INDUSTRIAL STATISTICS AND OPERATIONAL MANAGEMENT 6 : FORECASTING TECHNIQUES Dr. Ravi Mahendra Gor Associate Dean ICFAI Business School ICFAI HOuse, Nr. GNFC INFO Tower S. G. Road Bodakdev Ahmedabad-380054 Ph. : 079-26858632 (O); 079-26464029 (R); 09825323243 (M) E-mail: ravigor@hotmail. om Contents Introduction Some applications of forecasting Defining forecasting General steps in the forecasting process Qualitative techniques in forecasting Time series methods The Naive Methods Simple Moving Average Method Weighted Moving Average Exponential Smoothing Evaluating the forecast accuracy Trend Projections Linear Regression Analysis Least Squares Method for Linear Regression Decomposition of the time series Selecting A Suitable Forecasting Method More on Forecast Errors Review Exercise CHAPTER 6 FORECASTING TECHNIQUES 6. Introduction: Every manager would like to know exact nature of future events to accordingly take action or plan his action when sufficient time is in hand to implement the plan. The effectiveness of his plan depends upon the level of accuracy with which future events are known to him. But every manager plans for future irrespective of the fact whether future events are exactly known or not. That implies, he does try to forecast future to the best of his Ability, Judgment and Experience. Virtually all management decisions depend on forecasts.
Managers study sales forecasts, for example, to take decisions on working capital needs, the size of the work force, inventory levels, the scheduling of production runs, the location of facilities, the amount of advertising and sales promotion, the need to change prices, and many other problems. For our purpose forecasting can be defined as attempting to predict the future by using qualitative or quantitative methods. In an informal way, forecasting is an integral part of all human activity, but from the business point of view increasing attention is being given to formal forecasting systems which are continually being refined.
Some forecasting systems involve very advanced statistical techniques beyond the scope of this book, so are not included. All forecasting methodologies can be divided into three broad headings i. e. forecasts based on: What people have done What people Examples: say examples: Time Series Analysis Regression Analysis Surveys Questionnaires What people do examples: Testing Marketing Reaction tests The data from past activities are cheapest to collect but may be outdated and past behavior is not necessarily indicative of future behavior.
Data derived from surveys are more expensive to obtain and needs critical appraisal – intentions as expressed in surveys and questionnaires are not always translated into action. Finally, the data derived from recording what people actually do are the most reliable but also the most expensive and occasionally it is not feasible for the data to be obtained. Forecasting is a process of estimating a future event by casting forward past data. The past data are systematically combined in a predetermined way to obtain the estimate of the future.
Prediction is a process of estimating a future event based on subjective considerations other than just past data; these subjective considerations need not be combined in a predetermined way. Thus forecast is an estimate of future values of certain specified indicators relating to a decisional/planning situation, In some situations forecast regarding single indicator is sufficient, where as, in some other situations 142 forecast regarding several indicators is necessary. The number of indicators and the degree of detail required in the forecast depends on the intended use of the forecast.
There are two basic reasons for the need for forecast in any field. 1. Purpose – Any action devised in the PRESENT to take care of some contingency accruing out of a situation or set of conditions set in future. These future conditions offer a purpose / target to be achieved so as to take advantage of or to minimize the impact of (if the foreseen conditions are adverse in nature) these future conditions. 2. Time – To prepare plan, to organize resources for its implementation, to implement; and complete the plan; all these need time as a resource.
Some situations need very little time, some other situations need several years of time. Therefore, if future forecast is available in advance, appropriate actions can be planned and implemented ‘intime’. 6. 2 Some Applications of Forecasting: Forecasts are vital to every business organization and for every significant management decision. We now will discuss some areas in which forecasting is widely used. Sales Forecasting Any company in selling goods needs to forecast the demand for those goods.
Manufactures need to know how much to produce. Wholesalers and retailers need to know now much to stock. Substantially understanding demand is likely to lead to many lost sales, unhappy customers, and perhaps allowing the competition to gain the upper hand in the marketplace. On the other hand, significantly overestimating demand also is very costly due to (1) excessive inventory costs, (2) forced price reductions, (3) unneeded production or storage capacity, and (4) lost opportunities to market more profitable goods.
Successful marketing and production managers understand very well the importance of obtaining good sales forecasts. For the production managers these sales forecast are essential to help trigger the forecast for production which in turn triggers the forecasting of the raw materials needed for production. Forecasting the need for raw materials and spare parts Although effective sales forecasting is a key for virtually any company, some organizations must rely on other types of forecasts as well. A prime example involves forecasts of the need for raw materials and spare parts.
Many companies need to maintain an inventory of spare parts to enable them to quickly repair either own equipment or their products sold or leased to customers. Forecasting Economic Trends With the possible exception of sales forecasting, the most extensive forecasting effort is devoted to forecasting economic trends on a regional, national, or even international level. Forecasting Staffing Needs For economically developed countries there is a shifting emphasis from manufacturing to services. Goods are being produced outside the country (where labor is chapter) and then imported.
At the same time, an increasing number of business firms are specializing in providing a service of some kind (e. g. , travel, tourism, entertainment, legal aid, health services, financial, educational, design, maintenance, etc. ). For such a company forecasting “sales” becomes forecasting the demand for services, which then translates into forecasting staffing needs to provide those services. 143 Forecasting in education environment A good education institute typically plans its activities and areas concentration for the coming years based on the forecasted demand for its different activities.
The institute may come out with a forecast that the future requirements of its students who graduate may be more in particular sector. This may call for the reorientation of the syllabus and faculty, development of suitable teaching materials/cases, recruitment of new faculty with specific sector-oriented background, experience and teaching skills. Alternatively, the management may decide that the future is more secure with the conventional areas of operation and it may continue with the original syllabus, etc. Forecasting in a rural setting Cooperative milk producers’, union operates in a certain district.
The products it manufactures, the production capacities it creates, the manpower it recruits, and many more decisions are closely linked with the forecasts of the milk it may procure and the different milk products it may see. Milk being a product which has a ready market, is not difficult to sell. Thus demand forecasting for products may not be a very dominant issue for the organization. However, the forecast of milk procurement is a crucial issue as raw milk is a highly perishable commodity and building up of adequate processing capacity is important for the dairy.
The milk procurement forecast also forms an important input to the production planning process which includes making decisions on what to produce, how much and when to produce. Ministry of Petroleum The officials of this crucial ministry have to make decisions on the quantum of purchase to be made for various types of crude oils and petroleum products from different sources across the oil-exporting nations for the next few years. They also have to decide as to how much money has to be spent on development of indigenous sources.
These decisions involve/need information on the future demand of different types of petroleum products and the likely change in the prices and the availability of crude oil and petroleum products in the country and the oil-exporting nations. This takes us back to the filed of forecasting. Department of Technology The top officials of this department want to make decisions on the type of information technology to recommend to the union government for the next decade. But they are not very clear on the directions which will be taken by this year rapidly changing field.
They decided to entrust this task to the information system group of a national management institute. The team leader decided to forecast the changing technology in this area with the help of a team of information technology experts throughout the country. This is again a forecasting problem although of a much different type. This field of forecasting is known as technological forecasting. Forecasting is the basis of corporate long-run planning. In the functional areas of finance and accounting, forecasts provide the basis for budgetary planning and cost control.
Marketing relies on sales forecasting to plan new products, compensate sales personnel, and make other key decisions. Productions and operations personnel use forecasts to make periodic decisions involving process selection, capacity planning, and facility layout, as well as for continual decisions about production planning, scheduling, and inventory. As we have observed in the aforementioned examples, forecasting forms an important input in many business and social science-related situations. 144 6. 3 Defining Forecasting:
A forecast is an estimate of a future event achieved by systematically combining and casting forward in predetermined way data about the past. It is simply a statement about the future. It is clear that we must distinguish between forecast per se and good forecasts. Good forecast can be quite valuable and would be worth a great deal. Long-run planning decisions require consideration of many factors: general economic conditions, industry trends, probable competitor’s actions, overall political climate, and so on. Forecasts are possible only when a history of data exists.
An established TV manufacturer can use past data to forecast the number of picture screens required for next week’s TV assembly schedule. But suppose a manufacturer offers a new refrigerator or a new car, he cannot depend on past data. He cannot forecast, but has to predict. For prediction, good subjective estimates can be based on the manager’s skill experience, and judgment. One has to remember that a forecasting technique requires statistical and management science techniques. In general, when business people speak of forecasts, they usually mean some combination of both forecasting and prediction.
