HW05-T-2023S-Solutions

MEEN 260 Introduction to Engineering Experimentation Homework 5: Non-linear Regression and Sensor Characteristics Assigned: February 21, 2023 Due: February 26, 2023, 11:59 PM General Instructions: Follow the posted Homework Guidelines when completing your assignment. You may scan your hand-written pages, or use a tablet writing device to convert your document to PDF format. Note that each problem must have: • A brief restatement of the problem in your own words (i.e. NOT a screen image of the problem statement) • A clear, legible, organized solution in symbolic form. Include any references to data or values used in your calculations. • Final answers identified (boxed, underlined) with appropriate significant digits and units

Problem #1: A ping-pong is dropped from a prescribed height ( x i ) and allowed to bounce on a hard surface. The time required to complete a ‘bounce’ (the time between one vertical maximum and the next) is recorded ( t i ). All measured values are assumed to be known to five significant digits regardless of rounding in the data table. Initial height, cm ( x i ) Time to bounce, s ( t i ) 12 0.47 9 0.4 6 0.33 3 0.23 You assume bounce time t i is the response variable, and that bounce time will fit the equation for parabolic motion having an unknown constant, k (in s 2 /cm) that must be determined from the measured data: Eq. 1) ࠵? ! = #࠵?࠵? ! ) Based on the given information, perform the following tasks: a) Define modified predictor/response variables (࠵? ! ∗ , ࠵? ! ∗ ) so that you can use a linear regression technique to determine the ‘best-fit’ value of k . Write the linearized equation using the modified predictor/response variables. There are multiple methods to linearize the equation, but four possible transform equations are considered, the first and last pairs representing similar approaches: Transformation A: ࠵? ∗ = √࠵? , ࠵? ∗ = ࠵? ࠵? ∗ = ࠵?࠵? ∗ , ࠵? = √࠵? Transformation B: ࠵? ∗ = ࠵?, ࠵? ∗ = ࠵? # ࠵? ∗ = ࠵?࠵? ∗ , ࠵? = ࠵? Transformation C: ࠵? ∗ = ࠵?࠵?࠵?(࠵?), ࠵? ∗ = 2࠵?࠵?࠵?(࠵?) ࠵? ∗ = ࠵? ∗ + ࠵? , ࠵? = 10 $ Transformation D: ࠵? ∗ = ࠵?࠵?(࠵?), ࠵? ∗ = 2࠵?࠵?(࠵?) ࠵? ∗ = ࠵? ∗ + ࠵? , ࠵? = ࠵? $

