Experiments with noise

In our paper, we provide some experimental evidence that our algorithm is robust to relatively high noise levels and small time warping. We talk about noise levels regarding a noise range, according to the standard deviation of the original series. In figure 6 some examples of Gaussian noise series for each noise range.


Figure 6: Ten increasingly noisier versions of a time series (original). To the original version is added random Gaussian noise with a range of 10% up to 100% of the standard deviation of the original time series.

We execute MrMotif on the original time series with 10000 series of length 1024 and record the top-10 patterns. Then, we execute the algorithm for each noisy version, and record the precision/recall with respect to the original version's top-10. Results are shown on figure 7.


Figure 7: Variation of the Precision and Recall of each increasingly noisier version of the original 10000 series dataset.

We observe that precision and recall present high values up to noise greater than 60% standard deviation of the original series. These precision and recall values mean that almost no top-10 patterns are missed, and almost no patterns are incorrectly identified. Figure 6 shows that noise levels greater than 60% significantly differ from the original series.

In table 2 we show the results (the ten most frequent patterns) of the execution of MrMotif for the ten datasets. We show results only for resolution 4.

original 10% 20% 30% 40%
{0 , 0 , 0 , 1 , 2 , 3 , 3 , 3 } {3 , 3 , 3 , 2 , 1 , 0 , 0 , 0 } {0 , 0 , 0 , 1 , 2 , 3 , 3 , 3 } {3 , 3 , 3 , 2 , 1 , 0 , 0 , 0 } {0 , 0 , 0 , 1 , 2 , 3 , 3 , 3 }
{3 , 3 , 3 , 2 , 1 , 0 , 0 , 0 } {0 , 0 , 0 , 1 , 2 , 3 , 3 , 3 } {3 , 3 , 3 , 2 , 1 , 0 , 0 , 0 } {0 , 0 , 0 , 1 , 2 , 3 , 3 , 3 } {3 , 3 , 3 , 2 , 1 , 0 , 0 , 0 }
{3 , 3 , 2 , 2 , 1 , 0 , 0 , 0 } {0 , 0 , 0 , 1 , 2 , 2 , 3 , 3 } {0 , 0 , 1 , 1 , 2 , 3 , 3 , 3 } {3 , 3 , 3 , 2 , 1 , 1 , 0 , 0 } {3 , 3 , 3 , 2 , 1 , 1 , 0 , 0 }
{0 , 0 , 0 , 1 , 2 , 2 , 3 , 3 } {3 , 3 , 2 , 2 , 1 , 0 , 0 , 0 } {0 , 0 , 0 , 1 , 2 , 2 , 3 , 3 } {0 , 0 , 0 , 1 , 2 , 2 , 3 , 3 } {0 , 0 , 1 , 1 , 2 , 2 , 3 , 3 }
{3 , 3 , 3 , 2 , 1 , 1 , 0 , 0 } {3 , 3 , 3 , 2 , 1 , 1 , 0 , 0 } {3 , 3 , 3 , 2 , 1 , 1 , 0 , 0 } {3 , 3 , 2 , 2 , 1 , 0 , 0 , 0 } {3 , 3 , 2 , 2 , 1 , 1 , 0 , 0 }
{0 , 0 , 1 , 1 , 2 , 3 , 3 , 3 } {0 , 0 , 1 , 1 , 2 , 3 , 3 , 3 } {3 , 3 , 2 , 2 , 1 , 0 , 0 , 0 } {3 , 3 , 2 , 2 , 1 , 1 , 0 , 0 } {0 , 0 , 0 , 1 , 2 , 2 , 3 , 3 }
{0 , 0 , 1 , 1 , 2 , 2 , 3 , 3 } {0 , 0 , 1 , 1 , 2 , 2 , 3 , 3 } {3 , 3 , 2 , 2 , 1 , 1 , 0 , 0 } {0 , 0 , 1 , 1 , 1 , 2 , 3 , 3 } {3 , 3 , 2 , 2 , 1 , 0 , 0 , 0 }
{3 , 3 , 2 , 1 , 1 , 1 , 0 , 0 } {3 , 3 , 2 , 1 , 1 , 1 , 0 , 0 } {0 , 0 , 1 , 1 , 2 , 2 , 3 , 3 } {0 , 0 , 1 , 1 , 2 , 3 , 3 , 3 } {3 , 3 , 2 , 1 , 1 , 1 , 0 , 0 }
{3 , 3 , 2 , 2 , 1 , 1 , 0 , 0 } {3 , 3 , 2 , 2 , 1 , 1 , 0 , 0 } {0 , 0 , 1 , 1 , 1 , 2 , 3 , 3 } {0 , 0 , 1 , 1 , 2 , 2 , 3 , 3 } {0 , 0 , 1 , 1 , 2 , 3 , 3 , 3 }
{0 , 0 , 1 , 1 , 1 , 2 , 3 , 3 } {3 , 3 , 2 , 2 , 2 , 1 , 0 , 0 } {3 , 3 , 2 , 1 , 1 , 1 , 0 , 0 } {3 , 3 , 2 , 1 , 1 , 1 , 0 , 0 } {0 , 0 , 1 , 1 , 1 , 2 , 3 , 3 }