Forecasts are often classified according to time period and use. In general, short-term (up to one year) forecasts guide current operations. Medium-term (one to three years) and long-term (over five years) forecasts support strategic and competitive decisions. Bear in mind that a perfect forecast is usually impossible. Too many factors in the business environment cannot be predicted with certainty. Therefore, rather than search for the perfect forecast, it is far more important to establish the practice of continual review of forecasts and to learn to live with inaccurate forecasts.
This is not to say that we should not try to improve the forecasting model or methodology, but that we should try to find and use the best forecasting method available, within reason. Because forecasts deal with past data, our forecasts will be less reliable the further into the future we predict. That means forecast accuracy decreases as time horizon increases. The accuracy of the forecast and its costs are interrelated. In general, the higher the need for accuracy translates to higher costs of developing forecasting models. So how much money and manpower is budgeted for forecasting?
What possible benefits are accrued from accrued from accurate forecasting? What are possible cost of inaccurate forecasting? The best forecast are not necessarily the most accurate or the least costly. Factors as purpose and data availability play important role in determining the desired accuracy of forecast. When forecasting, a good strategy is to use two or three methods and look at them for the commonsense view. Will expected changes in the general economy affect the forecast? Are there changes in industrial and private consumer behaviors? Will there be a shortage of essential complementary items?
Continual review and updating in light of new data are basic to successful forecasting. In this chapter we look at qualitative and quantitative forecasting and concentrate primarily on several quantitative time series techniques. 145 The following figure illustrates various methods of forecasting. Forecasting Techniques Qualitative Methods Time Series Methods Naive Methods Moving Average Exponential Smoothing Trend Projections Causal Methods Regression Analysis Grass Roots Market Research Panel Consensus Historical Analogy Delphi Method Fig. 6. 1 Different Forecasting Methods 6. General Steps In The Forecasting Process The general steps in the forecasting process are as follows: 1) Identify the general need 2) Select the Period (Time Horizon) of Forecast 3) Select Forecast Model to be used: For this, knowledge of various forecasting models, in which situations these are applicable, how reliable each one of them is; what type of data is required. On these considerations; one or more models can be selected. 4) Data Collection: With reference to various indicators identified-collect data from various appropriate sources-data which is compatible with the model(s) selected in step(3).
Data should also go back that much in past, which meets the requirements of the model. 5) Prepare forecast: Apply the model using the data collected and calculate the value of the forecast. 6) Evaluate: The forecast obtained through any of the model should not be used, as it is, blindly. It should be evaluated in terms of ‘confidence interval’ – usually all good forecast models have methods of calculating upper value and the lower value within which the given forecast is expected to remain with 146 certain specified level of probability.
It can also be evaluated from logical point of view whether the value obtained is logically feasible? It can also be evaluated against some related variable or phenomenon. Thus, it is possible, some times advisable to modify the statistically forecasted’ value based on evaluation. 6. 5 Qualitative Techniques In Forecasting Grass Roots Grass roots forecasting builds the forecast by adding successively from the bottom. The assumption here is that the person closest to the customer or end use of the product knows its future needs best.
Though this is not always true, in many instances it is a valid assumption, and it is the basis for this method. Forecasts at this bottom level are summed and given to the next higher level. This is usually a district warehouse, which then adds in safely stocks and any effects of ordering quantity sizes. This amount is then fed to the next level, which may be a regional warehouse. The procedure repeat until it becomes an input at the top level, which, in the case of a manufacturing firm, would be the input to the production system.
Market Research: Firms often hire outside companies that specialize in market research to conduct this type of forecasting. You may have been involved in market surveys through a marketing class. Certainly you have not escaped telephone calls asking you about product preferences, your income, habits, and so on. Market research is used mostly for product research in the sense of looking for new product ideas, likes and dislikes about existing products, which competitive products within a particular class are preferred, and so on.
Again, the data collection methods are primarily surveys and interviews. Panel Consensus: In a panel consensus, the idea that two heads are better than one is extrapolated to the idea that a panel of people from a variety of positions can develop a more reliable forecast than a narrower group. Panel forecasts are developed through open meetings with free exchange of ideas form all levels of management and individuals. The difficulty with this open style is that lower employee levels are intimidated by higher levels of management.
For example, a salesperson in a particular product line may have a good estimate of future product demand but may not speak up to refute a much different estimate given by the vice president of marketing. The Delphi technique (which we discuss shortly) was developed to try to correct this impairment to free exchange. When decisions in forecasting are at a broader, higher level (as when introducing a new product line or concerning strategic product decisions such as new marketing areas) the term executive judgment is generally used. The term is self-explanatory: a higher level of management is involved.
Historical Analogy: The historical analogy method is used for forecasting the demand for a product or service under the circumstances that no past demand data are available. This may specially be true if the product happens to be new for the organization. However, the organization may have marketed product(s) earlier which may be similar in some features to the new product. In such circumstances, the marketing personnel use the historical analogy between the two products and derive the demand for the new product using the historical data of the earlier 147 product.
The limitations of this method are quite apparent. They include the questionable assumption of the similarity of demand behaviors, the changed marketing conditions, and the impact of the substitutability factor on the demand. Delphi Method: As we mentioned under panel consensus, a statement or opinion of a higher-level person will likely be weighted more than that of a lower-level person. The worst case in where lower level people feel threatened and do not contribute their true beliefs. To prevent this problem, the Delphi method conceals the identity of the individuals participating in the study.
Everyone has the same weight. A moderator creates a questionnaire and distributes it to participants. Their responses are summed and given back to the entire group along with a new set of questions. The Delphi method was developed by the Rand Corporation in the 1950s. The step-by-step procedure is 1) Choose the experts to participate. There should be a variety of knowledgeable people in different areas. 2) Through a questionnaire (or e-mail), obtain forecasts (and any premises or qualification captions for the forecasts) from all participants. ) Summarize the results and redistribute them to the participants along with appropriate new questions. 4) Summarize again, refining forecasts and conditions, and again develop new questions. 5) Repeat Step 4 if necessary. Distribute the final results to all participants. The Delphi technique can usually achieve satisfactory results in three rounds. The time required is a function of the number of participants, how much work is involved for them to develop their forecasts, and their speed in responding. We now discuss the quantitative methods of forecasting 6. 6 Time-Series Methods
In many forecasting situations enough historical consumption data are available. The data may relate to the past periodic sales of products, demands placed on services like transportation, electricity and telephones. There are available to the forecaster a large number of methods, popularly known as the time series methods, which carry out a statistical analysis of past data to develop forecasts for the future. The underlying assumption here is that past relationships will continue to hold in the future. The different methods differ primarily in the manner in which the past values are related to the forecasted ones.
A time series refers to the past recorded values of the variables under consideration. The values of the variables under consideration in a time-series are measured at specified intervals of time. These intervals may be minutes, hours, days, weeks, months, etc. In the analysis of a time series the following four time-related factors are important. 148 1) Trends: These relate to the long-term persistent movements/tendencies/changes in data like price increases, population growth, and decline in market shares. An example of a decreasing linear trend is shown in Fig. 6. 2 Market share
Market share Time Time Market share Market share Time Fig. 6. 2 Time (2) Seasonal variations: There could be periodic, repetitive variations in time-series which occur because of buying or consuming patterns and social habits, during different times of a year. The demand for products like soft drinks, woolens and refrigerators, also exhibits seasonal variations. An illustration of seasonal variations is provided in Fig. 6. 3 Sales Sales Time Fig. 6. 3 Time (3) Cyclical variations: These refer to the variations in time series which arise out of the phenomenon of business cycles.
The business cycle refers to the periods of expansion followed by periods of contraction. 149 The period of a business cycle may vary from one year to thirty years. The duration and the level of resulting demand variation due to business cycles are quite difficult to predict. (4) Random or irregular variations: These refer to the erratic fluctuations in the data which cannot be attributed to the trend, seasonal or cyclical factors. In many cases, the root cause of these variations can be isolated only after a detailed analysis of the data and the accompanying explanations, if any.
