Time Series Analysis and Forecasting

Some Historic Forecasts Harry Warner - Warner Bros. Pictures, 1927 “Who the hell wants to hear actors talk?”

Some Historic Forecasts Lord Kelvin, President of the Royal Society, 1895 “Heavier-than-air flying machines are impossible”

Some Historic Forecasts U.S. Admiral William Leahy in 1945 "The atom bomb will never go off - and I speak as an expert in explosives."

Some Historic Forecasts Business Week, August 2, 1968 "With over 50 foreign cars already on sale here, the Japanese auto industry isn't likely to carve out a big slice of the US market."

Some Historic Forecasts Popular Mechanics, forecasting the relentless march of science, 1949 "Where a calculator on the ENIAC is equipped with 19,000 vacuum tubes and weighs 30 tons, computers in the future may have only 1,000 vacuum tubes and perhaps only weigh 1.5 tons."

Some Historic Forecasts Thomas J. Watson, chairman of the board of IBM. “I think there's a world market for about five computers."

Some Historic Forecasts Ken Olson, president of Digital Equipment Corp. 1977 "There is no reason anyone would want a computer in their home."

Some Historic Forecasts American Radio pioneer Lee DeForest, 1926 "While theoretically and technically television may be feasible, commercially and financially it is an impossibility."

Forecasting Methods Judgement Methods: Salesforce composite: a compilation of estimates by salespeople (or dealers) of expected sales in their territories, with necessary adjustments. Scenario methods: narratives that describe an assumed future expressed as a sequence of snapshots. Delphi technique: a successive series of estimates independently developed by a group of “experts” each member of which, at each step in the process, uses a summary of the group’s previous results to formulate new estimates.

Forecasting Methods Time-series (extrapolation) methods: Moving average: averaging recent values to predict future outcomes. Exponential smoothing: forecasting the next period by calculating a weighted average of the recent data. Time-series decomposition: isolating trend, seasonal and cyclical components from a data series, and using outcome from each to make predictions.

Forecasting Methods Causal methods: Regression models: driving an equation that minimizes the variance between predictions and actual outcomes given a set of predictor (independent) variables. Econometric models: forecasts from an integrated system of simultaneous equations that represent relationships among elements of the economy.

Forecasting Methods In order to increase forecast accuracy Combine two or more forecasts obtained by different methods Forecasts could come from different forecasting method types (e.g. judgmental, causal etc.) or from the same type Forecast errors might cancel one another

Components of Time Series Data Yt t Base case – all observations are equal resulting in a horizontal line

Components of Time Series Data Yt t Trend – observations increase or decrease regularly through time Trend is the long-term component that represents the growth or decline in the time-series over an extended period of time

Components of Time Series Data Yt t Linear trend – observations increase or decrease by the same amount through time

Components of Time Series Data Yt t exponential trend – observations increase or decrease by the same percentage amount through time

Components of Time Series Data Yt t Cycle – the wavelike fluctuations around the trend

Components of Time Series Data Yt t Seasonal – pattern of change that repeats itself time after time

Components of Time Series Data Yt t Seasonal component with trend

Components of Time Series Data Yt t Noise – the unpredictable and irregular component

Notation Y : the variable we want to forecast Yt : observed value of the variable at time t T : the number of historical observations (t=1,2,...,T) Ft : the forecast for the variable for time t Et : the forecast error for time t (Et = Yt – Ft)

Notation Mean Absolute Error Root Mean Square Error Mean Absolute Percentage Error

Notation Let us use the following forecasting model on the given data set: Ft+1 = Yt (naive method) MAE = ? RMSE = ? MAPE = ?

Regression-Based Models Linear Trend: Ft = b0 + b1t Exponential Trend: ln (Ft) = b0 + b1t Seasonality (k seasons): Ft = b0 + b1S1 + b2S2 + ... + bk-1Sk-1

Exercise Use data set in P02_38.xls to come up with a regression model to use in forecasting.

Simple Averages A simple average uses the mean of all the relevant historical observations as the foreacast for the next period. Y1 Y2 Y3 Yt-1 Yt

Simple Averages

Simple Averages

Simple Averages

Simple Averages

Simple Averages

Simple Averages

Simple Averages

Simple Averages The method of simple averages is an appropriate technique when the forces generating the series to be forecast have stabilized, and the environment in which the series exists is generally unchanging. Examples: quantity of sales resulting from a consistent level of salesperson effort quantity of sales of a product in the mature stage of its life cycle number of appointments per week requested of a denstist whose patient base is fairly constant

Moving Averages A moving average of order k is the mean value of k consecutive observations. Equal weights are assigned to each observation. Each new data point is included in the average as it becomes available, and the earliest data point is discarded. Moving averages method does not handle trend or seasonality very well, although it does better than the simple average method.

Moving Averages

Moving Averages

Moving Averages

Moving Averages

Moving Averages

Moving Averages

Moving Averages

Moving Averages

Moving Averages

Moving Averages The analyst must use judgement when determining how many days, weeks, months or quarters on which to base the moving average. As the value of k increases, the forecasts become smoother. The smaller the k, the more weight is given to recent periods. The greater the k, the less weight is given to recent periods.

Double Moving Averages One way of forecasting time series data that have linear trend is to use double moving averages.

Double Moving Averages M(t) – M’(t)

Double Moving Averages

Exponential Smoothing Methods Exponential smoothing is a procedure for continually revising a forecast in the light of more recent experience. (new forecast) =  (new observation) + (1 - ) (old forecast)  : smoothing constant (0 ≤  ≤ 1) Simple exponential smoothing – no trend and seasonality Holt’s method – trend but no seasonality Winters’ method – trend and seasonality

Simple Exponential Smoothing When  is close to zero, the new forecast will be very similar to the old one. If it is desired that predictions be stable and random variations smoothed, a small value of  is required. If a rapid response to a real change in the pattern of observations is desired, a larger value of  is appropriate.

Simple Exponential Smoothing

Simple Exponential Smoothing

Simple Exponential Smoothing

Simple Exponential Smoothing

Holt’s Method level: base for the time-series trend: change in level from one period to the next

Holt’s Method

Holt’s Method

Additive vs. Multiplicative Seasonality additive seasonality multiplicative seasonality additive seasonality with trend multiplicative seasonality with trend

Additive vs. Multiplicative Seasonality In additive seasonal model, the appropriate seasonal index is added to the base of the forecast. In a multiplicative model, the appropriate seasonal index is multiplied with the base of the forecast.

Winters’ Model (Additive) actual = base + seasonal index

Winters’ Model (Multiplicative) actual = base x seasonal index

Winters’ Model

Time-Series Decomposition Yt = Tt x St x It Irregular component Seasonal index Actual observation Trend component Step 1: Deseasonalize data using centered moving averages Step 2: Estimate the trend component using regression Step 3: Estimate the seasonal indices Step 4: Construct forecast