Time-Series Forecast Models  A time series is a sequence of evenly time-spaced data points, such as daily shipments, weekly sales, or quarterly earnings.  Forecasting time-series data implies that forecasts are predicted only from the past values of that variable, and that other variables, no matter how potentially valuable, are ignored. Monthly Sales ( in units ) Jan Feb Mar Apr May Jun Jul Aug Sept Oct Nov Dec Data Point or (observation) MGMT E-5070

Decomposition of a Time Series  Analyzing time series means breaking down past data into components and then project- ing them into the future  A time series typically has four components: trend, seasonality, cycles, and random variation TIME SERIES MODELS ATTEMPT TO PREDICT THE FUTURE BY USING HISTORICAL DATA

Decomposition of a Time Series  Trend ( T )  Trend ( T ) is the gradual upward or downward movement of the data over time. Seasonality ( S )  Seasonality ( S ) is a pattern of the demand fluc- tuation above or below the trend line that repeats at regular intervals. Cycles ( C )  Cycles ( C ) are patterns in annual data that occur every several years. They are usually tied into the business cycle. Random variations ( R )  Random variations ( R ) are blips in the data that are caused by chance and unusual situations. They follow no discernible pattern.

Time Series & Components TREND COMPONENT SEASONAL PEAKS ACTUAL DEMAND LINE YEAR 1 YEAR 2 YEAR 3 YEAR 4 TIME AVERAGE DEMAND OVER 4 YEARS PRODUCT OR SERVICE DEMAND

Time Series & Components RANDOM VARIATIONS  Forecasters usually assume that the random variations are averaged out over time.  These random errors are often assumed to be normally distributed with a mean of zero. IT IS ALSO ASSUMED THAT RANDOM VARIATIONS DO NOT HEAVILY INFLUENCE DEMAND DO NOT HEAVILY INFLUENCE DEMAND

The Moving Average Model  Assumes demand will stay fairly steady over time.  A two-month moving average forecast is found by summing the demand during the past two periods and dividing by “ 2 ”.  With each passing period, the most recent demand is added to the sum; the earliest demand is dropped. This smooths out short-term irregularities in the data series.  It has no trend, seasonal, or cyclical components.

The Moving Average Model ( demands in previous n periods ) n n IS THE NUMBER OF PERIODS IN THE MOVING AVERAGE Forecast = Σ

The Moving Average Model Year DemandForecast TWO - PERIOD EXAMPLE / 2 = / 2 = / 2 = 130

The Moving Average Model Year DemandForecast FOUR - PERIOD EXAMPLE / 4 = / 4 = 132.5

Weighted Moving Average Model Weighted Moving Average Model  Makes the forecast more responsive to changes.  Used when there is a trend or pattern. Weights place more emphasis on recent values.  Deciding the weights requires some experience and good luck! SEVERAL WEIGHTS SHOULD BE TRIED, AND THE ONES WITH THE LOWEST FORECAST ERROR SHOULD BE SELECTED THE LOWEST FORECAST ERROR SHOULD BE SELECTED

Weighted Moving Average Model ∑ ( weight in period i )( actual value in period) ∑ ( weights )

Weighted Moving Average Model THREE - PERIOD (120) + 1 (100) + 1 (110) 10 == PeriodWeightDemand Most recent nd Most recent rd Most recent th Period Forecast 117 units ‘10’ represents the sum of the weights

Weighted Moving Average Model THREE - PERIOD (140) + 1 (120) + 1 (100) 10 == PeriodWeightDemand Most recent nd Most recent rd Most recent th Period Forecast 134 units

Exponential Smoothing Model THE NEW FORECAST LAST FORECASTED DEMAND α 1 - α += The new forecast is equal to the old forecast adjusted by a fraction of the error ( last period actual demand – last period forecast ). The smoothing coefficient ( α ) is a weight for the last actual demand. LAST ACTUAL DEMAND First Order or Primary Version A moving average technique that only requires the last period actual demand and the last period forecasted demand for input.

Exponential Smoothing Example ASSUMING THAT α =.7, THE NEXT FORECAST IS:.7 ( 100 units ) + ( )( 110 units ) = 103 units Last Forecast Last Actual Demand

Exponential Smoothing Example ASSUMING THAT α =.7, THE NEXT NEW FORECAST IS:.7 ( 120 units ) + ( )( 103 units ) = units Last Forecast Last Actual Demand

The Smoothing Coefficient  The symbol is alpha ( α )  It can assume any value between 0 and 1 inclusive  It places a weight on the last actual period demand  The value of alpha resulting in the lowest forecast error is selected for the model.

Smoothing Coefficient Selection  This range (.0 –.3 ) places the heaviest weight on the historical demand periods.  The intent is to make the forecast reflect the long - term stability of the product’s demand, as well as to minimize short-term fluctuations that could distort future forecasts.  It is appropriate for products whose demand patterns are extremely stable over time and expected to remain so. LOW - RANGE

Smoothing Coefficient Selection  This range (.4 –.6 ) splits weights between historical and most recent demand periods.  The intent is to make the forecast reflect the importance of each.  It is appropriate for products whose demand patterns are only slightly unstable. MEDIUM - RANGE

Smoothing Coefficient Selection  This range (.7 – 1.0 ) places the heaviest weight on the most recent demand periods.  The intent is to make the forecast largely reflect the most recent demand experience.  It is appropriate for products that are entirely new, and for products whose demand patterns are unstable. HIGH - RANGE

Trend Projection Model A REGRESSION MODEL OVER TIME This technique fits a trend line through a series of historical data points and then projects that trend line into the future for both medium and long-range forecasting. WE WILL FOCUS ON STRAIGHT-LINE TRENDS FOR NOW

Trend Projection Model TIME ( X ) DEMAND ( Y ) THIS ALSO IMPLIES THAT THE MEAN SQUARED ERROR (MSE) IS MINIMIZED MSE IS A MEASURE OF FORECAST ERROR We identify a straight line that minimizes the sum of the squares of the vertical distances from the regression line to each of the actual observations. THE TREND LINE A REGRESSION MODEL OVER TIME THE FITTED REGRESSION LINE

Trend Projection Model Y = a + b X ^ Y-AXIS INTERCEPT : THE POINT ON THE VERTICAL AXIS THAT THE REGRESSION LINE CROSSES THE SLOPE OF THE LEAST-SQUARES LINE: THE RATE OF CHANGE IN ‘Y’ GIVEN CHANGE IN TIME ‘X’ X AXIS Y AXIS ORIGIN THE SPECIFIED VALUE OF ‘X’ ( TIME ) THE PREDICTED VALUE ( FORECAST ) THE FITTED REGRESSION LINE

Trend Projection Model Y = a + b ( X ) Y = ( 11 ) units = ^ 11 th YEAR FORECASTY - INTERCEPTSLOPE11 th YEAR ^ EXAMPLE THIS TREND PROJECTION MODEL IS IDENTIFIED BY COMPUTER