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Suppose we wish to calculate seasonal factors and a trend, then calculate the forecasted sales for July in year 5. The first step in the seasonal forecast will be to compute monthly indices using the past four-year sales. For example, for January the index is: S(Jan) = D(Jan)/D = 208.6/181.84 = 1.14, where D(Jan) is the mean of all four January months, and D is the grand mean of all past four-year sales.Similar calculations are made for all other months. Indices are summarized in the last row of the above table. Notice that the mean (average value) for the monthly indices adds up to 12, which is the number of periods in a year for the monthly data. Next, a linear trend often is calculated using the annual sales: Y = 1684 + 200.4T, The main question is whether this equation represents the trend. Determination of the Annual Trend for the Numerical Example Year No: Actual Sales Linear Regression Quadratic Regression 1 1972 1884 1981 2 2016 2085 1988 3 2160 2285 2188 4 2592 2486 2583 Often fitting a straight line to the seasonal data is misleading. By constructing the scatter-diagram, we notice that a Parabola might be a better fit. Using the Polynomial Regression JavaScript, the estimated quadratic trend is:Y = 2169 - 284.6T + 97T2Predicted values using both the linear and the quadratic trends are presented in the above tables. Comparing the predicted values of the two models with the actual data indicates that the quadratic trend is a much superior fit than the linear one, as often expected.We can now forecast the next annual sales; which, corresponds to year 5, or T = 5 in the above quadratic equation: Y = 2169 - 284.6(5) + 97(5)2 = 3171 sales for the following year. The average monthly sales during next year is, therefore: 3171/12 = 264.25. Finally, the forecast for month of July is calculated by multiplying the average monthly sales forecast by the July seasonal index, which is 0.79; i.e., (264.25).(0.79) or 209. You might like to use the Seasonal Index JavaScript to check your hand computation. As always you must first use Plot of the Time Series as a tool for the initial characterization process.For testing seasonality based on seasonal index, you may like to use the Test for Seasonality JavaScript.Trend Removal and Cyclical Analysis: The cycles can be easily studied if the trend itself is removed. This is done by expressing each actual value in the time series as a percentage of the calculated trend for the same date. The resulting time series has no trend, but oscillates around a central value of 100.Decomposition Analysis: It is the pattern generated by the time series and not necessarily the individual data values that offers to the manager who is an observer, a planner, or a controller of the system. Therefore, the Decomposition Analysis is used to identify several patterns that appear simultaneously in a time series. A variety of factors are likely influencing data. It is very important in the study that these different influences or components be separated or decomposed out of the 'raw' data levels. In general, there are four types of components in time series analysis: Seasonality, Trend, Cycling and Irregularity. Xt = St . Tt. Ct . IThe first three components are deterministic which are called "Signals", while the last component is a random variable, which is called "Noise". To be able to make a proper forecast, we must know to what extent each component is present in the data. Hence, to understand and measure these components, the forecast procedure involves initially removing the component effects from the data (decomposition). After the effects are measured, making a forecast involves putting back the components on forecast estimates (recomposition). The time series decomposition process is depicted by the following flowchart: Definitions of the major components in the above flowchart:Seasonal variation: When a repetitive pattern is observed over some time horizon, the series is said to have seasonal behavior. Seasonal effects are usually associated with calendar or climatic changes. Seasonal variation is frequently tied to yearly cycles.Trend: A time series may be stationary or exhibit trend over time. Long-term trend is typically modeled as a linear, quadratic or exponential function. Cyclical variation: An upturn or downturn not tied to seasonal variation. Usually results from changes in economic conditions.Seasonalities are regular fluctuations which are repeated from year to year with about the same timing and level of intensity. The first step of a times series decomposition is to remove seasonal effects in the data. Without deseasonalizing the data, we may, for example, incorrectly infer that recent increase patterns will continue indefinitely; i.e., a growth trend is present, when actually the increase is 'just because it is that time of the year'; i.e., due