| Title: | Multiresolution Forecasting |
|---|---|
| Description: | Forecasting of univariate time series using feature extraction with variable prediction methods is provided. Feature extraction is done with a redundant Haar wavelet transform with filter h = (0.5, 0.5). The advantage of the approach compared to typical Fourier based methods is an dynamic adaptation to varying seasonalities. Currently implemented prediction methods based on the selected wavelets levels and scales are a regression and a multi-layer perceptron. Forecasts can be computed for horizon 1 or higher. Model selection is performed with an evolutionary optimization. Selection criteria are currently the AIC criterion, the Mean Absolute Error or the Mean Root Error. The data is split into three parts for model selection: Training, test, and evaluation dataset. The training data is for computing the weights of a parameter set. The test data is for choosing the best parameter set. The evaluation data is for assessing the forecast performance of the best parameter set on new data unknown to the model. This work is published in Stier, Q.; Gehlert, T.; Thrun, M.C. Multiresolution Forecasting for Industrial Applications. Processes 2021, 9, 1697. <https://doi.org/10.3390/pr9101697>. |
| Authors: | Quirin Stier [aut, cre, ctr],
Michael Thrun [ths, cph, rev, fnd, ctb]
|
| Maintainer: | Quirin Stier <[email protected]> |
| License: | GPL-3 |
| Version: | 0.1.6 |
| Built: | 2025-02-18 04:26:09 UTC |
| Source: | https://github.com/quirinms/mrfr |
Forecasting of univariate time series using feature extraction with variable prediction methods is provided. Feature extraction is done with a redundant Haar wavelet transform with filter h = (0.5, 0.5). The advantage of the approach compared to typical Fourier based methods is an dynamic adaptation to varying seasonalities. Currently implemented prediction methods based on the selected wavelets levels and scales are a regression and a multi-layer perceptron. Forecasts can be computed for horizon 1 or higher. Model selection is performed with an evolutionary optimization. Selection criteria are currently the AIC criterion, the Mean Absolute Error or the Mean Root Error. The data is split into three parts for model selection: Training, test, and evaluation dataset. The training data is for computing the weights of a parameter set. The test data is for choosing the best parameter set. The evaluation data is for assessing the forecast performance of the best parameter set on new data unknown to the model. This work is published in Stier, Q.; Gehlert, T.; Thrun, M.C. Multiresolution Forecasting for Industrial Applications. Processes 2021, 9, 1697. <https://doi.org/10.3390/pr9101697>. The package consists of a multiresolution forecasting method using a redundant Haar wavelet transform based on the manuscript [Stier et al., 2021] which is currently in press. One-step and multi-step forecasts are computable with this method. Nested and non-nested cross validation is possible.
Forecasting of univariate time series using feature extraction with variable prediction methods is provided. Feature extraction is done with a redundant Haar wavelet transform with filter h = (0.5, 0.5). The advantage of the approach compared to typical Fourier based methods is an dynamic adaptation to varying seasonalities. Currently implemented prediction methods based on the selected wavelets levels and scales are a regression and a multi-layer perceptron. Forecasts can be computed for horizon 1 or higher. Model selection is performed with an evolutionary optimization. Selection criterias are currently the AIC criterion, the Mean Absolute Error or the Mean Root Error. The data is split into three parts for model selection: Training, test, and evaluation dataset. The training data is for computing the weights of a parameter set. The test data is for choosing the best parameter set. The evaluation data is for assessing the forecast performance of the best parameter set on new data unknown to the model.
| Package: | mrf |
| Type: | Package |
| Version: | 0.1.4 |
| Date: | 2021-09-20 |
| License: | CC BY-NC-SA 4.0 |
Quirin Stier
[Stier et al., 2021] Stier, Q.; Gehlert, T.; Thrun, M.C. Multiresolution Forecasting for Industrial Applications. Processes 2021, 9, 1697. https://doi.org/10.3390/pr9101697
This function decomposes a time series in its wavelet and smooth coefficients using the redundant Haar wavelet transform.
decomposition(UnivariateData, Aggregation = c(2, 4, 8, 16, 32))decomposition(UnivariateData, Aggregation = c(2, 4, 8, 16, 32))
UnivariateData |
[1:n] Numerical vector with n time series values |
Aggregation |
[1:Scales] Numerical vector of length 'Scales' carrying numbers whose index is associated with the wavelet level. The numbers indicate the number of values used for aggregation from the original time series. |
The resulting wavelet and smooth coefficients are stored in so called wavelet and smooth part levels. The smooth part level is created from the original times series by aggregation (average). This makes the times series in some sense smoother, hence the naming. Each individual smooth part level can be created from the original time series by aggregating over different number of values. The different smooth part levels are ordered, so that the number of values used for aggregation are ascending. A dyadic scheme is recommended (increasing sequences of the power of two). The dyadic scheme for 5 levels would require agg_per_lvl = c(2, 4, 8, 16, 32). So the first smooth part level would be the average of two points of the original time series, the second smooth part level would be the average of four points, and so on. This averaging is applied asymmetrical. That means, that the result of the average of a sequence of points is obtained for the last point in time of that sequence. So each smooth part level starts with a certain offset, since no average can be obtained for the first particular points in time. The wavelet levels are the differences between the original time series and the smooth levels. The first wavelet level is the difference of the original time series and the first smooth part level. The second wavelet level is the difference of the first and second smooth part level and so on.
List of
UnivariateData |
[1:n] Numerical vector with n time series values. |
WaveletCoefficients |
[Scales, n] Matrix with 'Scales' many wavelet scales row-wise with n columns corresponding to the time domain of a time series. |
SmoothCoefficients |
[Scales, n] Matrix with 'Scales' many smooth approximation scales row-wise with n columns corresponding to the time domain of a time series. |
Scales |
Number of wavelet levels. |
Quirin Stier
Aussem, A., Campbell, J., and Murtagh, F. Waveletbased Feature Extraction and Decomposition Strategies for Financial Forecasting. International Journal of Computational Intelligence in Finance, 6:5–12, 1998.
Renaud, O., Starck, J.-L., and Murtagh, F. Prediction based on a Multiscale De- composition. International Journal of Wavelets, Multiresolution and Information Processing, 1(2):217–232. doi:10.1142/S0219691303000153, 2003.
Murtagh, F., Starck, J.-L., and Renaud, O. On Neuro-Wavelet Modeling. Decision Support Systems, 37(4):475–484. doi:10.1016/S0167-9236(03)00092-7, 2004.
Renaud, O., Starck, J.-L., and Murtagh, F. Wavelet-based combined Signal Filter- ing and Prediction. IEEE Transactions on Systems, Man, and Cybernetics, Part B (Cybernetics), 35(6):1241–1251. doi:10.1109/TSMCB.2005.850182, 2005.
data(AirPassengers) plot(AirPassengers, type = "l", col = "black") dec_res = decomposition(as.vector(array(AirPassengers)), Aggregation = c(2,4)) plot(dec_res$SmoothCoefficients[2,4:length(dec_res$SmoothCoefficients[2,])], type = "l", col = "blue") lines(array(AirPassengers)[4:length(dec_res$SmoothCoefficients[2,])], col = "black")data(AirPassengers) plot(AirPassengers, type = "l", col = "black") dec_res = decomposition(as.vector(array(AirPassengers)), Aggregation = c(2,4)) plot(dec_res$SmoothCoefficients[2,4:length(dec_res$SmoothCoefficients[2,])], type = "l", col = "blue") lines(array(AirPassengers)[4:length(dec_res$SmoothCoefficients[2,])], col = "black")
Data from a European Network of Transmission System Operators for Electricity Accessed: 2020-08-20, 2019. Time series contains 3652 data points without missing values. Data describes electrict load for time range between 2006 and 2015
data(entsoe)data(entsoe)
A DataFrame with 3652 rows and 2 columns
data(entsoe) data = entsoe$valuedata(entsoe) data = entsoe$value
This function computes a model selection using a criterion (AIC, MRE).
model_selection(UnivariateData, Aggregation, Horizon = 14, Window = 365, Method = "r", crit = "AIC", itermax = 1, lower_limit = 1, upper_limit = 2, NumClusters = 1)model_selection(UnivariateData, Aggregation, Horizon = 14, Window = 365, Method = "r", crit = "AIC", itermax = 1, lower_limit = 1, upper_limit = 2, NumClusters = 1)
UnivariateData |
[1:n] Numerical vector with n values. |
Aggregation |
[1:Scales] Numerical vector carrying numbers whose index is associated with the wavelet level. The numbers indicate the number of time in points used for aggregation from the original time series. |
Horizon |
Number indicating horizon for forecast from 1 to horizon. |
Window |
Number indicating how many points are used for cross validation. |
Method |
String indicating which method to use. Available methods: 'r' = Autoregression. 'nn' = Neural Network. |
crit |
String indicating which criterion to use. Available criterion: AIC = Akaikes Information Criterion. MRE = Mean Root Error. |
itermax |
Number of iterations for evolutionary optimization method. |
lower_limit |
Lower limit for coefficients selected for each level |
upper_limit |
Higher limit for coefficients selected for each level |
NumClusters |
Number of clusters used for parallel computing. |
The evaluation function (optimization function) is built with a rolling forecasting origin (rolling_window function), which computes a h-step ahead forecast (for h = 1, ..., horizon) for window_size many steps. The input space is searched with an evolutionary optimization method. The deployed forecast method can be an autoregression or a neural network (multilayer perceptron with one hidden layer).
Error |
[1:Window, 1:Horizon] Numerical Matrix with 'Window' many rows entries indicating one time point with 'Horizon' many forecast errors. |
Best |
[1:Scales+1] Numerical vector with integers associated with the best found number of coefficients per wavelet scale (1:Scales) and number of coefficients for the smooth approximation level in the last entry. |
Quirin Stier
Hyndman, R. and Athanasopoulos, G. Forecasting: principles and practice. OTexts, 3 edition. 2018.
data(entsoe) model_selection(UnivariateData = entsoe$value, Aggregation = c(2,4), Horizon = 1, Window = 1, Method = "r", crit = "AIC", itermax = 1, upper_limit = 1, NumClusters = 1)data(entsoe) model_selection(UnivariateData = entsoe$value, Aggregation = c(2,4), Horizon = 1, Window = 1, Method = "r", crit = "AIC", itermax = 1, upper_limit = 1, NumClusters = 1)
This function creates a one step forecast using a multi layer perceptron with one hidden Layer. The number of input is the sum of all coefficients chosen with the parameter CoefficientCombination. The CoefficientCombination parameter controls the number of coefficients chosen for each wavelet and smooth part level individually.
mrf_elm_forecast(UnivariateData, Horizon, Aggregation, Threshold="hard", Lambda=0.05)mrf_elm_forecast(UnivariateData, Horizon, Aggregation, Threshold="hard", Lambda=0.05)
UnivariateData |
[1:n] Numerical vector with n values. |
Horizon |
Number indicating horizon for forecast from 1 to horizon. |
Aggregation |
[1:Scales] Numerical vector carrying numbers whose index is associated with the wavelet level. The numbers indicate the number of time in points used for aggregation from the original time series. |
Threshold |
Character indicating if Thresholding is done on the wavelet decomposition or not. Default: Threshold="hard". Possible entries: Threshold="hard" for hard thresholding. Threshold="soft" for soft thresholding. Any other input indicates no thresholding. |
Lambda |
Numeric value indicating the threshold for computing a hard or soft threshold on the wavelet decomposition. |
forecast |
Numerical value with one step forecast |
Quirin Stier
Aussem, A., Campbell, J., and Murtagh, F. Waveletbased Feature Extraction and Decomposition Strategies for Financial Forecasting. International Journal of Computational Intelligence in Finance, 6,5-12, 1998.
