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A Sparse Bayesian Learning for Diagnosis of Nonstationary and Spatially Correlated Faults:References

This paper is available on arxiv under CC 4.0 license.
Authors:
(1) Jihoon Chung;
(2)  Zhenyu (James) Kong. Table of Links Abstract & Introduction
Review of Related Work
Proposed Research Methodology
Numerical Case Studies
Real-World Simulation Case Studies
Conclusion
References II. REVIEW OF RELATED WORK The related existing studies of fault diagnosis in manufacturing systems are reviewed in Section II A. Then, the literature related to sparse Bayesian learning is provided in Section II-B. Afterward, the research gaps in the current work are identified in Section II-C. A. Fault Diagnosis Methodologies in Manufacturing Systems Numerous studies have focused on fault diagnosis methodologies for manufacturing systems, utilizing the fault-quality model outlined in Eq. (1). [14] developed a PCA-based orthogonal diagonalization strategy to transform the measurement data. It enabled the estimation of the variance of KCCs in a multistation assembly system. [13] presented a fault diagnosis method in the multistation assembly systems integrating the state space model of the process and matrix perturbation theory. [25], [26] proposed a fault diagnosis method in the machining process considering the process physics regarding how fixtures generate the patterns. Using this method, root cause identification was conducted sequentially. The approaches mentioned above assume that the number of measurements is greater than the number of KCCs (i.e., M > N in Eq. (1)). However, this assumption may not always be consistent with industrial practice. These approaches become ineffective when this assumption is violated because the fault-quality linear model leads to an underdetermined system, resulting in a nonunique solution. To overcome an underdetermined system in the fault-quality linear model, sparse learning can be utilized, which has gained considerable attention in fault diagnosis and detection within manufacturing systems. For fault diagnosis in the manufacturing system, [27] developed a fault diagnosis method using dictionary learning and sparse representationbased classification. [28] proposed a novel root cause diagnostic framework satisfying the assumption that sparse inputs affect the process output. [29] proposed a group-sparsity learning approach for bearing fault diagnosis. In addition to fault diagnosis and detection in the general manufacturing system, sparse learning has been widely utilized to address the issue of an underdetermined system in the multistation assembly system. In particular, sparse Bayesian learning has been widely utilized to incorporate the sparsity of process faults as the prior distribution. [5] proposed a fault diagnosis approach by enhancing the relevant vector machine to detect process faults using the sparse estimate of the variance change of KCCs. [8] developed a spatially correlated sparse Bayesian learning to consider the spatial correlation of KCCs in sparse estimation. The work identifies the KCCs with mean shifts. [10] proposed a Bayesian model to identify the KCCs that have variance increases in a multistation manufacturing system using the sparse variance component prior. [9] developed a novel sparse Bayesian learning to figure out the KCCs with mean shifts by considering the temporal correlation of KCCs and the prior knowledge of process faults. B. Sparse Bayesian Learning After the introduction of sparse Bayesian learning (SBL) by [30], many researchers have extended this approach significantly. For instance, [31] was the first to apply SBL to sparse estimation for the single measurement vector model given in Eq. (1). Subsequently, [32] further extended it to the MMV model (Eq. (2)) by developing the MSBL algorithm under the common support assumption. The notable advantage of SBL and MSBL is that their global minimum is always the sparsest solution, while that of the minimization-based sparse algorithms [17] [33] is usually not the sparsest solution [16], [34]. Based on the MMV model, many previous studies exploit the spatial correlation in solution vectors (i.e., rows in X in Eq. (2)). [35] proposed a block structure to exploit the intra-block correlation for sparse estimation. [36] developed a Bayesian method for recovery of block-sparse solution whose block-sparse structures are entirely unknown. [37] modeled the spatial structure of the solution as Markov dependency by the Beta process. Besides considering the spatial correlation of sparse solutions, work that considers nonstationary sparse solutions has been studied recently under the MMV model in the SBL framework. [38] developed a method using the Dirichlet process to cluster the measurements into groups with common sparsity patterns. Compared to [38], [2] proposed a more general method having two sparsity components: a commonly shared sparsity and an individual sparsity to deal with outliers that deviated from the uniform sparsity pattern in each group. C. Research Gap Analysis The research presented in Section II-A focuses on utilizing the sparsity of process faults for accurate fault diagnosis when low dimensional measurements exist in actual manufacturing systems. However, there is a lack of efforts to identify process faults by considering the spatial correlation and the nonstationary process faults, which is common in industrial practice. Section II-B introduces methods in SBL. It introduces the work concerning the spatial correlation in the solution vector and dealing with the nonstationary sparse solution individually. However, it still lacks the work that uses both properties simultaneously. Therefore, this paper proposes a novel SBL method considering both properties for accurate fault diagnosis. This paper is available on arxiv under CC 4.0 license. Authors: (1) Jihoon Chung; (2)  Zhenyu (James) Kong. This paper is available on arxiv under CC 4.0 