Csaba Toth, Harald Oberhauser
We develop a Bayesian approach to learning from sequential data by using Gaussian processes (GPs) with so-called signature kernels as covariance functions. This allows to make sequences of different length comparable and to rely on strong theoretical results from stochastic analysis. Signatures capture sequential structure with tensors that can scale unfavourably in sequence length and state space dimension. To deal with this, we introduce a sparse variational approach with inducing tensors. We then combine the resulting GP with LSTMs and GRUs to build larger models that leverage the strengths of each of these approaches and benchmark the resulting GPs on multivariate time series (TS) classification datasets. Code available at https://github.com/tgcsaba/GPSig.
| Task | Dataset | Metric | Value | Model |
|---|---|---|---|---|
| Time Series Classification | SHAPES | Accuracy | 1 | GP-Sig |
| Time Series Classification | SHAPES | NLL | 0.011 | GP-Sig |
| Time Series Classification | SHAPES | Accuracy | 1 | GP-Sig-GRU |
| Time Series Classification | SHAPES | NLL | 0.012 | GP-Sig-GRU |
| Time Series Classification | SHAPES | Accuracy | 1 | GP-KConv1D |
| Time Series Classification | SHAPES | NLL | 0.012 | GP-KConv1D |
| Time Series Classification | SHAPES | Accuracy | 1 | GP-Sig-LSTM |
| Time Series Classification | SHAPES | NLL | 0.014 | GP-Sig-LSTM |
| Time Series Classification | SHAPES | Accuracy | 1 | GP-LSTM |
| Time Series Classification | SHAPES | NLL | 0.016 | GP-LSTM |
| Time Series Classification | SHAPES | Accuracy | 0.867 | GP-GRU |
| Time Series Classification | SHAPES | NLL | 0.168 | GP-GRU |
| Time Series Classification | NetFlow | Accuracy | 0.945 | GP-KConv1D |
| Time Series Classification | NetFlow | NLL | 0.168 | GP-KConv1D |
| Time Series Classification | NetFlow | Accuracy | 0.937 | GP-Sig |
| Time Series Classification | NetFlow | NLL | 0.189 | GP-Sig |
| Time Series Classification | NetFlow | Accuracy | 0.931 | GP-Sig-LSTM |
| Time Series Classification | NetFlow | NLL | 0.218 | GP-Sig-LSTM |
| Time Series Classification | NetFlow | Accuracy | 0.928 | GP-LSTM |
| Time Series Classification | NetFlow | NLL | 0.251 | GP-LSTM |
| Time Series Classification | NetFlow | Accuracy | 0.926 | GP-GRU |
| Time Series Classification | NetFlow | NLL | 0.194 | GP-GRU |
| Time Series Classification | NetFlow | Accuracy | 0.921 | GP-Sig-GRU |
| Time Series Classification | NetFlow | NLL | 0.259 | GP-Sig-GRU |
| Time Series Classification | Libras | Accuracy | 0.923 | GP-Sig |
| Time Series Classification | Libras | NLL | 0.259 | GP-Sig |
| Time Series Classification | Libras | Accuracy | 0.921 | GP-Sig-LSTM |
| Time Series Classification | Libras | NLL | 0.32 | GP-Sig-LSTM |
| Time Series Classification | Libras | Accuracy | 0.899 | GP-Sig-GRU |
| Time Series Classification | Libras | NLL | 0.346 | GP-Sig-GRU |
| Time Series Classification | Libras | Accuracy | 0.776 | GP-LSTM |
| Time Series Classification | Libras | NLL | 0.911 | GP-LSTM |
| Time Series Classification | Libras | Accuracy | 0.742 | GP-GRU |
| Time Series Classification | Libras | NLL | 1.11 | GP-GRU |
| Time Series Classification | Libras | Accuracy | 0.698 | GP-KConv1D |
| Time Series Classification | Libras | NLL | 1.608 | GP-KConv1D |
| Time Series Classification | DigitShapes | Accuracy | 1 | GP-Sig-LSTM |
| Time Series Classification | DigitShapes | NLL | 0.008 | GP-Sig-LSTM |
| Time Series Classification | DigitShapes | Accuracy | 1 | GP-LSTM |
| Time Series Classification | DigitShapes | NLL | 0.013 | GP-LSTM |
