O-24 A Reexamination of SRM as a Means of Beer Color Specification A.J. deLange ASBC 2007 Annual Meeting June 19, 2007

Current and Proposed Methods of Beer Color Specification 1 cm Absorption Spectrum Normalize by A 430 ; Convert to Transmission Spectrum A 380 A 385 A 780 A 430 X 12.7 Compute Spectrum Deviation; Encode into SDCs 1 cm Absorption Spectrum SRM X 12.7 A cm Absorption Spectrum Convert to Transmission Spectrum A 380 A 385 A 780 Illuminant C (3) 10° CMFs White Point Compute X, Y, Z; Map to L*, a*, b* E 308 L* a* b* (3) Eigenvectors Average Normalized Spectrum Reconstruct Spectrum Scale to any Path; Convert to Transmission SRM SDC 1 SDC 2 SDC 3 Compute X, Y, Z; Map to any coord. E 308 Avg. Norm. Spec. (3) Eigenvectors Any Illuminant Any (3) CMFs Any White Point L* a* or u b* or v Beer-10C report Proposed report Beer-10A report / A 700 <.039? O.K.

Beer’s Law Coloring matter in beer appears to follow Beer’s Law –Absorption (log) is proportional to molar concentration Colorants are in fixed proportion in an ensemble of average beers If true, absorption spectra would be identical if normalized by absorption at one wavelength –Noted by Stone and Miller in 1949 when proposing SRM

Deviation From Average Miller and Stone studied 39 beers Used deviation from average (A 700 /A 430 ratio) to disqualify beers as being suitable for SRM –Test still in MOA Beer-10A We propose to quantify deviation, encode it, and augment SRM report with this information –Encoding by spectral deviation Principal Components SRM plus encoded deviation permits reconstruction of spectrum –Spectrum inserted into ASTM E 308 for visible color calculation under various conditions Tested on an ensemble of 59 beers with good results Worked with transmission spectra rather than absorption because they give better computed color accuracy

Spectrum Compression: 59 Beer Transmission Spectra (1 cm). Ensemble variance (sum of squares of difference between spectrum and average spectrum)  2 = 6.48 Blue spectra are fruit beers

Normalize absorption spectra by A 430 ; convert to Transmission:  2 = 0.29 (4.4% of original) Normalization: Convert transmission to absorption (take -log 10 ), divide by 430 nm value and convert back to transmission (antilog[-A]) Conventional Beers Fruit Beers

Transmission Spectra (normalized) deviation from average (  2 = 0.29 i.e. 4.4% of original) Singular value decomposition (SVD) of matrix of these data (eigen analysis of covariance matrix) yield eigen vectors used to compute Principal Components of individual spectra

Variation from 1st 2 PC’s taken out, average added back in:  2 = (0.025% of original) “Fuzziness” about average can be modeled by use of additional PC’s

Summary of Last Few Slides Normalizing by SRM removes 95% of variation (relative to average) in beer spectra First 2 Principal Components removes most of remainder (leaving but 0.025% of the original total) –As these PCs quantify deviation of individual beer spectrum from average let’s call them“spectrum deviation coefficients” (SDC) What’s left is the average plus 0.025% variation Thus, if we take the average and add the 2 SDC’s worth of variation back, then un-normalize by SRM we can reconstruct the transmission spectrum, T( ) –T( ) ~ Log -1 {(Log[Avg( ) + SDC 1 *E 1 ( ) + SDC 2 *E 2 ( )])/(SRM/12.7)} From reconstructed spectrum we can calculate actual colors. Question: how accurately?

CIELAB Color Difference,  E CIELAB Tristimulus Color: –Brightness L* ( ) –a*: green-red (~ -100 to 100) –b*: blue-yellow (~ -100 to 100) –Calculated from 81 spectral transmission measurements (380, 385, 390… 780nm per ASTM E 308) All L*ab colors relative to a reference “White Point” –White: L* = 100, a* = 0, b* = 0 Supposed to be uniform perceptual space Difference between 2 colors –  E = [(L 1 -L 2 ) 2 + (a 1 -a 2 ) 2 + (b 1 -b 2 ) 2 ] 1/2 (i.e. Euclidean Distance) –  E < 3 considered a “good match” General accuracy of press reproduction: > 2

Example Color Differences Center patch: ~16 SRM, 1 cm, Illum. C  a*  b*  E’s Adjacent in same row or column (excluding top row): 3; Adjacent diagonal (excluding top row): 4.2 Center to corner (excluding top row): 8.5 Top Row Only  L*  E this patch to lower right corner: 20.8

Ensemble Error in L*ab color calculated from average spectrum unnormalized by SRM (no PC correction) Calculate L*ab color from full spectrum; calculate lab color from average spectrum and SRM; plot difference