to regular seasonal peaks. To measure seasonal effects, we calculate a series of seasonal indexes. A practical and widely used method to compute these indexes is the ratio-to-moving-average approach. From such indexes, we may quantitatively measure how far above or below a given period stands in comparison to the expected or 'business as usual' data period (the expected data are represented by a seasonal index of 100%, or 1.0). Trend is growth or decay that is the tendencies for data to increase or decrease fairly steadily over time. Using the deseasonalized data, we now wish to consider the growth trend as noted in our initial inspection of the time series. Measurement of the trend component is done by fitting a line or any other function. This fitted function is calculated by the method of least squares and represents the overall trend of the data over time.Cyclic oscillations are general up-and-down data changes; due to changes e.g., in the overall economic environment (not caused by seasonal effects) such as recession-and-expansion. To measure how the general cycle affects data levels, we calculate a series of cyclic indexes. Theoretically, the deseasonalized data still contains trend, cyclic, and irregular components. Also, we believe predicted data levels using the trend equation do represent pure trend effects. Thus, it stands to reason that the ratio of these respective data values should provide an index which reflects cyclic and irregular components only. As the business cycle is usually longer than the seasonal cycle, it should be understood that cyclic analysis is not expected to be as accurate as a seasonal analysis. Due to the tremendous complexity of general economic factors on long term behavior, a general approximation of the cyclic factor is the more realistic aim. Thus, the specific sharp upturns and downturns are not so much the primary interest as the general tendency of the cyclic effect to gradually move in either direction. To study the general cyclic movement rather than precise cyclic changes (which may falsely indicate more accurately than is present under this situation), we 'smooth' out the cyclic plot by replacing each index calculation often with a centered 3-period moving average. The reader should note that as the number of periods in the moving average increases, the smoother or flatter the data become. The choice of 3 periods perhaps viewed as slightly subjective may be justified as an attempt to smooth out the many up-and-down minor actions of the cycle index plot so that only the major changes remain. Irregularities (I) are any fluctuations not classified as one of the above. This component of the time series is unexplainable; therefore it is unpredictable. Estimation of I can be expected only when its variance is not too large. Otherwise, it is not possible to decompose the series. If the magnitude of variation is large, the projection for the future values will be inaccurate. The best one can do is to give a probabilistic interval for the future value given the probability of I is known.Making a Forecast: At this point of the analysis, after we have completed the study of the time series components, we now project the future values in making forecasts for the next few periods. The procedure is summarized below.Step 1: Compute the future trend level using the trend equation.Step 2: Multiply the trend level from Step 1 by the period seasonal index to include seasonal effects.Step 3: Multiply the result of Step 2 by the projected cyclic index to include cyclic effects and get the final forecast result. Exercise your knowledge about how to forecast by decomposition method? by using a sales time series available at An Illustrative Application (a pdf file). Therein you will find a detailed workout numerical example in the context of the sales time series which consists of all components including a cycle. Smoothing Techniques: A time series is a sequence of observations, which are ordered in time. Inherent in the collection of data taken over time is some form of random variation. There exist methods for reducing of canceling the effect due to random variation. A widely used technique is "smoothing". This technique, when properly applied, reveals more clearly the underlying trend, seasonal and cyclic components.Smoothing techniques are used to reduce irregularities (random fluctuations) in time series data. They provide a clearer view of the true underlying behavior of the series. Moving averages rank among the most popular techniques for the preprocessing of time series. They are used to filter random "white noise" from the data, to make the time series smoother or even to emphasize certain informational components contained in the time series. Exponential smoothing is a very popular scheme to produce a smoothed time series. Whereas in moving averages the past observations are weighted