# intradayModel: Modeling and Forecasting Financial Intraday Signals

2023-05-19

The Hong Kong University of Science and Technology (HKUST)

Welcome to the

`intradayModel`

package! This vignette provides an overview of the package’s features and how to use them.`intradayModel`

uses state-space models to model and forecast financial intraday signal, with a focus on intraday trading volume. Our team is currently working on expanding the package to include more support for intraday volatility.

# Quick start

To get started, we load our package and sample data: the 15-minute intraday trading volume of AAPL from 2019-01-02 to 2019-06-28, covering 124 trading days. We use the first 104 trading days for fitting, and the last 20 days for evaluation of forecasting performance.

```
library(intradayModel)
data(volume_aapl)
1:5, 1:5] # print the head of data
volume_aapl[#> 2019-01-02 2019-01-03 2019-01-04 2019-01-07 2019-01-08
#> 09:30 AM 10142172 3434769 20852127 15463747 14719388
#> 09:45 AM 5691840 19751251 13374784 9962816 9515796
#> 10:00 AM 6240374 14743180 11478596 7453044 6145623
#> 10:15 AM 5273488 14841012 16024512 7270399 6031988
#> 10:30 AM 4587159 18041115 8686059 7130980 5479852
<- volume_aapl[, 1:104]
volume_aapl_training <- volume_aapl[, 105:124] volume_aapl_testing
```

Next, we fit a univariate state-space model using
`fit_volume( )`

function.

`<- fit_volume(volume_aapl_training) model_fit `

Once the model is fitted, we can analyze the hidden components of any
intraday volume based on all its observations. By calling
`decompose_volume( )`

function with
`purpose = "analysis"`

, we obtain the smoothed daily,
seasonal, and intraday dynamic components. It involves incorporating
both past and future observations to refine the state estimates.

```
<- decompose_volume(purpose = "analysis", model_fit, volume_aapl_training)
analysis_result
# visualization
<- generate_plots(analysis_result)
plots $log_components plots
```

To see how well our model performs on new data, we call
`forecast_volume( )`

function to do one-bin-ahead forecast on
the testing set.

```
<- forecast_volume(model_fit, volume_aapl_testing)
forecast_result
# visualization
<- generate_plots(forecast_result)
plots $original_and_forecast plots
```

Now that you have a quick start on using the package, let’s explore the details and dive deeper into its functionalities and features.

# Usage of the package

## Preliminary theory

Intraday observations of trading volume are divided into days, indexed by \(t\in\{1,\dots,T\}\). Each day is further divided into bins, indexed by \(i\in\{1,\dots,I\}\). To refer to a specific observation, we use the index \(\tau = I \times (t-1) + i\).

Our package uses a state-space model to extract several components of intraday volume. These components include the daily component, which adjusts the mean level of the time series; the seasonal component, which captures the U-shaped intraday periodic pattern; and the intraday dynamic component, which represents movements within a day.

The observed intraday volume can be written in a multiplicative combination of the components (Brownlees et al., 2011):

\[ \large \text{intraday volume} = \text{daily} \times \text{seasonal} \times \text{intraday dynamic} \times \text{noise}. \tag{1} \small \]

Alternatively, by taking the logarithm transform, the intraday volume can be also regarded as an addictive combination of these components:

\[ \large y_{\tau} = \eta_{\tau} + \phi_i + \mu_{t,i} + v_{t,i}. \tag{2} \small \]

The state-space model proposed by (Chen et al., 2016) is defined on Equation (2) as \[ \large \begin{aligned} \mathbf{x}_{\tau+1} &= \mathbf{A}_{\tau}\mathbf{x}_{\tau} + \mathbf{w}_{\tau},\\ y_{\tau} &= \mathbf{C}\mathbf{x}_{\tau} + \phi_{\tau} + v_\tau, \end{aligned} \tag{3} \small \] 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;

\(\mathbf{A}_{\tau} = \left[\begin{array}{l}a_{\tau}^{\eta}&0\\0&a^{\mu}\end{array} \right]\) is the state transition matrix with \(a_{\tau}^{\eta} = \begin{cases}a^{\eta}&\tau = kI, k = 1,2,\dots\\0&\text{otherwise};\end{cases}\)

\(\mathbf{C} = [1, 1]\) 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;

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

\(v_\tau \sim \mathcal{N}(0, r)\) 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 this model, \(\boldsymbol{\Theta} = \{a^{\eta}, a^{\mu}, (\sigma^{\eta})^2, (\sigma^{\mu})^2, r, \boldsymbol{\phi}, \mathbf{x}_0, \mathbf{V}_0 \}\) are treated as parameters.