| Time Series Classification | DigitShapes | Accuracy | 1 | GP-Sig |
| Time Series Classification | DigitShapes | NLL | 0.021 | GP-Sig |
| Time Series Classification | DigitShapes | Accuracy | 1 | GP-Sig-GRU |
| Time Series Classification | DigitShapes | NLL | 0.035 | GP-Sig-GRU |
| Time Series Classification | DigitShapes | Accuracy | 1 | GP-KConv1D |
| Time Series Classification | DigitShapes | NLL | 0.035 | GP-KConv1D |
| Time Series Classification | DigitShapes | Accuracy | 0.812 | GP-GRU |
| Time Series Classification | DigitShapes | NLL | 0.727 | GP-GRU |
| Time Series Classification | CharacterTrajectories | Accuracy | 0.991 | GP-Sig-LSTM |
| Time Series Classification | CharacterTrajectories | NLL | 0.031 | GP-Sig-LSTM |
| Time Series Classification | CharacterTrajectories | Accuracy | 0.979 | GP-Sig |
| Time Series Classification | CharacterTrajectories | NLL | 0.108 | GP-Sig |
| Time Series Classification | CharacterTrajectories | Accuracy | 0.941 | GP-KConv1D |
| Time Series Classification | CharacterTrajectories | NLL | 0.409 | GP-KConv1D |
| Time Series Classification | CharacterTrajectories | Accuracy | 0.925 | GP-Sig-GRU |
| Time Series Classification | CharacterTrajectories | NLL | 0.258 | GP-Sig-GRU |
| Time Series Classification | CharacterTrajectories | Accuracy | 0.233 | GP-LSTM |
| Time Series Classification | CharacterTrajectories | NLL | 2.506 | GP-LSTM |
| Time Series Classification | CharacterTrajectories | Accuracy | 0.114 | GP-GRU |
| Time Series Classification | CharacterTrajectories | NLL | 3.523 | GP-GRU |
| Time Series Classification | PenDigits | Accuracy | 0.955 | GP-Sig |
| Time Series Classification | PenDigits | NLL | 0.146 | GP-Sig |
| Time Series Classification | PenDigits | Accuracy | 0.953 | GP-LSTM |
| Time Series Classification | PenDigits | NLL | 0.185 | GP-LSTM |
| Time Series Classification | PenDigits | Accuracy | 0.951 | GP-GRU |
| Time Series Classification | PenDigits | NLL | 0.187 | GP-GRU |
| Time Series Classification | PenDigits | Accuracy | 0.946 | GP-KConv1D |
| Time Series Classification | PenDigits | NLL | 0.181 | GP-KConv1D |
| Time Series Classification | PenDigits | Accuracy | 0.928 | GP-Sig-LSTM |
| Time Series Classification | PenDigits | NLL | 0.289 | GP-Sig-LSTM |
| Time Series Classification | PenDigits | Accuracy | 0.902 | GP-Sig-GRU |
| Time Series Classification | PenDigits | NLL | 0.399 | GP-Sig-GRU |
| Time Series Classification | AUSLAN | Accuracy | 0.983 | GP-Sig-LSTM |
| Time Series Classification | AUSLAN | NLL | 0.106 | GP-Sig-LSTM |
| Time Series Classification | AUSLAN | Accuracy | 0.978 | GP-Sig-GRU |
| Time Series Classification | AUSLAN | NLL | 0.123 | GP-Sig-GRU |
| Time Series Classification | AUSLAN | Accuracy | 0.949 | GP-GRU |
| Time Series Classification | AUSLAN | NLL | 0.248 | GP-GRU |
| Time Series Classification | AUSLAN | Accuracy | 0.925 | GP-Sig |
| Time Series Classification | AUSLAN | NLL | 0.55 | GP-Sig |
| Time Series Classification | AUSLAN | Accuracy | 0.88 | GP-LSTM |
| Time Series Classification | AUSLAN | NLL | 0.65 | GP-LSTM |
| Time Series Classification | AUSLAN | Accuracy | 0.784 | GP-KConv1D |
| Time Series Classification | AUSLAN | NLL | 1.9 | GP-KConv1D |
| Time Series Classification | ArabicDigits | Accuracy | 0.994 | GP-Sig-GRU |
| Time Series Classification | ArabicDigits | NLL | 0.023 | GP-Sig-GRU |
| Time Series Classification | ArabicDigits | Accuracy | 0.992 | GP-Sig-LSTM |
| Time Series Classification | ArabicDigits | NLL | 0.047 | GP-Sig-LSTM |
| Time Series Classification | ArabicDigits | Accuracy | 0.986 | GP-GRU |