equally, Exponential Smoothing assigns exponentially decreasing weights as the observation get older. In other words, recent observations are given relatively more weight in forecasting than the older observations. Double exponential smoothing is better at handling trends. Triple Exponential Smoothing is better at handling parabola trends.Exponential smoothing is a widely method used of forecasting based on the time series itself. Unlike regression models, exponential smoothing does not imposed any deterministic model to fit the series other than what is inherent in the time series itself.Simple Moving Averages: The best-known forecasting methods is the moving averages or simply takes a certain number of past periods and add them together; then divide by the number of periods. Simple Moving Averages (MA) is effective and efficient approach provided the time series is stationary in both mean and variance. The following formula is used in finding the moving average of order n, MA(n) for a period t+1, MAt+1 = [Dt + Dt-1 + ... +Dt-n+1] / n where n is the number of observations used in the calculation.The forecast for time period t + 1 is the forecast for all future time periods. However, this forecast is revised only when new data becomes available.You may like using Forecasting by Smoothing Javasript, and then performing some numerical experimentation for a deeper understanding of these concepts.Weighted Moving Average: Very powerful and economical. They are widely used where repeated forecasts required-uses methods like sum-of-the-digits and trend adjustment methods. As an example, a Weighted Moving Averages is:Weighted MA(3) = w1.Dt + w2.Dt-1 + w3.Dt-2where the weights are any positive numbers such that: w1 + w2 + w3 = 1. A typical weights for this example is, w1 = 3/(1 + 2 + 3) = 3/6, w2 = 2/6, and w3 = 1/6. You may like using Forecasting by Smoothing JavaScript, and then performing some numerical experimentation for a deeper understanding of the concepts.An illustrative numerical example: The moving average and weighted moving average of order five are calculated in the following table.WeekSales ($1000)MA(5)WMA(5)1105--2100--3105--495--51001011006959998710510010081201031079115107111101251171161112012011912120120119Moving Averages with Trends: Any method of time series analysis involves a different degree of model complexity and presumes a different level of comprehension about the underlying trend of the time series. In many business time series, the trend in the smoothed series using the usual moving average method indicates evolving changes in the series level to be highly nonlinear. In order to capture the trend, we may use the Moving-Average with Trend (MAT) method. The MAT method uses an adaptive linearization of the trend by means of incorporating a combination of the local slopes of both the original and the smoothed time series. The following formulas are used in MAT method:X(t): The actual (historical) data at time t.M(t) = å X(i) / n i.e., finding the moving average smoothing M(t) of order n, which is a positive odd integer number 3, for i from t-n+1 to t.F(t) = the smoothed series adjusted for any local trendF(t) = F(t-1) + a [(n-1)X(t) + (n+1)X(t-n) -2nM(t-1)], where constant coefficient a = 6/(n3 n). with initial conditions F(t) =X(t) for all t n, Finally, the h-step-a-head forecast f(t+h) is:F(t+h) = M(t) + [h + (n-1)/2] F(t).To have a notion of F(t), notice that the inside bracket can be written as:n[X(t) F(t-1)] + n[X(t-m) F(t-1)] + [X(t-m) X(t)],this is, a combination of three rise/fall terms. In making a forecast, it is also important to provide a measure of how accurate one can expect the forecast to be. The statistical analysis of the error terms known as residual time-series provides measure tool and decision process for modeling selection process.In applying MAT method sensitivity analysis is needed to determine the optimal value of the moving average parameter n, i.e., the optimal number of period m. The error time series allows us to study many of its statistical properties for goodness-of-fit decision. Therefore it is important to evaluate the nature of the forecast error by using the appropriate statistical tests. The forecast error must be a random variable distributed normally with mean close to zero and a constant variance across time. For computer implementation of the Moving Average with Trend (MAT) method one may use the forecasting (FC) module of WinQSB which is commercial grade stand-alone software. WinQSBs approach is to first select the model and then enter the parameters and the data. With the Help features in WinQSB there is no learning-curve one just needs a few minutes to master its useful features.Exponential Smoothing Techniques: One of the most successful forecasting methods is the exponential smoothing (ES) techniques. Moreover, it can be modified efficiently to use effectively for