Investigation of Viability of Genetic Programming for Symbolic Regression of Engineering - Math Problem Example

Investigation of Viability of Genetic Programming for Symbolic Regression of Engineering Your name Name of Assignment 3rd March, 2013 hi, Symbolic regression analysis/results in the paragraph below . What does the y stands for and also in page 6 can you please explain and i wanted to know the equation for Q. Because we don't need to use the same equation given we need to produce our own equation and equation we produce is to calculate the submerged flow as how it is asked in the brief on page 4 and 5. is it possible if the writer could send me the results for the matlab and excel please Introduction The controlling the flow or discharge of rivers and canals is essential in flood controlling, irrigation and navigation. This can be done by modelling using various mathematical software’s and equations. The common model is GPLAB genetic programming software created by Silva (2007). GPLAB genetic programming software is popular for approximating volumetric outflow rate of rivers channels and rivers in consideration with width and depth. In this paper will try to model the volumetric outflow rate of canal is computed from the following equation Q=bCd Where Q is the discharge (volumetric outflow rate) (m3/s), b is the width of the channel (m), Cd is the coefficient of discharge, yG is the height of the gate opening above the bed of the channel (m), g is the acceleration due to gravity (m/s2) y1 is the upstream depth of water above the bed of the channel (m) and y2 is the depth just downstream of the gate for free outflow. Represents velocity of flow when there is a restriction to the flow of water. The continuity equation states that the reduction in diameter would cause an increase in the fluid flow speed. This is where when the water depth some distance downstream of the gate (y3) is raised by blocking the channel during a flood, high tide levels. If this depth is large enough such that y3 > y2 then submerged outflow occurs Under submerged flow conditions equation A1 no longer applies. The submerged discharge, Q, is now a function of the “head difference” y1 – y3. The approach that has been used in this paper to obtain an empirical equation for the submerged discharge under a radial gate is to carry out a symbolic regression using genetic programming of observed head differences and discharges for different gate opening heights. This leads to use global expression for predicting the critical depth in channels with different cross sections and flow regimes, two separate analyses were performed to both verify the selected “best” expression and also investigate the underlying state of the expression’s coefficients. A dimensional analysis will be carried out to determine the relationship between the upstream depth of water of the channel, discharge and width of river. Any increase the flow speed to the point that viscous dissipation can no longer stabilize the flow, the macroscopic balance between mean flow inertia and viscous effects breaks down. At this point there is a transition from purely laminar flow to turbulence. Errors in the analysis can be minimized by quality control of the survey and are generally within acceptable tolerances. However, gross errors do sometimes occur and these need to be eliminated either by quality control procedures at the data input stage (for example by plotting all the cross-sections on the computer screen) or during the model calibration procedure. Errors in the hydrological data are not so easy to eliminate and may, therefore, need to be catered for by a sensitivity analysis. For example, if say a 100-year return period flood hydrograph has been generated, it may be subject to quite wide confidence limits. Thus it would be advisable to run the computational model with several flood hydrographs to establish the sensitivity of flood stage to flood discharge. The chosen cross-sections need to be representative of the river reach. Thus between cross-sections there should not be any large changes in cross-section. Also, the cross-sections should be drawn normal to the general flow direction. This is not a problem for the main channel, but is uncertain on the flood plains when using a one-dimensional model. It is very important that the discretized data preserves, as far as possible, both the river and flood plain reach lengths and the flood plain volumes. At the interface between the main channel and the flood channel, momentum is transferred across the interface by turbulent shearing action. In the main channel significant secondary flow cells develop. The whole width of the compound channel is also subjected to shearing action in both the horizontal and vertical planes GPLAB genetic programming will be used for solving this problem in control and flow. At beginning the models will be used to solve for upstream depth of water above the bed of the channel as well as the width of the channel. This programme will applied to entire experiment with changing form the height of the gate opening above the bed of the channel, the upstream depth of water above the bed of the channel to the depth just downstream of the gate for free outflow. GPLAB genetic programming is used