Renaud, O., Starck, J.-L., and Murtagh, F. Prediction based on a Multiscale De- composition. International Journal of Wavelets, Multiresolution and Information Processing, 1(2):217-232. doi:10.1142/S0219691303000153, 2003.
Murtagh, F., Starck, J.-L., and Renaud, O. On Neuro-Wavelet Modeling. Decision Support Systems, 37(4):475-484. doi:10.1016/S0167-9236(03)00092-7, 2004.
Renaud, O., Starck, J.-L., and Murtagh, F. Wavelet-based combined Signal Filter- ing and Prediction. IEEE Transactions on Systems, Man, and Cybernetics, Part B (Cybernetics), 35(6):1241-1251. doi:10.1109/TSMCB.2005.850182, 2005.
data(AirPassengers) len_data = length(as.vector(array(AirPassengers))) UnivariateData = as.vector(AirPassengers)[1:(len_data-1)] Aggregation = c(2,4) if(requireNamespace('nnfor', quietly = TRUE)){ forecast = mrf_elm_forecast(UnivariateData, Horizon=1, Aggregation) true_value = array(AirPassengers)[len_data] error = true_value - forecast }data(AirPassengers) len_data = length(as.vector(array(AirPassengers))) UnivariateData = as.vector(AirPassengers)[1:(len_data-1)] Aggregation = c(2,4) if(requireNamespace('nnfor', quietly = TRUE)){ forecast = mrf_elm_forecast(UnivariateData, Horizon=1, Aggregation) true_value = array(AirPassengers)[len_data] error = true_value - forecast }
Creates a multiresolution forecast for a given multiresolution model based on [Stier et al., 2021] which is currently in press. (mrf_train).
mrf_forecast(Model, Horizon=1)mrf_forecast(Model, Horizon=1)
Model |
List containing model specifications from mrf_train(). |
Horizon |
Number indicating horizon for forecast from 1 to horizon. |
List of
Forecast |
[1:Horizon] Numerical vector with forecast of horizon according to its index. |
Model |
List containing model specifications from mrf_train(). |
Quirin Stier
[Stier et al., 2021] Stier, Q.,Gehlert, T. and Thrun, M. C.: Multiresolution Forecasting for Industrial Applications, Processess, 2021.
data(AirPassengers) Data = as.vector(AirPassengers) len_data = length(Data) Train = Data[1:(len_data-2)] Test = Data[(len_data-1):len_data] # One-step forecast (Multiresolution Forecast) model = mrf_train(Train) one_step = mrf_forecast(model, Horizon=1) Error = one_step$Forecast - Test[1] # Multi-step forecast (Multiresolution Forecast) # Horizon = 2 => Forecast with Horizon 1 and 2 as vector model = mrf_train(Train, Horizon=2) multi_step = mrf_forecast(model, Horizon=2) Error = multi_step$Forecast - Testdata(AirPassengers) Data = as.vector(AirPassengers) len_data = length(Data) Train = Data[1:(len_data-2)] Test = Data[(len_data-1):len_data] # One-step forecast (Multiresolution Forecast) model = mrf_train(Train) one_step = mrf_forecast(model, Horizon=1) Error = one_step$Forecast - Test[1] # Multi-step forecast (Multiresolution Forecast) # Horizon = 2 => Forecast with Horizon 1 and 2 as vector model = mrf_train(Train, Horizon=2) multi_step = mrf_forecast(model, Horizon=2) Error = multi_step$Forecast - Test
Evaluates the best coefficient combination for a given aggregation scheme based on a rolling forecasting origin based on the manuscript [Stier et al., 2021] which is currently in press.
mrf_model_selection(UnivariateData, Aggregation, Horizon = 1, Window = 2, Method = "r", crit = "AIC", itermax = 1, lower_limit = 1, upper_limit = 2, NumClusters = 1, Threshold="hard", Lambda=0.05)mrf_model_selection(UnivariateData, Aggregation, Horizon = 1, Window = 2, Method = "r", crit = "AIC", itermax = 1, lower_limit = 1, upper_limit = 2, NumClusters = 1, Threshold="hard", Lambda=0.05)
UnivariateData |
[1:n] Numerical vector with n values. |
Aggregation |
[1:Scales] Numerical vector carrying numbers whose index is associated with the wavelet level. The numbers indicate the number of time in points used for aggregation from the original time series. |
Horizon |
Number indicating horizon for forecast from 1 to horizon. |
Window |
Number indicating how many points are used for cross validation. |
Method |
String indicating which method to use. Available methods: 'r' = Autoregression. 'nn' = Neural Network. 'elm' = Extreme Learning Machine. 'nnetar' = forecast::nnetar. Default: Method="r". |
crit |
String with criterion. Available criterions: "AIC" = Akaikes Info. Crit. "MAE" = Mean Abs. Error. "MRE" = Mean Root Error. Default: crit = "AIC". |
itermax |
Number of iterations used in the differential evolutionary optimization algorithm. Default: itermax = 1. |
lower_limit |
[1:Scales+1] Numeric vector: Lower limit for coefficients selected for each level. |
upper_limit |
[1:Scales+1] Numeric vector: Higher limit for coefficients selected for each level. |
NumClusters |
Number of clusters used for parallel computing. Default: NumClusters = 1. |
Threshold |
Character indicating if Thresholding is done on the wavelet decomposition or not. Default: Threshold="hard". Possible entries: Threshold="hard" for hard thresholding. Threshold="soft" for soft thresholding. Any other input indicates no thresholding. |
Lambda |
Numeric value indicating the threshold for computing a hard or soft threshold on the wavelet decomposition. |
The evaluation function (optimization function) is built with a rolling forecasting origin (rolling_window function), which computes a h-step ahead forecast (for h = 1, ..., horizon) for 'Window' many steps. The input space is searched with an evolutionary optimization method. The search is restricted to one fixed aggregation scheme (parameter: 'Aggregation'). The deployed forecast method can be an autoregression or a neural network (multilayer perceptron with one hidden layer).
CoefficientCombination |
[1:Scales+1] Numerical vector with numbers which are associated with wavelet levels. The last number is associated with the smooth level. Each number determines the number of coefficient used per level. The selection follows a specific scheme. Best combination of coefficients found by the model selection algorithm. |
Aggregation |
[1:Scales] Numerical vector carrying numbers whose index is associated with the wavelet level. The numbers indicate the number of time in points used for aggregation from the original time series. Best Aggregation scheme found by the model selection algorithm. |
Quirin Stier
[Stier et al., 2021] Stier, Q.,Gehlert, T. and Thrun, M. C.: Multiresolution Forecasting for Industrial Applications, Processess, 2021.
data(entsoe) UnivariateData = entsoe$value Aggregation = c(2,4) res = mrf_model_selection(UnivariateData, Aggregation, Horizon = 1, Window = 2, Method = "r", crit = "AIC", itermax = 1, lower_limit = 1, upper_limit = 2, NumClusters = 1) BestCoefficientCombination = res$CoefficientCombinationdata(entsoe) UnivariateData = entsoe$value Aggregation = c(2,4) res = mrf_model_selection(UnivariateData, Aggregation, Horizon = 1, Window = 2, Method = "r", crit = "AIC", itermax = 1, lower_limit = 1, upper_limit = 2, NumClusters = 1) BestCoefficientCombination = res$CoefficientCombination
This function creates a multi step forecast for all horizons from 1 to steps based on the manuscript [Stier et al., 2021] which is currently in press. The deployed forecast method can be an autoregression or a neural network (multilayer perceptron with one hidden layer). Multi step forecasts are computed recursively.
mrf_multi_step_forecast(UnivariateData, Horizon, Aggregation, CoefficientCombination=NULL, Method = "r", Threshold="hard", Lambda=0.05)mrf_multi_step_forecast(UnivariateData, Horizon, Aggregation, CoefficientCombination=NULL, Method = "r", Threshold="hard", Lambda=0.05)
UnivariateData |
[1:n] Numerical vector with n values. |
Horizon |
Number indicating horizon for forecast from 1 to horizon. |
CoefficientCombination |
[1:Scales+1] Numerical vector with numbers which are associated with wavelet levels. The last number is associated with the smooth level. Each number determines the number of coefficient used per level. The selection follows a specific scheme. |
Aggregation |
[1:Scales] Numerical vector carrying numbers whose index is associated with the wavelet level. The numbers indicate the number of time in points used for aggregation from the original time series. |
Method |
String indicating which method to use. Available methods: 'r' = Autoregression. 'nn' = Neural Network. 'elm' = Extreme Learning Machine. 'nnetar' = forecast::nnetar. Default: Method="r". |
Threshold |
Character indicating if Thresholding is done on the wavelet decomposition or not. Default: Threshold="hard". Possible entries: Threshold="hard" for hard thresholding. Threshold="soft" for soft thresholding. Any other input indicates no thresholding. |
Lambda |
Numeric value indicating the threshold for computing a hard or soft threshold on the wavelet decomposition. |
List of
multistep |
[1:Horizon] Numerical vector with forecast of horizon according to its index. |
Quirin Stier
[Stier et al., 2021] Stier, Q.,Gehlert, T. and Thrun, M. C.: Multiresolution Forecasting for Industrial Applications, Processess, 2021.
