时间正则化矩阵分解_Jupyter Notebook_Python_下载.zip

# Temporal Regularized Matrix Factorization Project was inspired by the paper: Yu, H. F., Rao, N., & Dhillon, I. S. (2016). Temporal regularized matrix factorization for high-dimensional time series prediction. In Advances in neural information processing systems (pp. 847-855). Which can be found there: http://www.cs.utexas.edu/~rofuyu/papers/tr-mf-nips.pdf ## 1. Problem description We have N timeseries of length T which are presented by matrix Y. We want to factorize it <a href="https://www.codecogs.com/eqnedit.php?latex=$Y&space;=&space;F\times&space;X$" target="_blank"><img src="https://latex.codecogs.com/gif.latex?$Y&space;=&space;F\times&space;X$" title="$Y = F\times X$" /></a>. To solve this problem we will minimize: <a href="https://www.codecogs.com/eqnedit.php?latex=$$\min\limits_{F,X}\sum\limits_{(i,t)\in\Omega}\left(Y_{it}-f_i^Tx_t\right)^2&plus;\lambda_fR_f(F)&plus;\lambda_xR_x(X).$$" target="_blank"><img src="https://latex.codecogs.com/gif.latex?$$\min\limits_{F,X}\sum\limits_{(i,t)\in\Omega}\left(Y_{it}-f_i^Tx_t\right)^2&plus;\lambda_fR_f(F)&plus;\lambda_xR_x(X).$$" title="$$\min\limits_{F,X}\sum\limits_{(i,t)\in\Omega}\left(Y_{it}-f_i^Tx_t\right)^2+\lambda_fR_f(F)+\lambda_xR_x(X).$$" /></a> By doing that we will find latent embedding vectors for timeseries and latent temporal embeddings for timepoints. One can further use this embeddings to forecast new data or to impute missings. ## 2. Package description Package consists of: - trmf : time series modelling - synthetic_data : data generation for experiments additionaly: - Metrics : metrics for experiments and validation - Forecast : simple models for testing experiments - RollingCV : rolling cross-validation implementation For usage information use help(trmf) ## 3. Experiments In experiments_[something].ipynb you can find some experiments on the package: 1) experiments_synthetic.ipynb: testing trmf model against other simple model on synthetic data **Lags = {1}** |horizon| 1 | 5 | 10 | 20 | |------|------|------|------|------| | Naive | **0.105**/**0.138** | **0.151**/**0.2** | **0.175**/**0.23** | 0.38/**0.477** | | Mean | 1.0/1.136 | 1.0/1.114 | 1.0/1.094 | 1.0/1.079 | | AutoRegression | 0.107/0.142 | 0.16/0.215 | 0.2/0.275 | 0.42/0.536 | | TRMF | 0.172/0.218 | 0.155/0.227 | 0.197/0.261 | **0.368**/0.48 | **Lags = {1,7}** |horizon| 1 | 5 | 10 | 20 | |------|------|------|------|------| | Naive | 0.82/0.956 | 1.072/1.292 | 0.893/1.119 | 1.051/1.303 | | Mean | 1.0/1.176 | 1.0/1.219 | 1.0/1.236 | 1.0/1.259 | | AutoRegression | **0.503**/**0.581** | **0.496**/**0.599** | 0.572/0.717 | **0.86/1.107** | | TRMF | 0.515/0.612 | 0.498/0.603 | **0.565/0.704** | 0.87/1.117 | **Lags = {1,7,14,28}** |horizon| 1 | 5 | 10 | 20 | |------|------|------|------|------| | Naive | 1.012/1.191 | 0.97/1.18 | 0.968/1.202 | 0.917/1.162 | | Mean | 1.0/1.164 | 1.0/1.218 | 1.0/1.206 | 1.0/1.197 | | AutoRegression | **0.618**/0.733 | **0.506**/**0.648** | **0.567**/**0.715** | 0.619/0.755 | | TRMF | 0.633/**0.662** | 0.544/0.676 | 0.578/0.726 | **0.582**/**0.72** | 2) experiments_electricity.ipynb: testing trmf model against other simple model on electricity data | horizon | 1 | 5 | 10 | 20 | |------|------|------|------|------| | Naive | **0.344/0.5** | 0.688/0.951 | 1.091/1.429 | 1.363/1.73 | | Mean | 1.0/1.19 | 1.0/1.201 | 1.0/1.204 | 1.0/1.188 | | AutoRegression | 0.427/0.557 | **0.612/0.831** | **0.627/0.876** | **0.58**/0.802 | | TRMF | 0.639/0.828 | 0.727/0.95 | 0.681/0.936 | 0.584/**0.799** | 3) experiments_crypto.ipynb: testing trmf model against other simple model on crypto-currency data | horizon | 1 | 5 | 10 | 20 | |------|------|------|------|------| | Naive | **0.158/0.36** | **0.228/0.574** | **0.29/0.644** | **0.347/0.842** | | Mean | 1.0/1.293 | 1.0/1.265 | 1.0/1.24 | 1.0/1.322 | | AutoRegression | 0.168/0.368 | 0.258/0.6 | 0.369/0.792 | 0.528/1.309 | | TRMF | 0.233/0.437 | 0.273/0.619 | 0.332/0.668 | 0.429/0.957 | 4) experiments_missings.ipynb: testing trmf model against other simple model on missing data imputation | missings | 5% | 10% | 25% | |----------|----|-----|-----| | Naive | 0.367/0.574 | 0.373/0.584 | 0.391/0.613 | | Mean | 129.506/150.367 | 108.291/125.712 | 89.242/103.586 | | TRMF | **0.359/0.516** | **0.36/0.519** | **0.361/0.52** | More details in notebooks. ## 4. Conclusion 1) TRMF model needs additional regularization on matrix W (sum of the row elements must be close to one). Otherwise, predictions for long periods will be unstable; 2) Every timeseries is better to be normalized before using TRMF; 3) TRMF is good on data imputation; 4) TRMF is good when data has missings. ## 5. Plan 1) Article analysis // done 2) Synthetic Data Generator // done 3) Basic realization of trmf with gradient descent // done 4) Documentation and help functions // done 5) Experiments on synthetic data (vs other models) // done 6) Rolling CV functionality // done 7) Experiments on electricity data (vs other models) // done 8) CryptoCurrency forecasting (vs other models) // done 9) Missing data handling // done 10) Missing data imputation experiments (vs other models) // done