license. Authors: (1) Jihoon Chung; (2)  Zhenyu (James) Kong. Table of Links Abstract & Introduction Review of Related Work Proposed Research Methodology Numerical Case Studies Real-World Simulation Case Studies Conclusion References Abstract & Introduction Abstract & Introduction Review of Related Work Review of Related Work Proposed Research Methodology Proposed Research Methodology Numerical Case Studies Numerical Case Studies Real-World Simulation Case Studies Real-World Simulation Case Studies Conclusion Conclusion References References II. REVIEW OF RELATED WORK The related existing studies of fault diagnosis in manufacturing systems are reviewed in Section II A. Then, the literature related to sparse Bayesian learning is provided in Section II-B. Afterward, the research gaps in the current work are identified in Section II-C. A. Fault Diagnosis Methodologies in Manufacturing Systems Numerous studies have focused on fault diagnosis methodologies for manufacturing systems, utilizing the fault-quality model outlined in Eq. (1). [14] developed a PCA-based orthogonal diagonalization strategy to transform the measurement data. It enabled the estimation of the variance of KCCs in a multistation assembly system. [13] presented a fault diagnosis method in the multistation assembly systems integrating the state space model of the process and matrix perturbation theory. [25], [26] proposed a fault diagnosis method in the machining process considering the process physics regarding how fixtures generate the patterns. Using this method, root cause identification was conducted sequentially. The approaches mentioned above assume that the number of measurements is greater than the number of KCCs (i.e., M > N in Eq. (1)). However, this assumption may not always be consistent with industrial practice. These approaches become ineffective when this assumption is violated because the fault-quality linear model leads to an underdetermined system, resulting in a nonunique solution. A. To overcome an underdetermined system in the fault-quality linear model, sparse learning can be utilized, which has gained considerable attention in fault diagnosis and detection within manufacturing systems. For fault diagnosis in the manufacturing system, [27] developed a fault diagnosis method using dictionary learning and sparse representationbased classification. [28] proposed a novel root cause diagnostic framework satisfying the assumption that sparse inputs affect the process output. [29] proposed a group-sparsity learning approach for bearing fault diagnosis. In addition to fault diagnosis and detection in the general manufacturing system, sparse learning has been widely utilized to address the issue of an underdetermined system in the multistation assembly system. In particular, sparse Bayesian learning has been widely utilized to incorporate the sparsity of process faults as the prior distribution. [5] proposed a fault diagnosis approach by enhancing the relevant vector machine to detect process faults using the sparse estimate of the variance change of KCCs. [8] developed a spatially correlated sparse Bayesian learning to consider the spatial correlation of KCCs in sparse estimation. The work identifies the KCCs with mean shifts. [10] proposed a Bayesian model to identify the KCCs that have variance increases in a multistation manufacturing system using the sparse variance component prior. [9] developed a novel sparse Bayesian learning to figure out the KCCs with mean shifts by considering the temporal correlation of KCCs and the prior knowledge of process faults. B. Sparse Bayesian Learning B. Sparse Bayesian Learning After the introduction of sparse Bayesian learning (SBL) by [30], many researchers have extended this approach significantly. For instance, [31] was the first to apply SBL to sparse estimation for the single measurement vector model given in Eq. (1). Subsequently, [32] further extended it to the MMV model (Eq. (2)) by developing the MSBL algorithm under the common support assumption. The notable advantage of SBL and MSBL is that their global minimum is always the sparsest solution, while that of the minimization-based sparse algorithms [17] [33] is usually not the sparsest solution [16], [34]. Based on the MMV model, many previous studies exploit the spatial correlation in solution vectors (i.e., rows in X in Eq. (2)). [35] proposed a block structure to exploit the intra-block correlation for sparse estimation. [36] developed a Bayesian method for recovery of block-sparse solution whose block-sparse structures are entirely unknown. [37] modeled the spatial structure of the solution as Markov dependency by the Beta process. Besides considering the spatial correlation of sparse solutions, work that considers nonstationary sparse solutions has been studied recently under the MMV model in the SBL framework. [38] developed a method using the Dirichlet process to cluster the measurements into groups with common sparsity patterns. Compared to [38], [2] proposed a more general method having two sparsity components: a commonly shared sparsity and an individual sparsity to deal with outliers that deviated from the uniform sparsity pattern in each group. C. Research Gap Analysis C. Research Gap Analysis The research presented in Section II-A focuses on utilizing the sparsity of process faults for accurate fault diagnosis when low dimensional measurements exist in actual manufacturing systems. However, there is a lack of efforts to identify process faults by considering the spatial correlation and the nonstationary process faults, which is common in industrial practice. Section II-B introduces methods in SBL. It introduces the work concerning the spatial correlation in the solution vector and dealing with the nonstationary sparse solution individually. However, it still lacks the work that uses both properties simultaneously. Therefore, this paper proposes a novel SBL method considering both properties for accurate fault diagnosis.

