Advanced Time Series Analysis

Introduction

Autoregression (AR) models are a fundamental tool in time series analysis, modeling a variable as a linear function of its past values. Kdb+'s efficiency in handling time series data makes it an ideal platform for building and analyzing AR models.

Understanding Autoregression

An AR(p) model is defined as:

Xt = c + φ1Xt-1 + φ2Xt-2 + ... + φpXt-p + εt

Where:

• Xt is the value of the time series at time t

• c is a constant

• φ1, φ2, ..., φp are the autoregressive coefficients

• εt is the error term (white noise)

Data Preparation

Code snippet

// Sample time series data
data:([]time:`times$;value:10f)

// Load time series data
data:read0 `:data/time_series.csv

// Calculate lagged values
data[`lag1]:lag value by time
data[`lag2]:lag value by time[2]

Building an AR Model

We can use statistical libraries like statsmodels to build AR models.

Python

import pandas as pd
from statsmodels.tsa.ar_model import AR

# Convert kdb+ data to pandas Series
time_series = pd.Series(data['value'], index=pd.to_datetime(data['time']))

# Build AR model
model = AR(time_series)
model_fit = model.fit()

# Print model summary
print(model_fit.summary())

Model Evaluation

Python

from sklearn.metrics import mean_squared_error

# Make predictions
predictions = model_fit.predict(start=len(time_series), end=len(time_series) + 10)

# Evaluate model performance
mse = mean_squared_error(time_series[-10:], predictions)
print(mse)

Model Selection

Determining the optimal order (p) for the AR model is crucial.

Python

from statsmodels.tsa.stattools import acf, pacf

# Calculate autocorrelation function (ACF) and partial autocorrelation function (PACF)
acf_values = acf(time_series)
pacf_values = pacf(time_series)

# Plot ACF and PACF to determine potential order

Forecasting

AR models can be used to forecast future values.

Python

# Forecast future values
forecast = model_fit.forecast(steps=12)

Stationarity

Stationarity is a key assumption for AR models.

Python

from statsmodels.tsa.stattools import adfuller

# Augmented Dickey-Fuller test
result = adfuller(time_series)
print('ADF Statistic: %f' % result[0])
print('p-value: %f' % result[1])

Incorporating Exogenous Variables

AR models can be extended to include exogenous variables (ARX models).

Python

# Assume an exogenous variable 'x'
data[`x]:(1 2 3 4 5)

# Build ARX model
model = AR(time_series, exog=data['x'])

Advanced Topics

• Non-linear AR models: Explore models like threshold autoregression (TAR) and exponential autoregression (EAR).

• Model selection criteria: Use AIC, BIC, or other criteria to compare models.

• Parameter estimation: Implement different estimation methods for AR models.

• Model diagnostics: Check for model assumptions and identify potential issues.

Conclusion

AR models are a powerful tool for time series analysis, and kdb+ provides an efficient platform for building and evaluating these models. By understanding the core concepts and applying the techniques outlined in this chapter, you can effectively analyze and forecast time series data.