after a simple calibration to provide correct figures. The model uses 3D flow structures that occur in rivers and channels which assist forecast the variation of depth of river bed as well as discharge of the river. It provides a tool for water level prediction for distributing flows within a cross section, and for predicting the lateral distributions of boundary shear stress. Symbolic regression analysis/results: The graph below shows output from Matlab iterations which was performed and consumed a lot of time. From the graph it can be noted that errors in calculating y is an alternative in approximating Q where Q = 5017.2h 1.6. The big fluctuation and vagaries from a strict average value for discharge before the gate could be due to the fact that since it is nearer to the inlet, there could be a lot external influences that causes slight changes in pressure head, thereby fluctuating the discharge of the flow The graphs above shows the best fit line out of canal one data it was obtained after 20 runs and the best model was selected. The residue graph next to the best fit was also plotted to show distribution for the data. The graph below shows discharge versus depth or height of the gate opening upstream. The structure of the graph shows clearly that the best fit line is collect and does not have much difference. The graph is plotted using the law data provided in excel. The volumetric outflow rate shown in the graph is supported by the fact that water is free outflow in usual conditions. It can be understood that the flow develops in a lamina manner, the curve at least linear. However there is a large difference between the two canals because of the type of brokerage that is applied. It was also affected by the depth conditions of the flow that is the riverbed and water service. The water velocity in y1 was different from that one in y2. The differences in height between y1 and y3 showed their increase in volumetric outflow. The mat lab output for the data has given a discharge curve as shown below. In the graph it can be noted that canal 1 has the following equations Q=3259h1.63 Q=2282.9h1.8 These coefficients that have been determined help to create linearity in the equation showing that the result is linear. According to the graph it indicates that under free outflow conditions the results will give a best fit that is linear with the results as a coefficient of discharge of 1.7. This however should be noted that the pressure difference between the two service levels and channel bed is taken care of by head difference figure (y1+y3). The discharge through the radial gate can be either laminar or turbulent. This is because of the difference in water levels which affects the discharge velocity. The values obtained for coefficient of discharge show interdependence between height and velocity. The pareto graph shows that there is two solutions to the equation which have the same optimum coefficient to solve the problem. The graph has given figures that have lower errors which is good. Since the gate with a hole in the middle is placed inside the duct in which fluid flows, when it reaches the blockade posed by a plate, the fluid is forced to converge to go through the small hole and the point of maximum convergence occurs at the vena contracta with a change in discharge and pressure. Beyond the vena contracta, since the fluid expands, there is once again a change in the velocity and pressure. By measuring the difference in fluid pressure between the pipe cross section and input nozzle, the volumetric and mass flow rates are obtained using the Bernoulli's equation. It is with these values that the Coefficient of Discharge corresponding to each pressure head difference was also calculated. the coefficient of discharge varies according to the change in the pressure head. It was then compared with the Reynold's number at each of the cross sections that were being observed. The flow of a fluid through a closed conduit can either be laminar or turbulent. This difference of flow characteristics is dependent on the flow velocity, density and viscosity of the fluid and the conduit diameter. Reynolds number have then been compared with the coefficient of discharge to show how they are interdependent and can be seen in Graph. Flow upto a Reynold's number of around 2100 is considered laminar flow and that above 4000 is considered turbulent flow. When the Reynolds number falls between these two extremes, it corresponds to a transition phase where, the flow can exhibit laminar, turbulent, or intermittent flow characteristics. The wide fluctuations and waviness of the graph could allude to the turbulence, or a phase of transition to turbulence flow property. Generations and fitness Generations and size Critical appraisal: When applying GP to calculate optimum coefficients that are application in symbolic expression, evolution of the initial data into many generations which requires more time to compute the answers. In the calculation to save time Derive 6 was employed which simplified the equation and obtain the optimum coefficients for model. The level of complexity of the analysis was simplified by the use of drive 6 which simplified the mathematical equation as well as providing optimum coefficient which was applied