# Method for outlier detection in noisy seasonal time series data?

I have around 1000 times series of around 1000 samples, where each sample is 5 minutes a part.

An example of a time series after performing seasonal decomposition is

As we can see the data is very noisy during the night.

So I am wondering

• What would be a good option for outlier detection in this case? Would any of the following methods make sense

• Fit a gaussian to residuals and estimate probability for each sample
• Some threshold for the number of "median absolute deviations" from the median each sample is allowed to have.
• Given a chosen method, what method would make sense to use to dynamically set the threshold depending on how noisy the data is during the night / day.

EDIT Some Sample data

[ 1.55,  1.22,  0.3 , -0.51, -0.17, -0.1 , -2.04, -2.64, -2.31, -0.45,  1.97,  0.71,  0.22, -0.46, -0.48, -0.24, -2.29, -2.06, -1.98, -0.22,  1.84,  0.3 , -0.19, -0.54, -0.37, -0.73, -2.26, -2.16, -1.99, -0.29,  1.36, -0.07, -0.2 , -0.48, -0.87, -0.55, -1.51, -1.75, -2.3 ,  0.12,  0.34, -0.24, -0.28, -0.9 , -0.83,  0.17, -0.62, -1.47, -0.84,  0.78,  1.47,  0.19, -0.1 , -0.31, -0.99, -0.65, -0.51, -1.08, -0.69,  0.2 ,  1.23, -0.49,  0.43, -0.55, -0.73,  0.32, -0.65, -0.72, -0.24,  0.25,  1.5 , -0.08, -0.03, -0.08, -0.38, -0.34,  0.23, -1.07, -0.12,  0.05,  1.3 ,  0.38,  0.02, -0.81, -0.45, -0.54, -0.21, -0.54, -0.18, -0.08,  0.93, -0.69, -0.22, -0.76, -0.31, -0.31, -0.54, -0.54, -0.47,  0.46,  0.54, -0.32,  0.14, -0.32, -0.47, -0.14,  0.12, -0.94, -0.62, -0.24,  0.75,  0.02, -0.62, -0.59, -0.09, -0.62, -0.58, -1.21, -1.1 , -0.58, -0.32, -0.79, -0.35, -0.75, -1.08, -0.52, -0.86, -1.07, -1.78, -0.77,  0.1 ,  0.35, -0.26, -0.56, -0.26, -0.57, -0.66, -1.26, -1.69,  0.58, -0.18, -0.  , -0.36, -0.41, -0.38, -0.85, -0.79, -0.68, -0.99, -0.38, -0.19, -0.5 , -0.23, -0.62,  0.04, -0.47,  0.3 , -1.26, -0.5 ,  0.51, -0.31, -0.15, -0.23, -1.14, -0.3 , -0.33, -0.23, -0.76, -0.9 ,  0.14, -0.05, -0.09,  0.22, -0.19, -0.27, -0.29, -0.58, -1.27, -1.16,  0.07, -0.36, -0.23, -0.22, -0.02, -0.57, -0.9 , -0.08, -0.95, -0.52,  0.63, -0.11,  0.17, -0.49,  0.83,  0.18,  0.14,  0.58,  0.63,  0.94,  1.75,  0.72,  1.19,  0.51,  0.58, -0.43, -1.05, -1.55, -2.91, -2.72, -3.18, -3.39, -2.45,  0.07,  0.02, -1.82, -3.78, -2.91, -3.49, -3.24, -2.55, -0.67,  0.83,  1.87,  2.77,  0.34,  0.17,  1.46,  0.96,  1.55,  1.33, -0.6 ,  0.52,  2.44,  3.07,  0.31,  0.24,  1.23,  0.92,  1.43,  1.15, -0.73,  0.7 ,  2.05,  2.26,  0.53, -0.1 ,  1.01,  0.41,  1.4 ,  1.24, -0.68,  0.74,  2.07,  1.56,  1.09, -0.32,  1.17,  0.55,  1.7 ,  1.06, -0.49,  0.64,  3.1 ,  1.55,  