Signal-to-Noise (S/N) Ratios for Static and Dynamic Systems
MSD AND S/N RATIOS NOTES AND RECOMMENDATION ON USE OF S/N RATIOS (Static condition) Recommendation: If you are not looking for a specific objective, then SELECT S/N ratio based on Mean Squared Deviation (MSD). MSD expression combines variation around the given target and is consistent with Taguchi’s quality objective. S/N based on variance is independent of target value and points to variation around the target. S/N based on variance and mean combines the two effects with the target at 0. The purpose is to increase this ratio ((Vm-Ve)/(nxVe)) and thus a + sign is used in front of Log() for S/N. Also, since for an arbitrary target value, (Vm-Ve) may be negative, target=0 is used for the calculation of Vm. Expressions for all types of S/N ratios are shown on the next screen. RELATIONSHIPS AMONG OBSERVED RESULTS, MSD, AND S/N RATIOS(Static condition) MSD = ( (Y1-Y0)^2 + (Y2-Y0)^2 + …. (Yn-Y0)^2 )/n for NOMINAL IS BEST Variance: Ve = (SSt – SSm)/(n-1) …………….. for NOMINAL IS BEST Variance and Mean = (SSm – Ve)/(n*Ve) (with TARGET=0) where SSt = Y1^2 + Y2^2 and SSm = (Y1 + Y2 +..)^2/n MSD = ( Y1^2 + Y2^2 + ………………. Yn^2 )/n for SMALLER IS BETTER MSD = ( 1/Y1^2 + 1/Y2^2 + …………. 1/Yn^2 )/n for BIGGER IS BETTER S/N = – 10 x Log(MSD)…………….. for all characteristics S/N = + 10 x Log(Ve or Ve and Mean) .. for NOMINAL IS BEST only. Note: Symbol (^2) indicates the value is SQUARED. DYNAMIC CHARACTERISTIC (Conduct of experiments and analysis of results) Reference text: Taguchi Methods by Glen S. Peace, Addison Wesley Publishing Company, Inc. NY, 1992 (Pages 338-363) WHAT IS DYNAMIC CHARACTERISTIC? A system is considered to exhibit dynamic characteristics when the strength of a particular factor has a direct effect on the response. Such a factor with a direct influence on the result is called a SIGNAL factor. SIGNAL FACTOR- is an input to the system. Its value/level may change. CONTROL FACTOR – is also an input to the system. Values/level is fixed at the optimum level for the best performance. NOISE FACTOR – is an uncontrollable factor. Its level is random during the actual performance. STATIC SYSTEM GOAL – is to determine the combination of control factor levels that produces the best performance when exposed to the influence of the varying levels of noise factors. DYNAMIC SYSTEM GOAL – is to find the combination of control factor levels that produce different levels of performance in direct proportion to the signal factor, but produce minimum variation due to the noise factors at each level of the signal. Example: Fabric dyeing process Control factor: Types of dyes, Temperature, PH number, etc. Signal factor: Quantity of dye Noise factor: Amount of starch CONDUCTING EXPERIMENTS WITH DYNAMIC CHARACTERISTICSWhen carrying out experiments, the proper order and sequence of samples tested under each trial condition must be maintained. The number of samples required for each trial condition will depend on the number of levels of signal factor, noise conditions, and repetitions for each cell (a fixed condition of noise and signal factor). Step 1. Design experiment with control factors by selecting your design type (manual or automatic design) from the main screen menu.Step 2. Print a description of trial conditions by selecting the PRINT option.Step 3. Enter your descriptions and experiment notes on the DYNAMIC CHARACTERISTICS screen. * You will need to describe signal and noise factors and their levels. You will also have to decide on the number of levels of signal and noise factors. BUT MOST IMPORTANTLY, you will have to choose the nature of the ideal function (Straight line representing the behavior Response vs. Signal) applicable to your system. Step 4. Strictly follow the prescribed test conditions.Step 5. Enter results in the order and locations (run#) prescribed using the RESULTS option from the main menu. SIGNAL-TO-NOISE RATIO EQUATIONS (alternate dynamic characteristic equations)Signal factors may not always be clearly defined or known. For common industrial experiments, one or more attributes may be applicable: * TRUE VALUE KNOWN * INTERVAL BETWEEN FACTOR LEVELS KNOWN * FACTOR LEVEL RATIOS KNOWN * FACTOR LEVEL VALUES VAGUE Depending on the circumstances of the input signal values and the resulting response data, different signal-to-noise (S/N) ratio equations are available. ZERO POINT PROPORTIONAL – Select this response type of equation when the response line passes through the origin. The signal may be known, unknown, or vague. REFERENCE POINT PROPORTIONAL – This response type of equation should be the choice when the