in the analysis. This helped in analysing model and its goodness of fit. The same data was analyzed used genetic programming to obtain optimum values for regression coefficient, the model was found to have an error of 9.6%. The accuracy of the model was confirmed by the second data canal 2 which also had an absolute error of 9%. Both model had a tree debt of ten levels with one hundred fifty nodes. The model performed 1% better than the benchmark for the planned investigation. From the investigation general programming appeared to give less complex and comparable information with certain degree of accuracy. An error of 9.1% for canal 1 and 9% for canal 2 data is a confirmation that the discharge rate that is being used to control flooding in the area is accurate. From results it can also be depicted that 85% to 95% of the data fear within the required range. The mean root of squared errors from the model should that the performance of general programming was high. The fitness graph that was made also showed that most data failed within the standard error of 0.84% the difference between canal 1 and canal 2 data in the model is minimum thus showing that the regression analysis performed well. The is also a number of values which does not fall within the plotted sections as show in the graph and has a percentage of 0.5%. The other methods of achieving the same objectives include Fast Fourier Transform approach is preferred due to its efficiency and the fact that it takes a much shorter time to achieve the same results. The use of the Fast Fourier Transform is what has enabled many electronic techniques to be possible and practical. This is because, unlike the previous approaches, the Fast Fourier Transform only utilizes hundreds of times fewer lines of code, mostly a few dozen (Brigham 1988). It should however be noted that, while being an efficient approach, the Fast Fourier Transform is also the most complicated algorithm in “DSP”. This chapter outlines the nature of the Fast Fourier Transform analysis and also how it works to achieve the aforementioned objectives. An important feature to note concerning the decomposition process is that it is what makes it possible for the samples in the original signal to be reordered. However, the re-ordering of these samples is supposed to follow a specific pattern which is determined by the binary equivalents of each sample. This algorithm that involves the rearrangements of the order of the N time domain samples through the counting in binaries that have been flipped from left to right. After the bit reversal sorting stage of the “FFT” algorithm, the next step is the finding of the frequency spectra which belongs to the 1 point time domain signals at the end of the last decomposition phase. This is a very easy process since the frequency spectrum of a 1 point signals is equal to itself and therefore there is virtually nothing to be done at this stage. Also, it should be noted that the final 1 point signals are no longer time domain signals but rather, a frequency spectrum. Lastly, the N frequency spectra are then supposed to be recombined in the reverse order that is exactly similar to the order followed during the decomposition of the time domain. Since bit reversal formula is not applicable for this recombination, the reverse process is performed one stage at a time. The reverse process is known as synthesis and this is done stage wise from the single point spectra and finally forming the 16 point frequency spectrum. The different flow diagrams used to represent this synthesis is referred to as a butterfly and it is also what forms the basic computational element of the “FFT” as it transforms two complex points into two different but still complex points. The process of synthesizing the frequency domain from the separate single signals undergoes three main essential loops which are also concentric in nature. The outermost loop is the one that runs through the Log2N stages while the middle loop is the one which moves through each of the individual frequency spectra that are in the stage currently being worked on. The final loop which is also the innermost is what now uses the butterfly diagram mentioned earlier in the calculation of the points that are in each frequency spectra. The three loops are the three main stages that constitute the transformation of a given data from the time domain data into the frequency domain data and vice versa. There are also special cases whereby the “FFT” algorithm is only applicable to periodic signals if one wants the sampled input to also become periodic. This phenomenon is referred to as spectral leakage. In the case, which happens frequently, whereby the sampled signal is not periodic, there will result discontinuities in the periodic signal that has been processed by the “FFT”. This discontinuity can also occur when an integer number of periods are not sampled. The eventual effect of this is that the energy that is contained in the signal will end up leaking from the signal frequency bin into the other frequency bins that are adjacent to it, hence the term spectral leakage. The result of this leakage is errors in the amplitudes of the frequency spectrum display. Due to these amplitude errors, the overall effect will be a