# US 11,308,414 B2 · Multi-step ahead forecasting using complex-valued vector autoregregression > Agents: this data is available as JSON from the patentgrep API and MCP server — see https://patentgrep.com/llms.txt - Number: US 11,308,414 B2 - Document ID: US11308414B2 - Status: Granted - Kind: B2 - Application number: 16/157,529 - Filed: 2018-10-11 - Published: 2020-04-16 - Granted: 2022-04-19 - Earliest claimed: 2018-10-11 - Assignees: International Business Machines Corporation - Applicants: International Business Machines Corporation - Inventors: Hiroshi Kajino - Web page: https://patentgrep.com/patent/US11308414B2 ## CPC - G06N 7/08: using chaos models or non-linear system models - G06F 11/0736: {in functional embedded systems, i.e. in a data processing system designed as a combination of hardware and software dedicated to performing a certain function (testing or monitoring of automated control systems G05B23/02)} - G06F 11/0751: {Error or fault detection not based on redundancy (power supply failures G06F1/30; network fault management H04L41/06)} - G06F 11/0787: {Storage of error reports, e.g. persistent data storage, storage using memory protection} - G06N 5/00: Computing arrangements using knowledge-based models - G06N 20/00: Machine learning - G06N 5/01: Dynamic search techniques; Heuristics; Dynamic trees; Branch-and-bound - G06N: COMPUTING ARRANGEMENTS BASED ON SPECIFIC COMPUTATIONAL MODELS - G06F: ELECTRIC DIGITAL DATA PROCESSING (computer systems based on specific computational models G06N) ## Abstract Computer-implemented methods, computer program products, and systems are provided for multi-step ahead forecasting. A method includes configuring, by a processor device, a Vector Autoregression (VAR) model to generate a multi-step-ahead forecast based on previous observations. The previous observations are predictors and the multi-step-ahead forecast is a response to the predictors. The method further includes training, by the processor device, the VAR model using complex-valued weight parameters to avoid a training result relating to any of a divergence and a convergence to zero. ## Claims ### Claim 1 A computer-implemented method for multi-step ahead forecasting, comprising: configuring, by a processor device, a Vector Autoregression (VAR) model to generate a multi-step-ahead forecast based on previous observations, where the previous observations are predictors and the multi-step-ahead forecast is a response to the predictors; and training, by the processor device, the VAR model using complex-valued weight parameters having real and imaginary values to avoid a training result relating to any of a divergence and a convergence to zero, the imaginary values used as inputs to the VAR model to generate successive predictions while only the real values are output from the VAR model without the imaginary values as the successive predictions. ### Claim 2 (depends on 1) The computer-implemented method of claim 1, wherein the multi-step-ahead forecast is generated further based on a multi-step cumulative error. ### Claim 3 (depends on 1) The computer-implemented method of claim 1, wherein said training step comprises learning the VAR model using a multi-step cumulative error as an objective function of the VAR model. ### Claim 4 (depends on 3) The computer-implemented method of claim 3, wherein the multi-step cumulative error accumulates an error of at least one variable of the VAR model across multiple time steps. ### Claim 5 (depends on 3) The computer-implemented method of claim 3, wherein said learning step updates the complex-valued weight parameters to a negative gradient of the multi-step cumulative error. ### Claim 6 (depends on 1) The computer-implemented method of claim 1, wherein an output of the VAR model is taken from a real portion of a complex-valued output from the VAR model, while ignoring an imaginary portion of the complex-valued output from the VAR model. ### Claim 7 (depends on 1) The computer-implemented method of claim 1, further comprising performing an inference step to generate a final version of the multi-step-ahead forecast based on a set of previous observations by extracting only a real portion of the multi-step ahead forecast without an imaginary portion. ### Claim 8 (depends on 1) The computer-implemented method of claim 1, further comprising controlling an operation of an assembly line machine to avoid an impending predicted failure, responsive to the multi-step-ahead forecast. ### Claim 9 (depends on 1) The computer-implemented method of claim 1, wherein the VAR model is trained using a loss function having complex-valued weight parameters with an imaginary term. ### Claim 10 (depends on 1) The computer-implemented method of claim 1, wherein each of the predictors include a respective real portion and a respective imaginary portion, and wherein the multi-step ahead forecast is based only on the respective real portion of the predictors. ### Claim 11 A computer program product for multi-step ahead forecasting, the computer program product comprising a non-transitory computer readable storage medium having program instructions embodied therewith, the program instructions executable by a computer to cause the computer to perform a method comprising: configuring, by a processor device, a Vector Autoregression (VAR) model to generate a multi-step-ahead forecast based on previous observations, where the previous observations are predictors and the multi-step-ahead forecast is a response to the predictors; and training, by the processor device, the VAR model using complex-valued weight parameters having