Sparse Bayesian Learning: Review of Related Work

Scientific American, Volume XXIV., No. 12, March 18, 1871 by Various, is part of the HackerNoon Books Series. You can jump to any chapter in this book . To Manufacturers of Knitted Goods. here To Manufacturers of Knitted Goods. is now in England, completing arrangements so as to be able to supply his American friends with his improved Knitting Machines with greater dispatch, and with all the latest improvements. He would beg to call especial attention to JOHN KENT The Improved Rib Top Frame, now so well known, and acknowledged to be the best rib top frame ever built, for speed and quality of goods produced. Price, delivered free in New York, $520, currency. The Improved Circular Web Frame, for drawers and shirts, built of any size and gage. Price for a 4-head set, 17 inch to 20 inch diameter, $810, currency, delivered free in New York. The Circular Stocking Frame, from 2 in. to 5½ in. diameter. These circular frames, with my last improvements, are as near perfection as possible. The Patent Full-fashioned Shirt, Drawers and Stocking Frames produce the most perfect goods ever made by steam-power machinery, and cost fifty per cent less to keep in repair than any other Knitting Machine. Built 10 to 24 gage, and from 30 to 140 inch wide, to order. The Improved Circular Looping Frame, for putting on shirt cuffs, drawers bands, clearing the top of circular shirts, &c., built to order, of any size, from 2 in. to 22 in. diameter, and of any gage. Steel Needles and Sinkers to pattern. Persons wishing to order while Mr. Kent is in England, will please address JOHN KENT, Nottingham, up to April 12th, or, if they prefer, may send through depot. Address JOHN KENT, 348 Pearl st., New York. PUMPS. —For Description, Price Lists etc., of the Best Centrifugal Pump ever invented, with Overwhelming Testimony in its favor, send for new illustrated pamphlet (40 pp.) to Messrs. HEALD, SISCO & CO., Baldwinsville, N. Y. BRICK PRESSES. **FOR RED AND FIRE BRICK.
\ Factory 309 S. Fifth street, Philadelphia, Pa. S. P. MILLER. ST. JOSEPH, Mo., Nov. 10, 1870. T. R. BAILEY & VAIL, Lockport, N. Y.: Gentlemen:—The Lathe you shipped me has arrived, and I have it in full operation. It works perfectly, and I think it the best lathe made in the world for Bedstead and Chair work. I would recommend it to any one desirous of obtaining such a lathe. Yours truly, H. R. BRISTOL. WOOLEN MILL FOR SALE. With House and 3 acres of land, Seymour, Ct., (Naugatuck Valley,) 2 miles from R. R. depot. Never-failing stream. 3 ft. fall, dam and wheel in good condition. Inquire of JAS. ORMSBEE, on the premises. Portable & Stationary Steam Engines AND HOISTING ENGINES. A good article at low prices. Every machine warranted. Send for descriptive Price List. H. B. BIGELOW & CO., New Haven, Conn. PATENT BANDSAW MACHINES Of the most approved kinds, of various sizes , to saw bevel as well as square, without inclining the table, by , 452 to 456 Tenth ave., New York. Price $250, $275, $350, and $400. At present (Oct. 16), there are in operation, in this city alone, 88 of our machines. Send for circular. Manufacture, also, an improved saw-filing apparatus; price, $30. Have also on hand a large stock of best French Bandsaw Blades. FIRST & PRYIBIL L. & J. W. FEUCHTWANGER, 55 Cedar st., New York, Chemists, Manufacturers, and Importers of Specialities, Silicates, Soda and Potash, Chloride of Calcium, Peroxide of Manganese, Hydrofluoric Acid, Metallic Oxides, Steel and Glass Makers' and Potters' Articles, Publishers of Treatises on "Soluble Glass," "Gems," and "Fermented Liquors." PIMLICO BRACES, SOMETHING NEW. THIS invention is based on a strictly scientific principle, and is a valuable improvement on old style suspenders. It is simple in construction, and combines the qualities of Brace and Suspender. They are unequaled for elegance, durability and comfort. Manufactured at the Monumental Silk Works, Baltimore. JOHN M. DAVIES & CO., Sole Agents, 384 & 386 B'd'y, N. Y. Dr. J. ARMSTRONG'S (Patent) Improved Heater, Filter, Lime Extractor, and Condenser Combined, For Steam Boilers. MANUFACTURED BY Toledo, Ohio. Armstrong & Starr, Send for Circulars. Formerly Armstrong & Welsh. —An Engine, 12x36 in. cylinder, and two Boilers, 4x15 feet, in good order, will be sold cheap. FOR SALE. J. J. TAYLOR & CO., 68 Courtlandt st., New York. Trapping and Fishing. All about it. Sent free. Address "Hunter," Hinsdale, N. H. HUNTING, First Premium awarded by Am. Inst., 1870 T. H. McALLISTER, Optician, 49 Nassau st., N. Y. Rare and Beautiful Flowers —AND— CHOICE VEGETABLES Can always be obtained by Sowing The Seventeenth Annual Edition of their celebrated "SEED CATALOGUE AND GUIDE TO THE FLOWER AND KITCHEN GARDEN," is now ready for distribution. It contains Four Beautifully Colored Lithographs, and about 300 choice Engravings of favorite Flowers and Vegetables, 136 pages of closely-printed matter, and a list of Twenty-five Hundred species and varieties of Flower and Vegetable Seeds, with explicit directions for their culture, and much other useful information upon the subject of Gardening. A copy will be mailed to all applicants inclosing 25 cts. Regular customers supplied gratis. Address B. K. BLISS & SONS, Nos. 23 Park Place, and 20 Murray st., P. O. Box No. 5712. New York. Science for the Million. THE Boston Journal of Chemistry, Devoted to the Science of Home Life, the Arts, Agriculture, and Medicine. JAMES R. NICHOLS, M.D., WILLIAM A. ROLFE, A.M., } Editors. ONE DOLLAR PER YEAR. A paper which commends itself at once to Physicians, Druggists, Chemists, Teachers, Farmers, Mechanics—in short, to Professional and Practical Men of every class. The Domestic Recipes and Formulæ for Art Processes are of themselves worth many times the cost of subscription. Specimen Copies Sent Free. Address **BOSTON JOURNAL OF CHEMISTRY,
