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Trend-Adjusted Time Series Models with an Application to Gold Price Forecasting

Sina Kazemdehbashi

TL;DR

The paper tackles volatility in time series forecasting by integrating trend information into the forecast through Trend-Adjusted Time Series (TATS). It decomposes forecasting into a trend-predicting binary classifier and a value forecaster, then uses an adjustment function to align the base forecast with the predicted trend, formalizing conditions under which this yields lower error. Theoretical results show that when the trend-prediction accuracy of the binary model exceeds the trend-detection accuracy of the base forecaster, TATS improves forecasting performance, with an approximate lower bound on error reduction. Empirically, on daily gold-price data, TATS outperforms standard LSTM and Bi-LSTM baselines, and trend-detection accuracy is highlighted as a critical evaluation metric alongside MSE/MAE. The work offers a practical framework for improving forecasts in volatile domains and points to enhancements in trend predictors and adjustment mechanisms for broader applicability.

Abstract

Time series data play a critical role in various fields, including finance, healthcare, marketing, and engineering. A wide range of techniques (from classical statistical models to neural network-based approaches such as Long Short-Term Memory (LSTM)) have been employed to address time series forecasting challenges. In this paper, we reframe time series forecasting as a two-part task: (1) predicting the trend (directional movement) of the time series at the next time step, and (2) forecasting the quantitative value at the next time step. The trend can be predicted using a binary classifier, while quantitative values can be forecasted using models such as LSTM and Bidirectional Long Short-Term Memory (Bi-LSTM). Building on this reframing, we propose the Trend-Adjusted Time Series (TATS) model, which adjusts the forecasted values based on the predicted trend provided by the binary classifier. We validate the proposed approach through both theoretical analysis and empirical evaluation. The TATS model is applied to a volatile financial time series (the daily gold price) with the objective of forecasting the next days price. Experimental results demonstrate that TATS consistently outperforms standard LSTM and Bi-LSTM models by achieving significantly lower forecasting error. In addition, our results indicate that commonly used metrics such as MSE and MAE are insufficient for fully assessing time series model performance. Therefore, we also incorporate trend detection accuracy, which measures how effectively a model captures trends in a time series.

Trend-Adjusted Time Series Models with an Application to Gold Price Forecasting

TL;DR

The paper tackles volatility in time series forecasting by integrating trend information into the forecast through Trend-Adjusted Time Series (TATS). It decomposes forecasting into a trend-predicting binary classifier and a value forecaster, then uses an adjustment function to align the base forecast with the predicted trend, formalizing conditions under which this yields lower error. Theoretical results show that when the trend-prediction accuracy of the binary model exceeds the trend-detection accuracy of the base forecaster, TATS improves forecasting performance, with an approximate lower bound on error reduction. Empirically, on daily gold-price data, TATS outperforms standard LSTM and Bi-LSTM baselines, and trend-detection accuracy is highlighted as a critical evaluation metric alongside MSE/MAE. The work offers a practical framework for improving forecasts in volatile domains and points to enhancements in trend predictors and adjustment mechanisms for broader applicability.

Abstract

Time series data play a critical role in various fields, including finance, healthcare, marketing, and engineering. A wide range of techniques (from classical statistical models to neural network-based approaches such as Long Short-Term Memory (LSTM)) have been employed to address time series forecasting challenges. In this paper, we reframe time series forecasting as a two-part task: (1) predicting the trend (directional movement) of the time series at the next time step, and (2) forecasting the quantitative value at the next time step. The trend can be predicted using a binary classifier, while quantitative values can be forecasted using models such as LSTM and Bidirectional Long Short-Term Memory (Bi-LSTM). Building on this reframing, we propose the Trend-Adjusted Time Series (TATS) model, which adjusts the forecasted values based on the predicted trend provided by the binary classifier. We validate the proposed approach through both theoretical analysis and empirical evaluation. The TATS model is applied to a volatile financial time series (the daily gold price) with the objective of forecasting the next days price. Experimental results demonstrate that TATS consistently outperforms standard LSTM and Bi-LSTM models by achieving significantly lower forecasting error. In addition, our results indicate that commonly used metrics such as MSE and MAE are insufficient for fully assessing time series model performance. Therefore, we also incorporate trend detection accuracy, which measures how effectively a model captures trends in a time series.
Paper Structure (12 sections, 2 theorems, 15 equations, 3 figures, 5 tables)

This paper contains 12 sections, 2 theorems, 15 equations, 3 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Literature Review
  3. Methodology
  4. Value Forecaster and Trend Predictor
  5. Adjustment Function Analysis
  6. $\alpha$ as a small positive value
  7. $\alpha$ as a hyperparameter
  8. Experiment Results
  9. Trend Predictor (Binary Classifiers)
  10. Value Forecaster (Base Time Series Models)
  11. TATS model
  12. Conclusion

Key Result

Proposition 3.1

Let $\alpha$ in the adjustment function (eq:basic-adjustement-definition) be a positive and sufficiently small constant such that $y_{t-1} + \hat{c}_t \alpha \approx y_{t-1}$. Then, the TATS model reduces the forecasting error of its value forecaster if $P(D^B) > P(D^T)$.

Figures (3)

  • Figure 1: Trend-Adjusted Time Series (TATS) Model
  • Figure 2: (a) Both the trend predictor (TP) and value forecaster (VF) predict the trend correctly; (b) VF predicts the trend correctly, but TP predicts incorrectly; (c) Both TP and VF predict the trend incorrectly; (d) VF predicts the trend incorrectly, while TP predicts correctly. In all cases, $\alpha$ is positive and sufficiently small ($y_{t-1} + \hat{c}_t \alpha \simeq y_{t-1}$).
  • Figure 3: The actual time series (real data) and the forecasts generated by two different models in Example 1. Both models have the same MSE; however, Model 1 successfully captures the time series trend (directional movements).

Theorems & Definitions (4)

  • Proposition 3.1
  • proof
  • Corollary 3.2
  • proof