Such variations can be due to a wide variety of factors like sudden weather changes, strike or a communal clash. Since these are truly random in nature, their future occurrence and the resulting impact on demand are difficult to predict. The effect of these events can be eliminated by smoothing the time series data. A graphical example of the random variations is given in Fig. 6. 4. Demand Time Fig. 6. 4 The historical time series, as obtained from the past records, contains all the four factors described earlier. One of the major tasks is to isolate each of the components, as elegantly as possible.
This process of desegregating the time series is called decomposition. The main objective here is to isolate the trend in time series by eliminating the other components. The trend line can then be used for projecting into the future. The effect of the other components on the forecast can be brought about by adding the corresponding cyclical, seasonal and irregular variations. In most short-term forecasting situations the elimination of the cyclical component is not attempted. Also, it is assumed that the irregular variations are small and tend to cancel each other out over time.
Thus, the major objective, in most cases, is to seek the removal of seasonal variations from the time series. This process is known as deseasonalization of the time series data. There are a number of time-series-based methods. Not all of them involve explicit decomposition of the data. The methods extend from mathematically very simple to fairly complicated ones. Let us also see some of the time series models are based on the trend lines of the data. The constant-level models assume no trend at all in the data. The time series is assumed to have a relatively constant mean.
The forecast for any period in the future is a horizontal line. Linear trend models forecast a straight-line trend for any period in the future. Refer Fig. 6. 2 Exponential trends forecast that the amount of growth will increase continuously. At long horizons, these trends become unrealistic. Thus models with a damped trend have been developed for longer-range forecasting. The amount of trend extrapolated declines each period in a damped trend model. Eventually, the trend dies out and the forecasts become a horizontal line. Refer Fig 6. The additive seasonal pattern assumes that the seasonal fluctuations are of constant size. 150 The multiplicative pattern assumes that the seasonal fluctuations are proportional to the data. As the trend increases, the seasonal fluctuations get larger. Refer Fig 6. 3 6. 6. 1 The Naive Methods The forecasting methods covered under this category are mathematically very simple. The simplest of them uses the most recently observed value in the time series as the forecast for the next period. Effectively, this implies that all prior observations are not considered.
Another method of this type is the ‘free-hand projection method’. This includes the plotting of the data series on a graph paper and fitting a free-hand curve to it. This curve is extended into the future for deriving the forecasts. The ‘semi-average projection method’ is another naive method. Here, the time-series is divided into two equal halves, averages calculated for both, and a line drawn connecting the two semi averages. This line is projected into the future and the forecasts are developed. Illustration 6. 1: Consider the demand data for 8 years as given.
Use these data for forecasting the demand for the year 1991 using the three naive methods described earlier. Year 1983 1984 1985 1986 1987 1988 1989 1990 100 105 103 107 109 110 115 117 Actual sales Solution: The forecasted demand for 1991, using the last period method = actual sales in 1990 = 117 units. The forecasted demand for 1991, using the free-hand projection method = 119 units. (Please check the results using a graph papers! ) The semi-averages for this problem will be calculated for the periods 1983-86 and 1987-90. The resultant semiaverages are 103. 75 and 112. 75.
A straight line joining these points would lead to a forecast for the year 1991. The value of this forecast will be = 120 units 151 Sales 120 115 110 105 100 Year 83 84 85 86 87 88 89 90 91 Fig. 6. 5 6. 6. 2 Simple Moving Average Method When demand for a product is neither growing nor declining rapidly, and if it does not have seasonal characteristics, a moving average can be useful can be useful in removing the random fluctuations for forecasting. Although moving averages are frequently centered, it is more convenient to use past data to predict the following period directly.
To illustrate, a centered five-month average of January, February, March, April and May gives an average centered on March. However, all five months of data must already exist. If our objective is to forecast for June, we must project our moving average- by some means- from March to June. If the average is not centered but is at forward end, we can forecast more easily, through we may lose some accuracy. Thus, if we want to forecast June with a five-month moving average, we can take the average of January, February, March, April and May. When June passes, the forecast for July would be the average of February, March, April, May and June.
Although it is important to select the best period for the moving average, there are several conflicting effects of different period lengths. The longer the moving average period, the more the random elements are smoothed (which may be desirable in many cases). But if there is a trend in the data-either increasing or decreasing-the moving average has the adverse characteristic of lagging the trend. Therefore, while a shorter time span produces more oscillation, there is a closer following of the trend. Conversely, a longer time span gives a smoother response but lags the trend.
The formula for a simple moving average is +A +A + …… + A A t? 2 t ? 3 t ? n F = t ? 1 t n where, Ft = Forecast for the coming period, n = Number of period to be averaged and At-1, At-2, At-3 and so on are the actual occurrences in the in the past period, two periods ago, three periods ago and so on respectively. 152 Illustration 6. 2: The data in the first two columns of the following table depict the sales of a company. The first two columns show the month and the sales. The forecasts based on 3, 6 and 12 month moving average and shown in the next three columns.
The 3 month moving average of a month is the average of sales of the preceding three months. The reader is asked to verify the calculations himself. Past sales of generators Month Actual units sold Forecasts produced by 3 month moving average 6 month moving average 12 month moving average January February March April May June July August September October November December January 450 440 460 410 380 400 370 360 410 450 470 490 460 (450+440+460)/3 = 450 (440+460+410)/3 = 437 (460+410+380)/3 = 417 397 383 377 380 407 443 470 423 410 397 388 395 410 425 424
The 6 month moving average is given by the average of the preceding 6 months actual sales. For the month of July it is calculated as July’s forecast = ( Sum of the actual sales from January to June ) / 6 = ( 450 + 440 + 460 + 410 + 380 + 400 ) / 6 = 423 ( rounded ) For the forecast of January by the 12 month moving average we sum up the actual sales from January to December of the preceding year and divide it by 12. Note: 1. A moving average can be used as a forecast as shown above but when graphing moving averages it is important to realize, that being averages, they must be plotted at the mid point of the period to which they relate. . Twelve-monthly moving averages or moving annual totals form part of a commonly used diagram, called the Z chart. It is called a Z chart because the completed diagram is shaped like a Z. The top part of the Z is formed by the moving annual total, the bottom part by the individual monthly figures and the sloping line by the cumulative monthly figures. 153 Illustration 6. 3: Using the data given in the Illustration 1 forecast the demand for the period 1987 to 1991 using a. b. 3- year moving average and 5- year moving average
If we want to check the error in our forecast as Error = Actual observed value – Forecasted value find which one gives a lower error in the forecast. Year Demand Three year moving average forecast error 4. 4 4. 0 3. 7 6. 4 5. 7 – Five year moving average forecast 5. 2 8. 2 8. 2 error 1983 1984 1985 1986 1987 1988 1989 1990 1991 100 105 103 107 109 110 115 117 – 102. 6 105. 0 106. 3 108. 6 111. 3 114. 0 104. 8 106. 8 108. 8 111. 6 Here we observe that the forecast always lags behind the actual values. The lag is greater for the 5-year moving average based forecasts.
Characteristics of moving averages a. The different moving averages produce different forecasts. b. The greater the number of periods in the moving average, the greater the smoothing effect. c. If the underlying trend of the past data is thought to be fairly constant with substantial randomness, then a greater number of periods should be chosen. d. Alternatively, if there is though to be some change in the underlying state of the data, more responsiveness is needed, therefore fewer periods should be included in the moving average. Limitations of moving averages a.
Equal weighting is given to each of the values used in the moving average calculation, whereas it is reasonable to suppose that the most recent data is more relevant to current conditions. b. An n period moving average requires the storage of n – 1 values to which is added the latest observation. This may not seem much of a limitation when only a few items are considered, but it becomes a significant factor when , for example, a company carries 25,000 stock items each of which requires a moving average calculation involving say 6 months usage data to be recorded. 54 c. d. The moving average calculation takes no account of data outside the period of average, so full use is not made of all the data available. The use of the unadjusted moving average as a forecast can cause misleading results when there is an underlying seasonal variation. 6. 6. 3 Weighted Moving Average Whereas the simple moving average gives equal weight to each component of the moving average database, a weighted moving average allows any weights to be placed on each element, providing, of course, that the sum of all weights equals 1.