\ 150 Congress st., Boston. FOOT LATHES, And all kinds of small tools. Illustrated catalogue free. GOODNOW & WIGHTMAN, 23 Cornhill, Boston, Mass. AGENTS WANTED.—To sell Stephens' which embraces a Rule, Level, Square, Plumb, Bevel, Slope Level, T Square, etc., in one compact tool. These instruments retail at $3.50 each, and energetic salesmen can make money by selling them among mechanics. We warrant them in every particular, as the construction and graduation is faultless. Send for descriptive circular, cuts, and terms. Patent Combination Rule, STEPHENS & CO., Riverton, Conn. FELT. THE BEST, CHEAPEST and MOST DURABLE non-conductor known, for sale by the Original Manufacturer, at the BOILER FELTING WORKS, 46 Courtland st., New York. Universal Wood Worker. For Agricultural, Railroad, Car, Carriage, and Wagon Works, Planing Mill, Sash, Door and Blind, Bedstead, Cabinet and Furniture Factories. McBETH, BENTEL & MARGEDANT, Hamilton, O. , Inventor's Exchange 245 Broadway, N. Y., "Am. Agriculturist" Building. TANGIBLE INVENTIONS NEGOTIATED. No goods received unless ordered. B. F. KEMP, Proprietor. MACHINISTS' TOOLS, at greatly reduced prices. Also, some Woodworth Planers and Second-hand Tools. 97 to 113 R. R. ave., Newark, N. J. E. & R. J. GOULD, successors to Gould Machine Co. N. B. PATENTED Articles introduced. Also, State and County Rights sold for Inventors. STONE, PUGH & CO., 55 N. 6th st., Philadelphia. G OLDEN HILL Seminary for young ladies, Bridgeport, Conn. Miss Emily Nelson, Principal. 1826 USE THE VEGETABLE PULMONARY BALSAM. 1870 The old standard remedy for Coughs, Colds, Consumption. "Nothing Better." Cutler Bros. & Co., Boston. T HE CALVERT IRON ROLLING MILLS are offered at private sale. These mills are situated in the city of Baltimore, and cover 1½ acres of ground. The Machinery is of the most approved description, for making all sizes of round and square bar iron, from ¼ in. to 3 in. diameter, and flat bars of all widths, up to 7 inches. The buildings are ample and commodious. In addition to the Rolling Mills are two brick buildings (50x125 feet and 40x90 feet), now containing an 80 H.P. Engine, and Spike Machinery, but which could be used for the manufacture of Nails, Horseshoes, or any other branch of heavy hardware. This property offers an unusual opportunity to capitalists, and will be sold at a reasonable price. For further description address MARSHALL  P. SMITH, P. O. Box 1158, Baltimore, Md. BURDON IRON WORKS.—Manufacturers of Pumping Engines for Water Works, High & Low Pressure Engines, Portable Engines and Boilers, of all kinds, Sugar Mills, Screw, Lever, Drop, & Hydraulic Presses, Machinery in general. HUBBARD & WHITTAKER, 102 Front st., Brooklyn. ENGINES AND MACHINERY FOR SALE, at a great sacrifice. Two new Steam Engines, 12 and 20 horse power; 1 Faribain's Riveting Machine; 1 large Power Shears; 1 ditto Table Punch; 2 ditto Flange Punches; 1 set Power Bending Rolls; together with a large lot of Turning Lathes, Drilling Machines, Machinists' and Smiths' Hand Tools, Pulleys, Hangers, and 6 Fairbanks' Platform Scales. Send for catalogue, or apply at the South Brooklyn Steam Engine Works, cor. Imlay and Summit sts., Brooklyn. SHINGLE AND HEADING MACHINE— Law's Patent with Trevor & Co.'s Improvements. The Simplest and Best in use. Also, Shingle, Heading and Stave Jointers, Equalizers, Heading Turners, Planers etc. AddressTREVOR & CO., Lockport, N. Y. AGENTS WANTED—($225 A MONTH) by the AMERICAN KNITTING MACHINE CO. Boston, Mass., or St. Louis, Mo. THE UNITED STATES Brick Machine IS THE BEST IN THE WORLD, BECAUSE IT MAKES THE , , and GREATEST NUMBER THE BEST THE CHEAPEST BRICKS. It is the Perfection of Simplicity. It is Durable, and not likely to get out of Repair. See Scientific American, Sept. 17, 1870. For Descriptive Circular apply to F. C. WELLS, President, Room 13, 98 Madison st.,\CHICAGO. MACHINES can be seen in operation at the Company's Works, Chicago; at rear 59 Ann st., New York city; and at Novelty Iron Works, corner of Delord and Peter sts., New Orleans. PREPARED Asphalte Roofing Felt. This new prepared production is ready coated, and can be applied on the roof without further trouble. It is easy of application, and does not require any repairs for a long time. It is more durable than some slates, and has been found a suitable substitute for iron or tin roofs. It has a sanded or stony surface, which renders it Exposed to the most intense fire, and sparks falling upon it, it will not propagate the fire. Under the influence of the sun it will not run, which makes it specially adapted to hot climates. Its easy application and pleasing appearance have made it a favorite roofing material throughout all the Indies and other