Download Regression Analysis of Membrane Protein Fusion: Linear & Quadratic Methods and more Study Guides, Projects, Research Mathematics in PDF only on Docsity! Math 181 Name (Group member’s names): March 3, 2008 Project # I Requirements: The solution for the project must be written in clean and clear report (You are recommended to type it in computer using word or Latex). You may use computer but you must claim what kind of software you used. For each group, all of the members’ names must be included in a single report. 1. (30 points) Membrane protein fusion experiments is proceeded in lab. Technicians measure the fluorescence intensity with respect to the time (seconds) to record the fusion process. Simple t-SNARE and v-SNARE proteins are used so that there are no regulation effects in the experiment. People believe the fluorescence intensity is function of time. The data collected in the experiment is listed in the table below. Use regression method to recover the appropriate functions. (You may use Excel or other programming softwares.) t 2 5 10 15 25 30 50 80 120 FI 2.341 2.714 3.464 4.473 7.113 7.665 10.236 12.98 13.55 (i) Find the line regression formula for the fluorescence intensity (FI) with respect to time. (In this problem, you are not allowed to use any software. Derive the formula by hand) Plot the figure and the collected data to compare your result. (ii) Find the quadratic regression formula for the fluorescence intensity (FI) with re- spect to time. Plot the figure and the collected data to compare your result. (iii) For each t in the data set, tell the value predicted by the quadratic formula de- rived in (ii). Then calculate the residual for each point, and the total residual. (iv) Predict the FI for the 5th minutes. Solution: (i) According to linear regression theory, the regression function is in the form of y = a0 + a1x. Then, the corresponding coefficients are 9∑ i=1 ti = 337; 9∑ i=1 t2i = 25179, 9∑ i=1 FIi = 64.536; 9∑ i=1 FIiti = 3703.962 so that the matrix expression of the linear regression is( n ∑ ti∑ ti ∑ t2i ) ( a0 a1 ) = ( ∑ FIi∑ FIiti ) ⇔ ( 9 337 337 25179 ) ( a0 a1 ) = ( 64.536 3703.962 ) . So, solving this linear system{ 9a0 + 337a1 = 64.536 337a0 + 25179a1 = 3703.962 1 2 by elimination, one has a0 = 3.33254, a1 = 0.1025. Therefore, the linear regression of these data is (1) FI = 3.3325 + 0.1025t. The comparison of the data and regression linear line is refer to figure 1. (ii) The quadratic regression line for the fluorescence intensity is given by the EXCEL that (2) FI = −0.0011t2 + 0.229t + 1.6214. The comparison of the data and regression quadratic function is refer to figure 1. (iii) By the formula (2), the predicted values by the quadratic formula derived in (ii) are listed in the following table t FI FIp(t) r(t) 2 2.341 2.075 0.070756 5 2.714 2.7389 0.00062001 10 3.464 3.8014 0.11383876 15 4.473 4.8089 0.11282881 25 7.113 6.6589 0.20620681 30 7.665 7.5014 0.02676496 50 10.236 10.3214 0.00729316 80 12.98 12.9014 0.00617796 120 13.55 13.2614 0.08328996 where FIp(t) = −0.0011t2 + 0.229t + 1.6214 for each t and the pointwise residual is given by r(ti) = |FIi − FIp(ti)|2. The total residual is therefore easily to be calculated as Rtotal = 9∑ i=1 r(ti) = ∑ |FIp(ti)− FIi|2 = 0.62777643. Remark: You may use residual r(ti) = |FIi − FIp(ti)| as well, this is L∞ regression. It is called uniform approximation. You may observe that this is larger than the one in the solution, which means it is not an appropriate residual (error) we should look at. (iv) For t = 5 minutes, it is equivalently that t = 300 seconds. By linear regression (1), the prediction is FI l = 54.5825; while by the quadratic regression (2), the prediction is FIq = −28.68. Note the fluorescence intensity could not be less than zero, we have following conclusions: (a) quadratic regression is much accurate than the linear regression, as the residuals is smaller; (b) the quadratic regression function is valid locally, i.e. as t→∞, quadratic regres- sion fails, while the linear regression is always valid.