For example, a department store may find that in a four-month period, the best forecast is derived by using 40 percent of the actual sales for the most recent month, 30 percent of two months ago, 20 percent of three months ago, and 10 percent of four months ago. If actual sales experience was Month 1 100 the forecast for month 5 would be Month 2 90 Month 3 105 Month 4 95 Month 5 ? F5 = 0. 40(95) + 0. 30(105) + 0. 20(90) + 0. 10(100) = 38 + 31. 5+ 18+ 10 = 97. 5 The formula for the weighted moving average is F =w A +w A +w A + …….. + w A t 1 t ? 1 2 t ? 2 3 t ? 3 n t? Where Ft = Forecast for the coming period, n = the total number of periods in the forecast. wi = the weight to be given to the actual occurrence for the period t-i Ai = the actual occurrence for the period t-i Although many periods may be ignored (that is, their weights are zero) and the weighting scheme may be in any order (for example, more distant data may have greater weights than more recent data), the sum of all the weights must equal 1. n ? w =1 i i =1 Suppose sales for month 5 actually turned out to be 110. Then the forecast for month 6 would be F6 = 0. 40(110) + 0. 30(95) + 0. 20(105) + 0. 10(90) = 44 + 28. 5 + 21 + 9 = 102. Choosing Weights : Experience and trial and error are the simplest ways to choose weights. As a general rule, the most recent past is the most important indicator of what to expect in the future, and, therefore, it should get higher weighting. The past month’s revenue or plant capacity, for example, would be a better estimate for the coming month than the revenue or plant capacity of several months ago. 155 However, if the data are seasonal, for example, weights should be established accordingly. For example, sales of air conditioners in May of last year should be weighted more heavily than sales of air conditioners in December.
The weighted moving average has a definite advantage over the simple moving average in being able to vary the effects of past data. However, it is more inconvenient and costly to use than the exponential smoothing method, which we examine next. 6. 6. 4 Exponential Smoothing In the previous methods of forecasting (simple and weighted moving average), the major drawback is the need to continually carry a large amount of historical data. (This is also true for regression analysis techniques, which we soon will cover) As each new piece of data is added in these methods, the oldest observation is dropped, and the new forecast is calculated.
In many applications (perhaps in most), the most recent occurrences are more indicative of the future than those in the more distant past. If this premise is valid – “that the importance of data diminishes as the past becomes more distant” – then exponential smoothing may be the most logical and easiest method to use. The reason this is called exponential smoothing is that each increment in the past is decreased by (1-? ). If ? is 0. 05 for example, weights for various period would be as follows (? is defined below): Weighting at ? = 0. 05 Most recent weighting = ? (1- ? ) 1 Data one time period older = ? (1- ? 2 Data two time periods older = ? (1- ? ) 3 Data three time periods older = ? (1- ? ) Therefore, the exponents 0, 1, 2, 3 and so on give it its name. i. e. the most current values receive a decreasing weighting. The exponential smoothing technique is a weighted moving average system and the underlying principle is that the 0 0. 0500 0. 0475 0. 0451 0. 0429 The method involves the automatic weighting of past data with weights that decrease exponentially with time, New Forecast = Old Forecast + a proportion of the forecast error The simplest formula is New forecast = Old forecast + ? (Latest Observation – Old Forecast) where ? alpha) is the smoothing constant. Or more mathematically, Ft = Ft-1 + ? (At-1 – Ft-1) i. e Where Ft = ? At-1 + (1- ? ) Ft-1 Ft = The exponentially smoothed forecast for period t Ft-1 = The exponentially smoothed forecast made for the prior period At-1 = The actual demand in the prior period ? = The desired response rate, or smoothing constant 156 The smoothing constant The value of ? can be between 0 and 1. The higher value of ? (i. e. the nearer to 1), the more sensitive the forecast becomes to current conditions, whereas the lower the value, the more stable the forecast will be, i. e. t will react less sensitively to current conditions. An approximate equivalent of alpha values to the number of periods’ moving average is given below: ? value 0. 1 0. 25 0. 33 0. 5 Approximate periods in equivalent Moving average 19 7 5 3 The total of the weights of observations contributing to the new forecast is 1 and the weight reduces exponentially progressively from the alpha value for the latest observation to smaller value for the older observations. For example, if the alpha value was 0. 3 and June’s sales were being forecast, then June’s forecast is produced from averaging past sales weighted as follows. 0. (May’s Sales) + 0. 21 (April’s Sales) + 0. 147 (March’s Sales) + 0. 1029 (February Sales) + 0. 072 (January Sales) + 0. 050 (December Sales), etc. In the above calculation, the reader will observe that ? (1- ? ) = 0. 3, ? (1- ? ) = 0. 21, ? (1- ? ) = 0. 147 0 1 2 ? (1- ? )3 = 0. 1029 and so on. From this it will be noted that the weightings calculated approach a total of 1. Exponential smoothing is the most used of all forecasting techniques. It is an integral part of virtually all computerized forecasting programs, and it is widely used in ordering inventory in retail firms, wholesale companies, and service agencies.
Exponential smoothing techniques have become well accepted for six major reasons: 1. 2. 3. 4. 5. 6. Exponential models are surprisingly accurate Formulating an exponential model is relatively easy The user can understand how the model works Little computation is required to use the model Computer storage requirement are small because of the limited use of historical data Tests for accuracy as to how well the model is performing are easy to compute In the exponential smoothing method, only three pieces of data are needed to forecast the future: the most recent forecast, the actual demand that ccurred for that forecast period and a smoothing constant alpha (? ). This smoothing constant determines the level of smoothing and the speed of reaction to differences between forecasts and actual occurrences. The value for the constant is determined both by the nature of the product and by the manager’s sense of what constitutes a good response rate. For example, if a firm produced a standard item with relatively stable demand, the reaction rate to difference between actual and forecast demand would tend to be 157 small, perhaps just 5 or 10 percentage points.
However, if the firm were experiencing growth, it would be desirable to have a higher reaction rate, perhaps 15 to 30 percentage points, to give greater importance to recent growth experience. The more rapid the growth, the higher the reaction rate should be. Sometimes users of the simple moving average switch to exponential smoothing but like to keep the forecasts about the same as the simple moving average. In this case, ? is approximated by 2 ? (n+1), where n is the number of time periods. To demonstrate the method once again, assume that the long-run demand for the product under study is relatively stable and a smoothing constant (? of 0. 05 is considered appropriate. If the exponential method were used as a continuing policy, forecast would have been made for last month. Assume that last month’s forecast (Ft-1) was 1,050 units. If 1,000 actually were demanded, rather than 1,050, the forecast for this month would be Ft =Ft-1 + a (At-1 – Ft-1) = 1,050 + 0. 05 (1,000 – 1,050) = 1,050 + 0. 05 (-50) = 1,047. 5 units Because the smoothing coefficient is small, the reaction of the new forecast to an error of 50 units is to decrease the next month’s forecast by only 2. 5 units. Illustration 6. : The data are given in the first two columns and the forecasts have been prepared using the values of ? as 0. 2 and 0. 8. Month January February March April May June July August September October November December January Actual units sold 450 440 460 410 380 400 370 360 410 450 470 490 460 450 ** Exponential Forecasts ? = 0. 2 450** 450 + 0. 8(440-450) =442 442 + 0. 8(460-442) =456. 4 456. 4 + 0. 8(410-456. 4) =419. 28 387. 86 397. 57 375. 51 363. 102 400. 62 440. 12 464. 02 484. 80 ? = 0. 8 450 + 0. 2 ( 440-450) = 448 448 + 0. 2 ( 460-448) = 450. 4 450. 4 + 0. 2 (410 – 450. 4) = 442. 32 429. 86 423. 9 413. 11 402. 49 403. 99 413. 19 424. 55 437. 64 ** In the above example, no previous forecast was available. So January sales were used as February’s forecast. The reader is advised to check the calculations for himself 158 It is apparent that the higher ? value, 0. 8, produces a forecast which adjusts more readily to the most recent sales. Extensions of exponential smoothing The basic principle of exponential smoothing has been outlined above, but to cope with various problem such as seasonal factors strongly, rising or failing demand, etc many developments to the basic model have been made.