colonies. Being not cumbrous for transport, it is of invaluable service to settlers and farmers in far remote districts. When used for temporary purposes it may be taken off and applied again to another construction. It replaces common Asphalting on Terraces, Lobbies, Counting-houses, Office Floors, etc.; is a great preservative against dampness and vermin, and equalizes the temperature. It is 32 inches wide, and made in rolls of 25 yards each. Send for circular to UNINFLAMMABLE and FIRE-PROOF. E. H. MARTIN, 70 Maiden Lane and 9 Liberty St., N. Y. J. J. H. GREGORY'S Seed Catalogue. My Annual Illustrated Catalogue, containing a list of many new and rare Vegetables, some of which are not found in any other catalogue, and all the standard vegetables of the farm and garden (over one hundred of which I grow on my three seed farms), with a carefully selected list of flower seed, will be sent free to all. All my seed is sold under three warrants: 1st. That all money sent shall reach me. 2d. That all seed ordered shall reach the purchaser. 3d. That my seeds shall be fresh and true to name. JAMES J. H. GREGORY, Marblehead, Mass. THE NEW WILSON Under-Feed Shuttle\SEWING MACHINES! $25 cheaper than any other! For Simplicity, Durability and Beauty they stand For Stitching, Hemming, Tucking, Felling, Quilting, Cording, Binding, Braiding, Gathering, Gathering & sewing on gathers, For particulars address unrivalled! they are unexcelled! Wilson Sewing Machine Co., Cleveland, O., or St. Louis, Mo. , **NEW and 2d-HAND.—
\ Send for Circular. Chas. PLACE & CO., 60 Vesey st., New York. MACHINERY MACHINISTS. Illustrated Catalogue and Price List of all kinds of small Tools and Materials sent free to any address. GOODNOW & WIGHTMAN, 23 Cornhill, Boston, Mass. P. BLAISDELL & CO. MANUFACTURERS of the "BLAISDELL" PATENT DRILL PRESSES, with quick return motion, Agricultural Drills, Improved Engine Lathes, from 12 in. to 28 in. swing, Planers, Gear Cutters, Boring Mills, Hand Lathes, and other first-class Machinists' Tools. Jackson st., Worcester, Mass. Patent BEDSTEAD FASTENING. The and ever invented. Best, Cheapest, Strongest FASTENING Rights for States and Territories for sale. Address JOHN DOMINGOS and                     BENJAMIN ESSIG, Sacramento, Cal. BENT, GOODNOW & CO., Boston, Mass., Publishers of , sell Patent Rights and goods of all kinds. Orders solicited. "PATENT STAR" AGENTS WANTED. Send stamp for copy. TO THE WORKING CLASS.—We are now prepared to furnish all classes with constant employment at home, the whole of the time or for the spare moments. Business new, light and profitable. Persons of either sex easily earn from 50c. to $5 per evening, and a proportional sum by devoting their whole time to the business. Boys and girls earn nearly as much as men. That all who see this notice may send their address, and test the business, we make this unparalleled offer: To such as are not well satisfied, we will send $1 to pay for the trouble of writing. Full particulars, a valuable sample which will do to commence work on, and a copy of —one of the largest and best family newspapers published—all sent free by mail. Reader, if you want permanent, profitable work, address The People's Literary Companion D. C. ALLEN & CO., Augusta, Maine. IMPORTANT TO  MACHINISTS.—The Best Metal for all Machine Uses is the MARTIN STEEL, made by The New Jersey Steel and Iron Co., Trenton, N. J. This steel is made by an entirely different process from any other and is tougher than wrought iron. It can be turned without annealing, being entirely free from hard spots. Every one who uses it pronounces it just what they have long wanted, for a multitude of uses, such as Crank Pins, Lathe Spindles and Screws, Cotton Machinery Rollers, Saw and Fan Spindles, etc., etc. Also, particularly adapted for Firebox Plates. Prices low. Send for further information, or a sample, stating use to which it is to be applied. OTIS' SAFETY HOISTING\Machinery. **OTIS, BROS. & CO.
\ No. 309 BROADWAY, NEW YORK. Union Emery Wheels. Solid and with Stone Center. UNION STONE CO., Boston, Mass. Branch Office, 93 Liberty st., N. Y. General Agents for the Am. Twist Drill Co.'s Superior Grinder and other Emery Wheel Machinery and Tools. Send for Circular. \ WOODBURY'S PATENT** Planing and Matching and Molding Machines, Gray & Wood's Planers, Self-oiling Saw Arbors, and other wood working machinery. S. A. WOODS, Send for Circulars. } 91 Liberty street, N. Y.; 67 Sudbury street, Boston. RI**CHARDSON, MERIAM & CO.,
\ Manufacturers of the latest improved Patent Daniels' and Woodworth Planing Machines, Matching, Sash, and molding, Tenoning, Mortising, Boring, Shaping, Vertical, and Circular Re-sawing Machines, Saw Mills Saw Arbors, Scroll Saws, Railway, Cut-off, and Rip-saw Machines, Spoke and Wood Turning Lathes, and various other kinds of Wood-working Machinery. Catalogues and price lists sent on application. Manufactory, Worcester, Mass. Warehouse, 107 Liberty st., New York. 17 1 Reynolds' TURBINE WATER WHEELS. The Oldest and Newest. All others only imitations of each other in their strife after complications to confuse the public. We do not boast but quietly excel them all in staunch reliable, economical power. Beautiful pamphlet free. Geo. TALLCOT, 96 Liberty st., New York. Gearing, Shafting.