Transformation B Transformation C Initial height, cm ( x i ) Time to bounce, s ( t i ) 12 0.47 9 0.4 6 0.33 3 0.23 a) Transformed Variable definitions: x*= xi, y* = (ti^2) Transformed Equation definition: y*=k*x*= ax* CURVE-FIT TABLE FOR TRANSFORMED EQUATION GOODNESS-OF-FIT CALCULATION FOR PREDICTIVE EQUATION b) & d) xi = xi ti = yi x*i y*i x*i y*i x*i^2 t^i=y^i t-bar=ybar (ti-tbar)^2 t^i-tbar)^2 (ti - t^i)^2 12 0.47 12 0.2209 2.6508 144 0.46680474 0.3575 0.01265625 0.01194753 1.02097E-05 9 0.4 9 0.16 1.44 81 0.40426476 0.3575 0.00180625 0.00218694 1.81882E-05 6 0.33 6 0.1089 0.6534 36 0.3300808 0.3575 0.00075625 0.00075181 6.52835E-09 3 0.23 3 0.0529 0.1587 9 0.23340237 0.3575 0.01625625 0.01540022 1.15761E-05 Sums 30 1.43 30 0.5427 4.9029 270 1.43455268 1.43 0.031475 0.0302865 3.99806E-05 SST SSE SSR The equation for best-fit value of slope y = ax is: c) a = Sum(x*i y*i)/Sum(x*i^2) a = 0.01815889 k = a= 0.01815889 e) r^2 =(SSE)/(SSE+SSR) 0.99868166 r = sqrt(r^2) 0.99934061 r' = sqrt(1=r^2) 0.03630892 R^2 = 1-(SSR/SST) 0.99872977 Initial height, cm ( x i ) Time to bounce, s ( t i ) 12 0.47 9 0.4 6 0.33 3 0.23 a) Transformed Variable Definitions: x*= log(xi), y* = 2log(ti) Transformed Equation Definition: y*= x* + b* CURVE-FIT TABLE FOR TRANSFORMED EQUATION GOODNESS-OF-FIT CALCULATION FOR PREDICTIVE EQUATION b) & d) xi = xi ti = yi x*i y*i x*i y*i x*i^2 t^i=y^i t-bar=ybar (ti-tbar)^2 t^i-tbar)^2 (ti - t^i)^2 12 0.47 1.07918125 -0.6558043 -0.7077317 1.16463216 0.46462599 0.3575 0.01265625 0.01147598 2.888E-05 9 0.4 0.95424251 -0.79588 -0.7594625 0.91057877 0.40237791 0.3575 0.00180625 0.00201403 5.65445E-06 6 0.33 0.77815125 -0.9629721 -0.749338 0.60551937 0.32854019 0.3575 0.00075625 0.00083867 2.13106E-06 3 0.23 0.47712125 -1.2765443 -0.6090664 0.22764469 0.23231299 0.3575 0.01625625 0.01567179 5.34994E-06 Sums 30 1.43 3.28869626 -3.6912007 -2.8255986 2.90837499 1.42785707 1.43 0.031475 0.03000046 4.20155E-05 SST SSE SSR The equation for best-fit value of slope y = 1x + b is: c) b* = [ -1*Sum(x*i)+Sum(y*i) ]/N = log(k) b* -1.7449743 k = 10^b 0.01798978 e) r^2 =(SSE)/(SSE+SSR) 0.99860146 r = sqrt(r^2) 0.99930049 r' = sqrt(1=r^2) 0.03739699 R^2 = 1-(SSR/SST) 0.99866512

Transformation D f) Return to the original non-linear equation and develop an expression for the optimum value of k that minimizes the residuals of the non-linear model (Eq. 1). Your expression should involve only sums and products of the measured non-linear predictor response variables (࠵? ! , ࠵? ! ) What does your result indicate about the linearized approximation for the optimum value of k ? Although the original equation is non-linear, because it is simple and has only one free parameter, we may attempt to determine a best-fit equation for ‘ k ’ by minimizing the sum-of-squares of the residual using the non-linear expression. Initial height, cm ( x i ) Time to bounce, s ( t i ) 12 0.47 9 0.4 6 0.33 3 0.23 a) x*= ln(xi), y* = 2ln(ti) y*=1x* + b* CURVE-FIT TABLE FOR TRANSFORMED EQUATION GOODNESS-OF-FIT CALCULATION FOR PREDICTIVE EQUATION b) & d) xi = xi ti = yi x*i y*i x*i y*i x*i^2 t^i=y^i t-bar=ybar (ti-tbar)^2 t^i-tbar)^2 (ti - t^i)^2 12 0.47 2.48490665 -1.5100452 -3.7523213 6.17476106 0.46462599 0.3575 0.01265625 0.01147598 2.888E-05 9 0.4 2.19722458 -1.8325815 -4.026593 4.82779584 0.40237791 0.3575 0.00180625 0.00201403 5.65445E-06 6 0.33 1.79175947 -2.2173252 -3.9729135 3.210402 0.32854019 0.3575 0.00075625 0.00083867 2.13106E-06 3 0.23 1.09861229 -2.9393519 -3.2292082 1.20694896 0.23231299 0.3575 0.01625625 0.01567179 5.34994E-06 Sums 30 1.43 7.57250299 -8.4993038 -14.981036 15.4199079 1.42785707 1.43 0.031475 0.03000046 4.20155E-05 SST SSE SSR The equation for best-fit value of slope y = 1x + b is: c) b* = [ -1*Sum(x*i)+Sum(y*i) ]/N = log(k) b* -4.0179517 k = e^b 0.01798978 e) r^2 =(SSE)/(SSE+SSR) 0.99860146 r = sqrt(r^2) 0.99930049 r' = sqrt(1=r^2) 0.03739699 R^2 = 1-(SSR/SST) 0.99866512

