Linear transformation, its application in the paper industry.

Application: Linear transformation simplifies data analysis. Mills can use the methods described here to simplify and improve drainage analysis.

Change of measuring unit is often encountered in daily life. For example, length is expressed in nanometer (nm) rather than in meter (m); this is a simple change in scale. Similarly, temperature can be expressed in degrees Celsius rather than in degrees Fahrenheit. In this second case, the scale is changed and the origin is shifted. That is, the new units are obtained by a linear transformation of the original units.

What is linear transformation? It is a way to express one variable (y) in a different variable (y') such that the relationship between y and y' can be expressed as y' = [a.sub.0] + [a.sub.1]y where [a.sub.0] and [a.sub.1] are constants. Examples of descriptive measures are mean and variance.

In wet-end chemistry, a polymer is often used to increase retention. We use the Britt dynamic drainage jar (DDJ) to measure filtrate consistency and retention The filtrate consistency should decrease with the addition of the polymer (retention aid). So, for a given polymer dosage, a convenient and commonly used definition of retention is y' = 100% * (b - y) / b where y' is retention, y is filtrate consistency and b is the reference point, as defined below. Thus, we can think of the amount of polymer added as our independent variable x, the filtrate consistency as our basic dependent variable y, and the retention as a transformation y' of y. Note that y' is a linear transformation of y (let [a.sub.0] be 1 and [a.sub.1] be -1/b). This is significant because it allows us alternative ways to define the reference point b. If the reference point is the total-solids consistency of the furnish, retention is called total-solids retention. In the mill, this is often called first-pass retention (FPR). If the reference point is the fines consistency of the furnish, retention is called fines retention.

We also show an example where apparent linear transformation is not. In this case, if we choose to express our result in drainage rate rather than in drainage time, the mean, the variance, and the regression coefficient of determination (obtained through linear regression) will not be likely the same.

Surya is with Ondeo-Nalco Pacific," you may him at email [e-mail]. Chua and Khoo are with the National University of Singapore; email Chua at [e-mail]; email Khoo at [e-mail].