These include double and triple exponential smoothing and correction for trend and delay factors, etc. These are outside the scope of the present book, so are not covered. Characteristics of exponential smoothing a) c) Greater weight is given to more recent data Less data needs to be stored than with the longer period moving averages. data becomes available and so it is frequently incorporated as an integral part of stock control and production control systems. e) f) To cope with various problems (trend, seasonal factors, etc) the basic model needs to be modified Whatever form of xponential smoothing is adopted, changes to the model to suit changing conditions can simply be made by altering the ? value. g) The selection of the smoothing constant ? is done through trial-error by the researcher/analyst. It is done by testing several values of ? (within the range 0 to 1) and selecting one which gives a forecast with the least error (one can take standard error). It has been found that values in the range 0. 1 to 0. 3 provide a good starting point. b) All past data are incorporated there is no cut-off point as with moving averages d) Like moving averages it is an adaptive forecasting system.
That is, it adapts continually as new Illustration 6. 5: Data on production of an item for 30 periods are tabulated below. Determine which value of the smoothing constant (? ), out of 0. 1 and 0. 3, will lead to the best simple exponential smoothing model. The first 15 values can be used for initialization of the model and then check the error in the forecast as asked after the table. Month 1 2 3 4 5 6 7 8 9 10 Use the Production (in tones) 30. 50 28. 80 31. 50 29. 90 31. 40 33. 50 25. 70 32. 10 29. 10 30. 80 Month 11 12 13 14 15 16 17 18 19 20 Production (in tones) 25. 70 30. 90 31. 50 28. 10 30. 80 29. 50 29. 0 30. 00 29. 90 31. 50 Month 21 22 23 24 25 26 27 28 29 30 Production (in tones) 27. 60 29. 90 30. 20 35. 50 30. 80 28. 80 30. 80 32. 20 31. 20 29. 80 total squared error or the mean squared error as the basis of comparison of the performances. The total squared error is the sum of the squares of all the error figures for the period selected (here only the last 15 figures because the first 15 periods will initialize the forecast). Their mean is the mean squared error. 159 Solution: The following table give the forecasted values for the two different values of the smoothing constant for the first 15 periods. = 0. 1 Forecast 30. 50 30. 33 30. 45 30. 39 30. 49 30. 79 30. 28 30. 47 30. 33 30. 38 29. 91 30. 01 30. 16 29. 95 30. 04 30. 50 29. 90 30. 44 30. 28 3. 62 31. 48 29. 75 30. 45 30. 05 30. 27 28. 90 29. 50 30. 10 29. 50 29. 89 ? = 0. 3 Forecast Month 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Production ( in tonnes) 30. 50 28. 80 31. 50 29. 90 31. 40 33. 50 25. 70 32. 10 29. 10 30. 80 25. 70 30. 90 31. 50 28. 10 30. 80 The following table now gives the forecasted values and also checks the errors in the forecast for the last 15 periods. Month
Production ( in tonnes) Forecast 30. 04 29. 98 29. 96 29. 97 29. 96 30. 12 29. 86 29. 87 29. 90 30. 46 30. 49 30. 32 30. 37 30. 56 30. 62 ? = 0. 1 error -0. 54 -0. 18 0. 04 -0. 07 1. 54 -2. 52 0. 04 0. 33 5. 60 0. 34 -1. 69 0. 48 1. 83 0. 64 -0. 82 Squared error 0. 29 0. 03 0. 0 0. 0 2. 37 6. 33 0. 0 0. 11 31. 35 0. 12 2. 87 0. 23 3. 34 0. 42 0. 67 48. 13 =48. 13/15 =3. 20 forecast 29. 89 29. 77 29. 78 29. 85 29. 86 30. 35 29. 53 29. 64 29. 81 31. 52 31. 30 30. 55 30. 63 31. 10 31. 13 ? = 0. 3 Squared error 0. 15 0. 0 0. 05 0. 0 2. 68 7. 58 0. 14 0. 1 32. 40 0. 51 6. 25 0. 06 2. 48 0. 01 1. 76 54. 41 =54. 41/15 =3. 62 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 29. 50 29. 80 30. 00 29. 90 31. 50 27. 60 29. 90 30. 20 35. 50 30. 80 28. 80 30. 80 32. 20 31. 20 29. 80 TOTAL SQUARED ERROR MEAN SQUARED ERROR 160 The results here indicate that the forecast accuracy is better for ? = 0. 1 as compared to 0. 3. They also indicate that a search around 0. 1 may lead to a more refined single exponential smoothing model. 6. 7 Evaluating the forecast accuracy: There are many ways to measure forecast accuracy.
Some of these measures are the mean absolute forecast error, called the MAD (Mean Absolute Deviation), the mean absolute percentage error (MAPE) and the mean square error (MSE) Error = Actual Observed value – Forecasted value Absolute Percentage Error = (Error / Actual Observed Value) ? 100 MAD = the average of the absolute errors MAPE = the average of the Absolute Percentage Errors MSE = the average of the squared errors It is common for two forecasting models to be ranked differently depending on the accuracy measure used. For example, model A may give a smaller MAD but a larger MSE than model B. Why?
Because the MAD gives equal weight to each error. The MSE gives more weight to large errors because they are squared. It is up to the manager, not the management scientist to decide which accuracy measure is most appropriate for his or her application. The MSE is most often used in practice. Given a preferred accuracy measure, how do we know when our forecasts are good, bad, or indifferent? One way to answer this question is to compare the accuracy of a given model with that of a benchmark model. A handy benchmark is the naive model, which assumes that the value of the series next period will be the same as it is this period; i. ,say Ft+1 = Xt where F is the forecast and X is the observed value. The subscript t is an index for the time period. The current period is t + 1. The first step in any forecasting problem should be to use the naive model to compute the benchmark accuracy. A model which cannot beat the naive model should be discarded. Checking model accuracy against that of the naive model may seem to be a waste of time, but unless we do so, it is easy to choose an inappropriate forecasting model. The mean error measures are computed only for the last half of the data. The forecasting models are evaluated by dividing the data in to two parts.
The first part is used to fit the forecasting model. Fitting consists of running the model through the first part of the data to get “warmed up. ” We call the fitting data the warm-up sample. The second part of the data is used to test the model and is called the forecasting sample. Accuracy in the warm-up sample is really irrelevant. Accuracy in the forecasting sample is more important because the pattern of the data often changes over time. The forecasting sample is used to evaluate how well the model tracks such changes. This point will be explained in detail in the next few sections.
There are no statistical rules on where to divide the data into warm-up samples and forecasting samples. There may not be enough data to have two samples. A good rule of thumb is to put at least six non seasonal data points or two complete seasons of seasonal data in the warm-up sample. If there are fewer data than this, there is no need to bother with two samples. In a long time series, it is common practice simply to divide the data in half. 161 Illustration 6. 6: Data regarding the sales of a particular item in the 12 time periods is given below. The manager wants to forecast 1 time period ahead in order to plan properly.
Determine the forecasts using (a) Naive method (b) 3 period moving average (c) simple exponential smoothing taking ? = 0. 1. Also compute the errors MAD, MAPE, and MSE to check the forecasting accuracy for the last six periods. Solution: (a) The following table shows the naive forecasting model. In this model the next period’s forecast is the present period’s actual observed value Time period (T) 1 2 3 4 5 6 7 8 9 10 11 12 13 Demand (D) 28 27 33 25 34 33 35 30 33 35 27 29 Forecast (F) 28 27 33 25 34 33 35 30 33 35 27 29 Absolute Error (E = ¦D – F ¦) 2 5 3 2 8 2 22 Absolute Percentage error (E/D) ? 100 5. % 16. 7% 9. 1% 5. 7% 29. 6% 6. 9% 73. 7% Squared Error (E2) 4 25 9 4 64 4 110 Total of time periods 7 to 12 to check forecasting accuracy MAD = 22 / 6 = 3. 7 MAPE = 73. 7% / 6 = 12. 3% MSE = 110 / 6 = 18. 3 162 (b) The following table shows the 3 period moving average model. Time period (T) Demand (D) 28 27 33 25 34 1 2 3 4 5 33 6 35 7 30 8 33 9 35 10 27 11 29 12 13 MSE ( periods 7 to 12 ) = ( 4. 32 +42 +0. 32 +2. 32 + 5. 72 +2. 72)/6 = 13. 3 Forecast by 3 period moving average (F) (28+27+33)/3 = 29. 3 (27+33+25)/3 =28. 3 30. 7 30. 7 34. 0 32. 7 32. 7 32. 7 31. 7 30. 3 Absolute Error (E) 4. 3 5. 7 2. 4. 3 4. 0 0. 3 2. 3 5. 7 2. 7 – (c) The simple exponential smoothing model with smoothing constant ? = 0. 1 is presented below. Time period (T) 1 2 3 4 5 6 7 8 9 10 11 12 13 Demand (D) 28 27 33 25 34 33 35 30 33 35 27 29 – Forecast by exponential smoothing with ? = 0. 1 (F) 30 30 + 0. 1(28-30) = 29. 8 29. 8 + 0. 1(27-29. 8) = 29. 5 29. 5 + 0. 1(33-29. 5) = 29. 9 29. 4 29. 9 30. 2 30. 7 30. 6 30. 8 31. 2 30. 8 30. 6 Absolute Error (E) 2 2. 8 3. 5 4. 9 4. 6 3. 1 4. 8 0. 7 2. 4 4. 2 4. 2 1. 8 MSE ( periods 7 to 12 ) = ( 4. 82 +0. 72 +2. 42 +4. 22 + 4. 22 +1. 82)/6 = 11. 3 163 6. 8 Trend Projections:
This time-series forecasting method fits a trend line to a series of historical data points and then projects the line into the future for medium- to long range forecasts. There are several mathematical trend equations that can be developed viz. linear, exponential, quadratic etc. Here we will concentrate only on the linear trends. Of the components of a time series, secular trend represents the long-term direction of the series. One way to describe the trend component is to fit a line visually to a set of points on a graph. Any given graph, however, is subject to slightly different interpretations by different individuals.
