Mandelbrot's Stochastic Time Series Models

Watkins, Nicholas W.

doi:10.1029/2019ea000598

Cited by 5 publications

(4 citation statements)

References 56 publications

(67 reference statements)

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“…By applying a fractional order difference filter, the residual obtain is uncorrelated with lags of its variables. Mandelbrot [38] suggested the use of range over standard deviation R/S statistics called ''rescaled range'', which used by hydrologist Harold Hurst [39] in the Hurst exponent. The main concept of R/S analysis is to analyze rescaled cumulative deviation from the mean.…”

Section: B Arfima Modelmentioning

confidence: 99%

Fractional Neuro-Sequential ARFIMA-LSTM for Financial Market Forecasting

et al. 2020

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Forecasting of fast fluctuated and high-frequency financial data is always a challenging problem in the field of economics and modelling. In this study, a novel hybrid model with the strength of fractional order derivative is presented with their dynamical features of deep learning, long-short term memory (LSTM) networks, to predict the abrupt stochastic variation of the financial market. Stock market prices are dynamic, highly sensitive, nonlinear and chaotic. There are different techniques for forecast prices in the time-variant domain and due to variability and uncertain behavior in stock prices, traditional methods, such as data mining, statistical approaches, and non-deep neural networks models are not suited for prediction and generalized forecasting stock prices. While autoregressive fractional integrated moving average (ARFIMA) model provides a flexible tool for classes of long-memory models. The advancement of machine learningbased deep non-linear modelling confirms that the hybrid model efficiently extracts profound features and model non-linear functions. LSTM networks are a special kind of recurrent neural network (RNN) that map sequences of input observations to output observations with capabilities of long-term dependencies. A novel ARFIMA-LSTM hybrid recurrent network is presented in which ARFIMA model-based filters having the linear tendencies better than ARIMA model in the data and passes the residual to the LSTM model that captures nonlinearity in the residual values with the help of exogenous dependent variables. The model not only minimizes the volatility problem but also overcome the over fitting problem of neural networks. The model is evaluated using PSX company data of the stock market based on RMSE, MSE and MAPE along with a comparison of ARIMA, LSTM model and generalized regression radial basis neural network (GRNN) ensemble method independently. The forecasting performance indicates the effectiveness of the proposed AFRIMA-LSTM hybrid model to improve around 80% accuracy on RMSE as compared to traditional forecasting counterparts.

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Section: B Arfima Modelmentioning

confidence: 99%

Fractional Neuro-Sequential ARFIMA-LSTM for Financial Market Forecasting

et al. 2020

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“…where H L (denoted by "H" in their papers) is not the usual Hurst exponent H of Mandelbrot's fBm and fGn (e.g. [13,38]), which we used above.…”

Section: Fractionally Integrating the Standard Fractional Non-markovi...mentioning

confidence: 99%

On Generalized Langevin Dynamics and the Modelling of Global Mean Temperature

Watkins

Chapman

Chechkin

et al. 2021

Springer Proceedings in Complexity

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Climate science employs a hierarchy of models, trading the tractability of simplified energy balance models (EBMs) against the detail of Global Circulation Models. Since the pioneering work of Hasselmann, stochastic EBMs have allowed treatment of climate fluctuations and noise. However, it has recently been claimed that observations motivate heavy-tailed temporal response functions in global mean temperature to perturbations. Our complementary approach exploits the correspondence between Hasselmann's EBM and the original mean-reverting stochastic model in physics, Langevin's equation of 1908. We propose mapping a model well known in statistical mechanics, the Mori-Kubo Generalised Langevin Equation (GLE) to generalise the Hasselmann EBM. If present, long range memory then simplifies the GLE to a fractional Langevin equation (FLE). We describe the corresponding EBMs that map to the GLE and FLE, briefly discuss their solutions, and relate them to Lovejoy et al's new Fractional Energy Balance Model (FEBE).

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“…To establish the relationship between the FIFHE and Lovejoy's FEBE we first note that the H defined in [24] is not that of Mandelbrot's fractional Brownian motion(e.g. [13,36]), but is instead Mandelbrot's H − 1/2. We can use the wellknown [36] relationship H = 1/2 + d to identify Lovejoy's H as the memory parameter d used in much of the mathematical literature on stochastic processes which exhibit LRD (e.g.…”

Section: Fractionally Integrating the Standard Fractional Non-markovi...mentioning

confidence: 99%

“…[13,36]), but is instead Mandelbrot's H − 1/2. We can use the wellknown [36] relationship H = 1/2 + d to identify Lovejoy's H as the memory parameter d used in much of the mathematical literature on stochastic processes which exhibit LRD (e.g. [13]).…”

Section: Fractionally Integrating the Standard Fractional Non-markovi...mentioning

confidence: 99%

On Generalized Langevin Dynamics and the Modelling of Global Mean Temperature

Watkins¹,

Chapman²,

Chechkin³

et al. 2020

Preprint

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Climate science employs a hierarchy of models, trading the tractability of simplified energy balance models (EBMs) against the detail of Global Circulation Models. Since the pioneering work of Hasselmann, stochastic EBMs have allowed treatment of climate fluctuations and noise. However, it has recently been claimed that observations motivate heavy-tailed temporal response functions in global mean temperature to perturbations. Our complementary approach exploits the correspondence between Hasselmanns EBM and the original mean-reverting stochastic model in physics, Langevins equation of 1908. We propose mapping a model well known in statistical mechanics, the Mori-Kubo Generalised Langevin Equation (GLE) to generalise the Hasselmann EBM. If present, long range memory then simplifies the GLE to a fractional Langevin equation (FLE). We describe the corresponding EBMs that map to the GLE and FLE, briefly discuss their solutions, and relate them to Lovejoys new Fractional Energy Balance Model.

show abstract

Mandelbrot's Stochastic Time Series Models

Cited by 5 publications

References 56 publications

Fractional Neuro-Sequential ARFIMA-LSTM for Financial Market Forecasting

Fractional Neuro-Sequential ARFIMA-LSTM for Financial Market Forecasting

On Generalized Langevin Dynamics and the Modelling of Global Mean Temperature

On Generalized Langevin Dynamics and the Modelling of Global Mean Temperature

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