# Time-series Forecasting

A time-series is a set of quantities related to a process taken at regular intervals. For example the water outbound flow from a lake, the price of a stock, the lung cancer mortality are classical time series examples.

The time series forecasting procedures implemented in IPredict are useful for logistic series (like the demand for a specific item or service), for financial series (like the price of the Microsoft share), Nature series (like the temperature measured in Central Park) and health related time series (like the lung cancer mortality).

We intend our forecasts to be very accurate but we do not expect them to be error free so time series forecasting must be applied and gives better results when:

• We intend to produce short term forecasts.
• The time series has ‘good’ statistical properties (it is not too short and it is not too variable).

We will use the following notation:

• $X_t$ to represent the observed quantity at time t.
• $X'_t$ to represent the forecast at time t.
• $e_t = X_t - X'_t$ to measure the forecast error at time t.

In time series forecasting we assume that the data consists of data patterns that are the subject matter of our interest and noise that is uninteresting and random or simply cannot be forecasted. Typical patterns we will analyze are the trend, the seasonality and the cycle.

# Types of Trend

The trend is the consistent behavior of the series to move in a specified direction, towards new highs or towards new lows.

# Seasonality and Cycles

Seasonality is the consistent behavior of the series to move through high and low periods that are directly related to the time (and more specifically to the time of the year, for example in summer you sell more air conditioning than in winter and this is consistently true in the north of the world every year). The cycle is a periodic movement not related to the calendar.

 Forecasting Methods Holt Winter’s, Series Decomposition and Wavelet Benchmarks Time Series Forecasting Use of the Moving Average in Time-series Forecasting Forecasting Concepts Denoising Techniques Error Statistics Computational Performance Fast Fourier Transform Moving Averages Kernel Smoothing Active Moving Average Savitsky-Golay Smoothing Fractal Projection Downloading Financial Data from Yahoo Multiple Regression Digital Signal Processing Principal Component Analysis Curve Analysis Options Pricing with Black-Scholes Markowitz Optimal Portfolio Time-series preprocessing