We can also fit a trend line by the method of least squares. In our discussion, we will concentrate on the method of least squares because visually fitting a line to a time to series is not a completely dependable process. Reasons for Studying Trends There are three reasons why it is useful to study secular trends: 1. 2. The study of secular trends allows us to describe a historical pattern. Studying secular trends permits us to project past patterns, or trends, into the future. In many situations, studying the secular trend of a time series allows us to eliminate the trend component from the series. 3. 6. 8. 1 Linear Regression Analysis
Regression can be defined as a functional relationship between two or more correlated variables. It is used to predict one variable given the other. The relationship is usually developed from observed data. The data should be plotted first to see if they appear linear or if at least parts of the data are linear. Linear regression refers to the special class of regression where the relationship between variables forms a straight line. The linear regression line is of the form Y = a + bX, where Y is the value of the dependent variable that we are solving for, a is the Y intercept, b is the slope, and X is the independent variable. In time series analysis, X is units of time) Linear regression is useful for long-term forecasting of major occurrences and aggregate planning. For example, linear regression would be very useful to forecast demands for product families. Even though demand for individual products within a family may vary widely during a time period, demand for the total product family is surprisingly smooth. The major restriction in using linear regression forecasting is, as the name implies, that past data and future projections are assumed to fall about a straight line.
Although this does limit its application, sometimes, if we use a shorter period of time, linear regression analysis can still be used. For example, there may be short segments of the longer period that are approximately linear. Linear regression is used both for time series forecasting and for casual relationship forecasting. When the dependent variable (usually the vertical axis on the graph) changes as a result of time (plotted on the horizontal axis), it is time series analysis.
When the dependent variable changes because of the change in another variable, this is a casual relationship (such as the demand of cold drinks increasing with the temperature). We illustrate the linear trend projection with a hand fit regression line. 164 Illustration 6. 7 : A firms sales for a product line during the 12 quarters of the past three years were as follows. Quarter 1 2 3 4 5 6 Sales 600 1550 1500 1500 2400 3100 7 8 9 10 11 12 Quarter Sales 2600 2900 3800 4500 4000 4900 Forecast the sales for the 13, 14, 15 and 16th quarters using a hand-fit regression equation.
Solution: The procedure is quite simple: Lay a straightedge across the data points until the line seems to fit well, and draw the line. This is the regression line. The next step is to determine the intercept a and slope b. The following fig shows a plot of the data and the straight line we drew through the points. 7000 6000 5000 4000 3000 2000 1000 0 1 3 5 7 9 11 13 15 The intercept a, where the line cuts the vertical axis, appears to be about 400. The slope b is the “rise” divided by the “run” (the change in the height of some portion of the line divided by the number of units in the horizontal axis).
Any two points can be used, but two points some distance apart give the best accuracy because of the errors in reading values from the graph. We use values for the 1st and 12th quarters. By reading from the points on the line, the Y values for quarter 1 and quarter 12 are about 750 and 4,950. Therefore. b = (4950 – 750) / (12- 1) = 382 The hand-fit regression equation is therefore Y = 400 + 382X 142 The forecasts for quarters 13 to 16 are Quarter 13 14 15 16 Forecast 400 + 382(13) = 5366 400 + 382(14) = 5748 400 + 382(15) = 6130 400 + 382(16) = 6512
These forecasts are based on the line only and do not identify or adjust for elements such as seasonal or cyclical elements. 6. 8. 2 Least Squares Method for Linear Regression: The least squares equation for linear regression is Y = a + bX Where, Y = Dependent variable computed by the equation y = The actual dependent variable data point (used below) a = y intercept , b = Slope of the line, X = Time period The least squares method tries to fit the line to the data that minimize the sum of the sum of the squares of the vertical distance between each data point and its corresponding point on the line.
If a straight line is drawn through general area of the points, the difference between the point and the line is y – Y. The sum of the squares of the differences between the plotted data points and the line points is (y1 – Y1)2 + (y2 – Y2)2 + ….. + (y12 – Y12)2 The best line to use is the one that minimizes this total. As before, the straight line equation is Y = a + bX Previously we determined a and b from the graph. In the least squares method, the equations for a and b are and a = y ? bX Where b = ? Xy ? n X . y 2 2 ? X ? nX a = Y intercept , b = slope of the line , n = number of data points.
We discuss the procedure to fit a straight line by the least squares method with the help of the following illustration and then we will compare the results obtained by hand fitting and the fitting by the method of least squares. Illustration 6. 8: Forecast the sales for the 13, 14, 15 and 16th quarters for the data given in illustration 7 using the least squares method. Also calculate the standard error of the estimate. 143 Solution: The following table shows the computations carried out for the 12 data points. X 1 2 3 4 5 6 7 8 9 10 11 12 Total :78 600 1550 1500 1500 2400 3100 2600 2900 3800 4500 4000 4900 33350 Xy 600 3100 4500 6000 12000 18600 18200 23200 34200 45000 44000 58800 X2 1 4 9 16 25 36 49 64 81 100 121 144 y2 360000 2402500 2250000 2250000 5760000 9610000 6760000 8410000 14440000 20250000 16000000 24010000 112502500 Y 801. 3 1160. 9 1520. 5 1880. 1 2239. 7 2599. 4 2959. 0 3318. 6 3678. 2 4037. 8 4397. 4 4757. 1 268200 b = 359. 61 650 X = 6. 5 y = 2779. 17 a = 441. 66 Y = 441. 66 + 359. 6 X Syx = 363. 9 The reader is advised to verify the calculations of a and b on his own.
Note that the final equation for Y shows an intercept of 441. 6 and a slope of 359. 6. The slope shows that for every unit change in X, Y changes by 359. 6. Strictly based on the equation, forecasts for periods 13 through 16 would be Y13 = 441. 6 + 359. 6(13) = 5116. 4 Y14 = 441. 6 + 359. 6(14) = 5476. 0 Y15 = 441. 6 + 359. 6(15) = 5835. 6 Y16 = 441. 6 + 359. 6(16) = 6195. 2 7000 6000 5000 4000 3000 2000 1000 0 1 3 5 7 9 11 13 15 144 The reader is also advised to verify the results for the forecasts for the above two illustrations 7 and 8.
The standard error of the estimate is computed as SyX which is given as follows S (600 ? 801. 3)2 + (1550 ? 1160. 9)2 + …….. (4900 ? 4757. 1)2 yX 12 = 363. 9 = We have plotted the graph of the values of the actual demand, forecasted demand values by the two methods of hand fitting and the method of least squares. Following is the graph. 6000 5000 4000 3000 2000 1000 0 1 3 5 7 9 11 Series1 Series2 Series3 Here, Series 1 is the actual observed demand values, Series 2 are the values based on calculations by the method of least squares and Series 3 are the values by hand fitting the trend line.