data(AirPassengers) len_data = length(array(AirPassengers)) UnivariateData = as.vector(AirPassengers)[1:(len_data-1)] # One-step forecast (Multiresolution Forecast) one_step = mrf_multi_step_forecast(UnivariateData = UnivariateData, Horizon = 2, CoefficientCombination = c(1,1,1), Aggregation = c(2,4), Method="r") # Multi-step forecast (Multiresolution Forecast) # Horizon = 2 => Forecast with Horizon 1 and 2 as vector multi_step = mrf_multi_step_forecast(UnivariateData = UnivariateData, Horizon = 2, CoefficientCombination = c(1,1,1), Aggregation = c(2,4), Method="r")data(AirPassengers) len_data = length(array(AirPassengers)) UnivariateData = as.vector(AirPassengers)[1:(len_data-1)] # One-step forecast (Multiresolution Forecast) one_step = mrf_multi_step_forecast(UnivariateData = UnivariateData, Horizon = 2, CoefficientCombination = c(1,1,1), Aggregation = c(2,4), Method="r") # Multi-step forecast (Multiresolution Forecast) # Horizon = 2 => Forecast with Horizon 1 and 2 as vector multi_step = mrf_multi_step_forecast(UnivariateData = UnivariateData, Horizon = 2, CoefficientCombination = c(1,1,1), Aggregation = c(2,4), Method="r")
This function creates a one step forecast using a multi layer perceptron with one hidden Layer. The number of input is the sum of all coefficients chosen with the parameter CoefficientCombination. The CoefficientCombination parameter controls the number of coefficients chosen for each wavelet and smooth part level individually.
mrf_neuralnet_one_step_forecast(UnivariateData, CoefficientCombination, Aggregation, Threshold="hard", Lambda=0.05)mrf_neuralnet_one_step_forecast(UnivariateData, CoefficientCombination, Aggregation, Threshold="hard", Lambda=0.05)
UnivariateData |
[1:n] Numerical vector with n values. |
CoefficientCombination |
[1:Scales+1] Numerical vector with numbers which are associated with wavelet levels. The last number is associated with the smooth level. Each number determines the number of coefficient used per level. The selection follows a specific scheme. |
Aggregation |
[1:Scales] Numerical vector carrying numbers whose index is associated with the wavelet level. The numbers indicate the number of time in points used for aggregation from the original time series. |
Threshold |
Character indicating if Thresholding is done on the wavelet decomposition or not. Default: Threshold="hard". Possible entries: Threshold="hard" for hard thresholding. Threshold="soft" for soft thresholding. Any other input indicates no thresholding. |
Lambda |
Numeric value indicating the threshold for computing a hard or soft threshold on the wavelet decomposition. |
forecast |
Numerical value with one step forecast |
Quirin Stier
Aussem, A., Campbell, J., and Murtagh, F. Waveletbased Feature Extraction and Decomposition Strategies for Financial Forecasting. International Journal of Computational Intelligence in Finance, 6,5-12, 1998.
Renaud, O., Starck, J.-L., and Murtagh, F. Prediction based on a Multiscale De- composition. International Journal of Wavelets, Multiresolution and Information Processing, 1(2):217-232. doi:10.1142/S0219691303000153, 2003.
Murtagh, F., Starck, J.-L., and Renaud, O. On Neuro-Wavelet Modeling. Decision Support Systems, 37(4):475-484. doi:10.1016/S0167-9236(03)00092-7, 2004.
Renaud, O., Starck, J.-L., and Murtagh, F. Wavelet-based combined Signal Filter- ing and Prediction. IEEE Transactions on Systems, Man, and Cybernetics, Part B (Cybernetics), 35(6):1241-1251. doi:10.1109/TSMCB.2005.850182, 2005.
data(AirPassengers) len_data = length(as.vector(array(AirPassengers))) UnivariateData = as.vector(AirPassengers)[1:(len_data-1)] CoefficientCombination = c(1,1,1) Aggregation = c(2,4) if(requireNamespace('monmlp', quietly = TRUE)){ forecast = mrf_neuralnet_one_step_forecast(UnivariateData, CoefficientCombination, Aggregation) true_value = array(AirPassengers)[len_data] error = true_value - forecast }data(AirPassengers) len_data = length(as.vector(array(AirPassengers))) UnivariateData = as.vector(AirPassengers)[1:(len_data-1)] CoefficientCombination = c(1,1,1) Aggregation = c(2,4) if(requireNamespace('monmlp', quietly = TRUE)){ forecast = mrf_neuralnet_one_step_forecast(UnivariateData, CoefficientCombination, Aggregation) true_value = array(AirPassengers)[len_data] error = true_value - forecast }
This function creates a one step forecast using a multi layer perceptron with one hidden Layer. The number of input is the sum of all coefficients chosen with the parameter CoefficientCombination. The CoefficientCombination parameter controls the number of coefficients chosen for each wavelet and smooth part level individually.
mrf_nnetar_forecast(UnivariateData, Horizon, Aggregation, Threshold="hard", Lambda=0.05)mrf_nnetar_forecast(UnivariateData, Horizon, Aggregation, Threshold="hard", Lambda=0.05)
UnivariateData |
[1:n] Numerical vector with n values. |
Horizon |
Number indicating horizon for forecast from 1 to horizon. |
Aggregation |
[1:Scales] Numerical vector carrying numbers whose index is associated with the wavelet level. The numbers indicate the number of time in points used for aggregation from the original time series. |
Threshold |
Character indicating if Thresholding is done on the wavelet decomposition or not. Default: Threshold="hard". Possible entries: Threshold="hard" for hard thresholding. Threshold="soft" for soft thresholding. Any other input indicates no thresholding. |
Lambda |
Numeric value indicating the threshold for computing a hard or soft threshold on the wavelet decomposition. |
forecast |
Numerical value with one step forecast |
Quirin Stier
Aussem, A., Campbell, J., and Murtagh, F. Waveletbased Feature Extraction and Decomposition Strategies for Financial Forecasting. International Journal of Computational Intelligence in Finance, 6,5-12, 1998.
Renaud, O., Starck, J.-L., and Murtagh, F. Prediction based on a Multiscale De- composition. International Journal of Wavelets, Multiresolution and Information Processing, 1(2):217-232. doi:10.1142/S0219691303000153, 2003.
Murtagh, F., Starck, J.-L., and Renaud, O. On Neuro-Wavelet Modeling. Decision Support Systems, 37(4):475-484. doi:10.1016/S0167-9236(03)00092-7, 2004.
Renaud, O., Starck, J.-L., and Murtagh, F. Wavelet-based combined Signal Filter- ing and Prediction. IEEE Transactions on Systems, Man, and Cybernetics, Part B (Cybernetics), 35(6):1241-1251. doi:10.1109/TSMCB.2005.850182, 2005.
data(AirPassengers) len_data = length(as.vector(array(AirPassengers))) UnivariateData = as.vector(AirPassengers)[1:(len_data-1)] Aggregation = c(2,4) if(requireNamespace('nnfor', quietly = TRUE)){ forecast = mrf_nnetar_forecast(UnivariateData, Horizon=1, Aggregation) true_value = array(AirPassengers)[len_data] error = true_value - forecast }data(AirPassengers) len_data = length(as.vector(array(AirPassengers))) UnivariateData = as.vector(AirPassengers)[1:(len_data-1)] Aggregation = c(2,4) if(requireNamespace('nnfor', quietly = TRUE)){ forecast = mrf_nnetar_forecast(UnivariateData, Horizon=1, Aggregation) true_value = array(AirPassengers)[len_data] error = true_value - forecast }
This function creates a one step forecast using the multiresolution forecasting framework based on the manuscript [Stier et al., 2021] which is currently in press.
mrf_one_step_forecast(UnivariateData, Aggregation, CoefficientCombination=NULL, Method="r", Threshold="hard", Lambda=0.05)mrf_one_step_forecast(UnivariateData, Aggregation, CoefficientCombination=NULL, Method="r", Threshold="hard", Lambda=0.05)
UnivariateData |
[1:n] Numerical vector with n values. |
CoefficientCombination |
[1:Scales+1] Numerical vector with numbers which are associated with wavelet levels. The last number is associated with the smooth level. Each number determines the number of coefficient used per level. The selection follows a specific scheme. |
Aggregation |
[1:Scales] Numerical vector carrying numbers whose index is associated with the wavelet level. The numbers indicate the number of time in points used for aggregation from the original time series. |
Method |
String indicating which method to use. Available methods: 'r' = Autoregression. 'nn' = Neural Network. 'elm' = Extreme Learning Machine. 'nnetar' = forecast::nnetar. Default: Method="r". |
Threshold |
Character indicating if Thresholding is done on the wavelet decomposition or not. Default: Threshold="hard". Possible entries: Threshold="hard" for hard thresholding. Threshold="soft" for soft thresholding. Any other input indicates no thresholding. |
Lambda |
Numeric value indicating the threshold for computing a hard or soft threshold on the wavelet decomposition. |
forecast |
Numerical value with one step forecast |
Quirin Stier
[Stier et al., 2021] Stier, Q.,Gehlert, T. and Thrun, M. C.: Multiresolution Forecasting for Industrial Applications, Processess, 2021.
data(AirPassengers) len_data = length(array(AirPassengers)) UnivariateData = as.vector(AirPassengers)[1:(len_data-1)] forecast = mrf_one_step_forecast(UnivariateData=UnivariateData, CoefficientCombination=c(1,1,1), Aggregation=c(2,4)) true_value = array(AirPassengers)[len_data] error = true_value - forecastdata(AirPassengers) len_data = length(array(AirPassengers)) UnivariateData = as.vector(AirPassengers)[1:(len_data-1)] forecast = mrf_one_step_forecast(UnivariateData=UnivariateData, CoefficientCombination=c(1,1,1), Aggregation=c(2,4)) true_value = array(AirPassengers)[len_data] error = true_value - forecast
This function computes the weights for the autoregression depending on the given wavelet decomposition. It uses ordinary least square method for optimizing a linear equation system.
mrf_regression_lsm_optimization(points_in_future, lsmatrix)mrf_regression_lsm_optimization(points_in_future, lsmatrix)
points_in_future |
n many values of the time series, for which there is an equation from a prediction scheme. |
lsmatrix |
Matrix carrying predictive equations associated with a specific value of the time series. |
List of
weights |
Array of weights carrying the solution for a matrix problem, which was solves with ordinary least squares. |
Quirin Stier
Aussem, A., Campbell, J., and Murtagh, F. Waveletbased Feature Extraction and Decomposition Strategies for Financial Forecasting. International Journal of Computational Intelligence in Finance, 6,5-12, 1998.
Renaud, O., Starck, J.-L., and Murtagh, F. Prediction based on a Multiscale De- composition. International Journal of Wavelets, Multiresolution and Information Processing, 1(2):217-232. doi:10.1142/S0219691303000153, 2003.
Murtagh, F., Starck, J.-L., and Renaud, O. On Neuro-Wavelet Modeling. Decision Support Systems, 37(4):475-484. doi:10.1016/S0167-9236(03)00092-7, 2004.
