visibility
but also the temporal information of relative magnitude between the data
points. We study the profiles of the six triadic time series. The six motif
occurrence frequencies are derived for uncorrelated time series, which are
approximately linear functions of the length of the time series. The
corresponding motif profile thus converges to a constant vector
$(0.2,0.2,0.1,0.2,0.1,0.2)$. These analytical results have been verified by
numerical simulations. For fractional Gaussian noises, numerical simulations
unveil the nonlinear dependence of motif occurrence frequencies on the Hurst
exponent. Applications of the time series motif analysis uncover that the motif
occurrence frequency distributions are able to capture the different dynamics
in the heartbeat rates of healthy subjects, congestive heart failure (CHF)
subjects, and atrial fibrillation (AF) subjects and in the price fluctuations
of bullish and bearish markets. Our method shows its potential power to
classify different types of time series and test the time irreversibility of
time series.