The following table compares the forecasted values for the 13th, 14th ,15th and the 16th quarters based on illustrations 7 and 8. quarter 13 14 15 16 Forecasts by the hand fit line (line going above in the figure) Series 3 5366 5748 6130 6512 Forecasts by the least squares method (line beneath the hand fit line) series 2 5116. 4 5476. 0 5835. 6 6195. 2 145 Converting the time for ease in calculations: Normally, we measure the independent variable time in terms such as weeks, months, and years. Fortunately, we can convert these traditional measures of time to a form that simplifies the computation, we call this process coding.
To use coding here, we find the mean time and then subtract that value from each of the sample times. Suppose our time series consists of only three points, 1992, 1993, and 1994. If we had to place these numbers in the above least squares equations we would find the resultant calculations tedious. Instead, we can transform the values 1992, 1993, and 1994 into corresponding values of -1,0, and 1, where 0 represents the mean (1993), -1 represents the first year (1992 -1993= -1), and 1 represents the last year (1994 – 1993 = 1) or alternatively to 1, 2 and 3.
We need to consider two cases when we are coding time values. The first is a time series with an odd number of elements, as in the previous example. The second is a series with an even number of elements. Consider the following table. In part (A), on the left, we have an odd number of years. Thus, the process is the same as the one we just described, using the years 1992, 1993, and 1994. In part (B), on the right, we have an even number of elements. In cases like this, when we find the mean and subtract it from each element, the fraction 1/2 becomes part of the answer.
To simplify the coding process and to remove the 1/2, we multiply each time element by 2. We will denote the “coded,” or translated, time with a lowercase x. Part (A) Odd number of entries Part (B) Even number of entries Coded time(x) X 1989 1990 1991 1992 1993 1994 1995 X? X 1989 – 1992 = 1990 – 1992 = 1991 – 1992 = 1992 – 1992 = 1993 – 1992 = 1994 – 1992 = 1995 – 1992 = X 1990 1991 1992 1993 1994 1995 X? X 1990 – 1992. 5 = 1991 – 1992. 5 = 1992 – 1992. 5 = 1993 – 1992. 5 = 1994 – 1992. 5 = 1995 – 1992. 5 = X? X? 2 -2. 5 ? 2 = -1. 5 ? 2 = -0. 5 ? 2 = 0. 5 ? 2 = 1. 5 ? = 2. 5 ? 2 = Coded time(x) -5 -3 -1 1 3 5 -3 -2 -1 0 1 2 3 ? X = 13944 X = 1992 ? X = 11955 X = 1992. 5 146 We have two reasons for this translation of time. First, it eliminates the need to square numbers as large as 1992, 1993, 1994, and so on. This method also sets the mean year, x , equal to zero and allows us to simplify the least squares equations of the Y intercept, a and the slope b as follows. b= ? xy 2 ? x a= y where x represents the coded values. Illustration 6. 9: The following table gives the number of items produced in a factory between 1988 and 1995.
Year Production 1988 98 1989 105 1990 116 1991 119 1992 135 1993 156 1994 177 1995 208 Determine the equation that will describe the secular trend of production. Also project the production for 1996. Solution: The following table calculates the necessary values. X 1988 1989 1990 1991 1992 1993 1994 1995 Y 98 105 116 119 135 156 177 208 X? X -3. 5 -2. 5 -1. 5 -0. 5 0. 5 1. 5 2. 5 3. 5 x = X ? X ? 2 -7 -5 -3 -1 1 3 5 7 xY -686 -525 -348 -119 135 468 885 1456 x2 49 25 9 1 1 9 25 49 2 ? X = 15932 ? Y = 1114 X = 1991. 5 Y = 139. 25 ? xY = 1266 ? x = 168
With these values, we can now compute b and a to find the slope and the Y intercept for the line describing the trend in production. b= ? xy 2 ? x b = 1266 / 168 = 7. 536 and a = 139. 25 a= y 147 Thus, the general linear equation describing the secular trend in production is Y = a + bx = 139. 25 + 7. 536 x where Y = dependent variable computed by the equation OR the estimated production calculated x = coded time value representing the number of half year intervals ( a – sign indicates half year intervals before 1991. 5, a + sign indicates half year intervals after 1991. ) Now suppose we want to estimate the production for the year 1996. First, we must convert 1996 to the value of the coded time ( in half year intervals) x = 1996 – 1991. 5 = 4. 5 years = 9 half year intervals Substituting this value in the equation for the secular trend, we get Y = 139. 25 + 7. 536 (9) = 139. 25 + 67. 82 = 207 Therefore we estimate the production for the year 1996 as 207. Note: If the number of elements in our time series would have been odd, our procedure would have been the same except that we would have dealt with 1 year intervals. 6. Decomposition of the time series We have seen earlier that observations taken over time (i. e. time series) often contain the four following characteristics : (a) A long-term trend (denoted by T) (b) Seasonal variations (denoted by S) (c) Cyclical variations (denoted by C) (d) Random or residual variations (denoted by R) The methods covered so far do not make any attempt to isolate the individual factors, namely, seasonally, trend, cyclical and random variations, in the time series. But there are many situations where such a breaking down of the time series is possible and necessary.
The decomposition methods basically operate on the principle that a time series is composed of the four factors stated earlier. 148 The decomposition methods assume the time series value at time t to be a function of the different components, i. e. , where Dt = f (Tt , St , Ct , Rt ) Tt = trend value at period t St = seasonal component at period t Ct = cyclical component at period t, and Rt = random variation at period t. To make reasonably accurate forecasts, it is often necessary to separate out the above characteristics (i. e. T, S, C and R) from the raw data.
This is known as time series decomposition or often just time series analysis. The separated elements are then combined to produce a forecast. The functional form for the series used may either be additive or multiplicative. The multiplicative form (most commonly used) is written as follows: Dt = Tt ? St ? Ct ? Rt Here the components are expressed as percentages or proportions The additive model takes the form Dt = Tt + St + Ct + Rt Here the components are expressed as absolute values The multiplicative model is commonly used in practice and is more appropriate if the characteristics interact, e. g. here a higher trend value increases the seasonal variation. The additive model is more suitable if the component factors are independent, e. g. where the amount of seasonal variation is not affected by the value of the trend. Of the four elements the most important are the first two; the trend and seasonal variation, so this book concentrates on these two. Seasonal Factor (Or Index) A seasonal factor is the amount of correction needed in a time series to adjust for the season of the year. Basically, the seasonal factor (or index) is the ratio of the amount sold during each season divided by the average for all seasons.
The following examples show how seasonal indices are determined and used to forecast. Illustration 6. 10: Assume that in the past years, a firm sold an average of 1000 units of air conditioners each year. On the average, 200 units were sold from the period January to March, 350 in April to June, 300 in July to September and 150 in October to December. Calculate the seasonal indices for each season (quarter). If the expected demand for the next year is believed to be 1100, forecast the demand in each quarter. 149 Solution: First we find the average sales for each season and then divide the sales of each season by that average.
Past sales Jan-Mar Apr-June July-Sept Oct-Dec Total 200 350 300 150 1000 Average sales for each season = Total / 4 = 1000 / 4 250 250 250 250 200 / 250 =0. 8 350 / 250 = 1. 4 300 / 250 = 1. 2 150 / 250 = 0. 6 Seasonal factor Using these factors, if we expected demand for next year to be 1100 units, we would forecast the demand to occur as shown in the following table. First we distribute the demand equally among all the quarters and then divide those amounts by the seasonal index obtained in the previous step. Expected demand For next year Jan-Mar Apr-June July-Sept Oct-Dec Total 1100
Average sales for each season = 1100 / 4 275 275 275 275 Seasonal Factor ? ? ? ? 0. 8 1. 4 1. 2 0. 6 = = = = Next years seasonal forecast 220 385 330 165 The seasonal factors are updated periodically as the new data are available. This is how we can forecast based on the past seasonal data and knowing the future expected demand. One issue in the above illustration was that the expected demand for the next year was known to be 1100. When not known, we can compute the seasonal indices by even using a hand-fir straight line. This procedure is given in the following illustration.