| Time Series Classification | ArabicDigits | NLL | 0.066 | GP-GRU |
| Time Series Classification | ArabicDigits | Accuracy | 0.985 | GP-LSTM |
| Time Series Classification | ArabicDigits | NLL | 0.082 | GP-LSTM |
| Time Series Classification | ArabicDigits | Accuracy | 0.984 | GP-KConv1D |
| Time Series Classification | ArabicDigits | NLL | 0.05 | GP-KConv1D |
| Time Series Classification | ArabicDigits | Accuracy | 0.979 | GP-Sig |
| Time Series Classification | ArabicDigits | NLL | 0.071 | GP-Sig |
| Time Series Classification | JapaneseVowels | Accuracy | 0.986 | GP-GRU |
| Time Series Classification | JapaneseVowels | NLL | 0.052 | GP-GRU |
| Time Series Classification | JapaneseVowels | Accuracy | 0.986 | GP-KConv1D |
| Time Series Classification | JapaneseVowels | NLL | 0.067 | GP-KConv1D |
| Time Series Classification | JapaneseVowels | Accuracy | 0.985 | GP-Sig-GRU |
| Time Series Classification | JapaneseVowels | NLL | 0.053 | GP-Sig-GRU |
| Time Series Classification | JapaneseVowels | Accuracy | 0.982 | GP-LSTM |
| Time Series Classification | JapaneseVowels | NLL | 0.061 | GP-LSTM |
| Time Series Classification | JapaneseVowels | Accuracy | 0.982 | GP-Sig |
| Time Series Classification | JapaneseVowels | NLL | 0.069 | GP-Sig |
| Time Series Classification | JapaneseVowels | Accuracy | 0.981 | GP-Sig-LSTM |
| Time Series Classification | JapaneseVowels | NLL | 0.08 | GP-Sig-LSTM |
| Time Series Classification | UWave | Accuracy | 0.97 | GP-Sig-LSTM |
| Time Series Classification | UWave | NLL | 0.113 | GP-Sig-LSTM |
| Time Series Classification | UWave | Accuracy | 0.968 | GP-Sig-GRU |
| Time Series Classification | UWave | NLL | 0.121 | GP-Sig-GRU |
| Time Series Classification | UWave | Accuracy | 0.964 | GP-Sig |
| Time Series Classification | UWave | NLL | 0.14 | GP-Sig |
| Time Series Classification | UWave | Accuracy | 0.947 | GP-KConv1D |
| Time Series Classification | UWave | NLL | 0.189 | GP-KConv1D |
| Time Series Classification | UWave | Accuracy | 0.87 | GP-LSTM |
| Time Series Classification | UWave | NLL | 0.745 | GP-LSTM |
| Time Series Classification | UWave | Accuracy | 0.763 | GP-GRU |
| Time Series Classification | UWave | NLL | 1.168 | GP-GRU |
| Time Series Classification | ECG | Accuracy | 0.848 | GP-Sig |
| Time Series Classification | ECG | NLL | 0.356 | GP-Sig |
| Time Series Classification | ECG | Accuracy | 0.832 | GP-Sig-GRU |
| Time Series Classification | ECG | NLL | 0.431 | GP-Sig-GRU |
| Time Series Classification | ECG | Accuracy | 0.816 | GP-Sig-LSTM |
| Time Series Classification | ECG | NLL | 0.402 | GP-Sig-LSTM |
| Time Series Classification | ECG | Accuracy | 0.782 | GP-LSTM |
| Time Series Classification | ECG | NLL | 0.496 | GP-LSTM |
| Time Series Classification | ECG | Accuracy | 0.76 | GP-KConv1D |
| Time Series Classification | ECG | NLL | 0.543 | GP-KConv1D |
| Time Series Classification | ECG | Accuracy | 0.734 | GP-GRU |
| Time Series Classification | ECG | NLL | 0.601 | GP-GRU |
| Time Series Classification | PEMS | Accuracy | 0.82 | GP-Sig |
| Time Series Classification | PEMS | NLL | 0.52 | GP-Sig |
| Time Series Classification | PEMS | Accuracy | 0.794 | GP-KConv1D |
| Time Series Classification | PEMS | NLL | 0.537 | GP-KConv1D |
| Time Series Classification | PEMS | Accuracy | 0.775 | GP-Sig-GRU |
| Time Series Classification | PEMS | NLL | 1.1 | GP-Sig-GRU |
| Time Series Classification | PEMS | Accuracy | 0.769 | GP-GRU |
| Time Series Classification | PEMS | NLL | 0.784 | GP-GRU |
| Time Series Classification | PEMS | Accuracy | 0.763 | GP-Sig-LSTM |