Renaud, O., Starck, J.-L., and Murtagh, F. Wavelet-based combined Signal Filter- ing and Prediction. IEEE Transactions on Systems, Man, and Cybernetics, Part B (Cybernetics), 35(6):1241-1251. doi:10.1109/TSMCB.2005.850182, 2005.
data(AirPassengers) len_data = length(array(AirPassengers)) CoefficientCombination = c(1,1,1) Aggregation = c(2,4) UnivariateData = as.vector(AirPassengers) # Decomposition dec_res <- wavelet_decomposition(UnivariateData, Aggregation) # Training trs_res <- wavelet_training_equations(UnivariateData, dec_res$WaveletCoefficients, dec_res$SmoothCoefficients, dec_res$Scales, CoefficientCombination, Aggregation) arr_future_points = trs_res$points_in_future matrix = trs_res$lsmatrix # Optimization method weights = mrf_regression_lsm_optimization(arr_future_points, matrix) # Forecast scheme = wavelet_prediction_equation(dec_res$WaveletCoefficients, dec_res$SmoothCoefficients, CoefficientCombination, Aggregation) forecast = weightsdata(AirPassengers) len_data = length(array(AirPassengers)) CoefficientCombination = c(1,1,1) Aggregation = c(2,4) UnivariateData = as.vector(AirPassengers) # Decomposition dec_res <- wavelet_decomposition(UnivariateData, Aggregation) # Training trs_res <- wavelet_training_equations(UnivariateData, dec_res$WaveletCoefficients, dec_res$SmoothCoefficients, dec_res$Scales, CoefficientCombination, Aggregation) arr_future_points = trs_res$points_in_future matrix = trs_res$lsmatrix # Optimization method weights = mrf_regression_lsm_optimization(arr_future_points, matrix) # Forecast scheme = wavelet_prediction_equation(dec_res$WaveletCoefficients, dec_res$SmoothCoefficients, CoefficientCombination, Aggregation) forecast = weights
This function creates a one step forecast using an autoregression method. The ccps parameter controls the number of coefficients chosen for each wavelet and smooth part level individually.
mrf_regression_one_step_forecast(UnivariateData, CoefficientCombination, Aggregation, Threshold="hard", Lambda=0.05)mrf_regression_one_step_forecast(UnivariateData, CoefficientCombination, Aggregation, Threshold="hard", Lambda=0.05)
UnivariateData |
[1:n] Numerical vector with n values. |
CoefficientCombination |
[1:Scales+1] Numerical vector with numbers which are associated with wavelet levels. The last number is associated with the smooth level. Each number determines the number of coefficient used per level. The selection follows a specific scheme. |
Aggregation |
[1:Scales] Numerical vector carrying numbers whose index is associated with the wavelet level. The numbers indicate the number of time in points used for aggregation from the original time series. |
Threshold |
Character indicating if Thresholding is done on the wavelet decomposition or not. Default: Threshold="hard". Possible entries: Threshold="hard" for hard thresholding. Threshold="soft" for soft thresholding. Any other input indicates no thresholding. |
Lambda |
Numeric value indicating the threshold for computing a hard or soft threshold on the wavelet decomposition. |
forecast |
Numerical value with one step forecast |
Quirin Stier
Aussem, A., Campbell, J., and Murtagh, F. Waveletbased Feature Extraction and Decomposition Strategies for Financial Forecasting. International Journal of Computational Intelligence in Finance, 6,5-12, 1998.
Renaud, O., Starck, J.-L., and Murtagh, F. Prediction based on a Multiscale De- composition. International Journal of Wavelets, Multiresolution and Information Processing, 1(2):217-232. doi:10.1142/S0219691303000153, 2003.
Murtagh, F., Starck, J.-L., and Renaud, O. On Neuro-Wavelet Modeling. Decision Support Systems, 37(4):475-484. doi:10.1016/S0167-9236(03)00092-7, 2004.
Renaud, O., Starck, J.-L., and Murtagh, F. Wavelet-based combined Signal Filter- ing and Prediction. IEEE Transactions on Systems, Man, and Cybernetics, Part B (Cybernetics), 35(6):1241-1251. doi:10.1109/TSMCB.2005.850182, 2005.
data(AirPassengers) len_data = length(as.vector(array(AirPassengers))) UnivariateData = as.vector(AirPassengers)[1:(len_data-1)] CoefficientCombination = c(1,1,1) Aggregation = c(2,4) forecast = mrf_regression_one_step_forecast(UnivariateData, CoefficientCombination, Aggregation) true_value = array(AirPassengers)[len_data] error = true_value - forecastdata(AirPassengers) len_data = length(as.vector(array(AirPassengers))) UnivariateData = as.vector(AirPassengers)[1:(len_data-1)] CoefficientCombination = c(1,1,1) Aggregation = c(2,4) forecast = mrf_regression_one_step_forecast(UnivariateData, CoefficientCombination, Aggregation) true_value = array(AirPassengers)[len_data] error = true_value - forecast
Computes requirements for given model using insights of various published papers and the manuscript [Stier et al., 2021] which is currently in press.
mrf_requirement(UnivariateData, CoefficientCombination, Aggregation)mrf_requirement(UnivariateData, CoefficientCombination, Aggregation)
UnivariateData |
[1:n] Numerical vector with n values. |
CoefficientCombination |
[1:Scales+1] Numerical vector with numbers which are associated with wavelet levels. The last number is associated with the smooth level. Each number determines the number of coefficient used per level. The selection follows a specific scheme. |
Aggregation |
[1:Scales] Numerical vector carrying numbers whose index is associated with the wavelet level. The numbers indicate the number of time in points used for aggregation from the original time series. |
List of
MinLen |
Integer minimum required length for model. |
StartTraining |
Integer indicating the index of time series at which the training equations can be built up. |
NumberWeights |
Number of weights required for building model. |
NumberEquations |
Number of equations which can be built with given data. |
Quirin Stier
[Stier et al., 2021] Stier, Q.,Gehlert, T. and Thrun, M. C.: Multiresolution Forecasting for Industrial Applications, Processess, 2021.
Aussem, A., Campbell, J., and Murtagh, F. Waveletbased Feature Extraction and Decomposition Strategies for Financial Forecasting. International Journal of Computational Intelligence in Finance, 6,5-12, 1998.
Renaud, O., Starck, J.-L., and Murtagh, F. Prediction based on a Multiscale De- composition. International Journal of Wavelets, Multiresolution and Information Processing, 1(2):217-232. doi:10.1142/S0219691303000153, 2003.
Murtagh, F., Starck, J.-L., and Renaud, O. On Neuro-Wavelet Modeling. Decision Support Systems, 37(4):475-484. doi:10.1016/S0167-9236(03)00092-7, 2004.
Renaud, O., Starck, J.-L., and Murtagh, F. Wavelet-based combined Signal Filter- ing and Prediction. IEEE Transactions on Systems, Man, and Cybernetics, Part B (Cybernetics), 35(6):1241-1251. doi:10.1109/TSMCB.2005.850182, 2005.
data(entsoe) UnivariateData = entsoe$value mrf_requirement(UnivariateData, c(2,3,4), c(2,4))data(entsoe) UnivariateData = entsoe$value mrf_requirement(UnivariateData, c(2,3,4), c(2,4))
This function computes a rolling forecasting origin for one- or multi-step forecasts with a specific method based on the manuscript [Stier et al., 2021] which is currently in press. Multi-step forecasts are computed recursively with the one step forecast method.
mrf_rolling_forecasting_origin(UnivariateData, Aggregation, CoefficientCombination=NULL, Horizon = 2, Window = 3, Method = "r", NumClusters = 1, Threshold="hard", Lambda=0.05)mrf_rolling_forecasting_origin(UnivariateData, Aggregation, CoefficientCombination=NULL, Horizon = 2, Window = 3, Method = "r", NumClusters = 1, Threshold="hard", Lambda=0.05)
UnivariateData |
[1:n] Numerical vector with n values. |
CoefficientCombination |
[1:Scales+1] Numerical vector with numbers which are associated with wavelet levels. The last number is associated with the smooth level. Each number determines the number of coefficient used per level. The selection follows a specific scheme. |
Aggregation |
[1:Scales] Numerical vector carrying numbers whose index is associated with the wavelet level. The numbers indicate the number of time in points used for aggregation from the original time series. |
Horizon |
Number indicating horizon for forecast from 1 to horizon. |
Window |
Number indicating how many points are used for cross validation. |
Method |
String indicating which method to use. Available methods: 'r' = Autoregression. 'nn' = Neural Network. 'elm' = Extreme Learning Machine. 'nnetar' = forecast::nnetar. Default: Method="r". |
NumClusters |
Number of clusters used for parallel computing. |
Threshold |
Character indicating if Thresholding is done on the wavelet decomposition or not. Default: Threshold="hard". Possible entries: Threshold="hard" for hard thresholding. Threshold="soft" for soft thresholding. Any other input indicates no thresholding. |
Lambda |
Numeric value indicating the threshold for computing a hard or soft threshold on the wavelet decomposition. |
Thus, h-step forecast for h = 1,..., horizon for window_size many steps can be computed. The forecasting method can be an autoregression or a neural network (multilayer perceptron). The CoefficientCombination parameter controls the number of coefficients chosen for each wavelet and smooth part level individually. The NumClusters parameter determines the number of cluster used for parallel computation. NumClusters = 1 performs a non parallel version. NumClusters is constrained by the maximum number of clusters available minus one to prevent the machine to be overchallenged.