Illustration 6. 11: Simply hand fit a straight line through the data points and measure the trend and the intercept from the graph. The history of the data is Year 2004 2004 2004 2004 Quarter 1 2 3 4 Sales 300 200 220 530 Year 2005 2005 2005 2005 Quarter 1 2 3 4 Sales 520 420 400 700 150 Find the seasonal factors for the quarters and using them forecast the sales in the quarters of 2006. Solution: First we plot the data points on the graph and then fit a straight line. (The reader can fit a straight line in a manner different from this.
Naturally then his estimates are likely to vary a bit. You can also fit a straight line by the method of least squares). 800 700 600 500 400 300 200 100 0 1 2 3 4 5 6 7 8 The equation for the trend line is Tt = 170 + 55t , which is derived from the intercept 170 and a slope of (610170) / 8 = 55. Next we derive a seasonal index by comparing the actual data with the trend line as shown below. Quarter Actual Calculated from the trend 170 + 55t 2004 Ration of Actual / Trend Seasonal factor (Average of the same quarters in both years) 1 2 3 4 2005 300 200 220 530 520 420 400 700 170 + 55(1) = 225 = 170 + 55(2) = 280 = 170 + 55(3) = 335 = 170 + 55(4) = 390 = 170 + 55(5) = 445 = 170 + 55(6) = 500 = 170 + 55(7) = 555 = 170 + 55(8) = 610 1. 33 0. 71 0. 66 1. 36 1. 17 0. 84 0. 72 1. 15 Q-1 : (1. 33 + 1. 17)/2 = 1. 25 Q-2 : (0. 71 + 0. 84)/2 = 0. 78 Q-3 : (0. 66 + 0. 72)/2 = 0. 69 Q-4 : (1. 36 + 1. 15)/2 = 1. 25 1 2 3 4 151 We can now compute the forecast for the quarters in 2006 including trend and the seasonal factors using the following equation: Forecast = Trend value ? seasonal factor Year 2006 Q1 Q2 Q3 Q4 trend value using the trend equation 170 + 55t Seasonal index for the quarter
Forecast including the trend and seasonal factors Tt = 170 + 55(9) = 665 = 170 + 55(10) = 720 = 170 + 55(11) = 775 = 170 + 55(12) = 830 1. 25 0. 78 0. 69 1. 25 St 831 562 535 1038 Tt ? St With this knowledge of seasonal factors we now go on and study the decomposition of time series and forecast into the future. The following illustration shows how the trend (T) and seasonal variation (S) are separated out from a time series and how the seasonal indices are calculated. We can calculate the seasonal indices for the time series data by two methods Method 1. By using a trend line from the data by the least squares method and Method 2.
By using the method of ratio to moving average. We illustrate the first method in the following illustration. Illustration 6. 12: Estimate the sales of air conditioners for the quarters 17, 18, 19, 20 i. e the four quarters of the year 2005 using the following data by the method of decomposition of time series. Sale of air conditioners in ‘000s Year 2001 2002 2003 2004 Quarter 1 20 21 23 27 Quarter 2 32 42 39 39 Quarter 3 62 75 77 92 Quarter 4 29 31 48 53 It is apparent that there is a strong seasonal element in the above data (low in Quarter 1 and high in Quarter 3) and that there is a generally upward trend.
The steps in analyzing the data and preparing a forecast are: Step 1: Calculate the trend in the data using the least squares method. Step 2: Estimate the sales for each quarter using the regression formula established in Step 1. Step 3: Calculate the percentage variation of each quarter’s actual sales from the estimates, obtained in Step 2. Step 4: Average the percentage variations from Step 3 for each quarter. This establishes the average seasonal variations. Step 5: Prepare forecast based on trend ? percentage seasonal variations. 152 Step 1.
We use the following least squares linear regression equations to compute the regression line Y = a + bX ? y = an + b ? x ? xy = a ? x + b? x X (quarters) Y ( sales ) 20 32 62 29 21 42 75 31 23 39 77 48 27 39 92 53 2 XY 20 64 186 116 105 252 525 248 207 390 847 576 351 546 1380 848 X2 1 4 9 16 25 36 49 64 81 100 121 144 169 196 225 256 2 Year 2001 1 2 3 4 5 6 7 8 Year 2002 Year 2003 9 10 11 12 13 14 15 16 Year 2004 ? X = 136 ? Y = 710 ? XY = 6661 ? X We get the following two equations 710 = 16a + 136b 6661 = 136a + 1496b = 1496 Solving them we get a = 28. 74 and b =1. 84. Thus, the trend line equation is Y = 28. 4 + 1. 84X Step 2 and Step 3: Use the trend line to calculate the estimated sales for each quarter. Express the actual value of sales as a percentage of this estimated sales. ( Remember that this is similar to finding the ratio of actual to trend in the above two illustrations ) 153 X (quarters) Year 2001 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Y ( sales ) (A) 20 32 62 29 21 42 75 31 23 39 77 48 27 39 92 53 Estimated sales Using trend line (B) 30. 58 32. 42 34. 26 36. 10 37. 94 39. 78 41. 62 43. 46 45. 30 47. 14 48. 98 50. 82 52. 66 54. 50 56. 34 58. 18 (actual/Estimated)% = (A/B)? 100 5 99 181 80 55 106 180 71 51 83 157 94 51 72 163 91 Year 2002 Year 2003 Year 2004 Step 4: Average the percentage variations to find the average seasonal variation. In % 2001 2002 2003 2004 Total ? 4= Q1 65 55 51 51 222 56% Q2 99 106 83 72 360 90% Q3 181 180 157 163 681 170% Q4 80 71 94 91 336 84% These then are the average variations expected from the trend for each of the quarters; for example, on average the first quarter of each year will be 56% of the value of the trend. Because the variations have been averaged, the amounts over 100% (Q3 in this example) should equal the amounts below 100%. Ql, Q2 and Q4 in this example); This can be checked by adding the average variations and verifying that they total 400% thus: 56% + 90% + 170% + 84% = 400%. On occasions, roundings in the calculations will make slight adjustments necessary to the average variations. We will discuss this in the next illustration. 154 Step 5. Prepare final forecasts based on the trend line estimates and the averaged seasonal variations. The seasonally adjusted forecast is calculated thus: Seasonally adjusted forecast = Trend estimate x Seasonal factor X (quarters) Year 2001 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Y ( sales ) 0 32 62 29 21 42 75 31 23 39 77 48 27 39 92 53 Estimated sales Using trend line 30. 58 32. 42 34. 26 36. 10 37. 94 39. 78 41. 62 43. 46 45. 30 47. 14 48. 98 50. 82 52. 66 54. 50 56. 34 58. 18 Seasonal factor 0. 56 0. 90 1. 7 0. 84 0. 56 0. 90 1. 7 0. 84 0. 56 0. 90 1. 7 0. 84 0. 56 0. 90 1. 7 0. 84 Seasonally adjusted forecast 17. 12 29. 18 58. 24 30. 32 21. 24 35. 80 70. 75 36. 51 25. 37 42. 43 83. 27 42. 69 29. 49 49. 05 95. 75 48. 87 Year 2002 Year 2003 Year 2004 Extrapolation using the trend and seasonal factors Once the formulae above have been calculated, they can be used to forecast (extrapolate) future sales.
Here it is required to estimate the sales for the year 2005 (i. e. Quarters 17, 18,19 and 20 in our series) this is done as follows: Year 2005 Q1 Q2 Q3 Q4 Number of the quarter in the time series (X) 17 18 19 20 Estimated trend Using the trend equation Y=28. 74 + 1. 84X ( T) = 28. 74 + 1. 84(17) = 60. 02 = 28. 74 + 1. 84(18) = 61. 86 = 28. 74 + 1. 84(19) = 63. 7 = 28. 74 + 1. 84(20) = 65. 54 Seasonal factor (S) 0. 56 0. 90 1. 7 0. 84 Forecasted sales T’s 33. 61 55. 67 108. 29 55. 05 155 120 100 80 60 40 20 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 The above figure shows the actual sales ( ), the trend line ( ¦ ) and the seasonally adjusted forecast ( ^ ). Notes : (a) Time series decomposition is not an adaptive forecasting system like moving averages and exponential smoothing. (b) Forecasts produced by such an analysis should always be treated with caution. Changing conditions and changing seasonal factors make long term forecasting a difficult task. (c) The above illustration has been an example of a multiplicative model. This is because the seasonal variations were expressed in percentage or proportionate terms. Similar steps would have been necessary if the additive