case whereby small frequency peaks that occur very close to larger ones will end up being obscured. Thus, in order to reduce the effects of this spectral leakage, a process known as windowing which basically utilizes window functions can be used The window function basically works by assigning a weighting coefficient to each of the input samples so that the samples which cause spectral leakage are reduced. For instance, all the samples that begin and end the sampling period will end up being reduced to zero and the resulting effect of this is that the discontinuities in the periodic sampled signal will end up being removed. The type of window to use for this purpose will be dependent upon the frequency content of the signal. Examples of windows include the Hanning window, the 5 term Blackman-Harris designed window and the Gaussian window just to name a few. Another error that can occur in a spectrum and which can be resolved by use of “FFT” algorithms is known as the scallop loss. It is caused by the discrete nature of the frequency spectrum whereby the signal is displayed as amplitude levels which are at discrete bins which are equally spaced. If the signal frequency coincides with the center of one of the discrete frequency bins, the correct peak level is displayed. If, however, the signal frequency is not at the centre of a frequency bin, then a reduced level is displayed, causing an error which can be up to 3 dB. Flat top windows have traditionally been used to reduce scallop loss, but these have a wide main lobe, which compromises resolution, and poor side lobe rejection (Halberstein 1966). Due to the dynamic nature of the “FFT”, it has been able to find very many applications. Most of the situations during the image processing and filtering utilize the Fast Fourier Transform. It is common to use the “FFT” while processing images from astronomy and sometimes microbiology. But it is often very useful when dealing with structures that are repetitive in nature such as crystals. The ability of Fast Fourier Transform to identify periodically repetitive components in a given lattice or image is what makes it a favourable method for removing irregular dirty spots or noise from images. Conclusion The aim was to predict volumetric outflow rate formula using Symbolic regression analysis in GPLAB genetic programming. Using the data provided various discharge rates determined as well as various coefficients successfully. The graphical representation of the volumetric outflow rate at various positions over the height helped understand the basis of how a radial gate is operated and how it affects the flow process. The coefficient of discharge of the flow regime over different operations of the rate was also successfully calculated. The graphical comparison of these values with Reynold's number was also done successfully. Further to that, the fact that the friction factors of the gate causes a marginal energy loss and thus a reduced pressure head difference could I turn be a limiting factor to properly evaluate the data and understand the flow process. Other models that were to be used in estimating the regression model for the problem was also discussed. References Abril, J.B. & Knight, D.W., 2004. Stage-discharge prediction for rivers in flood applying a depth-averaged model. Journal of Hydraulic Research, IAHR, 42 (6): 616-629. Aytek, A. & Kisi, O. , 2008. A genetic programming approach to suspended sediment modelling. Journal of Hydrology, Elsevier, 351 (3-4): 288-298. Batchelor, G. K., 1967, An Introduction to Fluid Dynamics, Cambridge: Cambridge University Press. Chadwick, A., Morfett, J. & Borthwick, M. 2004. Hydraulics in Civil and Environmental Engineering. London: Spon Press. Chaudhry, M.H., 2008. Open-Channel Flow. New York: Springer. Dey, S., 2001. Flow measurement by the end-depth method in inverted semicircular channels. Flow Measurement and Instrumentation, Elsevier, 12 (4): 253–258. Dooge, J. C. I. & O'Kane, J. P. 2003.Deterministic Methods in Systems Hydrology. Balkema, Lisse. Finnemore, E.J. & Franzini, J.B., 2002. Fluid mechanics with engineering applications. 10th edition, McGraw-Hill, Boston. Giustolisi, O. & Savic, D. A., 2006. A symbolic data-driven technique based on evolutionary polynomial regression, J. ydroinform., 8(3), 207–222, doi:10.2166/hydro. Halberstein, J.H., 1966. Recursive, complex Fourier analysis for real-time applications, Proc. IEEE. 54(6) 903-4 McGahey ,C.and Knight, D. & Samuels, P.G. 2009. Advice, methods and tools for estimating channel roughness. Proc. Instn. Civ. Engrs., Water Management, 162(6), 353-362. Novak, P. Guinot, V. Jeffrey, A. & Reeve, D. 2010. Hydraulic Modelling – An Introduction: Principles, Methods and Applications. Spon Press, London. Parasuraman, K., Elshorbagy, A., & Si, B. C., 2007. Estimating saturated hydraulic conductivity using genetic programming, Soil Sci. Soc. Am. J., 71, 1676–1684. Stravs, L. & Brilly, M., 2007. Development of a low-flow forecasting model using the M5 machine learning method, Hydrolog. Sci.J0, 52(3), 466–477. Zhao, M., Cheng, L., & Teng, B., 2007, Numerical Modeling of Flow and Hydrodynamic Forces around a Piggyback Pipeline near the Seabed, Journal of Waterway, Port, Coastal, and Ocean Engineering , pp.286-294. Read More