Forecasting Techniques CMA Part 1 However, if the actual values fluctuate substantially and randomly, we prefer a lower value for tt, because we do not want to adjust forecasts too much in response to random variations. In this case a larger alpha will create a greater MSE and a less accurate forecast because a larger alpha will cause more weight to be put on the random variations. For this reason, exponential smoothing as a forecasting technique is most useful when the time series is stable, without many fluctuations. As additional time series data is collected, the smoothing constant tt can be adjusted for future forecasts at any time. Note: Smoothing methods are useful for a stable time series that has no significant trend, cyclical, or seasonal effects. 2. Trend Projection and Regression Analysis When a time series is increasing or decreasing consistently, smoothing methods are not appropriate for forecasting. Instead, a time series that has a long-term upward or downward trend can be forecasted by means of trend projection. A trend projection is done with simple regression analysis, which forecasts values using information from all available observations. Simple linear regression analysis relies on two assumptions: • Variations in the dependent variable (i.e., what we are forecasting) are explainecl by variations in one single independent variable (i.e., the passage of time, if a time series is what we are fore casting). • The relationship between the independent variable (time or something else) and the dependent variable (sales or whatever we are forecasting based on the value of the independent variable) is li near. A linear relationship is one that will graph as a straight line. The line of best fit, as determined by simple linear regression, is a formalization of the way we would fit a line just by looking at it. To fit a line by looking at it, we would use a ruler and move it until we think we have minimized the differences between the pOints and the line we would draw with the ruler. This will usually be a straight line located approximately at the place where the same number of pOints are above the line as are below it. Similar to fitting a line visually, the goal in linear regression analysis is to take each of the differences between the individual values and the point on the regression (trend) line for that time period - called a deviation -square each deviation, then calculate the total of all the squares of the deviations; and have the result be as low as it can get. When this is the case, the total of the squares of the deviations is minimized, and the trend line is the line of best fit. That line can then be used for forecasting using extrapolation. To use regression analysis, we first graph the values of the time series and review the results. If the long term trend appears to be linear, we can use simple regression analysis to determine the trend value. Before actually using regression analysis for a forecast, however, we should perform correlation analysis to determine the strength of the linear relationship between the value of X and the value of Y in order to determine whether trend projection would be meaningful. Correlation analysis measures the relationship between two or more variables. This measurement shows how closely connected the variables are and the extent to which a change in one variable will result in a change in the other. Note: The X axis on the graph is the horizontal axis and the Y axis is the vertical axis. The X axis represents the independent variable, and the Y axis represents the dependent variable. In a time series regression analysis, the passage of time is the independent variable and is on the X axis. 48 Section A Forecasting Techniques The coefficient of correlation, represented by the letter R or r is a numerical measure that measures both the direction (positive or negative) and the strength of the linear association. In linear regression analysis using only one variable such as sales, the period of time serves as the independent variable - on the x axis while the sales level serves as the dependent variable - on the y axis. When a time series such as sales over a period of several years is graphed, the data pOints on the graph may show an upsloping linear pattern, a downsloping linear pattern, a nonlinear pattern (a curve) or no pattern at all. The pattern of the data pOints indicates the amount of correlation between the values on the x axis (time) and the values on the y axis (sales). This amount of correlation, or coefficient of correlation, r, is expressed as a number between -1 and +1. • A correlation coefficient, r, of +1 means there is a perfect positive (upsloping) linear relationship between each value for x and its corresponding value for y. • A correlation coefficient, r, of -1 means there is a perfect negative (downsloping) linear relation ship between each value for x and its corresponding value for y. • A coefficient of correlation, r, which is close to zero, usually means there is no, or very little, rela tionship between the variables. However, in some cases it may mean that there is a strong relationship, but it is not a linear one (you do not need to know how to recognize this, but just be aware that it may occur). • If the coefficient of correlation, r, is a positive number close to +1, such as .83, this indicates that the data pOints follow a linear pattern fairly closely, and the pattern is upsloping (i.e., sales are in creasing). • If the coefficient of correlation, r, is a negative number close to -1, such as -.77, this would indicate that the data pOints follow a linear pattern, although less closely than in the previous exam ple, and the pattern is downsloping instead of upsloping (i.e., sales are decreasing). The coefficient of correlation, r, does nothing to tell us how much of the variation in the independent variable is explained by changes in the dependent variable. It tells us only whether there is either a direct (upsloping) or inverse (downsloping) relationship between the variables and how strong that relationship is. Note: It is important to first look at the plotted data pOints on the graph when determining whether there is a relationship between the two variables. Do not rely on the value of the coefficient of correlation to tell you whether or not there is a relationship between the two variables, because the coefficient of correlation will not detect non-linear relationships. The coefficient of correlation, r, can be used to determine whether trend projection would be meaningful. • A high correlation coefficient, r, (i.e., a number close to either +1 or -1) would indicate that simple linear regression analysis would be useful as a way of making a projection using a trend line. • A low correlation coefficient, r, (close to 0) would indicate that a forecast made using simple regres sion analysis would not be very meaningful. Note: The coefficient of correlation, r, can be calculated in Excel by entering theXvalues in one column (say Column A, rows 1-10), the Yvalues in another column (say Column B, Rows 1-10), and entering the following formula in a blank cell : =CORREL(A1 :A10,B1 : B10) 49 Forecasting Techniques CMA Part 1 The statistical method used to perform simple regression analysis is called the Least Squares, also known as the Ordinary Least Squares method, or OlS. If we call the predicted value of y obtained from the fitted line y, then the prediction equation, or the equation of a linear regression line is: 9 = ax + b Where: y = the predicted value of y on the regression line corresponding to each value of x a = the slope of the line b = the y intercept, or the value of y when x is 0 x = the value of x on the x axis that corresponds to the value of y on the trend line. Note: This formula may be written in different ways, but x will always represent the independent variable and y is the constant. The coefficient of the Independent variable, or the variable coefficient, is whatever term is next to the x in the formula. This might be an a, as in the equation above, or the a and the b might be reversed and b might be next to the x. But regardless of which letter is next to the x, that term represents the amount of increase in y for each unit of increase in x, or the slope of the line. The constant coefficient is b in the equation above, and it represents the y intercept because this is the value of y when x is zero. The equation might also use a as the constant coefficient. The right side of the equation may present the terms in any order, too. The constant coefficient may come first. This equation, 9 = a + bx is exactly the same as this equation, 9 = ax + b. In the second equation, the coefficients are reversed and the order of the terms on the right side of the equation are reversed. Just remember to look for the x. The term next to it will be the variable coefficient, or the amount of increase in y for each unit of increase in x; and the term that is all by itself will be the constant coefficient and is the y intercept, or the value of y when x is zero. The symbol over the y in the formula is called a hat, and thus, it is read as ·y-hat. It means that we are looking at the predicted value, not at the actual value. Here is a graph that illustrates a regression analysis. We have made a couple of changes so that the trend line, the equation of the trend line, and the coefficient of determination, � or r, (more later on that) can
Posted on: Tue, 15 Oct 2013 05:55:12 +0000
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