GA approach for finding Rough Set decision rules based on bireducts

Feature selection plays an important role in knowledge discovery and data mining nowadays. In traditional rough set theory, feature selection using reduct - the minimal discerning set of attributes - is an important area. Nevertheless, the original definition of a reduct is restrictive, so in one of the previous research it was proposed to take into account not only the horizontal reduction of information by feature selection, but also a vertical reduction considering suitable subsets of the original set of objects. Following the work mentioned above, a new approach to generate bireducts using a multi--objective genetic algorithm was proposed. Although the genetic algorithms were used to calculate reduct in some previous works, we did not find any work where genetic algorithms were adopted to calculate bireducts. Compared to the works done before in this area, the proposed method has less randomness in generating bireducts. The genetic algorithm system estimated a quality of each bireduct by values of two objective functions as evolution progresses, so consequently a set of bireducts with optimized values of these objectives was obtained. Different fitness evaluation methods and genetic operators, such as crossover and mutation, were applied and the prediction accuracies were compared. Five datasets were used to test the proposed method and two datasets were used to perform a comparison study. Statistical analysis using the one-way ANOVA test was performed to determine the significant difference between the results. The experiment showed that the proposed method was able to reduce the number of bireducts necessary in order to receive a good prediction accuracy. Also, the influence of different genetic operators and fitness evaluation strategies on the prediction accuracy was analyzed. It was shown that the prediction accuracies of the proposed method are comparable with the best results in machine learning literature, and some of them outperformed it.

Letter to Louis J. Cahill from the Managing Editor [no indication of who he works for – same signature as is on the Knox, Harvie and Foss letter

Letter (1 typed page) to Louis J. Cahill from the Managing Editor [no indication of who he works for – same signature as is on the Knox, Harvie and Foss letter] saying that he has found various references to Samuel Zimmerman being active in the erection of the first suspension bridge and its Roebling successor, July 20, 1948.

Letter to S.D. Woodruff from William Colburn with the letterhead “Office of Detroit Bridge and Iron Works”

Letter to S.D. Woodruff from William Colburn with the letterhead “Office of Detroit Bridge and Iron Works” regarding stating that he closed up the business with Dewey’s. He states that “we” now have “warranty deeds” from them for exclusive rights for all time for hunting, shooting and trapping, Mar. 20, 1884.

Letter to S.D. Woodruff from J.W. Harper of the Department of Public Works, Toronto

Letter to S.D. Woodruff from J.W. Harper of the Department of Public Works, Toronto. He is sending a copy of an order in Council on the subject of certain charges made by Joshua Manly of Port Colborne against Mr. Woodruff and other persons connected with the Welland Canal, May 12, 1859.

Bill no.71: An act respecting the Public Works of Ontario

Bill no.71: An act respecting the Public Works of Ontario (10 ½ pages, printed). S.D. Woodruff has signed this copy of the bill and has made a note in pension “regulations for management”, 1868.

Receipt from Aetna Works, Sheffield for payment on account

Receipt from Aetna Works, Sheffield for payment on account, Nov. 26, 1873.

Receipt from Chance Brothers and Co. Glass Works near Birmingham, England

Receipt from Chance Brothers and Co. Glass Works near Birmingham, England regarding payment received for glass panes. This is accompanied by an envelope, April 6, 1875.

Receipt from St. Catharines Water Works for water rent

Receipt from St. Catharines Water Works for water rent, Aug. 18, 1887.

Receipt from St. Catharines Water Works for water rent

Receipt from St. Catharines Water Works for water rent, Oct. 1, 1887.

Certificate on the occasion of Samuel DeVeaux Woodruff’s retirement from the Water Works Commission of the City of St. Catharines

Certificate measuring 64 cm. x 48 cm. on the occasion of Samuel DeVeaux Woodruff’s retirement from the Water Works Commission of the City of St. Catharines. Mr. Woodruff served the commission from 1875 to 1899. He also served as chairman of the commission. This is signed by Lucius S. Oille, George C. Carlisle and Connolly B. Hare (members of the committee), Jan. 2, 1900.