2 6.1 3 3.5 or 2.9 a) Define modified predictor/response variables (࠵? ! ∗ , ࠵? ! ∗ ) so that you can use a linear regression technique to determine the ‘best-fit’ values of t and R o . Write the linearized equation using the modified predictor/response variables. This equation is converted to linear by taking natural logarithm of both sides: ࠵?࠵?(࠵?) = ࠵?࠵?<࠵? & ࠵? ’ ( ⁄ = ࠵?࠵?(࠵?) = ࠵?࠵?(࠵? & ) + ࠵? ࠵? > Defining the transform variables as: ࠵? ∗ = ࠵?࠵?(࠵?), ࠵? ∗ = ࠵? The transformed equation is: ࠵? ∗ = ࠵?࠵? ∗ + ࠵?, ࠵? = 1 ࠵? > , ࠵? = ࠵?࠵?(࠵? & ) Because the transformed equation is that of a line with both slope and intercept as free parameters, we can use the standard best-fit equations for both the slope and intercept. b) Create a new data table that includes columns for the original data, columns for the modified predictor/response variables, and columns for products of predictor/response variables needed to perform a linear regression using the modified variables. c) Calculate the best-fit value of t and R o from the linear regression of the modified variables (note that your regression calculation should be consistent with the linearized equation you state in part a). d) Using the value of k found in part c and Eq. 1, calculate: the sum-of-squares of the residuals ( SSR or S 2 r ), explained square variance ( SSE , or S 2 e ), total square variance ( SST , or S 2 t ) e) Calculate the correlation coefficient, r (note that ࠵? = √࠵? # ), the adjusted correlation coefficient, r’ ( ࠵? % = √1 − ࠵? # ), the coefficient of determination, R 2 As in problem #1, the best approach to solving the transformed equation and presenting goodness-of-fit is a spreadsheet calculation. Since the last data point R(3) = 3.5 or 2.9, both cases are solved:

Case A, R(3) = 3.5 Case B, R(3) = 2.9 t (hours) Radioactivit y (emissions/ ns) 0 13.8 1 7.9 2 6.1 3 3.5 a) Transformed variable definitions: x*= ti, y* = ln(Ri) Transformed equation definitions: y* = m*x* + b*, m* = 1/Tau, b*=ln(Ro) CURVE-FIT TABLE FOR TRANSFORMED EQUATION GOODNESS-OF-FIT CALCULATION FOR PREDICTIVE EQUATION b) & d) ti Ri x*i y*i x*i y*i x*i^2 R^i=y^i R-bar=y-bar (Ri-Rbar)^2 R^i-Rbar)^2 (Ri - R^i)^2 0 13.8 0 2.62466859 0 0 13.3870712 7.825 35.700625 30.9366355 0.17051023 1 7.9 1 2.06686276 2.06686276 1 8.64395572 7.825 0.005625 0.67068847 0.55347011 2 6.1 2 1.80828877 3.61657754 4 5.58135305 7.825 2.975625 5.03395162 0.26899465 3 3.5 3 1.25276297 3.75828891 9 3.60384793 7.825 18.705625 17.8181248 0.01078439 Sums 6 31.3 6 7.75258309 9.44172921 14 31.2162279 31.3 57.3875 54.4594004 1.00375939 SST SSE SSR c) m* = Slope, b*=intercept: the best-fit equations for slope and intercept are: m* = -0.4374291 b*= 2.5942894 Tau -2.2860848 Ro 13.3870712 e) r^2 =(SSE)/(SSE+SSR) 0.98190223 r = sqrt(r^2) 0.9909098 r' = sqrt(1=r^2) 0.13452794 R^2 = 1-(SSR/SST) 0.98250909 t (hours) Radioactivit y (emissions/ ns) 0 13.8 1 7.9 2 6.1 3 2.9 a) Transformed variable definitions: x*= ti, y* = ln(Ri) Transformed equation definitions: y* = m*x* + b*, m* = 1/Tau, b*=ln(Ro) CURVE-FIT TABLE FOR TRANSFORMED EQUATION GOODNESS-OF-FIT CALCULATION FOR PREDICTIVE EQUATION b) & d) ti Ri x*i y*i x*i y*i x*i^2 R^i=y^i R-bar=y-bar (Ri-Rbar)^2 R^i-Rbar)^2 (Ri - R^i)^2 0 13.8 0 2.62466859 0 0 13.900153 7.675 37.515625 38.75253 0.01003063 1 7.9 1 2.06686276 2.06686276 1 8.48292307 7.675 0.050625 0.6527397 0.33979931 2 6.1 2 1.80828877 3.61657754 4 5.17692027 7.675 2.480625 6.24040233 0.85207619 3 2.9 3 1.06471074 3.19413221 9 3.1593477 7.675 22.800625 20.3911157 0.06726123 Sums 6 30.7 6 7.56453086 8.87757251 14 30.7193441 30.7 62.8475 66.0367877 1.26916735 SST SSE SSR c) m* = Slope, b*=intercept: the best-fit equations for slope and intercept are: m* = -0.4938448 b*= 2.63189985 Tau -2.0249279 Ro 13.900153 e) r^2 =(SSE)/(SSE+SSR) 0.98114331 r = sqrt(r^2) 0.99052679 r' = sqrt(1=r^2) 0.13731965 R^2 = 1-(SSR/SST) 0.9798056

d) Repeat parts a-c for the data set of problem #2. Repeating a-c for problem #2, both data sets: t (hours) Radioactivit y (emissions/ ns) 0 13.8 1 7.9 2 6.1 3 3.5 Transformed variable definitions: x*= ti, y* = Ri Linear Equation: y =mx + b CURVE-FIT TABLE FOR TRANSFORMED EQUATION GOODNESS-OF-FIT CALCULATION FOR PREDICTIVE EQUATION ti Ri x*i y*i x*i y*i x*i^2 R^i=y^i R-bar=y-bar (Ri-Rbar)^2 R^i-Rbar)^2 (Ri - R^i)^2 0 13.8 0 13.8 0 0 12.73 7.825 35.700625 24.059025 1.1449 1 7.9 1 7.9 7.9 1 9.46 7.825 0.005625 2.673225 2.4336 2 6.1 2 6.1 12.2 4 6.19 7.825 2.975625 2.673225 0.0081 3 3.5 3 3.5 10.5 9 2.92 7.825 18.705625 24.059025 0.3364 Sums 6 31.3 6 31.3 30.6 14 31.3 31.3 57.3875 53.4645 3.923 SST SSE SSR a) Using Excel built-in functions for slope and intercept: m= -3.27 b= 12.73 b) r^2 =(SSE)/(SSE+SSR) 0.93164017 r = sqrt(r^2) 0.96521509 r' = sqrt(1=r^2) 0.26145714 R^2 = 1-(SSR/SST) 0.93164017 c) In this case, all three goodness-of-fit metrics show advantage of the exponential fit to the data compared to the linear fit t (hours) Radioactivit y (emissions/ ns) 0 13.8 1 7.9 2 6.1 3 2.9 Transformed variable definitions: x*= ti, y* = Ri Linear Equation: y =mx + b CURVE-FIT TABLE FOR TRANSFORMED EQUATION GOODNESS-OF-FIT CALCULATION FOR PREDICTIVE EQUATION ti Ri x*i y*i x*i y*i x*i^2 R^i=y^i R-bar=y-bar (Ri-Rbar)^2 R^i-Rbar)^2 (Ri - R^i)^2 0 13.8 0 13.8 0 0 12.85 7.675 37.515625 26.780625 0.9025 1 7.9 1 7.9 7.9 1 9.4 7.675 0.050625 2.975625 2.25 2 6.1 2 6.1 12.2 4 5.95 7.675 2.480625 2.975625 0.0225 3 2.9 3 2.9 8.7 9 2.5 7.675 22.800625 26.780625 0.16 Sums 6 30.7 6 30.7 28.8 14 30.7 30.7 62.8475 59.5125 3.335 SST SSE SSR a) Using Excel built-in functions for slope and intercept: m= -3.45 b= 12.85 b) r^2 =(SSE)/(SSE+SSR) 0.94693504 r = sqrt(r^2) 0.97310587 r' = sqrt(1=r^2) 0.23035833 R^2 = 1-(SSR/SST) 0.94693504 c) In this case, all three goodness-of-fit metrics show advantage of the exponential fit to the data compared to the linear fit