time series with seasonal patterns. It is also easy to adjust for past errors-easy to prepare follow-on forecasts, ideal for situations where many forecasts must be prepared, several different forms are used depending on presence of trend or cyclical variations. In short, an ES is an averaging technique that uses unequal weights; however, the weights applied to past observations decline in an exponential manner.Single Exponential Smoothing: It calculates the smoothed series as a damping coefficient times the actual series plus 1 minus the damping coefficient times the lagged value of the smoothed series. The extrapolated smoothed series is a constant, equal to the last value of the smoothed series during the period when actual data on the underlying series are available. While the simple Moving Average method is a special case of the ES, the ES is more parsimonious in its data usage. Ft+1 = a Dt + (1 - a) Ftwhere: Dt is the actual value Ft is the forecasted value a is the weighting factor, which ranges from 0 to 1 t is the current time period.Notice that the smoothed value becomes the forecast for period t + 1.A small a provides a detectable and visible smoothing. While a large a provides a fast response to the recent changes in the time series but provides a smaller amount of smoothing. Notice that the exponential smoothing and simple moving average techniques will generate forecasts having the same average age of information if moving average of order n is the integer part of (2-a)/a.An exponential smoothing over an already smoothed time series is called double-exponential smoothing. In some cases, it might be necessary to extend it even to a triple-exponential smoothing. While simple exponential smoothing requires stationary condition, the double-exponential smoothing can capture linear trends, and triple-exponential smoothing can handle almost all other business time series. Double Exponential Smoothing: It applies the process described above three to account for linear trend. The extrapolated series has a constant growth rate, equal to the growth of the smoothed series at the end of the data period.Triple Double Exponential Smoothing: It applies the process described above three to account for nonlinear trend.Exponenentially Weighted Moving Average: Suppose each day's forecast value is based on the previous day's value so that the weight of each observation drops exponentially the further back (k) in time it is. The weight of any individual isa(1 - a)k, where a is the smoothing constant.An exponenentially weighted moving average with a smoothing constant a, corresponds roughly to a simple moving average of length n, where a and n are related bya = 2/(n+1) OR n = (2 - a)/a.Thus, for example, an exponenentially weighted moving average with a smoothing constant equal to 0.1 would correspond roughly to a 19 day moving average. And a 40-day simple moving average would correspond roughly to an exponentially weighted moving average with a smoothing constant equal to 0.04878.This approximation is helpful, however, it is harder to update, and may not correspond to an optimal forecast.Smoothing techniques, such as the Moving Average, Weighted Moving Average, and Exponential Smoothing, are well suited for one-period-ahead forecasting as implemented in the following JavaScript: Forecasting by Smoothing.Holt's Linear Exponential Smoothing Technique: Suppose that the series yt is non-seasonal but does display trend. Now we need to estimate both the current level and the current trend. Here we define the trend Tt at time t as the difference between the current and previous level. The updating equations express ideas similar to those for exponential smoothing. The equations are: Lt = a yt + (1 - a) Ftfor the level and Tt = b ( Lt - Lt-1 ) + (1 - b) Tt-1for the trend. We have two smoothing parameters a and b; both must be positive and less than one. Then the forecasting for k periods into the future is: Fn+k = Ln + k. TnGiven that the level and trend remain unchanged, the initial (starting) values are T2 = y2 y1, L2 = y2, and F3 = L2 + T2 An Application: A companys credit outstanding has been increasing at a relatively constant rate over time:Applying the Holts techniques with smoothing with parameters a = 0.7 and b = 0.6, a graphical representation of the time series, its forecasts, together wit a few-step ahead forecasts, are depicted below: Year-end Past credit Yearcredit (in millions)1133215531654171519462317274831293131033311343 K-Period Ahead Forecast KForecast (in millions)1359.72372.63385.44398.3 Demonstration of the calculation procedure, with a = 0.7 and b = 0.6 L2 = y2 = 155, T2 = y2 - y1 = 155 133 = 22L3 = .7 y3 + (1 - .7) F3, T3 = .6 ( L3 - L2 ) + (1 - .6) T2 F4 = L3 + T3, F3 = L2 + T2 L3 = .7 y3 + (1 - .7) F3, T3 = .6 ( L3 - L2 ) + (1 - .6) T2 , F4 = L3 + T3 The Holt-Wint


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