Simple Linear Trend Model

Here we assume that the level and trend remain constant over the forecast horizon. Thus, X, = a + bf + e for f = 1,2,..n, where X, is the random variable denoting demand at period t, a, and b are constant level and trend and 8 is the random error. Given the observed values of X, = D, for f = 1. n, the forecast for period t is given by 

From the given data, we then have n linear equations relating the actual and forecasted demands. Following the approach given in Srinivasan (2010), we can use the least square regression method to estimate the level (a) and trend (b) parameters. The unconstrained optimization problem is to determine a and b such that 

Note that the minimization function given by Equation 2.5 is a convex function. Hence, a stationary point will give the absolute minimum. The stationary point can be obtained by setting the partial derivatives with respect to a and b to zero. Thus, we get the following two equations to solve for 

Solving the two linear equations, we determine the values of a and b. Then, the forecast for period f = n + 1 is given by 

Consider the monthly demand for a product for the past 6 months given in Table 2.4. The demand data clearly shows an increasing trend. The problem is to determine the forecast for month 7 using the simple linear trend model. 

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The main drawback of the simple linear trend model is that it assumes that the level and trend remain constant throughout the forecast horizon. 

The third fundamental component in the QSAR model is the mathematical algorithms. Many methods have been used, and in the last years, there has been an increase of the methods, and hence, quite probably this trend will continue, introducing many other methods [4—6]. Classical QSAR methods, used decades ago, were simple linear relationships. Corwin Hansch has been a pioneer of these methods [2]. An example can be the linear relationship between the fish toxicity and the partition coefficient between octanol and water, called Kow [3]. Kow, and its logarithm, called log P, is still the most popular chemical descriptor used in QSAR models for fish toxicity, and it is the base of software programs used by the US Environmental Protection Agency for fish toxicity [11]. The theoretical assumptions for the use of log P are that (1) octanol mimics the lipophylic component of the fish cell, and (2) the toxic effect is due to the adsorption of the chemical substance into the cell. 

Among the transients analyzed are an overspeed transient and startup. These transients result in mass flow rates and shaft speeds outside the ranges in the performance maps provided. To model transients that extend beyond the range covered by the performance maps, extrapolations to both lower and higher mass flow rates were performed. In addition, extrapolations to lower and higher shaft speeds were also included in the input model. Simple linear extrapolation was used. Use of these extrapolations demonstrates the system behavior trends and the ability of the computer code to provide reasonable calculations. If the JIMO project would have continued, more explicit performance maps for flow rates and shaft speeds beyond those currently available would have been required. 

One advantage of die quadratic expression is that it is veiy versatile and can match both linear F-mechanics model with a power law exponent of 1.5. 

The trend logio(CV) vs logjo(c) appears reasonably linear (compare this with Ref. 177 some points are from the method validation phase where various impurities were purposely increased in level). A linear regression line B) is used to represent Ae average trend (slope = -0.743). The target level for any given impurity is estimated by a simple model. Because the author-... 

The information discussed above can, however, yield only the overall trend in transfer quantities, and it would be unrealistic to expect, for example, that in aqueous mixtures, 6m (ion) is a linear function of mole fraction. Nevertheless it is noteworthy that the transfer quantities between pure solvents are not in agreement with trends expected from a simple Born model for ionic solvation. 

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