To Manufacturers of Knitted Goods

Scientific American, Volume XXIV., No. 12, March 18, 1871 by Various, is part of the HackerNoon Books Series. You can jump to any chapter in this book . Business and Personal here Business and Personal The Charge for Insertion under this head is One Dollar a Line. If the Notices exceed Four Lines, One Dollar and a Half per Line will be charged. The paper that meets the eye of manufacturers throughout the United States—Boston Bulletin, $4.00 a year. Advertisements 17c. a line. Half Interest for sale in established Machinery Depot, new and second-hand. Steam fitting connected. Small capital, with energy, required. Address T. V. Carpenter, Advertising Agent, Box 773, New York. See advertisement of a Woolen Mill for sale. A bargain. I am active, have a clear record, and some capital. How can I make some money? F. Carmill, Box 1268, Boston, Mass. Pattern Letters for Machinists, Molders, and Inventors, to letter patterns of castings, all sizes. Address H. W. Knight, Seneca Falls, N. Y. Improved mode of Graining Wood, pat. July 5, '70, by J. J. Callow, Cleveland, O. See illustrated S. A., Dec. 17, '70. Send stamp for circular. Can a round, spring-steel rod be drawn to any desired length, with a true taper to a point, with equal elasticity the whole length, and rolled temper? What is the price per hundred pounds, and where can they be procured? Answer "Sportsman," Malone, N. Y. Manufacturers of Foot Lathes and other light machinery please address Geo. B. Kirkham, 167 E. 33d st., N. Y. city. Business of importance! Safety Kerosene Lamps (Perkins & House's Patent). Explosion or breaking impossible; light equal to gas, and no odor. Families supplied and canvassers appointed, by Montgomery & Co., 42 Barclay st., New York, or Cleveland, O. All parties wanting a water wheel will learn something of interest by addressing P. H. Wait, Sandy Hill, N. Y., for a free circular of his Hudson River Champion Turbine. Ashcroft's Low Water Detector, $15; thousands in use; 17 year's experience. Can be applied for $1. Send for circular. E. H. Ashcroft, Boston, Mass. Wanted.—Machines for manufacturing Pails, Tubs, and Matches. Also, competent man to superintend construction of buildings, and manage all parts of business when complete. Address, with descriptive circulars, price, etc., No. 266 Lexington avenue, New York. Turbine Water Wheels, Portable and Stationary Engines, Gang and Circular Saw Mills, Rolling Mill Machinery, and Machinery for Axe Manufacturers, manufactured by Wm. P. Duncan, Bellefonte, Pa. For best Power Picket Header in use, apply to Wm. P. Duncan, Bellefonte, Pa. New Blind Wirer and Rod Cutter. B. C. Davis & Co., Binghamton, N. Y. Self-testing Steam Gage. There's a difference between a chronometer watch and a "bull's eye." Same difference between a self-tester and common steam gage. Send for Circular. E. H. Ashcroft, Boston, Mass. See advertisement of L. & J. W. Feuchtwanger, Chemists, N. Y. $3.50. Stephens' Patent Combination Rule, Level, Square, Plumb, Bevel, etc. See advertisement in another column. Agents wanted. American Boiler Powder Co., Box 315, Pittsburgh, Pa., make the only safe, sure, and cheap remedy for "Scaly Boilers." Orders solicited. Belting that is Belting.—Always send for the Best Philadelphia Oak-Tanned, to C. W. Arny, Manufacturer, 301 Cherry st., Phil'a. E. Howard & Co., Boston, make the best Stem-winding Watch in the country. Ask for it at all the dealers. Office 15 Maiden Lane, N. Y. For mining, wrecking, pumping, drainage, and irrigating machinery, see advertisement of Andrews' Patents in another column. The best place to get Working Models and parts is at T. B. Jeffery's, 160 South Water st., Chicago. Brown's Coalyard Quarry & Contractors' Apparatus for hoisting and conveying material by iron cable. W. D. Andrews & Bro, 414 Water st., N. Y. Improved Foot Lathes. Many a reader of this paper has one of them. Selling in all parts of the country, Canada, Europe, etc. Catalogue free. N. H. Baldwin, Laconia, N. H. Peteler Portable R. R. Co. contractors, graders. See adv'ment. E. P. Peacock, Manufacturer of Cutting Dies, Press Work. Patent Articles in Metals, etc. 55 Franklin st., Chicago. Peck's Patent Drop Press. Milo Peck & Co., New Haven, Ct. Millstone Dressing Diamond Machine—Simple, effective, durable. For description of the above see Scientific American, Nov. 27th, 1869. Also, Glazier's Diamonds. John Dickinson, 64 Nassau st., N. Y. Steel name stamps, figures, etc. E. H. Payn, M'f'r, Burlington, Vt. Cold Rolled-Shafting, piston rods, pump rods, Collins pat. double compression couplings, manufactured by Jones & Laughlins, Pittsburgh, Pa. Keuffel & Esser 116 Fulton st., N. Y., the best place to get 1st-class Drawing Materials, Swiss instruments, and Rubber Triangles and Curves. For Solid Wrought-iron Beams, etc., see advertisement. Address Union Iron Mills, Pittsburgh, Pa., for lithograph, etc. For the best Self-regulating Windmill in the world, to pump water for residences, farms, city buildings, drainage, and irrigation, address Con. Windmill Co., 5 College Place, New York. The Merriman Bolt Cutter—the best made. Send for circulars. H. B. Brown & Co., Fair Haven, Conn. Taft's Portable Hot Air, Vapor and Shower Bathing Apparatus. Address Portable Bath Co., Sag Harbor, N. Y. (Send for Circular.) Glynn's Anti-Incrustator for Steam Boilers—The only reliable preventive. No foaming, and does not attack metals of boilers. Price 25 cents per lb. C. D. Fredricks, 587 Broadway, New York. For Fruit-Can Tools, Presses, Dies for all Metals, apply to Bliss & Williams, successor to May & Bliss, 118, 120, and 122 Plymouth st., Brooklyn, N. Y. Send for catalogue. 2d-hand Worthington, Woodward and Novelty Pumps, Engines 25 to 100 H.P., 60 Horse Loc. Boiler. W. D. Andrews & Bro., 414 Water st., N. Y. Agents wanted, to sell the Star Bevel. It supersedes the old style. Send for Circular. Hallett & White, West Meriden, Conn. English and American Cotton Machinery and Yarns, Beam Warps and Machine Tools. Thos. Pray, Jr., 57 Weybosset st., Providence, R. I. For small, soft, Gray Iron Castings, Japanned, Tinned, or Bronzed, address Enterprise Manufacturing Company, Philadelphia. Conklin's Detachable Rubber Lip, for bowls, etc., works like a charm. For Rights, address O. P. Conklin, Worcester, Mass., or A. Daul, Philadelphia, Pa. To Ascertain where there will be a demand for new machinery or manufacturers' supplies read Boston Commercial Bulletin's Manufacturing News of the United States. Terms $4.00 a year. Facts for the Ladies. In 1870, Mrs. W. made, with her Wheeler & Wilson machine, 2,255 vests, besides doing her family sewing for six persons. The Pittsburgh, Pa., "Leader" says: "The firm of Geo. P. Rowell & Co. is the largest and best Advertising Agency in the United States, and we can cheerfully recommend it to the attention of those who desire to advertise their business scientifically and systematically in such a way; that is, to secure the largest amount of publicity for the least expenditure of money." at American Institute Fair for 1870, Pratt's Astral Oil was pronounced the safest and best. After an exhaustive trial, Its Varieties, Causes, Symptoms, and Cure. By E. P. Miller, M.D. Paper, 50cts.; Muslin, $1. Address Miller, Haynes & Co., 41 West Twenty-sixth st., New York city. Dyspepsia: How Wasted and How Preserved; or, Abuses of the Sexual Function, their Causes Effects and Means of Cure. By E. P. Miller M.D. Paper, 50cts. Address Miller, Haynes & Co., 41 West Twenty-sixth st., New York city. Vital Force: About HackerNoon Book Series: We bring you the most important technical, scientific, and insightful public domain books. This book is part of the public domain. Various (2006). Scientific American, Volume XXIV., No. 12, March 18, 1871. Urbana, Illinois: Project Gutenberg. Retrieved https://www.gutenberg.org/cache/epub/19180/pg19180-images.html This eBook is for the use of anyone anywhere at no cost and with almost no restrictions whatsoever. You may copy it, give it away or re-use it under the terms of the Project Gutenberg License included with this eBook or online at , located at . www.gutenberg.org https://www.gutenberg.org/policy/license.html

