Package 'intradayModel'

intradayModel: Modeling and Forecasting Financial Intraday Signals

Description

This package uses state-of-the-art state-space models to facilitate the modeling, analyzing and forecasting of financial intraday signals. It currently offers a univariate model for intraday trading volume, with new features on intraday volatility and multivariate models in development.

Functions

fit_volume, decompose_volume, forecast_volume, generate_plots

Data

volume_aapl, volume_fdx

Help

For a quick help see the README file: GitHub-README.

Author(s)

Shengjie Xiu, Yifan Yu and Daniel P. Palomar

---

Decompose Intraday Volume into Several Components

Description

This function decomposes the intraday volume into daily, seasonal, and intraday dynamic components according to (Chen et al., 2016). If purpose = “analysis” (aka Kalman smoothing), the optimal components are conditioned on both the past and future observations. Its mathematical expression is $\hat{x}_{\tau} = E[x_{\tau}|\{y_{j}\}_{j=1}^{M}]$, where $M$ is the total number of bins in the dataset.

If purpose = “forecast” (aka Kalman forecasting), the optimal components are conditioned on only the past observations. Its mathematical expression is $\hat{x}_{\tau+1} = E[x_{\tau+1}|\{y_{j}\}_{j=1}^{\tau}]$.

Three measures are used to evaluate the model performance:

• Mean absolute error (MAE): $\frac{1}{M}\sum_{\tau=1}^M|\hat{y}_{\tau} - y_{\tau}|$ ;

• Mean absolute percent error (MAPE): $\frac{1}{M}\sum_{\tau=1}^M\frac{|\hat{y}_{\tau} - y_{\tau}|}{y_{\tau}}$ ;

• Root mean square error (RMSE): $\sqrt{\sum_{\tau=1}^M\frac{\left(\hat{y}_{\tau} - y_{\tau}\right)^2}{M}}$ .

Usage

decompose_volume(purpose, model, data, burn_in_days = 0)

Arguments

purpose	String "analysis"/"forecast". Indicates the purpose of using the provided model.
model	A model object of class "volume_model" from fit_volume().
data	An n_bin * n_day matrix or an xts object storing intraday volume.
burn_in_days	Number of initial days in the burn-in period for forecast. Samples from the first burn_in_days are used to warm up the model and then are discarded.

Value

A list containing the following elements:

• original_signal: A vector of original intraday volume;

• smooth_signal / forecast_signal: A vector of smooth/forecast intraday volume;

• smooth_components /forecast_components: A list of smooth/forecast components: daily, seasonal, intraday dynamic, and residual components.

• error: A list of three error measures: mae, mape, and rmse.

Author(s)

Shengjie Xiu, Yifan Yu and Daniel P. Palomar

References

Chen, R., Feng, Y., and Palomar, D. (2016). Forecasting intraday trading volume: A Kalman filter approach. Available at SSRN 3101695.

Examples

library(intradayModel)
data(volume_aapl)
volume_aapl_training <- volume_aapl[, 1:20]
volume_aapl_testing <- volume_aapl[, 21:50]
model_fit <- fit_volume(volume_aapl_training, fixed_pars = list(a_mu = 0.5, var_mu = 0.05),
                        init_pars = list(a_eta = 0.5))

# analyze training volume
analysis_result <- decompose_volume(purpose = "analysis", model_fit, volume_aapl_training)

# forecast testing volume
forecast_result <- decompose_volume(purpose = "forecast", model_fit, volume_aapl_testing)

# forecast testing volume with burn-in
forecast_result <- decompose_volume(purpose = "forecast", model_fit, volume_aapl[, 1:50],
                             burn_in_days = 20)

---

Fit a Univariate State-Space Model on Intraday Trading Volume

Description

The main function for defining and fitting a univaraite state-space model on intraday trading volume. The model is proposed in (Chen et al., 2016) as

$\mathbf{x}_{\tau+1} = \mathbf{A}_{\tau}\mathbf{x}_{\tau} + \mathbf{w}_{\tau},$

$y_{\tau} = \mathbf{C}\mathbf{x}_{\tau} + \phi_{\tau} + v_\tau,$

where

• $\mathbf{x}_{\tau} = [\eta_{\tau}, \mu_{\tau}]^\top$ is the hidden state vector containing the log daily component and the log intraday dynamic component;