List of
Error |
[1:Window,1:Horizon] Numerical Matrix with 'Window' many row entries indicating one time point with 'Horizon' many forecast errors. |
Forecast |
[1:Window,1:Horizon] Numerical Matrix with 'Window' many row entries indicating one time point with 'Horizon' many forecasts. |
Quirin Stier
[Stier et al., 2021] Stier, Q.,Gehlert, T. and Thrun, M. C.: Multiresolution Forecasting for Industrial Applications, Processess, 2021.
data(AirPassengers) UnivariateData=as.vector(array(AirPassengers)) res = mrf_rolling_forecasting_origin(UnivariateData, CoefficientCombination = c(10,10,10), Aggregation = c(2,4), Horizon = 2, Window = 3, Method = "r", NumClusters = 1) Error = res$Error Forecast = res$Forecastdata(AirPassengers) UnivariateData=as.vector(array(AirPassengers)) res = mrf_rolling_forecasting_origin(UnivariateData, CoefficientCombination = c(10,10,10), Aggregation = c(2,4), Horizon = 2, Window = 3, Method = "r", NumClusters = 1) Error = res$Error Forecast = res$Forecast
Creates a multiresolution forecast model which can be used for forecasting with method 'mrf_forecast' based on the manuscript [Stier et al., 2021] which is currently in press.
mrf_train(Data, Horizon=1, Aggregation="auto", Method = "r", TimeSteps4ModelSelection=2, crit="AIC", InSample=FALSE, Threshold="hard", Lambda=0.05, NumClusters=1, itermax=1)mrf_train(Data, Horizon=1, Aggregation="auto", Method = "r", TimeSteps4ModelSelection=2, crit="AIC", InSample=FALSE, Threshold="hard", Lambda=0.05, NumClusters=1, itermax=1)
Data |
[1:n] Numerical vector with n values from the training data. |
Horizon |
Number indicating forecast horizon. Horizon = 1 means one-step forecast and Horizon > 1 means a one-step forecast and all multi-step forecasts from horizon 2 to 'Horizon'. Default: Horizon = 1. |
Aggregation |
[1:Scales] Numerical vector carrying numbers whose index is associated with the wavelet level. The numbers indicate the number of time in points used for aggregation from the original time series. Default: Aggregation = "auto". |
Method |
String indicating which method to use. Available methods: 'r' = Autoregression. 'nn' = Neural Network. 'elm' = Extreme Learning Machine. 'nnetar' = forecast::nnetar. Default: Method="r". |
TimeSteps4ModelSelection |
Number of time steps of data (newest part) on which a model selection is performed. Default: TimeSteps4ModelSelection = 2. |
crit |
String with criterion. Available criterions: "AIC" = Akaikes Info. Crit. "MAE" = Mean Abs. Error. "MRE" = Mean Root Error. Default: crit = "AIC". |
InSample |
Boolean, deciding if in-sample-forecast based on rolling forecasting origin is computed or not. TRUE = Computation of in-sample-forecast. FALSE = No computation. Default: InSample = FALSE |
Threshold |
Character indicating if Thresholding is done on the wavelet decomposition or not. Default: Threshold="hard". Possible entries: Threshold="hard" for hard thresholding. Threshold="soft" for soft thresholding. Any other input indicates no thresholding. |
Lambda |
Numeric value indicating the threshold for computing a hard or soft threshold on the wavelet decomposition. |
NumClusters |
Number of clusters used for parallel computing. Default: NumClusters = 1. |
itermax |
Number of iterations used in the differential evolutionary optimization algorithm. Default: itermax = 1. |
List with
Data |
[1:n] Numerical vector with n values from the training data. |
Method |
String indicating which method to use. |
Aggregation |
[1:Scales] Numerical vector carrying numbers whose index is associated with the wavelet level. The numbers indicate the number of time in points used for aggregation from the original time series. |
CoefficientCombination |
[1:Scales+1] Numerical vector with numbers which are associated with wavelet levels. The last number is associated with the smooth level. Each number determines the number of coefficient used per level. The selection follows a specific scheme. |
Horizon |
Number indicating forecast horizon. Horizon = 1 means one-step forecast and Horizon > 1 means a one-step forecast and all multi-step forecasts from horizon 2 to 'Horizon'. |
ModelError |
[1:TimeSteps4ModelSelection, 1:Horizon] Numerical matrix with one-/multi-steps in columns and the time steps rowwise. The error is according to the scheme of a rolling forecasting origin. The length depends on the minimum required length for constructing the wavelet model and the length of data. The newer part of the data is used for the model fit truncating the oldest data according to the minimum required length for constructing the model. |
ModelMAE |
Integer: Mean Absolute Error (MAE) computed for the in-sample-forecast resulting from a rolling forecasting origin. |
Quirin Stier
[Stier et al., 2021] Stier, Q.,Gehlert, T. and Thrun, M. C.: Multiresolution Forecasting for Industrial Applications, Processess, 2021.
data(AirPassengers) Data = as.vector(AirPassengers) len_data = length(Data) Train = Data[1:(len_data-2)] Test = Data[(len_data-1):len_data] # One-step forecast (Multiresolution Forecast) model = mrf_train(Train) one_step = mrf_forecast(model, Horizon=1) Error = one_step$Forecast - Test[1] # Multi-step forecast (Multiresolution Forecast) # Horizon = 2 => Forecast with Horizon 1 and 2 as vector model = mrf_train(Train, Horizon=2) multi_step = mrf_forecast(model, Horizon=2) Error = multi_step$Forecast - Testdata(AirPassengers) Data = as.vector(AirPassengers) len_data = length(Data) Train = Data[1:(len_data-2)] Test = Data[(len_data-1):len_data] # One-step forecast (Multiresolution Forecast) model = mrf_train(Train) one_step = mrf_forecast(model, Horizon=1) Error = one_step$Forecast - Test[1] # Multi-step forecast (Multiresolution Forecast) # Horizon = 2 => Forecast with Horizon 1 and 2 as vector model = mrf_train(Train, Horizon=2) multi_step = mrf_forecast(model, Horizon=2) Error = multi_step$Forecast - Test
This function creates a multi step forecast for all horizons from 1 to steps. The deployed forecast method can be an autoregression or a neural network (multilayer perceptron with one hidden layer). Multi step forecasts are computed recursively.
multi_step(UnivariateData, Horizon, CoefficientCombination, Aggregation, Method = "r")multi_step(UnivariateData, Horizon, CoefficientCombination, Aggregation, Method = "r")
UnivariateData |
[1:n] Numerical vector with n values. |
Horizon |
Number indicating horizon for forecast from 1 to horizon. |
CoefficientCombination |
[1:Scales+1] Numerical vector with numbers which are associated with wavelet levels. The last number is associated with the smooth level. Each number determines the number of coefficient used per level. The selection follows a specific scheme. |
Aggregation |
[1:Scales] Numerical vector carrying numbers whose index is associated with the wavelet level. The numbers indicate the number of time in points used for aggregation from the original time series. |
Method |
String indicating which method to use. Available methods: 'r' = Autoregression. 'nn' = Neural Network. |
List of
multistep |
[1:Horizon] Numerical vector with forecast of horizon according to its index. |
Quirin Stier
Aussem, A., Campbell, J., and Murtagh, F. Waveletbased Feature Extraction and Decomposition Strategies for Financial Forecasting. International Journal of Computational Intelligence in Finance, 6:5–12, 1998.
Renaud, O., Starck, J.-L., and Murtagh, F. Prediction based on a Multiscale De- composition. International Journal of Wavelets, Multiresolution and Information Processing, 1(2):217–232. doi:10.1142/S0219691303000153, 2003.
Murtagh, F., Starck, J.-L., and Renaud, O. On Neuro-Wavelet Modeling. Decision Support Systems, 37(4):475–484. doi:10.1016/S0167-9236(03)00092-7, 2004.
Renaud, O., Starck, J.-L., and Murtagh, F. Wavelet-based combined Signal Filter- ing and Prediction. IEEE Transactions on Systems, Man, and Cybernetics, Part B (Cybernetics), 35(6):1241–1251. doi:10.1109/TSMCB.2005.850182, 2005.
data(AirPassengers) len_data = length(array(AirPassengers)) multistep = multi_step(as.vector(AirPassengers)[1:(len_data-1)], 2, c(1,1,1), c(2,4), Method="r")data(AirPassengers) len_data = length(array(AirPassengers)) multistep = multi_step(as.vector(AirPassengers)[1:(len_data-1)], 2, c(1,1,1), c(2,4), Method="r")
This function computes a nested cross validation (with the rolling forecasting origin). The data is split into 3 datasets: training, test and evaluation dataset. The best model is selected on the test and its performance is measured on the evaluation dataset.
nested_cross_validation(UnivariateData, Horizon=14, EvaluationLength=2, TestLength=2, Method = "r", MultivariateData=NULL, NumMV=1, NumClusters = 1)nested_cross_validation(UnivariateData, Horizon=14, EvaluationLength=2, TestLength=2, Method = "r", MultivariateData=NULL, NumMV=1, NumClusters = 1)
UnivariateData |
[1:n] Numerical vector with n values. |
Horizon |
Number indicating horizon for forecast from 1 to horizon. |
EvaluationLength |
Number indicating how many points are used for cross validation for the evaluation dataset. |
TestLength |
Number indicating how many points are used for cross validation for the test dataset. |
Method |
String indicating which method to use. Available methods: 'r' = Autoregression. 'nn' = Neural Network. |
MultivariateData |
Not implemented yet. |
NumMV |
Not implemented yet. |
NumClusters |
Number of clusters used for parallel computing. |
The evaluation function (optimization function) is built with a rolling forecasting origin (rolling_window function), which computes a h-step ahead forecast (for h = 1, ..., horizon) for window_size many steps. The input space is searched with an evolutionary optimization method. The deployed forecast method can be an autoregression or a neural network (multilayer perceptron with one hidden layer).
Best |
[1:Scales+1] Numerical vector with integers associated with the best found number of coefficients per wavelet scale (1:Scales) and number of coefficients for the smooth approximation level in the last entry. |
Error |
[1:Window, 1:Horizon] Numerical Matrix with 'Window' many rows entries indicating one time point with 'Horizon' many forecast errors. |
Forecast |
[1:Window, 1:Horizon] Numerical Matrix with 'Window' many rows entries indicating one time point with 'Horizon' many forecasts. |
Quirin Stier
Hyndman, R. and Athanasopoulos, G. Forecasting: principles and practice. OTexts, 3 edition. 2018.
data(entsoe) res = nested_cross_validation(entsoe$value, Horizon = 2, EvaluationLength=2, TestLength=2, Method="r", MultivariateData=NULL, NumMV=1, NumClusters=1) BestCoefficientCombination = res$Best Error = res$Error Forecast = res$Forecastdata(entsoe) res = nested_cross_validation(entsoe$value, Horizon = 2, EvaluationLength=2, TestLength=2, Method="r", MultivariateData=NULL, NumMV=1, NumClusters=1) BestCoefficientCombination = res$Best Error = res$Error Forecast = res$Forecast
This function creates a one step forecast using a multi layer perceptron with one hidden Layer. The number of input is the sum of all coefficients chosen with the parameter CoefficientCombination. The CoefficientCombination parameter controls the number of coefficients chosen for each wavelet and smooth part level individually.
neuralnet_one_step(UnivariateData, CoefficientCombination, Aggregation)neuralnet_one_step(UnivariateData, CoefficientCombination, Aggregation)
UnivariateData |
[1:n] Numerical vector with n values. |
CoefficientCombination |
[1:Scales+1] Numerical vector with numbers which are associated with wavelet levels. The last number is associated with the smooth level. Each number determines the number of coefficient used per level. The selection follows a specific scheme. |
Aggregation |
[1:Scales] Numerical vector carrying numbers whose index is associated with the wavelet level. The numbers indicate the number of time in points used for aggregation from the original time series. |
forecast |
Numerical value with one step forecast |
Quirin Stier
Aussem, A., Campbell, J., and Murtagh, F. Waveletbased Feature Extraction and Decomposition Strategies for Financial Forecasting. International Journal of Computational Intelligence in Finance, 6:5–12, 1998.