real and imaginary values to avoid a training result relating to any of a divergence and a convergence to zero, the imaginary values used as inputs to the VAR model to generate successive predictions while only the real values are output from the VAR model without the imaginary values as multi-step-ahead forecasts as the successive predictions. ### Claim 12 (depends on 11) The computer program product of claim 11, wherein the multi-step-ahead forecast is generated further based on a multi-step cumulative error. ### Claim 13 (depends on 11) The computer program product of claim 11, wherein said training step comprises learning the VAR model using a multi-step cumulative error as an objective function of the VAR model. ### Claim 14 (depends on 13) The computer program product of claim 13, wherein the multi-step cumulative error accumulates an error of at least one variable of the VAR model across multiple time steps. ### Claim 15 (depends on 13) The computer program product of claim 13, wherein said learning step updates the complex-valued weight parameters to a negative gradient of the multi-step cumulative error. ### Claim 16 (depends on 11) The computer program product of claim 11, wherein an output of the VAR model is taken from a real portion of a complex-valued output from the VAR model, while ignoring an imaginary portion of the complex-valued output from the VAR model. ### Claim 17 (depends on 11) The computer program product of claim 11, wherein the method further comprises performing an inference step to generate a final version of the multi-step-ahead forecast based on a set of previous observations by extracting only a real portion of the multi-step ahead forecast without an imaginary portion. ### Claim 18 (depends on 11) The computer program product of claim 11, wherein the method further comprises controlling an operation of an assembly line machine to avoid an impending predicted failure, responsive to the multi-step-ahead forecast. ### Claim 19 (depends on 11) The computer program product of claim 11, wherein the VAR model is trained using a loss function having complex-valued weight parameters with an imaginary term. ### Claim 20 (depends on 11) The computer program product of claim 11, wherein each of the predictors include a respective real portion and a respective imaginary portion, and wherein the multi-step ahead forecast is based only on the respective real portion of the predictors. ### Claim 21 A computer processing system for multi-step ahead forecasting, comprising: a memory device for storing program code; and a processor device for running the program code to configure a Vector Autoregression (VAR) model to generate a multi-step-ahead forecast based on previous observations, where the previous observations are predictors and the multi-step-ahead forecast is a response to the predictors; and train the VAR model using complex-valued weight parameters having real and imaginary values to avoid a training result relating to any of a divergence and a convergence to zero, the imaginary values used as inputs to the VAR model to generate successive predictions while only the real values are output from the VAR model without the imaginary values as multi-step-ahead forecasts as the successive predictions. ### Claim 22 (depends on 21) The computer processing system of claim 21, wherein the computer processing system is comprised in an Advanced Driver-Assistance System, and the processor device is further configured to perform an action on a controlled motor vehicle to avoid one or more obstacles responsive to the multi-step-ahead forecast. ### Claim 23 A computer-implemented method for multi-step-ahead forecasting, comprising: configuring, by a processor device, a Vector Autoregression (VAR) model to generate a multi-step-ahead forecast based on previous observations, where the previous observations are predictors and the multi-step ahead forecast is the response to the predictors; learning, by the processor device, the VAR model using a multi-step cumulative error as an objective function of the VAR model; and training the VAR model using complex-valued weight parameters having real and imaginary values to avoid a training result relating to any of a divergence and a convergence to zero, the imaginary values used as inputs to the VAR model to generate successive predictions while only the real values are output from the VAR model without the imaginary values as multi-step-ahead forecasts as the successive predictions. ### Claim 24 (depends on 23) A non-transitory article of manufacture tangibly embodying a computer readable program which when executed causes a computer to perform the steps of claim 23. ### Claim 25 A computer processing system for multi-step-ahead forecasting, comprising: a memory for storing program code; and a processor device for running the program code to configure a Vector Autoregression (VAR) model to generate a multi-step-ahead forecast based on previous observations, where the previous observations are predictors and the multi-step ahead forecast is the response to the predictors; learn the VAR model using a multi-step cumulative error as an objective function of the VAR model; and training the VAR model using complex-valued weight parameters having real and imaginary values to avoid a training result relating to any of a divergence and a convergence to zero, the imaginary values used as inputs to the VAR model to generate successive predictions while only the real values are output from the VAR model without the imaginary values as multi-step-ahead forecasts as the successive predictions.