• $\mathbf{A}_{\tau} = \left[\begin{array}{cc}a_{\tau}^{\eta}&0\\0&a^{\mu}\end{array} \right]$ is the state transition matrix with $a_{\tau}^{\eta} = \left\{\begin{array}{cl}a^{\eta} & t = kI, k = 1,2,\dots\\0 & \textrm{otherwise};\end{array}\right.$

• $\mathbf{C} = [1, 1]$ is the observation matrix;

• $\phi_{\tau}$ is the corresponding element from $\boldsymbol{\phi} = [\phi_1,\dots, \phi_I]^\top$, which is the log seasonal component;

• $\mathbf{w}_{\tau} = [\epsilon_{\tau}^{\eta},\epsilon_{\tau}^{\mu}]^\top \sim \mathcal{N}(\mathbf{0}, \mathbf{Q}_{\tau})$ represents the i.i.d. Gaussian noise in the state transition, with a time-varying covariance matrix $\mathbf{Q}_{\tau} = \left[\begin{array}{cc}(\sigma_{\tau}^{\eta})^2&0\\ 0&(\sigma^{\mu})^2\end{array} \right]$ and $\sigma_\tau^{\eta} = \left\{\begin{array}{cl} \sigma^{\eta} & t = kI, k = 1,2,\dots\\0 & \textrm{otherwise};\end{array}\right.$

• $v_\tau \sim \mathcal{N}(0, r)$ is the i.i.d. Gaussian noise in the observation;

• $\mathbf{x}_1$ is the initial state at $\tau = 1$, and it follows $\mathcal{N}(\mathbf{x}_0, \mathbf{V}_0)$.

In the model, $\boldsymbol{\Theta} = \left\{a^{\eta},a^{\mu},\sigma^{\eta},\sigma^{\mu},r,\boldsymbol{\phi}, \mathbf{x}_0, \mathbf{V}_0\right\}$ are treated as parameters. The model is fitted by expectation-maximization (EM) algorithms. The implementation follows (Chen et al., 2016), and the accelerated scheme is provided in (Varadhan and Roland, 2008). The algorithm terminates when maxit is reached or the condition $\|\Delta \boldsymbol{\Theta}_i\| \le \textrm{abstol}$ is satisfied.

Usage

fit_volume(
  data,
  fixed_pars = NULL,
  init_pars = NULL,
  verbose = 0,
  control = NULL
)

Arguments

data	An n_bin * n_day matrix or an xts object storing intraday trading volume.
fixed_pars	A list of parameters' fixed values. The allowed parameters are listed below, "a_eta": $a^{\eta}$ of size 1 ; "a_mu": $a^{\mu}$ of size 1 ; "var_eta": $\sigma^{\eta}$ of size 1 ; "var_mu": $\sigma^{\mu}$ of size 1 ; "r": $r$ of size 1 ; "phi": $\boldsymbol{\phi} = [\phi_1,\dots, \phi_I]^\top$ of size $I$ ; "x0": $\mathbf{x}_0$ of size 2 ; "V0": $\mathbf{V}_0$ of size 2 * 2 .
init_pars	A list of unfitted parameters' initial values. The parameters are the same as fixed_pars. If the user does not assign initial values for the unfitted parameters, default ones will be used.
verbose	An integer specifying the print level of information during the algorithm (default 1). Possible numbers: "0": no output; "1": show the iteration number and $\|\Delta \boldsymbol{\Theta}_i\|$; "2": 1 + show the obtained parameters.
control	A list of control values of EM algorithm: acceleration: TRUE/FALSE indicating whether to use the accelerated EM algorithm (default TRUE); maxit: Maximum number of iterations (default 3000); abstol: Absolute tolerance for parameters' change $\|\Delta \boldsymbol{\Theta}_i\|$ as the stopping criteria (default 1e-4) log_switch: TRUE/FALSE indicating whether to record the history of convergence progress (defalut TRUE).

Value

A list of class "volume_model" with the following elements (if the algorithm converges):

• par: A list of parameters' fitted values.

• init: A list of valid initial values from users.

• par_log: A list of intermediate parameters' values if log_switch = TRUE.

• converged: A list of logical values indicating whether each parameter is fitted.

Author(s)

Shengjie Xiu, Yifan Yu and Daniel P. Palomar

References

Chen, R., Feng, Y., and Palomar, D. (2016). Forecasting intraday trading volume: A Kalman filter approach. Available at SSRN 3101695.