Renaud, O., Starck, J.-L., and Murtagh, F. Prediction based on a Multiscale De- composition. International Journal of Wavelets, Multiresolution and Information Processing, 1(2):217–232. doi:10.1142/S0219691303000153, 2003.
Murtagh, F., Starck, J.-L., and Renaud, O. On Neuro-Wavelet Modeling. Decision Support Systems, 37(4):475–484. doi:10.1016/S0167-9236(03)00092-7, 2004.
Renaud, O., Starck, J.-L., and Murtagh, F. Wavelet-based combined Signal Filter- ing and Prediction. IEEE Transactions on Systems, Man, and Cybernetics, Part B (Cybernetics), 35(6):1241–1251. doi:10.1109/TSMCB.2005.850182, 2005.
data(AirPassengers) len_data = length(array(AirPassengers)) forecast = neuralnet_one_step(as.vector(AirPassengers)[1:(len_data-1)], c(1,1,1), c(2,4)) true_value = array(AirPassengers)[len_data] error = true_value - forecastdata(AirPassengers) len_data = length(array(AirPassengers)) forecast = neuralnet_one_step(as.vector(AirPassengers)[1:(len_data-1)], c(1,1,1), c(2,4)) true_value = array(AirPassengers)[len_data] error = true_value - forecast
This function creates a one step forecast using the multiresolution forecasting framework.
onestep(UnivariateData, CoefficientCombination, Aggregation, Method="r")onestep(UnivariateData, CoefficientCombination, Aggregation, Method="r")
UnivariateData |
[1:n] Numerical vector with n values. |
CoefficientCombination |
[1:Scales+1] Numerical vector with numbers which are associated with wavelet levels. The last number is associated with the smooth level. Each number determines the number of coefficient used per level. The selection follows a specific scheme. |
Aggregation |
[1:Scales] Numerical vector carrying numbers whose index is associated with the wavelet level. The numbers indicate the number of time in points used for aggregation from the original time series. |
Method |
String indicating which method to use. Available methods: 'r' = Autoregression. 'nn' = Neural Network. |
forecast |
Numerical value with one step forecast |
Quirin Stier
Aussem, A., Campbell, J., and Murtagh, F. Waveletbased Feature Extraction and Decomposition Strategies for Financial Forecasting. International Journal of Computational Intelligence in Finance, 6:5–12, 1998.
Renaud, O., Starck, J.-L., and Murtagh, F. Prediction based on a Multiscale De- composition. International Journal of Wavelets, Multiresolution and Information Processing, 1(2):217–232. doi:10.1142/S0219691303000153, 2003.
Murtagh, F., Starck, J.-L., and Renaud, O. On Neuro-Wavelet Modeling. Decision Support Systems, 37(4):475–484. doi:10.1016/S0167-9236(03)00092-7, 2004.
Renaud, O., Starck, J.-L., and Murtagh, F. Wavelet-based combined Signal Filter- ing and Prediction. IEEE Transactions on Systems, Man, and Cybernetics, Part B (Cybernetics), 35(6):1241–1251. doi:10.1109/TSMCB.2005.850182, 2005.
data(AirPassengers) len_data = length(array(AirPassengers)) forecast = onestep(as.vector(AirPassengers)[1:(len_data-1)], c(1,1,1), c(2,4)) true_value = array(AirPassengers)[len_data] error = true_value - forecastdata(AirPassengers) len_data = length(array(AirPassengers)) forecast = onestep(as.vector(AirPassengers)[1:(len_data-1)], c(1,1,1), c(2,4)) true_value = array(AirPassengers)[len_data] error = true_value - forecast
This function delivers the required wavelet and smooth coefficients from the decomposition based on a prediction scheme.
prediction_scheme(WaveletCoefficients, SmoothCoefficients, CoefficientCombination, Aggregation)prediction_scheme(WaveletCoefficients, SmoothCoefficients, CoefficientCombination, Aggregation)
WaveletCoefficients |
[Scales, n] Matrix with 'Scales' many wavelet scales row-wise with n columns corresponding to the time domain of a time series. |
SmoothCoefficients |
[Scales, n] Matrix with 'Scales' many smooth approximation scales row-wise with n columns corresponding to the time domain of a time series. |
CoefficientCombination |
[1:Scales+1] Numerical vector with numbers which are associated with wavelet levels. The last number is associated with the smooth level. Each number determines the number of coefficient used per level. The selection follows a specific scheme. |
Aggregation |
[1:Scales] Numerical vector carrying numbers whose index is associated with the wavelet level. The numbers indicate the number of time in points used for aggregation from the original time series. |
future_point |
Numerical value carrying one step forecast. |
Quirin Stier
Aussem, A., Campbell, J., and Murtagh, F. Waveletbased Feature Extraction and Decomposition Strategies for Financial Forecasting. International Journal of Computational Intelligence in Finance, 6:5–12, 1998.
Renaud, O., Starck, J.-L., and Murtagh, F. Prediction based on a Multiscale De- composition. International Journal of Wavelets, Multiresolution and Information Processing, 1(2):217–232. doi:10.1142/S0219691303000153, 2003.
Murtagh, F., Starck, J.-L., and Renaud, O. On Neuro-Wavelet Modeling. Decision Support Systems, 37(4):475–484. doi:10.1016/S0167-9236(03)00092-7, 2004.
Renaud, O., Starck, J.-L., and Murtagh, F. Wavelet-based combined Signal Filter- ing and Prediction. IEEE Transactions on Systems, Man, and Cybernetics, Part B (Cybernetics), 35(6):1241–1251. doi:10.1109/TSMCB.2005.850182, 2005.
data(AirPassengers) len_data = length(array(AirPassengers)) ccps = c(1,1,1) agg_per_lvl = c(2,4) # Decomposition dec_res <- decomposition(as.vector(AirPassengers), Aggregation = agg_per_lvl) # Training trs_res <- training(dec_res$UnivariateData, dec_res$WaveletCoefficients, dec_res$SmoothCoefficients, dec_res$Scales, ccps, agg_per_lvl) arr_future_points = trs_res$points_in_future matrix = trs_res$lsmatrix # Optimization method weights = regression_lsm_optimization(arr_future_points, matrix) # Forecast scheme = prediction_scheme(dec_res$WaveletCoefficients, dec_res$SmoothCoefficients, ccps, agg_per_lvl) forecast = weightsdata(AirPassengers) len_data = length(array(AirPassengers)) ccps = c(1,1,1) agg_per_lvl = c(2,4) # Decomposition dec_res <- decomposition(as.vector(AirPassengers), Aggregation = agg_per_lvl) # Training trs_res <- training(dec_res$UnivariateData, dec_res$WaveletCoefficients, dec_res$SmoothCoefficients, dec_res$Scales, ccps, agg_per_lvl) arr_future_points = trs_res$points_in_future matrix = trs_res$lsmatrix # Optimization method weights = regression_lsm_optimization(arr_future_points, matrix) # Forecast scheme = prediction_scheme(dec_res$WaveletCoefficients, dec_res$SmoothCoefficients, ccps, agg_per_lvl) forecast = weights
This function computes the weights for the autoregression depending on the given wavelet decomposition. It uses ordinary least square method for optimizing a linear equation system.
regression_lsm_optimization(points_in_future, lsmatrix)regression_lsm_optimization(points_in_future, lsmatrix)
points_in_future |
n many values of the time series, for which there is an equation from a prediction scheme. |
lsmatrix |
Matrix carrying predictive equations associated with a specific value of the time series. |
List of
weights |
Array of weights carrying the solution for a matrix problem, which was solves with ordinary least squares. |
Quirin Stier
Aussem, A., Campbell, J., and Murtagh, F. Waveletbased Feature Extraction and Decomposition Strategies for Financial Forecasting. International Journal of Computational Intelligence in Finance, 6:5–12, 1998.
Renaud, O., Starck, J.-L., and Murtagh, F. Prediction based on a Multiscale De- composition. International Journal of Wavelets, Multiresolution and Information Processing, 1(2):217–232. doi:10.1142/S0219691303000153, 2003.
Murtagh, F., Starck, J.-L., and Renaud, O. On Neuro-Wavelet Modeling. Decision Support Systems, 37(4):475–484. doi:10.1016/S0167-9236(03)00092-7, 2004.
Renaud, O., Starck, J.-L., and Murtagh, F. Wavelet-based combined Signal Filter- ing and Prediction. IEEE Transactions on Systems, Man, and Cybernetics, Part B (Cybernetics), 35(6):1241–1251. doi:10.1109/TSMCB.2005.850182, 2005.
data(AirPassengers) len_data = length(array(AirPassengers)) ccps = c(1,1,1) agg_per_lvl = c(2,4) # Decomposition dec_res <- decomposition(as.vector(AirPassengers), Aggregation = agg_per_lvl) # Training trs_res <- training(dec_res$UnivariateData, dec_res$WaveletCoefficients, dec_res$SmoothCoefficients, dec_res$Scales, ccps, agg_per_lvl) arr_future_points = trs_res$points_in_future matrix = trs_res$lsmatrix # Optimization method weights = regression_lsm_optimization(arr_future_points, matrix) # Forecast scheme = prediction_scheme(dec_res$WaveletCoefficients, dec_res$SmoothCoefficients, ccps, agg_per_lvl) forecast = weightsdata(AirPassengers) len_data = length(array(AirPassengers)) ccps = c(1,1,1) agg_per_lvl = c(2,4) # Decomposition dec_res <- decomposition(as.vector(AirPassengers), Aggregation = agg_per_lvl) # Training trs_res <- training(dec_res$UnivariateData, dec_res$WaveletCoefficients, dec_res$SmoothCoefficients, dec_res$Scales, ccps, agg_per_lvl) arr_future_points = trs_res$points_in_future matrix = trs_res$lsmatrix # Optimization method weights = regression_lsm_optimization(arr_future_points, matrix) # Forecast scheme = prediction_scheme(dec_res$WaveletCoefficients, dec_res$SmoothCoefficients, ccps, agg_per_lvl) forecast = weights
This function creates a one step forecast using an autoregression method. The ccps parameter controls the number of coefficients chosen for each wavelet and smooth part level individually.
regression_one_step(UnivariateData, CoefficientCombination, Aggregation)regression_one_step(UnivariateData, CoefficientCombination, Aggregation)
UnivariateData |
[1:n] Numerical vector with n values. |
CoefficientCombination |
[1:Scales+1] Numerical vector with numbers which are associated with wavelet levels. The last number is associated with the smooth level. Each number determines the number of coefficient used per level. The selection follows a specific scheme. |
Aggregation |
[1:Scales] Numerical vector carrying numbers whose index is associated with the wavelet level. The numbers indicate the number of time in points used for aggregation from the original time series. |
forecast |
Numerical value with one step forecast |
Quirin Stier
Aussem, A., Campbell, J., and Murtagh, F. Waveletbased Feature Extraction and Decomposition Strategies for Financial Forecasting. International Journal of Computational Intelligence in Finance, 6:5–12, 1998.