response line does not go through the origin but through a known value of the signal or when signal values are wide apart or far away from the origin. When the signal values are known, zero point or reference pointproportion should be considered first. If neither is appropriate, the linear equation should be used. LINEAR EQUATION – is based on the least I response squares fit equation and should be used where neither zero nor reference point proportional equation is appropriate. Use it when signal values are close together and the response does not pass through the origin. WHEN IN DOUBT plot the response as a function of the signal factor values on a linear graph and examine the y-intercept. If it passes through the origin, use ZERO POINT. If the intercept is not through the origin but the line passes through a fixed point, use REF. POINT. In all other situations use LINEAR EQUATION. S/N Ratio Equation and Calculation Steps y = m + Beta (M – Mavg) + e Linear Eqn. (L) y = Beta M Zero Point (Z) y = Beta (M – Mstd.) + ystd Ref. Point (R) Where y = system response (QC), M = Signal factor Beta = slope of the ideal Eqn. Mavg = Average of signals ystd. = avg. response under reference/standard signal Mstd = reference/standard value of the signal strength Notations * = multiplication, ^ = raised to the power / = division by Response Components for Each Trial Condition (Layout shown only for trial#1 below) S I G N A L F A C T O R Signal lev 1 Signal lev 2 Signal lev 3 _________N1_______N2__________N1______N2____________N1_______N2______ Trl#1| y11, y12, y13, y14.. y21, y22, y23, y24.. y31, y32, y33, y34. Step 1: Determine r (Start with trial# 1) ro = Number of samples tested under each SIGNAL LEVEL (Number of NOISE LEVELSxSAMPLES per CELL) M1, M2, M3,.. Mk. Signal levels (strengths) N1 & N2 are two levels of the noise factor k = number of signal levels Mavg = (M1 + M2 + …. Mk)/k r = ro [ (M1-Mavg)^2 + (M2-Mavg)^2 +… + (Mk-Mavg)^2] … (L) r = ro [ (M1-Mstd)^2 + (M2-Mstd)^2 +… + (Mk-Mstd)^2] … (R) r = ro ( M1^2 + M2^2 + M3^2 … + Mk^2) …………. (Z) Step 2: Calculate of Slope Beta Beta = (1/r) [y1*(M1-Mavg) +y2*(M2-Mavg) +… + yk*(Mk-Mavg)] .. (L) Beta = (1/r) [y1*(M1-Mstd) +y2*(M2-Mstd) +… + yk*(Mk-Mstd)] .. (R) Beta = (1/r) ( y1*M1 + y2*M2 + … + yk*Mk) ……… (Z) Step 3: Determine Total Sum of Squares St = Sum [Sum (yij – yavg)] i= 1,2 .. k. j=1,2,.. ro .. (L) yavg = ystd for (R), yavg = 0 for (Z) Step 4: Calculate Variation Caused by the Linear Effect Sbeta = r Beta^2 …. for all equations Sbeta = (1/r) [y1*(M1-Mavg) +y2*(M2-Mavg) +.. + yk*(Mk-Mavg)]^2 .. (L) Sbeta = (1/r) [y1*(M1-Mstd) +y2*(M2-Mstd) +.. + yk*(Mk-Mstd)]^2 .. (R) Sbeta = (1/r) ( y1*M1 + y2*M2 + … + yk*Mk)^2 ……. (Z) Step 5: Calculate the Variations Associated with Error and Non-linearity Se = St – Sbeta … for all equations Step 5: Calculate Error Variance Ve = Se / [ k*ro – 2 ] …… (L) Ve = Se / [ k*ro – 1 ] …… (R and Z) Step 6: Calculate S/N Ratio Eta = 10 Log (Sbeta – Ve) / (r*Ve) … for all Eqns. Step 7: Repeat calculations for all other trials in the same manner. Example calculations: Case of LINEAR EQUATION (Expt. file: DC-AS400.QT4) The results of samples tested for trial#1 of an experiment with dynamic characteristics. There are three signal levels, two noise levels, and two repetitions per cell. M1 M2 M3 Noise 1 Noise 2 Noise 1 Noise 2 Noise 1 Noise 2 |_______________________|______________________|______________________Trl#1| 5.2 5.6 5.9 5.8 | 12.3 12.1 12.4 12.5| 22.4 22.6 22.5 22.2 Signal strengths: M1 = 1/3, M2 = 1, M3 = 3 CALCULATIONS FOR S/N: Mavg = (1/3 + 1 + 3 ) / 3 = 1.444 ro = 4 (2 simple/cell * 2 noise levels) r = 4[(1/3 – 1.444)^2 + (1 – 1.444)^2 + (3 – 1.444)^2] … (L) = 4( 1.2343 + 0.1971 + 2.421 ) = 15.41 y1 = 5.2 + 5.6 + 5.9 + 5.8 = 22.5 y2 = 12.3 + 12.1 + 12.4 + 12.5 = 49.3 y3 = 22.4 + 22.6 + 22.5 + 22.2 = 89.7 Beta = (1/r)[22.5*(1/3-1.444) + 49.3*(1-1.444) + 89.7*(3-1.444)] = (1/15.41) [ -24.9975 – 21.692 + 139.5732 ] = 92.8842/15.4101 = 6.01 Sbeta = r*Beta^2 = 15.4101 * 6.0274^2 = 556.82 yavg = [5.2 + 5.6 + …… + 22.2]/12 = 161.5/12 = 13.46 St = (5.2 – yavg)^2 + (5.6 – yavg)^2 + …..+ (22.2 – yavg)^2 = 68.23 + 61.78 + 57.15 + 58.67 + 1.346 + 1.85 + 1.123 + .921 + 79.92 + 83.54 + 81.72 + 76.387 = 572.65 Se = St – Sb = 572.65 – 556.82 = 15.83 Ve = Se / ( 12 – 2 ) = 15.83 /10 = 1.583 Eta = 10 Log (Sbeta – Ve) / (r*Ve) … for all Eqns. = 10 Log [(556.82 – 1.583)/(15.41*1.583)] = 10 Log(22.76) = 13.572 (S/N for the trial# 1 results ) Similarly, S/N ratios for all other trial conditions are calculated and analysis performed using NOMINAL IS THE BEST quality characteristic as normally done for the static systems.