50% 60% 70% 80% 90% 100%
{0 , 0 , 0 , 1 , 2 , 3 , 3 , 3 } {3 , 3 , 3 , 2 , 1 , 0 , 0 , 0 } {3 , 3 , 2 , 2 , 1 , 1 , 0 , 0 } {3 , 3 , 2 , 2 , 1 , 1 , 0 , 0 } {3 , 3 , 2 , 2 , 1 , 1 , 0 , 0 } {3 , 3 , 2 , 2 , 1 , 1 , 0 , 0 }
{3 , 3 , 3 , 2 , 1 , 0 , 0 , 0 } {0 , 0 , 1 , 1 , 2 , 2 , 3 , 3 } {0 , 0 , 1 , 1 , 2 , 2 , 3 , 3 } {0 , 0 , 1 , 1 , 2 , 2 , 3 , 3 } {3 , 3 , 2 , 1 , 1 , 1 , 0 , 0 } {0 , 0 , 1 , 1 , 2 , 2 , 3 , 3 }
{3 , 3 , 2 , 2 , 1 , 1 , 0 , 0 } {3 , 3 , 2 , 1 , 1 , 1 , 0 , 0 } {3 , 3 , 3 , 2 , 1 , 0 , 0 , 0 } {3 , 3 , 2 , 1 , 1 , 1 , 1 , 0 } {0 , 0 , 1 , 2 , 2 , 2 , 2 , 3 } {0 , 0 , 1 , 2 , 2 , 2 , 3 , 3 }
{0 , 0 , 1 , 1 , 2 , 3 , 3 , 3 } {3 , 3 , 2 , 2 , 2 , 1 , 0 , 0 } {0 , 0 , 1 , 2 , 2 , 2 , 3 , 3 } {0 , 0 , 1 , 1 , 1 , 2 , 3 , 3 } {0 , 0 , 1 , 1 , 2 , 3 , 2 , 3 } {3 , 2 , 2 , 2 , 2 , 1 , 0 , 0 }
{0 , 0 , 1 , 1 , 2 , 2 , 3 , 3 } {3 , 3 , 2 , 2 , 1 , 1 , 0 , 0 } {0 , 0 , 0 , 1 , 2 , 3 , 3 , 3 } {0 , 1 , 1 , 1 , 1 , 2 , 3 , 3 } {0 , 0 , 1 , 1 , 2 , 2 , 3 , 3 } {0 , 1 , 1 , 1 , 1 , 2 , 3 , 3 }
{0 , 0 , 0 , 1 , 2 , 2 , 3 , 3 } {0 , 0 , 0 , 1 , 2 , 2 , 3 , 3 } {3 , 3 , 2 , 1 , 1 , 1 , 0 , 0 } {0 , 0 , 1 , 1 , 2 , 2 , 2 , 3 } {0 , 0 , 1 , 1 , 2 , 3 , 3 , 2 } {0 , 1 , 1 , 1 , 2 , 2 , 2 , 3 }
{3 , 3 , 3 , 2 , 1 , 1 , 0 , 0 } {0 , 0 , 1 , 1 , 1 , 2 , 3 , 3 } {3 , 3 , 2 , 1 , 2 , 1 , 0 , 0 } {0 , 0 , 1 , 2 , 2 , 3 , 3 , 2 } {0 , 0 , 1 , 1 , 1 , 2 , 3 , 3 } {0 , 0 , 1 , 1 , 2 , 3 , 3 , 2 }
{0 , 0 , 1 , 1 , 1 , 2 , 3 , 3 } {0 , 0 , 1 , 2 , 2 , 2 , 3 , 3 } {3 , 3 , 2 , 2 , 1 , 0 , 0 , 0 } {3 , 3 , 2 , 1 , 1 , 1 , 0 , 0 } {3 , 2 , 3 , 2 , 1 , 1 , 0 , 0 } {3 , 3 , 2 , 2 , 1 , 0 , 0 , 1 }
{3 , 3 , 2 , 1 , 1 , 1 , 0 , 0 } {0 , 0 , 0 , 1 , 2 , 3 , 3 , 3 } {0 , 0 , 0 , 2 , 2 , 2 , 3 , 3 } {0 , 0 , 0 , 1 , 2 , 3 , 3 , 3 } {3 , 3 , 2 , 2 , 1 , 0 , 0 , 1 } {2 , 3 , 2 , 2 , 2 , 1 , 0 , 0 }
{3 , 3 , 2 , 2 , 1 , 0 , 0 , 0 } {3 , 3 , 3 , 2 , 1 , 1 , 0 , 0 } {3 , 2 , 2 , 2 , 2 , 1 , 0 , 0 } {3 , 2 , 2 , 2 , 2 , 1 , 0 , 0 } {3 , 3 , 2 , 1 , 1 , 1 , 1 , 0 } {3 , 3 , 2 , 1 , 1 , 1 , 1 , 0 }
Table 2: Top-10 patterns found for each execution of MrMotif.