## Datasets

Two data classes of intraday volume are supported:

a 2D numeric matrix of size

`(n_bin, n_day)`

;an xts object.

To help you get started, we provide two sample datasets: a
matrix-class `volume_aapl`

and an xts-class
`volume_fdx`

. Here, we elaborate on the later one.

```
data(volume_fdx)
head(volume_fdx)
#> FDX.Volume
#> 2019-07-01 09:30:00 78590
#> 2019-07-01 09:45:00 81203
#> 2019-07-01 10:00:00 52789
#> 2019-07-01 10:15:00 54344
#> 2019-07-01 10:30:00 47637
#> 2019-07-01 10:45:00 36240
tail(volume_fdx)
#> FDX.Volume
#> 2019-12-31 14:30:00 19284
#> 2019-12-31 14:45:00 18030
#> 2019-12-31 15:00:00 30946
#> 2019-12-31 15:15:00 45762
#> 2019-12-31 15:30:00 72011
#> 2019-12-31 15:45:00 219667
```

## Fitting

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

To fit a univariate state-space model on intraday volume, you should
use `fit_volume( )`

function. If you want to fix some
parameters to specific values, you can provide a list of values to
`fixed_pars`

. If you have prior knowledge of the initial
values for the unfitted parameters, you can provide it through
`init_pars`

. Besides, `verbose`

controls the level
of print, and more control options can be set via
`control`

.

The fitting process stops when either the maximum number of iterations is reached or the termination criteria is met \(\|\Delta \boldsymbol{\Theta}_i\| \le \text{abstol}\).

The following code shows how to fit the model to the FDX stock.

```
# set fixed value
<- list()
fixed_pars $"x0" <- c(13.33, -0.37)
fixed_pars
# set initial value
<- list()
init_pars $"a_eta" <- 1
init_pars
<- volume_fdx['2019-07-01/2019-11-30']
volume_fdx_training <- fit_volume(volume_fdx_training, verbose = 2, control = list(acceleration = TRUE))
model_fit #> Warning in intraday_xts_to_matrix(data): For input xts:
#> Remove trading days with missing bins: 2019-07-03, 2019-11-29.
#> iter:5 diff:0.002073476
#> iter:10 diff:0.003347168
#> iter:15 diff:0.0008842684
#> iter:20 diff:0.001107481
#> iter:25 diff:0.0003287878
#> iter:30 diff:0.0003875934
#> iter:35 diff:0.0001219829
#> Success! abstol test passed at 39 iterations.
#> --- obtained parameters ---
#> List of 8
#> $ a_eta : num 0.999
#> $ a_mu : num 0.839
#> $ var_eta: num 0.121
#> $ var_mu : num 0.0358
#> $ r : num 0.118
#> $ phi : num [1:26] 0.8415 0.4275 0.3783 0.216 0.0848 ...
#> $ x0 : num [1:2] 10.899 -0.303
#> $ V0 : num [1:2, 1:2] 6.76e-06 -6.90e-07 -6.90e-07 9.07e-06
#> ---------------------------
```

Trading days with missing bins are automatically removed. They are 2019-07-03 (Independence Day) and 2019-11-29 (Thanksgiving Day) which have early close.

## Decomposition

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

`decompose_volume( )`

function allows you to decomposes
the intraday volume into its daily, seasonal, and intraday dynamic
components.