Problem #4: Consider the attached datasheet for temperature sensor devices: a type-T Thermocouple , a platinum wire RTD, and a model #44004 semi-conductor Thermistor. Based on the data tables alone, determine: a) The units of the sensitivity ( S ) value for all three sensors. Note the input is temperature for all three sensors. The definition of sensitivity is: ࠵? = ࠵?࠵?࠵?࠵?࠵?࠵?࠵? ࠵?࠵?࠵?࠵?࠵?࠵? , ࠵?࠵? ࠵?࠵?࠵?࠵?࠵?࠵?࠵?࠵?࠵?࠵?࠵?࠵?࠵? ∆࠵?࠵?࠵?࠵?࠵?࠵? ∆࠵?࠵?࠵?࠵?࠵? By inspection of the values and units of the data sheets, we identify the output/inputs and sensitivity for all three thermos-electric sensors as: Input Output Sensitivity Thermocouple T Voltage mV/C RTD T Resistance Ohms/C Thermistor T Resistance Ohms/C b) The value of the sensitivity when the input temperature is 0 ° C for all three sensors. Sensitivity can be approximated by: D (OUTPUT)/ D T At 0 °C, we can consider small change from 0 to 1 °C: D (OUTPUT) = {Output at 1 °C - Output at 0 °C} D T = {1 °C - 0 °C} = 1 °C Output 0 C Output 1 C D(OUTPUT) Sensitivity Thermocouple (mV) 0.000 0.039 0.039 0.039 mV/C RTD (ohms) 100 100.39 0.390 0.390 Ohms/C Thermistor (ohms) 7355 6989 -366 -366 Ohms/C c) The value of the sensitivity when the input temperature is 20 ° C for all three sensors.

Computing expected and actual responses for all three sensors: Output 20C Pred Out 20C DOUT, act DOUT, pred Difference % Linearity Thermocouple (mV) 0.790 0.780 0.790 0.780 0.010 1.28% RTD (ohms) 107.79 107.80 7.79 7.80 -0.01 0.13% Thermistor (ohms) 2814 35 -4541 -7320 2779 37.96% e) What resolution of the electrical measurement is needed so that temperature resolution will be 0.1 °C at 0 °C for each sensor? The electrical resolution required can be computed by multiply sensitivity and required temperature resolution: ࠵?(࠵?࠵?࠵?࠵?࠵?࠵?) = ࠵? ∙ ࠵?࠵? For each sensor, the results are: dT in C Sensitivity @ 0C Resolution Thermocouple (mV) 0.1 0.039 0.0039 mV RTD (ohms) 0.1 0.390 0.039 ohms Thermistor (ohms) 0.1 -366 36.6 ohms f) What resolution of the electrical measurement is needed so that temperature resolution will be 0.1 °C at 20 °C for each sensor? Similarly at 20 °C, dT in C Sensitivity @ 0C Resolution Thermocouple (mV) 0.1 0.040 0.0040 mV RTD (ohms) 0.1 0.390 0.039 ohms Thermistor (ohms) 0.1 -124 12.4 ohms