Ensemble error in L*ab color calculated from SRM + 2 SDCs

Beer-10C L * ab Computation 1 cm Transmission Spectrum, 81 pts Illum. C Distribution +, 81 pts Point wise Multiply x matching function +, 81 pts y matching function +, 81 pts z matching function +, 81 pts Accum, Scale + Accum, Scale + Accum, Scale + x datay dataz data Point wise Multiply 1 ~ 380nm 81 ~ 780nm X Y Z (X/X r ) 1/3 (Y/Y r ) 1/3 (Z/Z r ) 1/3 ZrZr YrYr XrXr Reference White +    a* b* L* + = Tabulated in MOA Other illuminants, matching functions, reference whites allowed by E 308 For different path (E 308) take log, scale, take antilog

Beer-10C Illustrated

Beer -10C Word Chart Basis: ASTM E308 - Defines color measurement in US Take 81 spectrum measurements: 380 to 780 nm; 5 nm steps; 1 cm path or scale to 1 cm from any other path length (Lambert Law). Convert to transmission. Weight by spectral distribution of Illuminant C (tabulated values) Multiply point wise by each (3) color matching functions (table values of CIE 10° observer). Scale sums by 100/ to compute X, Y, Z Compute f x (X/X r ), f y (Y/Y r ), f z (Z/Z r ) –f(u) = u 1/3 (in E 308 f(u) is an offset linear function for u< ) –X r = , Y r = 100, Z r = (in E 308 these are calculated from illuminant spectral distribution function) Compute –L* = 116 f x (X/X r ) - 16 –a*= 500[f x (X/X r )- f y (Y/Y r )] –b*= 200[f y (Y/Y r ) - f z (Z/Z r )] Report L*, a* and b* (could report X, Y and Z or other tristim.)

Proposed MOA SDC Computation Average Spectrum +, 81 pts Point wise Subtract 1st Eigenfunction +, 81 pts 2nd Eigenfunction +, 81 pts 3rd Eigenfunction +, 81 pts Accum 1st data 2nd data 3rd data Point wise Multiply + = Tabulated in proposed MOA Eigenfunctions are those of covariance matrix of normalized, de-meaned spectrum ensemble “SDC” is, thus, a Principal Component of the input spectrum. 1 cm Absorption Spectrum, 81 pts 1 ~ 380nm 81 ~ 780nm Normalize (point wise divide) A st SDC > 2nd SDC > 3rd SDC SRM Convert to transmission (10 -A ) Reported Parameters:

Proposed Method Illustrated Note: Before application of matching function the tabulated average function is subtracted from normalized function. This is not shown on this chart.

New Method Word Chart Take 81 absorption (log) measurements: 380 to 780 nm, 5 nm steps, 1 cm path or scale (Lambert law) to 1 cm from any other path Compute SRM = 10*A 430 *2.54/2 = 12.7*A 430 Divide each point in spectrum by A 430 (absorption at 430 nm) Convert to transmission (change sign and take antilog) Subtract average transmission spectrum (from published table values) Multiply point wise by each of “matching functions” (published table values of ensemble eigenfunctions) and accumulate Report SRM and accumulated sums (SDC 1, SDC 2,  ) Notes: 1. Table values would be published as part of a new MOA 2. Matching functions are eigenfunctions of covariance matrix of “normalized”, de-meaned transmission spectra thus coefficients (SDC’s) are “Principal Components” of the beer’s spectrum.

Color Calculation from New Parameters 500 Average Spectrum +, 81 pts Point wise Add --> Aprox Norm. Spec. 1st Eigenfunction +, 81 pts 2nd Eigenfunction +, 81 pts 3rd Eigenfunction +, 81 pts Sum scaled eigenfunctions = deviation 3rd SDC + = Tabulated in proposed MOA 1 cm Absorption Spectrum, 81 pts 1 ~ 380nm 81 ~ 780nm Un-normalize (point wise multiply) A 430 1/12.7 2nd SDC1st SDC SRM Convert to absorption (-log 10 ) 81 Path, cm E A XYZ Luv Lab Illuminant Observer (CIE matching functions) etc Ref. XYZ Input Parameters:

Color Computation Word Chart Add point wise SDC 1 times first matching function + SDC 2 times second matching function (table values)… to average (tabulated values) spectrum –If no SDC values (i.e. SRM only) then just use average spectrum Convert to absorption (log) spectrum Compute A 430 = SRM/12.7 Multiply each point in spectrum by A 430 –This is the reconstructed 1 cm absorption spectrum Compute color per ASTM E 308 (or Beer 10C) –Scale to any path length –Weight by any illuminant –Use either 10° or 2° color matching functions –Relative to any white point

59 Beers in CIELAB Coordinates Raspberry Ale Kriek Beer colors are restricted: generally follow “corkscrew” in (in ~ dark) to page SDC’s model deviation from corkscrew

Summary Beer colors are a subspace of all colors; spectra are similar –This makes data compression possible SRM SDC’s (PCs) gives spectrum reconstruction sufficiently close for accurate tristimulus color calculation Calculation of SDC’s is as simple as calculation of tristim. –Can all be done in a spreadsheet like that for Beer 10C SRM + SDC’s is a candidate for new color reporting method Plenty to be done before a new MOA could be promulgated –Acceptance of concept –Verification of claim –Definition of ensemble and measurements for determination of average spectrum, eigen functions –Trials, collaborative testing….