With `purpose = "analysis"`

, it applies Kalman smoothing
to estimate the hidden states given all available observations up to a
certain point in time. The daily component and intraday dynamic
component at time \(\tau\) are the
smoothed state estimate conditioned on all the data, and denoted by
\(\mathbb{E}[\mathbf{x}_{\tau}|\{y_{j}\}_{j=1}^{M}]\),
where \(M\) is the total number of bins
in the dataset. Besides, the seasonal component has the value of \(\boldsymbol{\phi}\).

```
<- decompose_volume(purpose = "analysis", model_fit, volume_fdx_training)
analysis_result #> Warning in intraday_xts_to_matrix(data): For input xts:
#> Remove trading days with missing bins: 2019-07-03, 2019-11-29.
str(analysis_result)
#> List of 4
#> $ original_signal : num [1:2730] 78590 81203 52789 54344 47637 ...
#> $ smooth_signal : num [1:2730] 92764 65438 61063 53198 47103 ...
#> $ smooth_components:List of 4
#> ..$ daily : num [1:2730] 54116 54116 54116 54116 54116 ...
#> ..$ dynamic : num [1:2730] 0.739 0.789 0.773 0.792 0.8 ...
#> ..$ seasonal: num [1:2730] 2.32 1.53 1.46 1.24 1.09 ...
#> ..$ residual: num [1:2730] 0.847 1.241 0.865 1.022 1.011 ...
#> $ error :List of 3
#> ..$ mae : num 14233
#> ..$ mape: num 0.223
#> ..$ rmse: num 38111
#> - attr(*, "type")= chr [1:2] "analysis" "smooth"
```

Function `generate_plots( )`

visualizes the smooth
components and the smoothing performance.

```
<- generate_plots(analysis_result)
plots $log_components plots
```

`$original_and_smooth plots`

With `purpose = "forecast"`

, it applies Kalman forecasting
to estimate the one-bin-ahead hidden state based on the available
observations, which is mathematically denoted by \(\mathbb{E}[\mathbf{x}_{\tau+1}|\{y_{j}\}_{j=1}^{\tau}]\).
Details can be found in the next subsection.

This function also helps to evaluate the model performance with the following measures:

Mean absolute error (MAE): \(\frac{1}{M}\sum_{\tau=1}^M\lvert\hat{y}_\tau - y_\tau\rvert\).

Mean absolute percent error (MAPE): \(\frac{1}{M}\sum_{\tau=1}^M\frac{\lvert\hat{y}_\tau - y_\tau\rvert}{y_\tau}\).

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

## Forecasting

forecast_volume(model, data, burn_in_days = 0)

`forecast_volume( )`

function is a wrapper of
`decompose_volume(purpose = "forecast", ...)`

. It forecasts
the one-bin-ahead intraday volume on a new dataset. The one-bin-ahead
forecast is mathematically denoted by \(\hat{y}_{\tau+1} =
\mathbb{E}[y_{\tau+1}|\{y_{j}\}_{j=1}^{\tau}]\).

When encountering a new dataset with different statistical
characteristics or from different stocks, the state space model may not
initially start in an optimal state. To address this, the first
`burn_in_days`

days in the data can be utilized to warm up
the Kalman filter, allowing it to reach the desired state. These initial
days will be discarded after initialization.

```
# use training data for burn-in
<- forecast_volume(model_fit, volume_fdx, burn_in_days = 105)
forecast_result #> Warning in intraday_xts_to_matrix(data): For input xts:
#> Remove trading days with missing bins: 2019-07-03, 2019-11-29, 2019-12-24.
str(forecast_result)
#> List of 4
#> $ original_signal : num [1:520] 149293 136426 134342 75474 61054 ...
#> $ forecast_signal : num [1:520] 81290 77773 94069 89915 72067 ...
#> $ forecast_components:List of 4
#> ..$ daily : num [1:520] 37989 49345 57227 61320 59639 ...
#> ..$ dynamic : num [1:520] 0.922 1.028 1.126 1.181 1.11 ...
#> ..$ seasonal: num [1:520] 2.32 1.53 1.46 1.24 1.09 ...
#> ..$ residual: num [1:520] 1.837 1.754 1.428 0.839 0.847 ...
#> $ error :List of 3
#> ..$ mae : num 36242
#> ..$ mape: num 0.284
#> ..$ rmse: num 162071
#> - attr(*, "type")= chr "forecast"
```

Function `generate_plots( )`

visualizes the one-bin-ahead
forecast components and the forecasting performance.

```
<- generate_plots(forecast_result)
plots $log_components plots
```

`$original_and_forecast plots`

# Next steps

This guide gives an overview of the package’s main features. Check the manual for details on each function, including parameters and examples.

The current version only supports univariate state-space models for intraday trading volume. Soon, we’ll add models for intraday volatility and their multivariate versions. We hope you find these resources helpful and that our package will continue to be a valuable tool for your work.

# References

*Journal of Financial Econometrics*,

*9*(3), 489–518.

*Available at SSRN 3101695*.