Renaud, O., Starck, J.-L., and Murtagh, F. Prediction based on a Multiscale De- composition. International Journal of Wavelets, Multiresolution and Information Processing, 1(2):217–232. doi:10.1142/S0219691303000153, 2003.
Murtagh, F., Starck, J.-L., and Renaud, O. On Neuro-Wavelet Modeling. Decision Support Systems, 37(4):475–484. doi:10.1016/S0167-9236(03)00092-7, 2004.
Renaud, O., Starck, J.-L., and Murtagh, F. Wavelet-based combined Signal Filter- ing and Prediction. IEEE Transactions on Systems, Man, and Cybernetics, Part B (Cybernetics), 35(6):1241–1251. doi:10.1109/TSMCB.2005.850182, 2005.
data(AirPassengers) len_data = length(as.vector(array(AirPassengers))) forecast = regression_one_step(as.vector(AirPassengers)[1:(len_data-1)], c(1,1,1), c(2,4))data(AirPassengers) len_data = length(as.vector(array(AirPassengers))) forecast = regression_one_step(as.vector(AirPassengers)[1:(len_data-1)], c(1,1,1), c(2,4))
This function computes a rolling forecasting origin for one- or multi-step forecasts with a specific method. Multi step forecasts are computed recursively with the one step forecast method.
rolling_window(UnivariateData, CoefficientCombination, Aggregation, Horizon = 2, Window = 3, Method = "r", NumClusters = 1)rolling_window(UnivariateData, CoefficientCombination, Aggregation, Horizon = 2, Window = 3, Method = "r", NumClusters = 1)
UnivariateData |
[1:n] Numerical vector with n values. |
CoefficientCombination |
[1:Scales+1] Numerical vector with numbers which are associated with wavelet levels. The last number is associated with the smooth level. Each number determines the number of coefficient used per level. The selection follows a specific scheme. |
Aggregation |
[1:Scales] Numerical vector carrying numbers whose index is associated with the wavelet level. The numbers indicate the number of time in points used for aggregation from the original time series. |
Horizon |
Number indicating horizon for forecast from 1 to horizon. |
Window |
Number indicating how many points are used for cross validation. |
Method |
String indicating which method to use. Available methods: 'r' = Autoregression. 'nn' = Neural Network. |
NumClusters |
Number of clusters used for parallel computing. |
Thus, h-step forecast for h = 1,..., horizon for window_size many steps can be computed. The forecasting method can be an autoregression or a neural network (multilayer perceptron). The CoefficientCombination parameter controls the number of coefficients chosen for each wavelet and smooth part level individually. The NumClusters parameter determines the number of cluster used for parallel computation. NumClusters = 1 performs a non parallel version. NumClusters is constrained by the maximum number of clusters available minus one to prevent the machine to be overchallenged.
List of
Error |
[1:Window,1:Horizon] Numerical Matrix with 'Window' many row entries indicating one time point with 'Horizon' many forecast errors. |
Forecast |
[1:Window,1:Horizon] Numerical Matrix with 'Window' many row entries indicating one time point with 'Horizon' many forecasts. |
Quirin Stier
Hyndman, R. and Athanasopoulos, G. Forecasting: principles and practice. OTexts, 3 edition. 2018.
data(AirPassengers) res = rolling_window(as.vector(array(AirPassengers)), c(10,10,10), c(2,4), Horizon = 2, Window = 3, Method = "r", NumClusters = 1) Error = res$Error Forecast = res$Forecastdata(AirPassengers) res = rolling_window(as.vector(array(AirPassengers)), c(10,10,10), c(2,4), Horizon = 2, Window = 3, Method = "r", NumClusters = 1) Error = res$Error Forecast = res$Forecast
This function creates a single step for a rolling forecasting origin for a specifc one step forecast method. Multi step forecasts are computed recursively with the one step forecast method. Thus, h-step forecast for h = 1,..., horizon for window_size many steps can be computed. The forecasting method can be an autoregression or a neural network (multilayer perceptron). The ccps parameter controls the number of coefficients chosen for each wavelet and smooth part level individually.
rolling_window_single( i, data, ccps, agg_per_lvl, horizon = 14, window_size = 365, method = "r" )rolling_window_single( i, data, ccps, agg_per_lvl, horizon = 14, window_size = 365, method = "r" )
i |
number indicating index for parallel computation. |
data |
Time series with n values. |
ccps |
Vector with numbers which are associated with wavelet and smooth levels from decomposition. The last number is associated with the smooth part level. The preceding numbers are associated with the wavelet levels which are ordered increasingly. Each number determines the number of coefficient used per level. The coefficient selection follows a fixed scheme. |
agg_per_lvl |
Vector carrying numbers whose index is associated with the wavelet level. The numbers indicate the number of time in points used for aggregation from the original time series. |
horizon |
Number indicating horizon for forecast from 1 to horizon. |
window_size |
Number indicating how many points are used to create cross validation. |
method |
String indicating which method to use (r = Autoregression, nn = Neural Network). |
List of parameter with a 2D matrix of the forecast error.
Quirin Stier
This function computes the input for the training phase required for one step forecasts. This computational step is required for all one step forecast procedures contained in this package.
training(UnivariateData, WaveletCoefficients, SmoothCoefficients, Scales, CoefficientCombination, Aggregation)training(UnivariateData, WaveletCoefficients, SmoothCoefficients, Scales, CoefficientCombination, Aggregation)
UnivariateData |
[1:n] Numerical vector with n values. |
WaveletCoefficients |
[Scales, n] Matrix with 'Scales' many wavelet scales row-wise with n columns corresponding to the time domain of a time series. |
SmoothCoefficients |
[Scales, n] Matrix with 'Scales' many smooth approximation scales row-wise with n columns corresponding to the time domain of a time series. |
Scales |
Number of wavelet levels. |
CoefficientCombination |
[1:Scales+1] Numerical vector with numbers which are associated with wavelet levels. The last number is associated with the smooth level. Each number determines the number of coefficient used per level. The selection follows a specific scheme. |
Aggregation |
[1:Scales] Numerical vector carrying numbers whose index is associated with the wavelet level. The numbers indicate the number of time in points used for aggregation from the original time series. |
points_in_future |
n many values of the time series, for which there is an equation from a prediction scheme. |
lsmatrix |
Matrix carrying predictive equations associated with a specific value of the time series. |
Quirin Stier
Aussem, A., Campbell, J., and Murtagh, F. Waveletbased Feature Extraction and Decomposition Strategies for Financial Forecasting. International Journal of Computational Intelligence in Finance, 6:5–12, 1998.
Renaud, O., Starck, J.-L., and Murtagh, F. Prediction based on a Multiscale De- composition. International Journal of Wavelets, Multiresolution and Information Processing, 1(2):217–232. doi:10.1142/S0219691303000153, 2003.
Murtagh, F., Starck, J.-L., and Renaud, O. On Neuro-Wavelet Modeling. Decision Support Systems, 37(4):475–484. doi:10.1016/S0167-9236(03)00092-7, 2004.
Renaud, O., Starck, J.-L., and Murtagh, F. Wavelet-based combined Signal Filter- ing and Prediction. IEEE Transactions on Systems, Man, and Cybernetics, Part B (Cybernetics), 35(6):1241–1251. doi:10.1109/TSMCB.2005.850182, 2005.
data(AirPassengers) len_data = length(array(AirPassengers)) ccps = c(1,1,1) agg_per_lvl = c(2,4) # Decomposition dec_res <- decomposition(as.vector(AirPassengers), Aggregation = agg_per_lvl) # Training trs_res <- training(dec_res$UnivariateData, dec_res$WaveletCoefficients, dec_res$SmoothCoefficients, dec_res$Scales, ccps, agg_per_lvl) arr_future_points = trs_res$points_in_future matrix = trs_res$lsmatrix # Optimization method weights = regression_lsm_optimization(arr_future_points, matrix) # Forecast scheme = prediction_scheme(dec_res$WaveletCoefficients, dec_res$SmoothCoefficients, ccps, agg_per_lvl) forecast = weightsdata(AirPassengers) len_data = length(array(AirPassengers)) ccps = c(1,1,1) agg_per_lvl = c(2,4) # Decomposition dec_res <- decomposition(as.vector(AirPassengers), Aggregation = agg_per_lvl) # Training trs_res <- training(dec_res$UnivariateData, dec_res$WaveletCoefficients, dec_res$SmoothCoefficients, dec_res$Scales, ccps, agg_per_lvl) arr_future_points = trs_res$points_in_future matrix = trs_res$lsmatrix # Optimization method weights = regression_lsm_optimization(arr_future_points, matrix) # Forecast scheme = prediction_scheme(dec_res$WaveletCoefficients, dec_res$SmoothCoefficients, ccps, agg_per_lvl) forecast = weights
This function decomposes a time series in its wavelet and smooth coefficients using the redundant Haar wavelet transform.
wavelet_decomposition(UnivariateData, Aggregation = c(2, 4, 8, 16, 32), Threshold="hard", Lambda=0.05)wavelet_decomposition(UnivariateData, Aggregation = c(2, 4, 8, 16, 32), Threshold="hard", Lambda=0.05)
UnivariateData |
[1:n] Numerical vector with n time series values |
Aggregation |
[1:Scales] Numerical vector of length 'Scales' carrying numbers whose index is associated with the wavelet level. The numbers indicate the number of values used for aggregation from the original time series. |
Threshold |
Character indicating if Thresholding is done on the wavelet decomposition or not. Default: Threshold="hard". Possible entries: Threshold="hard" for hard thresholding. Threshold="soft" for soft thresholding. Any other input indicates no thresholding. |
Lambda |
Numeric value indicating the threshold for computing a hard or soft threshold on the wavelet decomposition. |
The resulting wavelet and smooth coefficients are stored in so called wavelet and smooth part levels. The smooth part level is created from the original times series by aggregation (average). This makes the times series in some sense smoother, hence the naming. Each individual smooth part level can be created from the original time series by aggregating over different number of values. The different smooth part levels are ordered, so that the number of values used for aggregation are ascending. A dyadic scheme is recommended (increasing sequences of the power of two). The dyadic scheme for 5 levels would require agg_per_lvl = c(2, 4, 8, 16, 32). So the first smooth part level would be the average of two points of the original time series, the second smooth part level would be the average of four points, and so on. This averaging is applied asymmetrical. That means, that the result of the average of a sequence of points is obtained for the last point in time of that sequence. So each smooth part level starts with a certain offset, since no average can be obtained for the first particular points in time. The wavelet levels are the differences between the original time series and the smooth levels. The first wavelet level is the difference of the original time series and the first smooth part level. The second wavelet level is the difference of the first and second smooth part level and so on.