Varadhan, R., and Roland, C. (2008). Simple and globally convergent methods for accelerating the convergence of any EM algorithm. Scandinavian Journal of Statistics, 35(2), 335–353.

Examples

library(intradayModel)
data(volume_aapl)
volume_aapl_training <- volume_aapl[, 1:20]

# fit model with no prior knowledge
model_fit <- fit_volume(volume_aapl_training)

# fit model with fixed_pars and init_pars
model_fit <- fit_volume(volume_aapl_training, fixed_pars = list(a_mu = 0.5, var_mu = 0.05),
                        init_pars = list(a_eta = 0.5))

# fit model with other control options
model_fit <- fit_volume(volume_aapl_training, verbose = 2, 
  control = list(acceleration = FALSE, maxit = 1000, abstol = 1e-4, log_switch = FALSE))

---

Forecast One-bin-ahead Intraday Volume

Description

This function forecasts one-bin-ahead intraday volume. Its mathematical expression is $\hat{y}_{\tau+1} = E[y_{\tau+1}|\{y_{j}\}_{j=1}^{\tau}]$. It is a wrapper of decompose_volume() with purpose = "forecast".

Usage

forecast_volume(model, data, burn_in_days = 0)

Arguments

model	A model object of class "volume_model" from fit_volume().
data	An n_bin * n_day matrix or an xts object storing intraday volume.
burn_in_days	Number of initial days in the burn-in period. Samples from the first burn_in_days are used to warm up the model and then are discarded.

Value

A list containing the following elements:

• original_signal: A vector of original intraday volume;

• forecast_signal: A vector of forecast intraday volume;

• forecast_components: A list of the three forecast components: daily, seasonal, intraday dynamic, and residual components.

• error: A list of three error measures: mae, mape, and rmse.

Author(s)

Shengjie Xiu, Yifan Yu and Daniel P. Palomar

References

Chen, R., Feng, Y., and Palomar, D. (2016). Forecasting intraday trading volume: A Kalman filter approach. Available at SSRN 3101695.

Examples

library(intradayModel)
data(volume_aapl)
volume_aapl_training <- volume_aapl[, 1:20]
volume_aapl_testing <- volume_aapl[, 21:50]
model_fit <- fit_volume(volume_aapl_training, fixed_pars = list(a_mu = 0.5, var_mu = 0.05),
                        init_pars = list(a_eta = 0.5))
                        
# forecast testing volume
forecast_result <- forecast_volume(model_fit, volume_aapl_testing)

# forecast testing volume with burn-in
forecast_result <- forecast_volume(model_fit, volume_aapl[, 1:50], burn_in_days = 20)

---

Plot Analysis and Forecast Result

Description

Generate plots for the analysis and forecast results.

Usage

generate_plots(analysis_forecast_result)

Arguments

analysis_forecast_result

Analysis/forecast result from decompose_volume() or forecast_volume().

Value

A list of patchwork objects:

• components: Plot of components of intraday volume;

• log_components: Plot of components of intraday volume in their log10 scale;

• original_and_smooth / original_and_forecast: Plot of the original and the smooth/forecast intraday volume.

Author(s)

Shengjie Xiu, Yifan Yu and Daniel P. Palomar

Examples

library(intradayModel)
data(volume_aapl)
volume_aapl_training <- volume_aapl[, 1:20]
volume_aapl_testing <- volume_aapl[, 21:50]

# obtain analysis and forecast result
model_fit <- fit_volume(volume_aapl_training, fixed_pars = list(a_mu = 0.5, var_mu = 0.05),
                        init_pars = list(a_eta = 0.5))
analysis_result <- decompose_volume(purpose = "analysis", model_fit, volume_aapl_training)
forecast_result <- forecast_volume(model_fit, volume_aapl_testing)

# plot the analysis and forecast result
generate_plots(analysis_result)
generate_plots(forecast_result)

---

15-min Intraday Volume of AAPL

Description

A 26 * 124 matrix including 15-min trading volume of AAPL from 2019-01-02 to 2019-06-28.

Usage

data(volume_aapl)

Format

A 26 * 124 matrix.

Source

barchart

---

15-min Intraday Volume of FDX

Description

An xts object including 15-min trading volume of FDX from 2019-07-01 to 2019-12-31.

Usage

data(volume_fdx)

Format

An xts object.

Source

barchart

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