Business and Personal

Population Stability Index (PSI) is a statistical measure commonly used in credit risk modeling and other fields to assess the stability of a population across different time periods or segments. It's particularly valuable in scenarios where changes in the population distribution could impact the effectiveness of a model. It is a fundamental assumption that the population on which a statistical model is applied to is similar to the population on which the model is developed at. The model performance relies on the closeness between the model development sample and the production data. Therefore, there is an intrinsic need to compare the two samples (development vs production) and assess whether a model needs to be recalibrated. The population change may occur for several reasons – change in the business strategies and policies or changes in external factors such as the political, economic and social environment. The model performance is impacted because of a change in the joint distribution p(y, X), where y is the model outcome and X is the design matrix that comprises the observations of the model features. Therefore, the entire model fit, i.e., the estimates of the model coefficients and the inherent functions get impacted. If this happens, then a decision must be made towards the recalibration or redevelopment of the model.  Before we dive deep into the PSI mathematical expression, we start with some key concepts on Entropy and KL divergence. Entropy and KL Divergence Entropy is a measure of uncertainty or randomness in a set of data. In machine learning, particularly in decision trees and information theory, entropy is used to quantify the impurity or disorder in a dataset. For instance, the splits of a decision tree for a classification problem are created to minimize the entropy. Entropy is used to determine the best attribute to split the data at each node of the decision tree. In the above definition of entropy, the negative sign ensures that Entropy is positive (because log of the probability will be negative as the probability will be < 1). Higher (lower) the value of P(X = X_i) lower (higher) is the entropy or randomness or uncertainty about X. Thus, ‘skewed’ distributions have lower entropy but ‘balanced’ distributions have higher entropy. The objective of splitting a variable at a node for a binary classification problem is to create more skewed distribution such that we can discriminate between an outcome value y = 1 and outcome value y = 0. Therefore, we move from a high entropy stage before splitting the variable to a lower entropy stage after splitting the variable.
If Entropy H(X) is high, then the uncertainty before knowing X is high. So after knowing X we have gained more information.
If Entropy H(X) is low, then the uncertainty before knowing X is low. So after knowing X we have gained less information.
We try to minimize entropy whilst building binary regression models. For example, we ‘bin’ or ‘discretize’ a model factor such the distribution of binary outcome becomes as skewed as possible across the different bins. Kullback-Liebler Divergence (KL Divergence) has been referred in several places in the literature (Lin, 2017), (Wu & Olson, 2010), (Li et al., 2008), etc. It measures the difference between two probability distributions p(x) and q(x). It quantifies how one probability distribution diverges from another. The distribution p(x) is considered the ‘true’ or ‘base’ distribution, and the distribution q(x) is considered the ‘untrue’ or ‘target’ distribution, so that KL Divergence represents some sort of loss due to using the ‘wrong’ distribution. KL divergence is non-negative and is zero if and only if p(x) and q(x) are identical, i.e., the same distribution. KL divergence measures the dissimilarity of the two probability distributions p(x) and q(x). If both the distributions are identical, KL Divergence is 0. We have a single random variable “X” and we have two different probability distributions for that variable. One is assumed to be the ‘true’ distribution and the other is the approximation of the ‘true’ distribution. So, we try “approximating” one distribution with the other and calculate the difference in the information about the event as a measure of the “divergence”. Note that the notation ‘P||Q’ implies P is approximated as Q.
Notice how the mathematical expressions for the individual terms of KL divergence are similar to that of Entropy or information which we introduced earlier. The second term inside the brackets is the information in p(x) about x. The first term inside the brackets is the information in an approximation distribution of p(x) as q(x) about x. That is if we approximate the target distribution p(x) by q(x); then the difference in the information between them is the relative entropy or KL Divergence index. The better our approximation the closer p(x) and q(x) and hence the KL divergence will be 0.
Even though KL Divergence measures divergence of q(x) from p(x), it is technically not a distance measure because the definition is not symmetric. Thus KL(P||Q) != KL(Q||P) Population Stability Index – Now that we have covered the mathematical expressions of Entropy and KL Divergence Index, we will see how they would help us in understanding the expression of Population Stability Index. Notice that we have added another term q(x) inside the brackets of KL divergence expression to get the mathematical expression for PSI. But the formulae look so similar that we can think of expressing PSI as a function of KL Divergence. Here is how we can do that. That is the PSI metric is actually the sum of the relative entropies of q(x) over p(x) and p(x) over q(x). In other words we are adding the information ‘gain’ by expressing p(x) using an approximation q(x) and vice-versa. The reason we do this is because we don’t know the ‘ground truth’. We assume that one of them is a ‘true’ distribution and the other is an ‘approximation’; and as we know that the KL divergence is not symmetric, we compute it from p(x) to q(x) and vice-versa and add them to get a final metric which can be judged to ascertain the difference between p(x) and q(x). The PSI calculation typically begins by dividing the data into bins or segments based on key variables used in the model.
PSI values close to zero indicate stability between the initial and new populations.
Higher PSI values suggest significant shifts in the population distribution, which may require further investigation or model recalibration. So how high should PSI be to make that judgement? The following table summarizes the PSI values which are ‘industry wide rules of thumb’. Interestingly, PSI is very similar to ‘Information Value’ used to measure the strength of the modelling variables in developing credit risk models. We will explore that soon in a different article. Population Stability Index (PSI) is a statistical measure commonly used in credit risk modeling and other fields to assess the stability of a population across different time periods or segments. It's particularly valuable in scenarios where changes in the population distribution could impact the effectiveness of a model. It is a fundamental assumption that the population on which a statistical model is applied to is similar to the population on which the model is developed at. The model performance relies on the closeness between the model development sample and the production data. Therefore, there is an intrinsic need to compare the two samples (development vs production) and assess whether a model needs to be recalibrated. The population change may occur for several reasons – change in the business strategies and policies or changes in external factors such as the political, economic and social environment. The model performance is impacted because of a change in the joint distribution p(y, X ), where y is the model outcome and X is the design matrix that comprises the observations of the model features. Therefore, the entire model fit, i.e., the estimates of the model coefficients and the inherent functions get impacted. If this happens, then a decision must be made towards the recalibration or redevelopment of the model.  