List of
UnivariateData |
[1:n] Numerical vector with n time series values. |
WaveletCoefficients |
[Scales, n] Matrix with 'Scales' many wavelet scales row-wise with n columns corresponding to the time domain of a time series. |
SmoothCoefficients |
[Scales, n] Matrix with 'Scales' many smooth approximation scales row-wise with n columns corresponding to the time domain of a time series. |
Scales |
Number of wavelet levels. |
Quirin Stier
Aussem, A., Campbell, J., and Murtagh, F. Waveletbased Feature Extraction and Decomposition Strategies for Financial Forecasting. International Journal of Computational Intelligence in Finance, 6,5-12, 1998.
Renaud, O., Starck, J.-L., and Murtagh, F. Prediction based on a Multiscale De- composition. International Journal of Wavelets, Multiresolution and Information Processing, 1(2):217-232. doi:10.1142/S0219691303000153, 2003.
Murtagh, F., Starck, J.-L., and Renaud, O. On Neuro-Wavelet Modeling. Decision Support Systems, 37(4):475-484. doi:10.1016/S0167-9236(03)00092-7, 2004.
Renaud, O., Starck, J.-L., and Murtagh, F. Wavelet-based combined Signal Filter- ing and Prediction. IEEE Transactions on Systems, Man, and Cybernetics, Part B (Cybernetics), 35(6):1241-1251. doi:10.1109/TSMCB.2005.850182, 2005.
data(AirPassengers) plot(AirPassengers, type = "l", col = "black") UnivariateData = as.vector(array(AirPassengers)) dec_res = wavelet_decomposition(UnivariateData, Aggregation = c(2,4)) plot(dec_res$SmoothCoefficients[2,4:length(dec_res$SmoothCoefficients[2,])], type = "l", col = "blue") lines(array(AirPassengers)[4:length(dec_res$SmoothCoefficients[2,])], col = "black")data(AirPassengers) plot(AirPassengers, type = "l", col = "black") UnivariateData = as.vector(array(AirPassengers)) dec_res = wavelet_decomposition(UnivariateData, Aggregation = c(2,4)) plot(dec_res$SmoothCoefficients[2,4:length(dec_res$SmoothCoefficients[2,])], type = "l", col = "blue") lines(array(AirPassengers)[4:length(dec_res$SmoothCoefficients[2,])], col = "black")
This function delivers the required wavelet and smooth coefficients from the decomposition based on a prediction scheme.
wavelet_prediction_equation(WaveletCoefficients, SmoothCoefficients, CoefficientCombination, Aggregation)wavelet_prediction_equation(WaveletCoefficients, SmoothCoefficients, CoefficientCombination, Aggregation)
WaveletCoefficients |
[Scales, n] Matrix with 'Scales' many wavelet scales row-wise with n columns corresponding to the time domain of a time series. |
SmoothCoefficients |
[Scales, n] Matrix with 'Scales' many smooth approximation scales row-wise with n columns corresponding to the time domain of a time series. |
CoefficientCombination |
[1:Scales+1] Numerical vector with numbers which are associated with wavelet levels. The last number is associated with the smooth level. Each number determines the number of coefficient used per level. The selection follows a specific scheme. |
Aggregation |
[1:Scales] Numerical vector carrying numbers whose index is associated with the wavelet level. The numbers indicate the number of time in points used for aggregation from the original time series. |
future_point |
Numerical value carrying one step forecast. |
Quirin Stier
Aussem, A., Campbell, J., and Murtagh, F. Waveletbased Feature Extraction and Decomposition Strategies for Financial Forecasting. International Journal of Computational Intelligence in Finance, 6,5-12, 1998.
Renaud, O., Starck, J.-L., and Murtagh, F. Prediction based on a Multiscale De- composition. International Journal of Wavelets, Multiresolution and Information Processing, 1(2):217-232. doi:10.1142/S0219691303000153, 2003.
Murtagh, F., Starck, J.-L., and Renaud, O. On Neuro-Wavelet Modeling. Decision Support Systems, 37(4):475-484. doi:10.1016/S0167-9236(03)00092-7, 2004.
Renaud, O., Starck, J.-L., and Murtagh, F. Wavelet-based combined Signal Filter- ing and Prediction. IEEE Transactions on Systems, Man, and Cybernetics, Part B (Cybernetics), 35(6):1241-1251. doi:10.1109/TSMCB.2005.850182, 2005.
data(AirPassengers) len_data = length(array(AirPassengers)) CoefficientCombination = c(1,1,1) Aggregation = c(2,4) UnivariateData = as.vector(AirPassengers) # Decomposition dec_res <- wavelet_decomposition(UnivariateData, Aggregation) # Training trs_res <- wavelet_training_equations(UnivariateData, dec_res$WaveletCoefficients, dec_res$SmoothCoefficients, dec_res$Scales, CoefficientCombination, Aggregation) arr_future_points = trs_res$points_in_future matrix = trs_res$lsmatrix # Optimization method weights = mrf_regression_lsm_optimization(arr_future_points, matrix) # Forecast scheme = wavelet_prediction_equation(dec_res$WaveletCoefficients, dec_res$SmoothCoefficients, CoefficientCombination, Aggregation) forecast = weightsdata(AirPassengers) len_data = length(array(AirPassengers)) CoefficientCombination = c(1,1,1) Aggregation = c(2,4) UnivariateData = as.vector(AirPassengers) # Decomposition dec_res <- wavelet_decomposition(UnivariateData, Aggregation) # Training trs_res <- wavelet_training_equations(UnivariateData, dec_res$WaveletCoefficients, dec_res$SmoothCoefficients, dec_res$Scales, CoefficientCombination, Aggregation) arr_future_points = trs_res$points_in_future matrix = trs_res$lsmatrix # Optimization method weights = mrf_regression_lsm_optimization(arr_future_points, matrix) # Forecast scheme = wavelet_prediction_equation(dec_res$WaveletCoefficients, dec_res$SmoothCoefficients, CoefficientCombination, Aggregation) forecast = weights
This function computes the input for the training phase required for one step forecasts. This computational step is required for all one step forecast procedures contained in this package.
wavelet_training_equations(UnivariateData, WaveletCoefficients, SmoothCoefficients, Scales, CoefficientCombination, Aggregation)wavelet_training_equations(UnivariateData, WaveletCoefficients, SmoothCoefficients, Scales, CoefficientCombination, Aggregation)
UnivariateData |
[1:n] Numerical vector with n values. |
WaveletCoefficients |
[Scales, n] Matrix with 'Scales' many wavelet scales row-wise with n columns corresponding to the time domain of a time series. |
SmoothCoefficients |
[Scales, n] Matrix with 'Scales' many smooth approximation scales row-wise with n columns corresponding to the time domain of a time series. |
Scales |
Number of wavelet levels. |
CoefficientCombination |
[1:Scales+1] Numerical vector with numbers which are associated with wavelet levels. The last number is associated with the smooth level. Each number determines the number of coefficient used per level. The selection follows a specific scheme. |
Aggregation |
[1:Scales] Numerical vector carrying numbers whose index is associated with the wavelet level. The numbers indicate the number of time in points used for aggregation from the original time series. |
points_in_future |
n many values of the time series, for which there is an equation from a prediction scheme. |
lsmatrix |
Matrix carrying predictive equations associated with a specific value of the time series. |
Quirin Stier
Aussem, A., Campbell, J., and Murtagh, F. Waveletbased Feature Extraction and Decomposition Strategies for Financial Forecasting. International Journal of Computational Intelligence in Finance, 6,5-12, 1998.
Renaud, O., Starck, J.-L., and Murtagh, F. Prediction based on a Multiscale De- composition. International Journal of Wavelets, Multiresolution and Information Processing, 1(2):217-232. doi:10.1142/S0219691303000153, 2003.
Murtagh, F., Starck, J.-L., and Renaud, O. On Neuro-Wavelet Modeling. Decision Support Systems, 37(4):475-484. doi:10.1016/S0167-9236(03)00092-7, 2004.
Renaud, O., Starck, J.-L., and Murtagh, F. Wavelet-based combined Signal Filter- ing and Prediction. IEEE Transactions on Systems, Man, and Cybernetics, Part B (Cybernetics), 35(6):1241-1251. doi:10.1109/TSMCB.2005.850182, 2005.
data(AirPassengers) len_data = length(array(AirPassengers)) CoefficientCombination = c(1,1,1) Aggregation = c(2,4) UnivariateData = as.vector(AirPassengers) # Decomposition dec_res <- wavelet_decomposition(UnivariateData, Aggregation) # Training trs_res <- wavelet_training_equations(UnivariateData, dec_res$WaveletCoefficients, dec_res$SmoothCoefficients, dec_res$Scales, CoefficientCombination, Aggregation) arr_future_points = trs_res$points_in_future matrix = trs_res$lsmatrix # Optimization method weights = mrf_regression_lsm_optimization(arr_future_points, matrix) # Forecast scheme = wavelet_prediction_equation(dec_res$WaveletCoefficients, dec_res$SmoothCoefficients, CoefficientCombination, Aggregation) forecast = weightsdata(AirPassengers) len_data = length(array(AirPassengers)) CoefficientCombination = c(1,1,1) Aggregation = c(2,4) UnivariateData = as.vector(AirPassengers) # Decomposition dec_res <- wavelet_decomposition(UnivariateData, Aggregation) # Training trs_res <- wavelet_training_equations(UnivariateData, dec_res$WaveletCoefficients, dec_res$SmoothCoefficients, dec_res$Scales, CoefficientCombination, Aggregation) arr_future_points = trs_res$points_in_future matrix = trs_res$lsmatrix # Optimization method weights = mrf_regression_lsm_optimization(arr_future_points, matrix) # Forecast scheme = wavelet_prediction_equation(dec_res$WaveletCoefficients, dec_res$SmoothCoefficients, CoefficientCombination, Aggregation) forecast = weights