0.88,  0.06,  0.89,  0.45,  1.48,  0.88, -0.22,  0.83,  2.43,  1.7 ,  0.58, -0.16,  0.93,  0.21,  1.04,  0.41, -0.27,  0.94,  1.73,  1.26, -0.51,  0.22,  0.92,  0.34,  0.52, -0.43, -0.3 ,  1.34,  1.53,  1.05,  0.84,  0.87,  1.88,  0.42,  0.57, -0.78, -0.51,  1.26,  1.11,  0.92,  1.3 ,  0.11,  1.71,  0.57,  0.27,  0.17, -0.62,  1.19, -0.19,  1.4 ,  1.03,  0.58,  1.27,  0.65, -0.13,  0.26,  0.76,  0.74,  0.28,  0.82,  0.57,  0.27,  1.12, -0.36,  0.16, -0.6 , -0.34, -0.16,  0.38,  0.35, -0.76,  0.09,  0.59, -0.64, -0.4 , -0.43,  0.63,  0.11,  0.84,  0.38, -0.04,  0.85,  0.47, -0.56, -0.16,  0.28,  0.84, -0.08,  0.32, -0.06, -0.08,  0.6 ,  0.01, -0.69, -0.25, -0.35,  0.45, -0.29,  0.37,  0.15, -0.4 ,  0.29,  0.21, -0.09, -0.46, -0.4 , -0.34,  0.43,  1.2 ,  0.13, -0.36, -0.3 , -0.2 , -0.46,  0.31,  0.28, -0.11,  0.01, -0.22, -0.4 , -0.6 ,  0.37, -0.78, -0.33,  0.38,  0.32, -0.24, -0.13, -0.45, -0.09, -0.48, -0.34, -0.91, -0.1 , -0.05,  0.13,  0.31,  0.04,  0.33,  0.38,  0.02,  0.11, -0.35, -0.2 , -0.87,  0.12, -0.12, -0.12,  0.49,  0.53, -0.02, -0.25, -0.15,  0.2 , -0.51, -0.42,  0.07,  0.25,  0.22,  0.18, -0.45,  0.95,  1.95, -0.64,  0.04,  0.46,  0.24,  0.08, -0.09,  0.08, -0.15,  0.34,  1.22,  0.17,  0.03,  0.21, -0.29,  0.43, -0.38,  0.57, -0.35,  1.24,  0.49, -1.05, -0.06,  0.08,  0.24,  0.66,  0.36,  0.2 , -0.38,  0.09,  0.08, -0.09, -0.61,  0.39,  0.11,  0.39, -0.3 , -0.08,  0.12,  0.84,  0.22,  0.03,  0.1 ,  0.03, -0.22, -0.29,  0.09,  0.38, -0.04,  0.51, -0.51, -0.36,  0.06,  0.56,  0.36, -0.86, -0.02, -0.85, -0.42, -0.47, -0.79, -0.73, -0.74, -0.07, -0.73, -0.19, -0.26,  0.57,  0.51,  0.46, -0.2 ,  1.57,  0.93,  0.59, -1.41, -1.45,  1.19,  3.97,  2.89,  0.89,  0.32,  1.15,  0.39, -0.95, -0.91, -1.77,  1.46,  2.67,  0.97, -0.84, -1.13, -1.14, -1.66, -2.38, -1.09, -2.1 ,  0.97,  2.14,  0.77, -0.7 , -1.46, -1.22, -2.03, -2.36, -0.39, -1.29,  1.1 ,  2.21,  0.59, -0.45, -1.22, -1.36, -2.45, -1.83, -0.15, -0.43,  1.04,  2.8 ,  0.5 , -0.56, -1.41, -1.53, -2.7 , -1.07, -0.79, -0.36,  1.14,  2.43,  0.41, -0.83, -1.12, -1.61, -2.87, -0.76, -0.87, -0.36,  1.42,  2.39, -0.2 , -0.32, -0.96, -1.85, -2.49, -0.85, -0.4 , -0.07,  1.61,  2.33, -0.5 , -0.64, -1.28, -2.18, -1.89, -0.93, -0.41,  0.24,  1.84,  2.83,  0.05, -0.34, -1.96, -2.28, -1.4 , -0.66,  0.24,  0.42,  1.88,  2.4 ,  0.55, -0.4 , -1.67, -1.56, -0.9 , -0.49,  0.75,  0.15,  2.02,  1.46,  0.12, -0.73, -1.46, -1.63, -1.1 , -0.1 ,  0.87, -0.37,  1.83,  0.97,  1.02,  0.04, -0.38, -0.65, -0.44,  0.06,  0.6 , -0.22,  1.38,  0.62,  0.37, -0.55, -0.76, -0.72, -0.4 ,  0.05,  1.1 ,  0.37,  1.06,  0.59,  0.08, -0.31, -0.57, -0.34, -1.21, -0.19,  0.48, -0.04,  1.12,  0.29,  0.15, -0.05, -0.8 , -0.52, -0.73,  0.  ,  0.48, -0.01,  0.11,  0.4 , -0.93, -0.55, -1.25, -0.67, -0.23, -0.04,  0.22,  0.48,  0.92,  0.7 , -0.12,  0.48, -0.89, -0.44,  0.03,  0.39,  0.65,  0.19,  0.94, -0.28,  0.29,  0.19, -0.96, -0.45, -0.18,  0.06,  0.81, -0.14,  0.15,  1.41,  0.53,  0.19, -0.44, -0.17, -0.16, -0.24, -0.68,  0.08,  0.73,  0.14,  0.31,  0.34,  0.52,  0.02,  0.21,  0.26, -0.  , -0.44,  0.96,  0.67,  0.64,  0.24,  0.95,  0.08,  0.23,  0.31,  0.03,  0.39,  1.1 ,  0.31, -0.26,  0.06,  0.13, -0.45,  0.12,  0.32,  0.47,  0.77,  0.94,  0.35, -0.24,  0.21,  0.16,  0.29,  0.52,  0.19,  0.34, -0.1 ,  0.05,  0.02,  0.01,  0.54,  0.37,  0.08, -0.  ,  0.48, -0.06,  0.13,  0.61,  0.67,  0.83, -0.05,  0.66, -0.3 , -0.33, -0.2 ,  0.57,  0.36,  0.45,  0.42,  0.94, -0.1 ,  0.26,  0.2 ,  0.44,  0.31,  0.48,  0.52,  0.13,  0.44,  1.03, -0.27,  0.05, -0.73,  0.13,  0.04, -0.17,  0.71, -0.16, -0.16, -0.15, -1.02,  0.02, -1.12,  0.22, -0.39,  0.69,  0.49,  1.04,  2.45,  2.91,  1.61,  2.46,  1.86,  1.34,  1.43,  0.62, -0.2 ,  0.02,  2.6 ,  2.92,  1.4 ,  0.28,  0.12, -0.96, -1.  , -1.8 , -2.84, -2.43, -0.13, -0.42, -0.19,  2.26,  1.86, -1.36, -0.97, -1.29, -2.39, -2.  , -0.22,  0.03,  0.07,  2.77,  1.66, -1.66, -0.97, -1.63, -2.11, -1.6 ,  0.04, -0.18,  0.12,  3.13,  1.08, -1.92, -1.12, -2.13, -2.48, -1.67,  0.01, -0.29,  0.47,  3.18,  0.43, -2.31, -1.19, -2.02, -2.49, -1.31,  0.38, -0.37,  0.73,  3.09, -0.07, -1.57, -1.34, -2.  , -2.22, -0.72,  0.11, -0.08,  1.44,  2.76, -0.09, -1.33, -1.19, -1.1 , -2.56, -0.42,  0.31, -0.79,  1.39,  1.89,  0.1 , -0.95, -1.2 , -0.65, -1.05,  0.38,  0.38, -0.58,  2.36,  1.69, -0.15, -0.88, -1.11, -0.89, -0.46, -0.29,  0.05, -0.44,  1.09,  1.71, -0.16, -0.19, -0.83, -0.79,  0.12,  0.59,  0.36,  0.23,  1.44,  0.54, -0.15, -0.28,  0.1 , -0.89,  0.52,  0.16,  0.2 , -0.11,  1.49,  1.06,  1.  , -0.15, -0.31, -0.  ,  0.76, -0.13,  0.41, -0.31,  0.96, -0.13,  0.15, -0.96, -0.1 , -0.51, -0.36,  0.14,  0.66, -0.5 ,  0.55, -0.06,  0.82, -0.07, -0.21, -0.39, -0.17,  0.08,  0.49, -0.44,  0.95,  0.31,  0.36, -0.47,  0.19,  0.06,  0.38,  0.84,  0.59,  0.4 ,  0.69,  0.55,  0.42, -0.96, -0.07, -0.35,  0.15,  0.5 ,  0.06, -0.35,  0.84,  0.29,  0.36, -0.12,  0.52,  0.2 ,  0.46,  0.96, -0.31,  0.04,  0.46,  0.28,  0.39,  0.11,  0.37,  0.21, -0.13,  0.99,  0.15, -0.27,  0.01,  0.48,  0.78,  0.44,  0.16, -0.18,  0.96,  1.14,  0.44,  0.67,  0.65,  0.26,  0.62,  0.6 ,  0.43, -0.09,  0.65,  1.3 ,  0.33, -0.54, -0.02, -0.04]


• Fit a HMM to learn the two state behaviour and detect outliers by examining $$P(O|X)$$ (probability of observation given hidden state).