Editing Formulae For Converting Between Grit and Microns

Numerous tables displaying measures of the size of abrasive particulates of various substances exist on the internet and elsewhere, and generally correspond to certain manufacturer's products. These tables include, but are not strictly limited to, grit and micron measures. The purpose of this Wiki entry is to quantify the relationship between grit and micron particulate measures, and to develop a mathematical relationship between the two which may then be used to convert between the units simply and quickly. Note that the formulae developed in the subsequent text should in no way be deemed definitive: the formulae are only as accurate as the data used to derive them, and it has been noted elsewhere that the available information tends to vary, sometimes significantly, between sources for a variety of reasons.

== Measures, Definitions, and Their Impact on Derivations ==

As a general definition, grit refers to the amount of particulates per unit volume of hone suspension media, and can be, in theory, any of a number of summary measures such as an average, a maximum or a minimum. Micron refers to the size of the abrasive particulates suspended in the hone media and can again be, in theory, an average, a maximum or a minimum. 

Particulate geometry has an important impact on observed measures of grit or micron. Geometries which suit high levels of compactability will clearly display higher measures of grit than those that are not as compactable. When considering micron measures, particulate geometry must be considered also (e.g. maximal particulate width vs radii of spherically symmetric particulates such as garnets found in Belgian stones).  

For the purposes of this article it will be assumed that the summary measure has no substantive effect on the derived formulae. However, it should be recognised that different summary measures may account for some of the variation observed between formulae derived using measures obtained from different manufacturers (not only in terms of the measure used by the manufacturer but also in terms of the particulate geometry used in manufacture). Furthermore, the exact definition of the measure used (mean, maximum, minimum etc.) will have a technical (although arguably marginal) impact on any statistically-derived formulae and its subsequent use. These impacts will not be considered here.

== Data Sources ==

Two sources of data were used. The first is taken from an online faceting site <ref>http://www.faceters.com/askjeff/answer67.shtml</ref>. The second is based on information given by Shapton <ref>http://shaptonstones.com/</ref>. 

== Analysis ==

For the purposes of illutration only the first set of data (faceting site) will be used. The analysis of the Shapton-based data follows in a similar manner.

=== Graphical Exploration ===

A graph of the raw data shows an exponential relationship (decay):

[[Image:graph1.jpg||280px|Micron Versus Grit]] 

This is confirmed by the almost linear relationship when the natural logarithm of each variable is taken:

[[Image:graph2.jpg||280px|Micron Versus Grit (log Scale)]] 

=== Plausible Model ===

Based on the graphs above, a plausible deterministic model for the realtionship betwee microns and grit is

<math>\mbox{Micron}= \beta_{0}\mbox{Grit}^{\beta_{1}}</math>

where <math>\beta_{0}</math> is a constant (to be estimated) representing the micron measure of a particulate with a grit of 1, and  <math>\beta_{1}</math> is a constant (to be estimated) representing the rate at which microns decrease as grit increases.

Standard statistical software packages such as SAS, Splus, Stata, R, and others have the capacity to fit nonlinear models such as the one postulated above easily and quickly using well-tested and understood fitting algorithms.

An alternative to non-linear fitting would be to take advantage of the mathematical form of the postulated model, which is amenable to a linearising transformation. Specifically:

<math>\log(\mbox{Micron}) = \log(\beta_{0})+ \beta_{1}\log(\mbox{Grit})</math>

yielding a standard linear relationship between the log-transformed variables.

Having postulated an initial plausible deterministic model we procede to add a random variation element (error) to the equation. Inspection of the graphical relationship presented above shows that while the data follows the general pattern of an exponential decay model, individual observations vary from this average behaviour. We assume these deviations occur randomly and independently and, moreover, follow a gaussian distribution with mean 0 and constant variance (to be determined). We also assume an additive measurement and error relationship viz:
 
<math>\mbox{Micron}_{i} = \beta_{0}\mbox{Grit}_{i}^{\beta_{1}} + \epsilon_{i}</math>

where <math>\epsilon_{i}</math> represent the error, as described above, associated with each observation/response pairing, <math>i=1, \ldots, n</math>.

=== Fitted Model and Assessment ===

Based on the faceting site data the relationship between Microns and Grit is estimated to be

Estimated Microns = 11764.71 x Grit^(-0.93589)

with approximate 95% Confidence intervals of (9146.074, 15133.100) for <math>\beta_{0}</math> and (-0.969, -0.903) for <math>\beta_{1}</math>.

The fidelity of the fitted line is demonstrated in the graph below (original data are the points, the estimated relationship is the line):

[[Image:graph3.jpg||280px|Micron Versus Grit (log Scale): Fitted Model]] 

=== Using the Model for Interpolatory Predictions ===

As an example, consider a Shapton 16k ceramic on glass stone. Suppose that interest lies in determining the micron rating of this stone. Using the formula above, we obtain:

Estimated Microns = 11764.71 x 16000^(-0.93589) = 1.37

An approximate 95% confidence interval for this prediction (details are omitted) shows that the micron rating for this stone lies somewhere in the interval (0.78, 2.41) with 95% confidence.

Similarly, inversion of the formula enables prediction of grit when micron rating is known. For example a 0.92 micron hone is predicted to have a grit of 24440.42, with an approximate 95%CI for the prediction of (13404.45, 46542.28).

=== Concluding Remarks ===

It is a relatively simple matter to develop statistical models which facilitate the conversion between Micron and Grit measures, which eliminate the need for reference tables to accomplish a similar task. Such models take into account the inherent variability accompanying these measures and allow reasonably precise interpolatory predictions, a process fraught with peril using tables.

It should be noted however that any models developed in the manner described above are local approximations only. While the model appears to fit the data well over the range of measured values, there is no guarantee such relationships hold outside this range. Furthermore, all statistical models are only as good as the data used to derive them: "All models are bad, but some are more useful than others."
@@ Line 75: / Line 75: @@
 It is a relatively simple matter to develop statistical models which facilitate the conversion between Micron and Grit measures, which eliminate the need for reference tables to accomplish a similar task. Such models take into account the inherent variability accompanying these measures and allow reasonably precise interpolatory predictions, a process fraught with peril using tables.
-It should be noted however that any models developed in the manner described above are local approximations only. While the model appears to fit the data well over the range of measured values, there is no guarantee such relationships hold outside this range. Furthermore, all statistical models are only as good as the data used to derive them: "All models are wrong, but some are useful." <ref>http://en.wikiquote.org/wiki/George_E._P._Box</ref>
+It should be noted however that any models developed in the manner described above are local approximations only. While the model appears to fit the data well over the range of measured values, there is no guarantee such relationships hold outside this range. Furthermore, all statistical models are only as good as the data used to derive them: "All models are bad, but some are more useful than others."
-== Notes ==
-<references/>
-[[Category:Honing]]