Before we dive deep into the PSI mathematical expression, we start with some key concepts on Entropy and KL divergence. X Entropy and KL Divergence Entropy and KL Divergence Entropy and KL Divergence Entropy is a measure of uncertainty or randomness in a set of data. In machine learning, particularly in decision trees and information theory, entropy is used to quantify the impurity or disorder in a dataset. For instance, the splits of a decision tree for a classification problem are created to minimize the entropy. Entropy is used to determine the best attribute to split the data at each node of the decision tree. In the above definition of entropy, the negative sign ensures that Entropy is positive (because log of the probability will be negative as the probability will be < 1). Higher (lower) the value of P(X = X_i) lower (higher) is the entropy or randomness or uncertainty about X. Thus, ‘skewed’ distributions have lower entropy but ‘balanced’ distributions have higher entropy. The objective of splitting a variable at a node for a binary classification problem is to create more skewed distribution such that we can discriminate between an outcome value y = 1 and outcome value y = 0. Therefore, we move from a high entropy stage before splitting the variable to a lower entropy stage after splitting the variable. If Entropy H(X) is high, then the uncertainty before knowing X is high. So after knowing X we have gained more information. If Entropy H(X) is low, then the uncertainty before knowing X is low. So after knowing X we have gained less information. We try to minimize entropy whilst building binary regression models. For example, we ‘bin’ or ‘discretize’ a model factor such the distribution of binary outcome becomes as skewed as possible across the different bins. Higher (lower) the value of P(X = X_i) lower (higher) is the entropy or randomness or uncertainty about X. Thus, ‘skewed’ distributions have lower entropy but ‘balanced’ distributions have higher entropy. The objective of splitting a variable at a node for a binary classification problem is to create more skewed distribution such that we can discriminate between an outcome value y = 1 and outcome value y = 0. Therefore, we move from a high entropy stage before splitting the variable to a lower entropy stage after splitting the variable. If Entropy H(X) is high, then the uncertainty before knowing X is high. So after knowing X we have gained more information. If Entropy H(X) is low, then the uncertainty before knowing X is low. So after knowing X we have gained less information. We try to minimize entropy whilst building binary regression models. For example, we ‘bin’ or ‘discretize’ a model factor such the distribution of binary outcome becomes as skewed as possible across the different bins. Kullback-Liebler Divergence (KL Divergence) has been referred in several places in the literature (Lin, 2017), (Wu & Olson, 2010), (Li et al., 2008), etc. It measures the difference between two probability distributions p(x) and q(x). It quantifies how one probability distribution diverges from another. The distribution p(x) is considered the ‘true’ or ‘base’ distribution, and the distribution q(x) is considered the ‘untrue’ or ‘target’ distribution, so that KL Divergence represents some sort of loss due to using the ‘wrong’ distribution. KL divergence is non-negative and is zero if and only if p(x) and q(x) are identical, i.e., the same distribution. KL divergence measures the dissimilarity of the two probability distributions p(x) and q(x). If both the distributions are identical, KL Divergence is 0. We have a single random variable “X” and we have two different probability distributions for that variable. One is assumed to be the ‘true’ distribution and the other is the approximation of the ‘true’ distribution. So, we try “approximating” one distribution with the other and calculate the difference in the information about the event as a measure of the “divergence”. Note that the notation ‘P||Q’ implies P is approximated as Q. Notice how the mathematical expressions for the individual terms of KL divergence are similar to that of Entropy or information which we introduced earlier. The second term inside the brackets is the information in p(x) about x. The first term inside the brackets is the information in an approximation distribution of p(x) as q(x) about x. That is if we approximate the target distribution p(x) by q(x); then the difference in the information between them is the relative entropy or KL Divergence index. The better our approximation the closer p(x) and q(x) and hence the KL divergence will be 0. Even though KL Divergence measures divergence of q(x) from p(x), it is technically not a distance measure because the definition is not symmetric. Thus KL(P||Q) != KL(Q||P) We have a single random variable “X” and we have two different probability distributions for that variable. One is assumed to be the ‘true’ distribution and the other is the approximation of the ‘true’ distribution. So, we try “approximating” one distribution with the other and calculate the difference in the information about the event as a measure of the “divergence”. Note that the notation ‘P||Q’ implies P is approximated as Q. Notice how the mathematical expressions for the individual terms of KL divergence are similar to that of Entropy or information which we introduced earlier. The second term inside the brackets is the information in p(x) about x. The first term inside the brackets is the information in an approximation distribution of p(x) as q(x) about x. That is if we approximate the target distribution p(x) by q(x); then the difference in the information between them is the relative entropy or KL Divergence index. The better our approximation the closer p(x) and q(x) and hence the KL divergence will be 0. Even though KL Divergence measures divergence of q(x) from p(x), it is technically not a distance measure because the definition is not symmetric. Thus KL(P||Q) != KL(Q||P) Population Stability Index – Population Stability Index Population Stability Index Now that we have covered the mathematical expressions of Entropy and KL Divergence Index, we will see how they would help us in understanding the expression of Population Stability Index. Notice that we have added another term q(x) inside the brackets of KL divergence expression to get the mathematical expression for PSI. But the formulae look so similar that we can think of expressing PSI as a function of KL Divergence. Here is how we can do that. That is the PSI metric is actually the sum of the relative entropies of q(x) over p(x) and p(x) over q(x). In other words we are adding the information ‘gain’ by expressing p(x) using an approximation q(x) and vice-versa. The reason we do this is because we don’t know the ‘ground truth’. We assume that one of them is a ‘true’ distribution and the other is an ‘approximation’; and as we know that the KL divergence is not symmetric, we compute it from p(x) to q(x) and vice-versa and add them to get a final metric which can be judged to ascertain the difference between p(x) and q(x). The PSI calculation typically begins by dividing the data into bins or segments based on key variables used in the model. PSI values close to zero indicate stability between the initial and new populations. Higher PSI values suggest significant shifts in the population distribution, which may require further investigation or model recalibration. The PSI calculation typically begins by dividing the data into bins or segments based on key variables used in the model. PSI values close to zero indicate stability between the initial and new populations. Higher PSI values suggest significant shifts in the population distribution, which may require further investigation or model recalibration. So how high should PSI be to make that judgement? The following table summarizes the PSI values which are ‘industry wide rules of thumb’. Interestingly, PSI is very similar to ‘Information Value’ used to measure the strength of the modelling variables in developing credit risk models. We will explore that soon in a different article.

Population Stability Index in Machine learning models

Mission Generation Using Classic Machine Learning and Recurrent Neural Networks in Zombie State

জম্বি রাজ্যে ক্লাসিক মেশিন লার্নিং এবং পুনরাবৃত্ত নিউরাল নেটওয়ার্ক ব্যবহার করে মিশন জেনারেশন

Missionsgenerierung mit klassischem maschinellem Lernen und rekurrierenden neuronalen Netzwerken im Zombie-Zustand

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Генерация миссий с использованием классического машинного обучения и рекуррентных нейронных сетей в состоянии зомби

Zombie State'te Klasik Makine Öğrenimi ve Tekrarlayan Sinir Ağlarını Kullanarak Görev Oluşturma

Tạo nhiệm vụ bằng cách sử dụng mạng lưới thần kinh định kỳ và học máy cổ điển ở trạng thái Zombie

在僵尸状态下使用经典机器学习和循环神经网络生成任务

Starting Simple: The Strategic Advantage of Baseline Models in Machine Learning

Empowering AWS DevOps With Python and Machine Learning

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