| Time Series Classification | PEMS | NLL | 0.704 | GP-Sig-LSTM |
| Time Series Classification | PEMS | Accuracy | 0.745 | GP-LSTM |
| Time Series Classification | PEMS | NLL | 1.194 | GP-LSTM |
| Time Series Classification | WalkvsRun | Accuracy | 1 | GP-Sig |
| Time Series Classification | WalkvsRun | NLL | 0.023 | GP-Sig |
| Time Series Classification | WalkvsRun | Accuracy | 1 | GP-GRU |
| Time Series Classification | WalkvsRun | NLL | 0.028 | GP-GRU |
| Time Series Classification | WalkvsRun | Accuracy | 1 | GP-Sig-LSTM |
| Time Series Classification | WalkvsRun | NLL | 0.03 | GP-Sig-LSTM |
| Time Series Classification | WalkvsRun | Accuracy | 1 | GP-Sig-GRU |
| Time Series Classification | WalkvsRun | NLL | 0.03 | GP-Sig-GRU |
| Time Series Classification | WalkvsRun | Accuracy | 1 | GP-LSTM |
| Time Series Classification | WalkvsRun | NLL | 0.048 | GP-LSTM |
| Time Series Classification | WalkvsRun | Accuracy | 1 | GP-KConv1D |
| Time Series Classification | WalkvsRun | NLL | 0.066 | GP-KConv1D |
| Time Series Classification | CMUsubject16 | Accuracy | 1 | GP-Sig-GRU |
| Time Series Classification | CMUsubject16 | NLL | 0.04 | GP-Sig-GRU |
| Time Series Classification | CMUsubject16 | Accuracy | 1 | GP-Sig-LSTM |
| Time Series Classification | CMUsubject16 | NLL | 0.088 | GP-Sig-LSTM |
| Time Series Classification | CMUsubject16 | Accuracy | 0.993 | GP-GRU |
| Time Series Classification | CMUsubject16 | NLL | 0.089 | GP-GRU |
| Time Series Classification | CMUsubject16 | Accuracy | 0.979 | GP-Sig |
| Time Series Classification | CMUsubject16 | NLL | 0.089 | GP-Sig |
| Time Series Classification | CMUsubject16 | Accuracy | 0.924 | GP-LSTM |
| Time Series Classification | CMUsubject16 | NLL | 0.27 | GP-LSTM |
| Time Series Classification | CMUsubject16 | Accuracy | 0.897 | GP-KConv1D |
| Time Series Classification | CMUsubject16 | NLL | 0.255 | GP-KConv1D |
| Time Series Classification | KickvsPunch | Accuracy | 0.9 | GP-Sig |
| Time Series Classification | KickvsPunch | NLL | 0.224 | GP-Sig |
| Time Series Classification | KickvsPunch | Accuracy | 0.9 | GP-Sig-LSTM |
| Time Series Classification | KickvsPunch | NLL | 0.301 | GP-Sig-LSTM |
| Time Series Classification | KickvsPunch | Accuracy | 0.82 | GP-Sig-GRU |
| Time Series Classification | KickvsPunch | NLL | 0.493 | GP-Sig-GRU |
| Time Series Classification | KickvsPunch | Accuracy | 0.7 | GP-KConv1D |
| Time Series Classification | KickvsPunch | NLL | 0.662 | GP-KConv1D |
| Time Series Classification | KickvsPunch | Accuracy | 0.62 | GP-LSTM |
| Time Series Classification | KickvsPunch | NLL | 0.696 | GP-LSTM |
| Time Series Classification | KickvsPunch | Accuracy | 0.6 | GP-GRU |
| Time Series Classification | KickvsPunch | NLL | 0.674 | GP-GRU |
| Time Series Classification | Wafer | Accuracy | 0.994 | GP-GRU |
| Time Series Classification | Wafer | NLL | 0.029 | GP-GRU |
| Time Series Classification | Wafer | Accuracy | 0.988 | GP-Sig-LSTM |
| Time Series Classification | Wafer | NLL | 0.048 | GP-Sig-LSTM |
| Time Series Classification | Wafer | Accuracy | 0.984 | GP-KConv1D |
| Time Series Classification | Wafer | NLL | 0.085 | GP-KConv1D |
| Time Series Classification | Wafer | Accuracy | 0.978 | GP-Sig-GRU |
| Time Series Classification | Wafer | NLL | 0.081 | GP-Sig-GRU |
| Time Series Classification | Wafer | Accuracy | 0.966 | GP-LSTM |
| Time Series Classification | Wafer | NLL | 0.105 | GP-LSTM |
| Time Series Classification | Wafer | Accuracy | 0.965 | GP-Sig |
| Time Series Classification | Wafer | NLL | 0.105 | GP-Sig |